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QUERY ROUTE
Query Route: summary
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**Elapsed Time: 1.38 seconds**
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RAG PARAMS
RAG Parameters: {'dynamically_expand': False, 'top_k': 120, 'actual_k': 120, 'satisfying_score': 0}
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VECTOR SEARCH RESULTS
Results: [{'id': '51085944-450d-4ac0-b8da-d50af746ab74',
 'metadata': {'chunk': 2.0,
              'file_name': 'www-fairwaycapecod-com-menu-breakfast-62826.txt',
              'is_dict': 'no',
              'text': '## Fairway Favorites\n'
                      '\n'
                      '**The Mountain**  \n'
                      'grilled homemade biscuit topped with two sausage '
                      'patties, two scrambled eggs, bacon, cheddar cheese and '
                      'sausage gravy â\x80¦ served with home fries  \n'
                      '14\n'
                      '\n'
                      '**Nannyâ\x80\x99s Potato Pancakes**  \n'
                      'two homemade potato pancakes and two eggs â\x80¦ served '
                      'with toast and choice of bacon, ham or sausage  \n'
                      '13.50\n'
                      '\n'
                      '**Corned Beef Hash and Eggs**  \n'
                      'our own recipe, served with two eggs, toast and home '
                      'fries  \n'
                      '14.50\n'
                      '\n'
                      '**Fairway Breakfast**  \n'
                      'three eggs with ham, bacon and sausageâ\x80¦ served '
                      'with toast and home fries  \n'
                      '12.50\n'
                      '\n'
                      '**Silver Skillet**  \n'
                      'sauteed onions, peppers, home fries and choice of '
                      'bacon, ham or sausage, all scrambled with three eggs '
                      'and topped with a blend of cheddar and mozzarellaâ\x80¦ '
                      'served with toast  \n'
                      '12.50\n'
                      '\n'
                      '**Country Breakfast**  \n'
                      'grilled homemade biscuit topped with two sausage '
                      'patties, two eggs and sausage gravy â\x80¦ served with '
                      'home fries  \n'
                      '12.50\n'
                      '\n'
                      '**Two by Two**  \n'
                      'two eggs, short stack of buttermilk pancakes or French '
                      'toast, and choice of two strips of bacon, two sausage '
                      'links or slice of ham  \n'
                      '12\n'
                      '\n'
                      '**Quinoa Bowl**  \n'
                      'Quinoa mixed with black beans, corn, roasted red '
                      'peppers and a touch of chipotle pepperâ\x80¦topped with '
                      'two eggs any style, fresh avocado, pico de gallo and '
                      'cheddar jack cheese.  \n'
                      '13\n'
                      '\n'
                      '**Avocado Toast**  \n'
                      'two slices of toast topped with mashed avocado and '
                      'poached eggs, finished with our house recipe everything '
                      'bagel seasoning and fresh micro greens.  \n'
                      '11\n'
                      '\n'
                      '## Benedicts\n'
                      '\n'
                      '_all served with home fries_\n'
                      '\n'
                      '**Traditional**  \n'
                      'English muffin topped with two poached eggs, Canadian '
                      'bacon, and hollandaise  \n'
                      '12.50\n'
                      '\n'
                      '**Eggs Mulligan**  \n'
                      'English muffin topped with poached eggs, our own corned '
                      'beef hash and hollandaise  \n'
                      '14.50\n'
                      '\n'
                      '**Nannyâ\x80\x99s**  \n'
                      'homemade potato pancakes topped with poached eggs and '
                      'hollandaiseâ\x80¦ served with choice of fresh fruit or '
                      'home fries  \n'
                      '13\n'
                      '\n'
                      '**Garden**  \n'
                      'homemade Italian herbed focaccia toast topped with two '
                      'poached eggs, spinach, tomatoes, mushrooms and '
                      'hollandaise  \n'
                      '12.50'},
 'score': 0.0,
 'values': []}, {'id': '4a0a5bff-c57f-4b2f-b879-1ce906d0611b',
 'metadata': {'chunk': 3.0,
              'file_name': 'www-fairwaycapecod-com-menu-breakfast-62826.txt',
              'is_dict': 'no',
              'text': '**Fried Green Tomatoes**  \n'
                      'a bed of spinach topped with two poached eggs, fried '
                      'green tomatoes, avocado, hollandaise and garnished with '
                      'basil   \n'
                      '12\n'
                      '\n'
                      '**Crab Cakes**  \n'
                      'grilled homemade crab cakes on a bed of fresh spinach, '
                      'topped with poached eggs and hollandaise  \n'
                      '18\n'
                      '\n'
                      '**Portuguese Benedict**  \n'
                      'homemade English muffin bread topped with two poached '
                      'eggs, grilled linguica, sautéed onions and peppers '
                      'with hollandaise  \n'
                      '16\n'
                      '\n'
                      '## Egg Combos\n'
                      '\n'
                      '_served with toast and home fries_\n'
                      '\n'
                      'add bacon, ham, chicken sausage or sausage 4 Linguica '
                      '6\n'
                      '\n'
                      'one egg 6 two eggs 7 three eggs 8\n'
                      '\n'
                      '## Omelets\n'
                      '\n'
                      '_Sautéed toppings cooked into the eggs, folded over, '
                      'and served with home fries and toast_ \n'
                      '\n'
                      '**Western**  \n'
                      'sautéed onions, peppers, ham and cheese  \n'
                      '14\n'
                      '\n'
                      '**Green Goodness**  \n'
                      'sautéed asparagus, zucchini, broccoli, spinach served '
                      'with homemade English muffin bread toast  \n'
                      '14\n'
                      '\n'
                      '**Athenian**  \n'
                      'spinach, tomatoes, sautéed onions and feta cheese  \n'
                      '14\n'
                      '\n'
                      '**Build Your Own**  \n'
                      'plain   \n'
                      '9\n'
                      '\n'
                      '**add your own ingredients**\n'
                      '\n'
                      'cheese: American, Swiss, cheddar, feta\n'
                      '\n'
                      '2 each\n'
                      '\n'
                      'ham  \n'
                      'bacon  \n'
                      'sausage  \n'
                      'tomato  \n'
                      'onions  \n'
                      'peppers  \n'
                      'mushrooms  \n'
                      'spinach\n'
                      '\n'
                      '2 each\n'
                      '\n'
                      'fresh avocado  \n'
                      'linguiça\n'
                      '\n'
                      '3 each\n'
                      '\n'
                      'hash\n'
                      '\n'
                      '4 each\n'
                      '\n'
                      '## Pancakes, French Toast and Waffles\n'
                      '\n'
                      '_add bacon, ham, sausage, chicken sausage 4 or linguica '
                      '6_ \n'
                      '_100% pure maple syrup 2_ \n'
                      '\n'
                      '**Cinnamon Roll French Toast**  \n'
                      'made with â\x80\x9cHoleâ\x80\x93Inâ\x80\x93Oneâ\x80\x9d '
                      'cinnamon roll  \n'
                      '10\n'
                      '\n'
                      '**Traditional French Toast**  \n'
                      'made with your choice of white, wheat, or raisin  \n'
                      'two 9 three 10.50\n'
                      '\n'
                      '**Belgian Waffle**  \n'
                      'made with our own recipe with malted flour.  \n'
                      'plain 9 warm apples 11\n'
                      '\n'
                      '**Buttermilk Pancakes**  \n'
                      'two 7 three 8.50\n'
                      '\n'
                      '**Fresh Blueberry Pancakes**  \n'
                      'two 9 three 10.50'},
 'score': 0.0,
 'values': []}, {'id': '863d270e-4e38-40c6-a22f-f3cabbe2d797',
 'metadata': {'chunk': 4.0,
              'file_name': 'www-fairwaycapecod-com-menu-breakfast-62826.txt',
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              'text': '**Fresh Blueberry Pancakes**  \n'
                      'two 9 three 10.50\n'
                      '\n'
                      '**Chocolate Chip Pancakes**  \n'
                      'two 9 three 10.50\n'
                      '\n'
                      '**Fresh Blueberry Crunch Pancakes**  \n'
                      'two 10 three 11.50\n'
                      '\n'
                      '## Breakfast Sandwiches\n'
                      '\n'
                      '**Classic**  \n'
                      'two eggs, cheese and choice of bacon, ham, sausage '
                      'patty or linguiça +3 â\x80¦ served on a brioche '
                      'roll  \n'
                      '8\n'
                      '\n'
                      'add home fries  \n'
                      '3\n'
                      '\n'
                      'substitute flour wrap, buttermilk biscuit or bagel  \n'
                      '1\n'
                      '\n'
                      '**Veggie Focaccia**  \n'
                      'scrambled eggs, spinach, mushrooms, tomatoes, sauteed '
                      'onions and cheddar cheese on focaccia breadâ\x80¦ '
                      'served with home fries  \n'
                      '10\n'
                      '\n'
                      '**California Focaccia**  \n'
                      'scrambled eggs, fresh avocado, fresh tomato and cheddar '
                      'cheese on focaccia breadâ\x80¦ served with home '
                      'fries  \n'
                      '10\n'
                      '\n'
                      '## Sides\n'
                      '\n'
                      '**Potato Pancakes**  \n'
                      'served with sour cream  \n'
                      '5  \n'
                      'add apple sauce 2\n'
                      '\n'
                      '**Biscuit And Gravy**  \n'
                      '8\n'
                      '\n'
                      '**Homemade Corned Beef Hash**  \n'
                      '8\n'
                      '\n'
                      '**Home Fries**  \n'
                      '3\n'
                      '\n'
                      '**Loaded Home Fries**  \n'
                      'loaded with chopped bacon, cheddar cheese, scallions '
                      'topped with hollandaise sauce  \n'
                      '8\n'
                      '\n'
                      '**Crab Cakes** \n'
                      '15\n'
                      '\n'
                      '**Breakfast Protein** \n'
                      'Bacon, Ham, Sausage, Canadian Bacon or Chicken sausage '
                      'â\x80\x93 4 Linguiça â\x80\x93 6\n'
                      '\n'
                      '**Seasonal Fresh Fruit Cup**  \n'
                      'cup 5\n'
                      '\n'
                      '**Oatmeal**  \n'
                      '5  \n'
                      'add Craisins®, raisins, walnuts or chocolate chips 1 '
                      'each  \n'
                      'add fresh blueberries 1.5\n'
                      '\n'
                      '[Fairway Favorites](#favorites) | '
                      '[Benedicts](#benedicts) | [Egg Combos](#eggcombos) | '
                      '[Omelets](#omelets) | [Pancakes & French '
                      'Toast](#pancakes) | [Breakfast Sandwiches](#sandwiches) '
                      '| [Sides](#sides)\n'
                      '\n'
                      'Follow us on Facebook\n'
                      '\n'
                      'Comments Box SVG iconsUsed for the like, share, '
                      'comment, and reaction icons\n'
                      '\n'
                      '[ Fairway Restaurant & Pizzeria '
                      '](https://facebook.com/354293134601213) \n'
                      '\n'
                      ' 14 hours ago'},
 'score': 0.0,
 'values': []}, {'id': 'e274e22b-cfc0-444c-8f65-59cd0c2480e4',
 'metadata': {'chunk': 5.0,
              'file_name': 'www-fairwaycapecod-com-menu-breakfast-62826.txt',
              'is_dict': 'no',
              'text': '14 hours ago \n'
                      '\n'
                      '[ ](https://facebook.com/354293134601213) \n'
                      '\n'
                      'Today, we pause to honor and thank our veterans for '
                      'their courage, dedication, and sacrifice. Your service '
                      'means the world to us. ð\x9f\x87ºð\x9f\x87¸ From all of '
                      'us at Fairway, Happy Veterans Day! ... [See MoreSee '
                      'Less](#) \n'
                      '\n'
                      '[View](https://scontent-iad3-2.xx.fbcdn.net/v/t39.30808-6/465907081%5F1137112068259222%5F696551884986646616%5Fn.jpg?stp=dst-jpg%5Fs720x720&%5Fnc%5Fcat=111&ccb=1-7&%5Fnc%5Fsid=127cfc&%5Fnc%5Fohc=bt-%5FxPu6sKwQ7kNvgHyMWFl&%5Fnc%5Fzt=23&%5Fnc%5Fht=scontent-iad3-2.xx&edm=AKIiGfEEAAAA&%5Fnc%5Fgid=AaQzlHjjBMgFxbUUdAhDSBi&oh=00%5FAYBodCsyQaAS0ym8GklrCWbrRkGMXBr5QYNpcprr-sGniA&oe=6738D9F7)\n'
                      '\n'
                      '0 Comments[Comment on '
                      'Facebook](https://www.facebook.com/354293134601213%5F1139471834689912)\n'
                      '\n'
                      'Find Us on Instagram\n'
                      '\n'
                      '[](https://www.instagram.com/fairway%5Frestaurant%5Fcapecod/) '
                      '#riseandshine #homemade #breakfast #capecod #yummy '
                      '#comfortfood #eastham #homemade #dessert #flavors '
                      '#openyearround #open #locals #dinner #steak #pasta '
                      '#tasty #family #friends #eat\n'
                      '\n'
                      'Get Our E-newsletter:\n'
                      '\n'
                      'Featuring: Specials, News, Events and More!'},
 'score': 0.0,
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 'metadata': {'chunk': 6.0,
              'file_name': 'www-fairwaycapecod-com-menu-breakfast-62826.txt',
              'is_dict': 'no',
              'text': 'Get Our E-newsletter:\n'
                      '\n'
                      'Featuring: Specials, News, Events and More!\n'
                      '\n'
                      '[ '
                      '](https://www.tripadvisor.com/Restaurant%5FReview-g41555-d385199-Reviews-Fairway%5FRestaurant%5FPizzeria-Eastham%5FCape%5FCod%5FMassachusetts.html)[ '
                      '](https://www.yelp.com/biz/fairway-restaurant-and-pizzeria-north-eastham)[](https://www.facebook.com/fairwayrestaurantcapecod)\n'
                      '\n'
                      'Give Us a Review\n'
                      '\n'
                      'â\x80\x9cExcellent place with great service. Fresh food '
                      'and large servings. Love breakfast but also '
                      'dinner.â\x80\x9d\n'
                      '\n'
                      '\\-Denise K.\n'
                      '\n'
                      'The Fairway Restaurant and Pizzeria  \n'
                      '4295 State Highway North Eastham, MA 02642\n'
                      '\n'
                      '[508.255.3893](tel:508-255-3893)\n'
                      '\n'
                      'HOURS:  \n'
                      'Breakfast: 6:30 AM â\x80\x93 12:00 PM  \n'
                      'Dinner:  \n'
                      '4:00 PM â\x80\x93 8:00 PM SUN-THURS  \n'
                      '4:00 PM â\x80\x93 9:00 PM FRI-SAT\n'
                      '\n'
                      '[Privacy '
                      'Policy](https://www.fairwaycapecod.com/privacy/)\n'
                      '\n'
                      '[Sitemap](https://www.fairwaycapecod.com/sitemap/)\n'
                      '\n'
                      '© 2024 The Hole In One Group | All Rights Reserved\n'
                      '\n'
                      'PreviousNext\n'
                      '\n'
                      'View on Facebook'},
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              'file_name': 'www-fairwaycapecod-com-menu-breakfast-62826.txt',
              'is_dict': 'no',
              'text': '[508-255-3893](tel:508-255-3893) \n'
                      '\n'
                      '[ ](https://www.fairwaycapecod.com/) \n'
                      '\n'
                      '* [Home](https://www.fairwaycapecod.com/)\n'
                      '* [Menus](https://www.fairwaycapecod.com/menu/)  \n'
                      '   * [î\x84\x81 Order '
                      'Online](https://direct.chownow.com/order/20872/locations/30204)  \n'
                      '   * '
                      '[Breakfast](https://www.fairwaycapecod.com/menu/breakfast/)  \n'
                      '   * '
                      '[Dinner](https://www.fairwaycapecod.com/menu/dinner/)  \n'
                      '   * [Weekly '
                      'Specials](https://www.fairwaycapecod.com/daily-specials/)  \n'
                      '   * [Family Style '
                      'Takeout](https://www.fairwaycapecod.com/menu/home-meal-replacement/)  \n'
                      '   * [Homemade '
                      'Desserts](https://www.fairwaycapecod.com/menu/homemade-desserts/)  \n'
                      '   * [From the '
                      'Tavern](https://www.fairwaycapecod.com/menu/from-the-tavern/)  \n'
                      '   * '
                      '[Beverages](https://www.fairwaycapecod.com/menu/beverages/)\n'
                      '* [ Order '
                      'Online](https://direct.chownow.com/order/20872/locations/30204)\n'
                      '* [About '
                      'Us](https://www.fairwaycapecod.com/about-us/)  \n'
                      '   * '
                      '[History](https://www.fairwaycapecod.com/about-us/history/)  \n'
                      '   * [Photo '
                      'Gallery](https://www.fairwaycapecod.com/photo-gallery/)  \n'
                      '   * [Customer '
                      'Reviews](https://www.fairwaycapecod.com/about-us/customer-reviews/)\n'
                      '* [Location](https://www.fairwaycapecod.com/location/)\n'
                      '* [Store](https://www.fairwaycapecod.com/store/)\n'
                      '* [Contact '
                      'Us](https://www.fairwaycapecod.com/contact/)  \n'
                      '   * '
                      '[Employment](https://www.fairwaycapecod.com/contact/employment/)\n'
                      '* [The Hole in One](https://theholecapecod.com/)'},
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              'text': 'Select Page \n'
                      '* [Home](https://www.fairwaycapecod.com/)\n'
                      '* [Menus](https://www.fairwaycapecod.com/menu/)  \n'
                      '   * [î\x84\x81 Order '
                      'Online](https://direct.chownow.com/order/20872/locations/30204)  \n'
                      '   * '
                      '[Breakfast](https://www.fairwaycapecod.com/menu/breakfast/)  \n'
                      '   * '
                      '[Dinner](https://www.fairwaycapecod.com/menu/dinner/)  \n'
                      '   * [Weekly '
                      'Specials](https://www.fairwaycapecod.com/daily-specials/)  \n'
                      '   * [Family Style '
                      'Takeout](https://www.fairwaycapecod.com/menu/home-meal-replacement/)  \n'
                      '   * [Homemade '
                      'Desserts](https://www.fairwaycapecod.com/menu/homemade-desserts/)  \n'
                      '   * [From the '
                      'Tavern](https://www.fairwaycapecod.com/menu/from-the-tavern/)  \n'
                      '   * '
                      '[Beverages](https://www.fairwaycapecod.com/menu/beverages/)\n'
                      '* [ Order '
                      'Online](https://direct.chownow.com/order/20872/locations/30204)\n'
                      '* [About '
                      'Us](https://www.fairwaycapecod.com/about-us/)  \n'
                      '   * '
                      '[History](https://www.fairwaycapecod.com/about-us/history/)  \n'
                      '   * [Photo '
                      'Gallery](https://www.fairwaycapecod.com/photo-gallery/)  \n'
                      '   * [Customer '
                      'Reviews](https://www.fairwaycapecod.com/about-us/customer-reviews/)\n'
                      '* [Location](https://www.fairwaycapecod.com/location/)\n'
                      '* [Store](https://www.fairwaycapecod.com/store/)\n'
                      '* [Contact '
                      'Us](https://www.fairwaycapecod.com/contact/)  \n'
                      '   * '
                      '[Employment](https://www.fairwaycapecod.com/contact/employment/)\n'
                      '* [The Hole in One](https://theholecapecod.com/)\n'
                      '\n'
                      'Breakfast\n'
                      '\n'
                      '[Fairway Favorites](#favorites) | '
                      '[Benedicts](#benedicts) | [Egg Combos](#eggcombos) | '
                      '[Omelets](#omelets) | [Pancakes & French '
                      'Toast](#pancakes) | [Breakfast Sandwiches](#sandwiches) '
                      '| [Sides](#sides)'},
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              'file_name': 'Data%20Mining%20Concepts%20and%20Techniques%20-%20Jiawei%20Han%2C%20Micheline%20Kamber%2C%20Jian%20Pei%20%28PDF%29.pdf',
              'page': 107.0,
              'text': 'inarydissimilarity.Ifobjectsiandjaredescribedbysymmetricbinaryattributes,thentheTable2.3ContingencyTableforBinaryAttributesObjectj10sum1qrq+rObjecti0sts+tsumq+sr+tp'},
 'score': 0.0,
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              'page': 108.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page71#332.4MeasuringDataSimilarityandDissimilarity71dissimilaritybetweeniandjisd(i,j)=r+sq+r+s+t.(2.13)Forasymmetricbinaryattributes,thetwostatesarenotequallyimportant,suchasthepositive(1)andnegative(0)outcomesofadiseasetest.Giventwoasymmetricbinaryattributes,theagreementoftwo1s(apositivematch)isthenconsideredmoresignifi-cantthanthatoftwo0s(anegativematch).Therefore,suchbinaryattributesareoftenconsidered“monary”(havingonestate).Thedissimilaritybasedontheseattributesiscalledasymmetricbinarydissimilarity,wherethenumberofnegativematches,t,isconsideredunimportantandisthusignoredinthefollowingcomputation:d(i,j)=r+sq+r+s.(2.14)Complementarily,wecanmeasurethedifferencebetweentwobinaryattributesbasedonthenotionofsimilarityinsteadofdissimilarity.Forexample,theasymmetricbinarysimilaritybetweentheobjectsiandjcanbecomputedassim(i,j)=qq+r+s=1−d(i,j).(2.15)Thecoefficientsim(i,j)ofEq.(2.15)iscalledtheJaccardcoefficientandispopularlyreferencedintheliterature.Whenbothsymmetricandasymmetricbinaryattributesoccurinthesamedataset,themixedattributesapproachdescribedinSection2.4.6canbeapplied.Example2.18Dissimilaritybetweenbinaryattributes.Supposethatapatientrecordtable(Table2.4)containstheattributesname,gender,fever,cough,test-1,test-2,test-3,andtest-4,wherenameisanobjectidentifier,genderisasymmetricattribute,andtheremainingattributesareasymmetricbinary.Forasymmetricattributevalues,letthevaluesY(yes)andP(positive)besetto1,andthevalueN(noornegative)besetto0.SupposethatthedistancebetweenobjectsTable2.4RelationalTableWherePatientsAreDescribedbyBinaryAttributesnamegenderfevercoughtest-1test-2test-3test-4JackMYNPNNNJimMYYNNNNMaryFYNPNPN........................'},
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              'page': 109.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page72#3472Chapter2GettingtoKnowYourData(patients)iscomputedbasedonlyontheasymmetricattributes.AccordingtoEq.(2.14),thedistancebetweeneachpairofthethreepatients—Jack,Mary,andJim—isd(Jack,Jim)=1+11+1+1=0.67,d(Jack,Mary)=0+12+0+1=0.33,d(Jim,Mary)=1+21+1+2=0.75.ThesemeasurementssuggestthatJimandMaryareunlikelytohaveasimilardiseasebecausetheyhavethehighestdissimilarityvalueamongthethreepairs.Ofthethreepatients,JackandMaryarethemostlikelytohaveasimilardisease.2.4.4DissimilarityofNumericData:MinkowskiDistanceInthissection,wedescribedistancemeasuresthatarecommonlyusedforcomputingthedissimilarityofobjectsdescribedbynumericattributes.ThesemeasuresincludetheEuclidean,Manhattan,andMinkowskidistances.Insomecases,thedataarenormalizedbeforeapplyingdistancecalculations.Thisinvolvestransformingthedatatofallwithinasmallerorcommonrange,suchas[−1,1]or[0.0,1.0].Consideraheightattribute,forexample,whichcouldbemeasuredineithermetersorinches.Ingeneral,expressinganattributeinsmallerunitswillleadtoalargerrangeforthatattribute,andthustendtogivesuchattributesgreatereffector“weight.”Normalizingthedataattemptstogiveallattributesanequalweight.Itmayormaynotbeusefulinaparticularapplication.MethodsfornormalizingdataarediscussedindetailinChapter3ondatapreprocessing.ThemostpopulardistancemeasureisEuclideandistance(i.e.,straightlineor“asthecrowflies”).Leti=(xi1,xi2,...,xip)andj=(xj1,xj2,...,xjp)betwoobjectsdescribedbypnumericattributes.TheEuclideandistancebetweenobjectsiandjisdefinedasd(i,j)=(cid:113)(xi1−xj1)2+(xi2−xj2)2+···+(xip−xjp)2.(2.16)Anotherwell-knownmeasureistheManhattan(orcityblock)distance,namedsobecauseitisthedistanceinblocksbetweenanytwopointsinacity(suchas2blocksdownand3blocksoverforatotalof5blocks).Itisdefinedasd(i,j)=|xi1−xj1|+|xi2−xj2|+···+|xip−xjp|.(2.17)BoththeEuclideanandtheManhattandistancesatisfythefollowingmathematicalproperties:Non-negativity:d(i,j)≥0:Distanceisanon-negativenumber.Identityofindiscernibles:d(i,i)=0:Thedistanceofanob'},
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              'page': 110.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page73#352.4MeasuringDataSimilarityandDissimilarity73Symmetry:d(i,j)=d(j,i):Distanceisasymmetricfunction.Triangleinequality:d(i,j)≤d(i,k)+d(k,j):Goingdirectlyfromobjectitoobjectjinspaceisnomorethanmakingadetouroveranyotherobjectk.Ameasurethatsatisfiestheseconditionsisknownasmetric.Pleasenotethatthenon-negativitypropertyisimpliedbytheotherthreeproperties.Example2.19EuclideandistanceandManhattandistance.Letx1=(1,2)andx2=(3,5)repre-senttwoobjectsasshowninFigure2.23.TheEuclideandistancebetweenthetwois√22+32=3.61.TheManhattandistancebetweenthetwois2+3=5.MinkowskidistanceisageneralizationoftheEuclideanandManhattandistances.Itisdefinedasd(i,j)=h(cid:113)|xi1−xj1|h+|xi2−xj2|h+···+|xip−xjp|h,(2.18)wherehisarealnumbersuchthath≥1.(SuchadistanceisalsocalledLpnorminsomeliterature,wherethesymbolpreferstoournotationofh.Wehavekeptpasthenumberofattributestobeconsistentwiththerestofthischapter.)ItrepresentstheManhattandistancewhenh=1(i.e.,L1norm)andEuclideandistancewhenh=2(i.e.,L2norm).Thesupremumdistance(alsoreferredtoasLmax,L∞normandastheChebyshevdistance)isageneralizationoftheMinkowskidistanceforh→∞.Tocomputeit,wefindtheattributefthatgivesthemaximumdifferenceinvaluesbetweenthetwoobjects.Thisdifferenceisthesupremumdistance,definedmoreformallyas:d(i,j)=limh→∞\uf8eb\uf8edp(cid:88)f=1|xif−xjf|h\uf8f6\uf8f81h=pmaxf|xif−xjf|.(2.19)TheL∞normisalsoknownastheuniformnorm.1223543321x2 '
                      '= (3,5)x1 = (1,2)Euclidean distance= (22 + 32)1/2 = '
                      '3.61Manhattan distance= 2 + 3 = 5Supremum distance= 5 – '
                      '2 = '
                      '3Figure2.23Euclidean,Manhattan,andsupremumdistancesbetweentwoobjects.'},
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              'page': 111.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page74#3674Chapter2GettingtoKnowYourDataExample2.20Supremumdistance.Let’susethesametwoobjects,x1=(1,2)andx2=(3,5),asinFigure2.23.Thesecondattributegivesthegreatestdifferencebetweenvaluesfortheobjects,whichis5−2=3.Thisisthesupremumdistancebetweenbothobjects.Ifeachattributeisassignedaweightaccordingtoitsperceivedimportance,theweightedEuclideandistancecanbecomputedasd(i,j)=(cid:113)w1|xi1−xj1|2+w2|xi2−xj2|2+···+wm|xip−xjp|2.(2.20)Weightingcanalsobeappliedtootherdistancemeasuresaswell.2.4.5ProximityMeasuresforOrdinalAttributesThevaluesofanordinalattributehaveameaningfulorderorrankingaboutthem,yetthemagnitudebetweensuccessivevaluesisunknown(Section2.1.4).Anexam-pleincludesthesequencesmall,medium,largeforasizeattribute.Ordinalattributesmayalsobeobtainedfromthediscretizationofnumericattributesbysplittingthevaluerangeintoafinitenumberofcategories.Thesecategoriesareorganizedintoranks.Thatis,therangeofanumericattributecanbemappedtoanordinalattributefhavingMfstates.Forexample,therangeoftheinterval-scaledattributetemperature(inCelsius)canbeorganizedintothefollowingstates:−30to−10,−10to10,10to30,repre-sentingthecategoriescoldtemperature,moderatetemperature,andwarmtemperature,respectively.LetMrepresentthenumberofpossiblestatesthatanordinalattributecanhave.Theseorderedstatesdefinetheranking1,...,Mf.“Howareordinalattributeshandled?”Thetreatmentofordinalattributesisquitesimilartothatofnumericattributeswhencomputingdissimilaritybetweenobjects.Supposethatfisanattributefromasetofordinalattributesdescribingnobjects.Thedissimilaritycomputationwithrespecttofinvolvesthefollowingsteps:1.Thevalueofffortheithobjectisxif,andfhasMforderedstates,representingtheranking1,...,Mf.Replaceeachxifbyitscorrespondingrank,rif∈{1,...,Mf}.2.Sinceeachordinalattributecanhaveadifferentnumberofstates,itisoftennecessarytomaptherangeofeachattributeonto[0.0,1.0]sothateachattributehasequalweight.Weperformsuchdatanormalizationbyreplacingtherankrifoftheithobjectinthefthatt'},
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              'page': 111.0,
              'text': 'ithobjectinthefthattributebyzif=rif−1Mf−1.(2.21)3.DissimilaritycanthenbecomputedusinganyofthedistancemeasuresdescribedinSection2.4.4fornumericattributes,usingziftorepresentthefvaluefortheithobject.'},
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              'page': 112.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page75#372.4MeasuringDataSimilarityandDissimilarity75Example2.21Dissimilaritybetweenordinalattributes.SupposethatwehavethesampledatashownearlierinTable2.2,exceptthatthistimeonlytheobject-identifierandthecontinuousordinalattribute,test-2,areavailable.Therearethreestatesfortest-2:fair,good,andexcellent,thatis,Mf=3.Forstep1,ifwereplaceeachvaluefortest-2byitsrank,thefourobjectsareassignedtheranks3,1,2,and3,respectively.Step2normalizestherankingbymappingrank1to0.0,rank2to0.5,andrank3to1.0.Forstep3,wecanuse,say,theEuclideandistance(Eq.2.16),whichresultsinthefollowingdissimilaritymatrix:\uf8ee\uf8ef\uf8ef\uf8ef\uf8f001.000.50.5001.00.50\uf8f9\uf8fa\uf8fa\uf8fa\uf8fb.Therefore,objects1and2arethemostdissimilar,asareobjects2and4(i.e.,d(2,1)=1.0andd(4,2)=1.0).Thismakesintuitivesensesinceobjects1and4arebothexcellent.Object2isfair,whichisattheoppositeendoftherangeofvaluesfortest-2.Similarityvaluesforordinalattributescanbeinterpretedfromdissimilarityassim(i,j)=1−d(i,j).2.4.6DissimilarityforAttributesofMixedTypesSections2.4.2through2.4.5discussedhowtocomputethedissimilaritybetweenobjectsdescribedbyattributesofthesametype,wherethesetypesmaybeeithernominal,sym-metricbinary,asymmetricbinary,numeric,orordinal.However,inmanyrealdatabases,objectsaredescribedbyamixtureofattributetypes.Ingeneral,adatabasecancontainalloftheseattributetypes.“So,howcanwecomputethedissimilaritybetweenobjectsofmixedattributetypes?”Oneapproachistogroupeachtypeofattributetogether,performingseparatedatamining(e.g.,clustering)analysisforeachtype.Thisisfeasibleiftheseanalysesderivecompatibleresults.However,inrealapplications,itisunlikelythataseparateanalysisperattributetypewillgeneratecompatibleresults.Amorepreferableapproachistoprocessallattributetypestogether,performingasingleanalysis.Onesuchtechniquecombinesthedifferentattributesintoasingledis-similaritymatrix,bringingallofthemeaningfulattributesontoacommonscaleoftheinterval[0.0,1.0].Supposethatthedatasetcontainspattributesofmixedtype.Thedissimilar'},
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              'page': 112.0,
              'text': 'edtype.Thedissimilarityd(i,j)betweenobjectsiandjisdefinedasd(i,j)=(cid:80)pf=1δ(f)ijd(f)ij(cid:80)pf=1δ(f)ij,(2.22)'},
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              'page': 113.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page76#3876Chapter2GettingtoKnowYourDatawheretheindicatorδ(f)ij=0ifeither(1)xiforxjfismissing(i.e.,thereisnomea-surementofattributefforobjectiorobjectj),or(2)xif=xjf=0andattributefisasymmetricbinary;otherwise,δ(f)ij=1.Thecontributionofattributeftothedissimilaritybetweeniandj(i.e.,d(f)ij)iscomputeddependentonitstype:Iffisnumeric:d(f)ij=|xif−xjf|maxhxhf−minhxhf,wherehrunsoverallnonmissingobjectsforattributef.Iffisnominalorbinary:d(f)ij=0ifxif=xjf;otherwise,d(f)ij=1.Iffisordinal:computetheranksrifandzif=rif−1Mf−1,andtreatzifasnumeric.Thesestepsareidenticaltowhatwehavealreadyseenforeachoftheindividualattributetypes.Theonlydifferenceisfornumericattributes,wherewenormalizesothatthevaluesmaptotheinterval[0.0,1.0].Thus,thedissimilaritybetweenobjectscanbecomputedevenwhentheattributesdescribingtheobjectsareofdifferenttypes.Example2.22Dissimilaritybetweenattributesofmixedtype.Let’scomputeadissimilaritymatrixfortheobjectsinTable2.2.Nowwewillconsideralloftheattributes,whichareofdifferenttypes.InExamples2.17and2.21,weworkedoutthedissimilaritymatricesforeachoftheindividualattributes.Theprocedureswefollowedfortest-1(whichisnominal)andtest-2(whichisordinal)arethesameasoutlinedearlierforprocessingattributesofmixedtypes.Therefore,wecanusethedissimilaritymatricesobtainedfortest-1andtest-2laterwhenwecomputeEq.(2.22).First,however,weneedtocomputethedissimilaritymatrixforthethirdattribute,test-3(whichisnumeric).Thatis,wemustcomputed(3)ij.Followingthecasefornumericattributes,weletmaxhxh=64andminhxh=22.ThedifferencebetweenthetwoisusedinEq.(2.22)tonormalizethevaluesofthedissimilaritymatrix.Theresultingdissimilaritymatrixfortest-3is\uf8ee\uf8ef\uf8ef\uf8ef\uf8f000.5500.451.0000.400.140.860\uf8f9\uf8fa\uf8fa\uf8fa\uf8fb.WecannowusethedissimilaritymatricesforthethreeattributesinourcomputationofEq.(2.22).Theindicatorδ(f)ij=1foreachofthethreeattributes,f.Weget,forexample,d(3,1)=1(1)+1(0.50)+1(0.45)3=0.65.Theresultingdissimilaritymatrixobtainedforthe'},
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              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page77#392.4MeasuringDataSimilarityandDissimilarity77datadescribedbythethreeattributesofmixedtypesis:\uf8ee\uf8ef\uf8ef\uf8ef\uf8f000.8500.650.8300.130.710.790\uf8f9\uf8fa\uf8fa\uf8fa\uf8fb.FromTable2.2,wecanintuitivelyguessthatobjects1and4arethemostsimilar,basedontheirvaluesfortest-1andtest-2.Thisisconfirmedbythedissimilaritymatrix,whered(4,1)isthelowestvalueforanypairofdifferentobjects.Similarly,thematrixindicatesthatobjects1and2aretheleastsimilar.2.4.7CosineSimilarityAdocumentcanberepresentedbythousandsofattributes,eachrecordingthefrequencyofaparticularword(suchasakeyword)orphraseinthedocument.Thus,eachdocu-mentisanobjectrepresentedbywhatiscalledaterm-frequencyvector.Forexample,inTable2.5,weseethatDocument1containsfiveinstancesofthewordteam,whilehockeyoccursthreetimes.Thewordcoachisabsentfromtheentiredocument,asindicatedbyacountvalueof0.Suchdatacanbehighlyasymmetric.Term-frequencyvectorsaretypicallyverylongandsparse(i.e.,theyhavemany0val-ues).Applicationsusingsuchstructuresincludeinformationretrieval,textdocumentclustering,biologicaltaxonomy,andgenefeaturemapping.Thetraditionaldistancemeasuresthatwehavestudiedinthischapterdonotworkwellforsuchsparsenumericdata.Forexample,twoterm-frequencyvectorsmayhavemany0valuesincommon,meaningthatthecorrespondingdocumentsdonotsharemanywords,butthisdoesnotmakethemsimilar.Weneedameasurethatwillfocusonthewordsthatthetwodocu-mentsdohaveincommon,andtheoccurrencefrequencyofsuchwords.Inotherwords,weneedameasurefornumericdatathatignoreszero-matches.Cosinesimilarityisameasureofsimilaritythatcanbeusedtocomparedocu-mentsor,say,givearankingofdocumentswithrespecttoagivenvectorofquerywords.Letxandybetwovectorsforcomparison.UsingthecosinemeasureasaTable2.5DocumentVectororTerm-FrequencyVectorDocumentteamcoachhockeybaseballsoccerpenaltyscorewinlossseasonDocument15030200200Document23020110101Document30702100300Document40100122030'},
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              'page': 115.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page78#4078Chapter2GettingtoKnowYourDatasimilarityfunction,wehavesim(x,y)=x·y||x||||y||,(2.23)where||x||istheEuclideannormofvectorx=(x1,x2,...,xp),definedas(cid:113)x21+x22+···+x2p.Conceptually,itisthelengthofthevector.Similarly,||y||istheEuclideannormofvectory.Themeasurecomputesthecosineoftheanglebetweenvec-torsxandy.Acosinevalueof0meansthatthetwovectorsareat90degreestoeachother(orthogonal)andhavenomatch.Thecloserthecosinevalueto1,thesmallertheangleandthegreaterthematchbetweenvectors.NotethatbecausethecosinesimilaritymeasuredoesnotobeyallofthepropertiesofSection2.4.4definingmetricmeasures,itisreferredtoasanonmetricmeasure.Example2.23Cosinesimilaritybetweentwoterm-frequencyvectors.Supposethatxandyarethefirsttwoterm-frequencyvectorsinTable2.5.Thatis,x=(5,0,3,0,2,0,0,2,0,0)andy=(3,0,2,0,1,1,0,1,0,1).Howsimilararexandy?UsingEq.(2.23)tocomputethecosinesimilaritybetweenthetwovectors,weget:xt·y=5×3+0×0+3×2+0×0+2×1+0×1+0×0+2×1+0×0+0×1=25||x||=(cid:112)52+02+32+02+22+02+02+22+02+02=6.48||y||=(cid:112)32+02+22+02+12+12+02+12+02+12=4.12sim(x,y)=0.94Therefore,ifwewereusingthecosinesimilaritymeasuretocomparethesedocuments,theywouldbeconsideredquitesimilar.Whenattributesarebinary-valued,thecosinesimilarityfunctioncanbeinterpretedintermsofsharedfeaturesorattributes.Supposeanobjectxpossessestheithattributeifxi=1.Thenxt·yisthenumberofattributespossessed(i.e.,shared)bybothxandy,and|x||y|isthegeometricmeanofthenumberofattributespossessedbyxandthenumberpossessedbyy.Thus,sim(x,y)isameasureofrelativepossessionofcommonattributes.Asimplevariationofcosinesimilarityfortheprecedingscenarioissim(x,y)=x·yx·x+y·y−x·y,(2.24)whichistheratioofthenumberofattributessharedbyxandytothenumberofattributespossessedbyxory.Thisfunction,knownastheTanimotocoefficientorTanimotodistance,isfrequentlyusedininformationretrievalandbiologytaxonomy.'},
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              'page': 116.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page79#412.6Exercises792.5SummaryDatasetsaremadeupofdataobjects.Adataobjectrepresentsanentity.Dataobjectsaredescribedbyattributes.Attributescanbenominal,binary,ordinal,ornumeric.Thevaluesofanominal(orcategorical)attributearesymbolsornamesofthings,whereeachvaluerepresentssomekindofcategory,code,orstate.Binaryattributesarenominalattributeswithonlytwopossiblestates(suchas1and0ortrueandfalse).Ifthetwostatesareequallyimportant,theattributeissymmetric;otherwiseitisasymmetric.Anordinalattributeisanattributewithpossiblevaluesthathaveameaningfulorderorrankingamongthem,butthemagnitudebetweensuccessivevaluesisnotknown.Anumericattributeisquantitative(i.e.,itisameasurablequantity)representedinintegerorrealvalues.Numericattributetypescanbeinterval-scaledorratio-scaled.Thevaluesofaninterval-scaledattributearemeasuredinfixedandequalunits.Ratio-scaledattributesarenumericattributeswithaninherentzero-point.Measurementsareratio-scaledinthatwecanspeakofvaluesasbeinganorderofmagnitudelargerthantheunitofmeasurement.Basicstatisticaldescriptionsprovidetheanalyticalfoundationfordatapreprocess-ing.Thebasicstatisticalmeasuresfordatasummarizationincludemean,weightedmean,median,andmodeformeasuringthecentraltendencyofdata;andrange,quan-tiles,quartiles,interquartilerange,variance,andstandarddeviationformeasuringthedispersionofdata.Graphicalrepresentations(e.g.,boxplots,quantileplots,quantile–quantileplots,histograms,andscatterplots)facilitatevisualinspectionofthedataandarethususefulfordatapreprocessingandmining.Datavisualizationtechniquesmaybepixel-oriented,geometric-based,icon-based,orhierarchical.Thesemethodsapplytomultidimensionalrelationaldata.Additionaltechniqueshavebeenproposedforthevisualizationofcomplexdata,suchastextandsocialnetworks.Measuresofobjectsimilarityanddissimilarityareusedindataminingapplicationssuchasclustering,outlieranalysis,andnearest-neighborclassification.Suchmea-suresofproximitycanbecomputedforeachattributetypestudiedinthischapt'},
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              'page': 116.0,
              'text': 'pestudiedinthischapter,orforcombinationsofsuchattributes.ExamplesincludetheJaccardcoefficientforasymmetricbinaryattributesandEuclidean,Manhattan,Minkowski,andsupremumdistancesfornumericattributes.Forapplicationsinvolvingsparsenumericdatavec-tors,suchasterm-frequencyvectors,thecosinemeasureandtheTanimotocoefficientareoftenusedintheassessmentofsimilarity.2.6Exercises2.1Givethreeadditionalcommonlyusedstatisticalmeasuresthatarenotalreadyillus-tratedinthischapterforthecharacterizationofdatadispersion.Discusshowtheycanbecomputedefficientlyinlargedatabases.'},
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              'page': 116.0,
              'text': 'pestudiedinthischapter,orforcombinationsofsuchattributes.ExamplesincludetheJaccardcoefficientforasymmetricbinaryattributesandEuclidean,Manhattan,Minkowski,andsupremumdistancesfornumericattributes.Forapplicationsinvolvingsparsenumericdatavec-tors,suchasterm-frequencyvectors,thecosinemeasureandtheTanimotocoefficientareoftenusedintheassessmentofsimilarity.2.6Exercises2.1Givethreeadditionalcommonlyusedstatisticalmeasuresthatarenotalreadyillus-tratedinthischapterforthecharacterizationofdatadispersion.Discusshowtheycanbecomputedefficientlyinlargedatabases.'},
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              'page': 117.0,
              'text': '# Chapter 2 Getting to Know Your Data\n'
                      '\n'
                      '## 2.2 Data Analysis\n'
                      '\n'
                      'Suppose that the data for analysis includes the '
                      'attribute `age`. The age values for the data tuples are '
                      '(in increasing order) 13, 15, 16, 16, 19, 20, 21, 22, '
                      '22, 25, 25, 30, 33, 33, 35, 35, 35, 36, 40, 45, 46, 52, '
                      '70.\n'
                      '\n'
                      '1. **What is the mean of the data?**\n'
                      '2. **What is the median?**\n'
                      '3. **What is the mode of the data?** Comment on the '
                      "data's modality (i.e., bimodal, trimodal, etc.).\n"
                      '4. **What is the midrange of the data?**\n'
                      '5. **Can you find (roughly) the first quartile (Q1) and '
                      'the third quartile (Q3) of the data?**\n'
                      '6. **Give the five-number summary of the data.**\n'
                      '7. **Show a boxplot of the data.**\n'
                      '8. **How is a quantile–quantile plot different from a '
                      'quantile plot?**\n'
                      '\n'
                      '## 2.3 Frequency Distribution\n'
                      '\n'
                      'Suppose that the values for a given set of data are '
                      'grouped into intervals. The intervals and corresponding '
                      'frequencies are as follows:\n'
                      '\n'
                      '| Age   | Frequency |\n'
                      '|-------|-----------|\n'
                      '| 1–5   | 200       |\n'
                      '| 6–15  | 450       |\n'
                      '| 16–20 | 300       |\n'
                      '| 21–50 | 1500      |\n'
                      '| 51–80 | 700       |\n'
                      '| 81–110| 44        |\n'
                      '\n'
                      '**Compute an approximate median value for the data.**\n'
                      '\n'
                      '## 2.4 Age and Body Fat Data\n'
                      '\n'
                      'Suppose that a hospital tested the age and body fat '
                      'data for 18 randomly selected adults with the following '
                      'results:\n'
                      '\n'
                      '| Age |    |    |   |   |   |   |   |   |   |   |   |   '
                      '|   |   |   |   |   |\n'
                      '|-----|----|----|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n'
                      '|     | 23 | 23 | 27| 27| 39| 41| 47| 49| 50|   |   |   '
                      '|   |   |   |   |   |\n'
                      '| %fat| 9.5| 26.5| 7.8| 17.8| 31.4| 25.9| 27.4| 27.2| '
                      '31.2|   |   |   |   |   |   |   |   |\n'
                      '\n'
                      '| Age |    |    |   |   |   |   |   |   |   |   |   |   '
                      '|   |   |   |   |   |\n'
                      '|-----|----|----|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n'
                      '|     | 52 | 54 | 54| 56| 57| 58| 58| 60| 61|   |   |   '
                      '|   |   |   |   |   |\n'
                      '| %fat| 34.6| 42.5| 28.8| 33.4| 30.2| 34.1| 32.9| 41.2| '
                      '35.7|   |   |   |   |   |   |   |   |\n'
                      '\n'
                      '1. **Calculate the mean, median, and standard deviation '
                      'of age and %fat.**\n'
                      '2. **Draw the boxplots for age and %fat.**\n'
                      '3. **Draw a scatter plot and a q-q plot based on these '
                      'two variables.**\n'
                      '\n'
                      '## 2.5 Dissimilarity of Objects\n'
                      '\n'
                      'Briefly outline how to compute the dissimilarity '
                      'between objects described by the following:\n'
                      '\n'
                      '- **Nominal attributes**\n'
                      '- **Asymmetric binary attributes**'},
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              'page': 117.0,
              'text': '# Chapter 2: Getting to Know Your Data\n'
                      '\n'
                      '## 2.2 \n'
                      '\n'
                      'Suppose that the data for analysis includes the '
                      'attribute **age**. The age values for the data tuples '
                      'are (in increasing order) 13, 15, 16, 16, 19, 20, 21, '
                      '22, 22, 25, 25, 30, 33, 33, 35, 35, 35, 36, 40, 45, 46, '
                      '52, 70.\n'
                      '\n'
                      '(a) What is the **mean** of the data? What is the '
                      '**median**?  \n'
                      '(b) What is the **mode** of the data? Comment on the '
                      "data's modality (i.e., bimodal, trimodal, etc.).  \n"
                      '(c) What is the **range** of the data?  \n'
                      '(d) Can you find (roughly) the first quartile (Q1) and '
                      'the third quartile (Q3) of the data?  \n'
                      '(e) Give the **five-number summary** of the data.  \n'
                      '(f) Show a **boxplot** of the data.  \n'
                      '(g) How is a **quantile–quantile plot** different from '
                      'a **quantile plot**?\n'
                      '\n'
                      '## 2.3 \n'
                      '\n'
                      'Suppose that the values for a given set of data are '
                      'grouped into intervals. The intervals and corresponding '
                      'frequencies are as follows:\n'
                      '\n'
                      '| age      | frequency |\n'
                      '|----------|-----------|\n'
                      '| 1–5      | 200       |\n'
                      '| 6–15     | 450       |\n'
                      '| 16–20    | 300       |\n'
                      '| 21–50    | 1500      |\n'
                      '| 51–80    | 700       |\n'
                      '| 81–110   | 44        |\n'
                      '\n'
                      'Compute an **approximate median** value for the data.\n'
                      '\n'
                      '## 2.4 \n'
                      '\n'
                      'Suppose that a hospital tested the age and body fat '
                      'data for 18 randomly selected adults with the following '
                      'results:\n'
                      '\n'
                      '| age | 23 | 23 | 27 | 27 | 39 | 41 | 47 | 49 | 50 |\n'
                      '|-----|----|----|----|----|----|----|----|----|----|\n'
                      '| %fat | 9.5 | 26.5 | 7.8 | 17.8 | 31.4 | 25.9 | 27.4 | '
                      '27.2 | 31.2 |\n'
                      '\n'
                      '| age | 52 | 54 | 54 | 56 | 57 | 58 | 58 | 60 | 61 |\n'
                      '|-----|----|----|----|----|----|----|----|----|----|\n'
                      '| %fat | 34.6 | 42.5 | 28.8 | 33.4 | 30.2 | 34.1 | 32.9 '
                      '| 41.2 | 35.7 |\n'
                      '\n'
                      '(a) Calculate the mean, median, and standard deviation '
                      'of age and %fat.  \n'
                      '(b) Draw the **boxplots** for age and %fat.  \n'
                      '(c) Draw a **scatter plot** and a **q-q plot** based on '
                      'these two variables.\n'
                      '\n'
                      '## 2.5 \n'
                      '\n'
                      'Briefly outline how to compute the dissimilarity '
                      'between objects described by the following:  \n'
                      '(a) **Nominal attributes**  \n'
                      '(b) **Asymmetric binary attributes**'},
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              'page': 118.0,
              'text': '## 2.7 Bibliographic Notes\n'
                      '\n'
                      '### 2.6\n'
                      '\n'
                      'Given two objects represented by the tuples (22, 1, 42, '
                      '10) and (20, 0, 36, 8):\n'
                      '\n'
                      '1. Compute the **Euclidean distance** between the two '
                      'objects.\n'
                      '2. Compute the **Manhattan distance** between the two '
                      'objects.\n'
                      '3. Compute the **Minkowski distance** between the two '
                      'objects, using \\( q = 3 \\).\n'
                      '4. Compute the **supremum distance** between the two '
                      'objects.\n'
                      '\n'
                      '### 2.7\n'
                      '\n'
                      'The median is one of the most important holistic '
                      'measures in data analysis. Propose several methods for '
                      'median approximation. Analyze their respective '
                      'complexity under different parameter settings and '
                      'decide to what extent the real value can be '
                      'approximated. Moreover, suggest a heuristic strategy to '
                      'balance between accuracy and complexity and then apply '
                      'it to all methods you have given.\n'
                      '\n'
                      '### 2.8\n'
                      '\n'
                      'It is important to define or select similarity measures '
                      'in data analysis. However, there is no commonly '
                      'accepted subjective similarity measure. Results can '
                      'vary depending on the similarity measures used. '
                      'Nonetheless, seemingly different similarity measures '
                      'may be equivalent after some transformation.\n'
                      '\n'
                      'Suppose we have the following 2-D data set:\n'
                      '\n'
                      '|   A₁ |   A₂ |\n'
                      '|------|------|\n'
                      '| x₁   | 1.5  | 1.7  |\n'
                      '| x₂   | 2.2  | 1.9  |\n'
                      '| x₃   | 1.6  | 1.8  |\n'
                      '| x₄   | 1.2  | 1.5  |\n'
                      '| x₅   | 1.5  | 1.0  |\n'
                      '\n'
                      '1. Consider the data as 2-D data points. Given a new '
                      'data point, \\( x = (1.4, 1.6) \\) as a query, rank the '
                      'database points based on similarity with the query '
                      'using Euclidean distance, Manhattan distance, supremum '
                      'distance, and cosine similarity.\n'
                      '2. Normalize the data set to make the norm of each data '
                      'point equal to 1. Use Euclidean distance on the '
                      'transformed data to rank the data points.\n'
                      '\n'
                      '### 2.7 Bibliographic Notes\n'
                      '\n'
                      'Methods for descriptive data summarization have been '
                      'studied in the statistics literature long before the '
                      'onset of computers. Good summaries of statistical '
                      'descriptive data mining methods include Freedman, '
                      'Pisani, and Purves [FP97] and Devore [Dev95].'},
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              'page': 118.0,
              'text': '# 2.7 Bibliographic Notes\n'
                      '\n'
                      '## 2.6\n'
                      'Given two objects represented by the tuples (22, 1, 42, '
                      '10) and (20, 0, 36, 8):\n'
                      '\n'
                      '1. Compute the Euclidean distance between the two '
                      'objects.\n'
                      '2. Compute the Manhattan distance between the two '
                      'objects.\n'
                      '3. Compute the Minkowski distance between the two '
                      'objects, using \\( q = 3 \\).\n'
                      '4. Compute the supremum distance between the two '
                      'objects.\n'
                      '\n'
                      '## 2.7\n'
                      'The median is one of the most important holistic '
                      'measures in data analysis. Propose several methods for '
                      'median approximation. Analyze their respective '
                      'complexity under different parameter settings and '
                      'decide to what extent the real value can be '
                      'approximated. Moreover, suggest a heuristic strategy to '
                      'balance between accuracy and complexity and then apply '
                      'it to all methods you have given.\n'
                      '\n'
                      '## 2.8\n'
                      'It is important to define or select similarity measures '
                      'in data analysis. However, there is no commonly '
                      'accepted subjective similarity measure. Results can '
                      'vary depending on the similarity measures used. '
                      'Nonetheless, seemingly different similarity measures '
                      'may be equivalent after some transformation.\n'
                      '\n'
                      'Suppose we have the following 2-D data set:\n'
                      '\n'
                      '|     |  A₁   |  A₂   |\n'
                      '|-----|-------|-------|\n'
                      '| x₁  |  1.5  |  1.7  |\n'
                      '| x₂  |  2.2  |  1.9  |\n'
                      '| x₃  |  1.6  |  1.8  |\n'
                      '| x₄  |  1.2  |  1.5  |\n'
                      '| x₅  |  1.5  |  1.0  |\n'
                      '\n'
                      '1. Consider the data as 2-D data points. Given a new '
                      'data point, \\( x = (1.4, 1.6) \\) as a query, rank the '
                      'database points based on similarity with the query '
                      'using Euclidean distance, Manhattan distance, supremum '
                      'distance, and cosine similarity.\n'
                      '2. Normalize the data set to make the norm of each data '
                      'point equal to 1. Use Euclidean distance on the '
                      'transformed data to rank the data points.'},
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              'page': 119.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page82#4482Chapter2GettingtoKnowYourDatastatistics-basedvisualizationofdatausingboxplots,quantileplots,quantile–quantileplots,scatterplots,andloesscurves,seeCleveland[Cle93].PioneeringworkondatavisualizationtechniquesisdescribedinTheVisualDis-playofQuantitativeInformation[Tuf83],EnvisioningInformation[Tuf90],andVisualExplanations:ImagesandQuantities,EvidenceandNarrative[Tuf97],allbyTufte,inadditiontoGraphicsandGraphicInformationProcessingbyBertin[Ber81],VisualizingDatabyCleveland[Cle93],andInformationVisualizationinDataMiningandKnowledgeDiscoveryeditedbyFayyad,Grinstein,andWierse[FGW01].MajorconferencesandsymposiumsonvisualizationincludeACMHumanFactorsinComputingSystems(CHI),Visualization,andtheInternationalSymposiumonInfor-mationVisualization.ResearchonvisualizationisalsopublishedinTransactionsonVisualizationandComputerGraphics,JournalofComputationalandGraphicalStatistics,andIEEEComputerGraphicsandApplications.Manygraphicaluserinterfacesandvisualizationtoolshavebeendevelopedandcanbefoundinvariousdataminingproducts.Severalbooksondatamining(e.g.,DataMiningSolutionsbyWestphalandBlaxton[WB98])presentmanygoodexamplesandvisualsnapshots.Forasurveyofvisualizationtechniques,see“Visualtechniquesforexploringdatabases”byKeim[Kei97].Similarityanddistancemeasuresamongvariousvariableshavebeenintroducedinmanytextbooksthatstudyclusteranalysis,includingHartigan[Har75];JainandDubes[JD88];KaufmanandRousseeuw[KR90];andArabie,Hubert,anddeSoete[AHS96].MethodsforcombiningattributesofdifferenttypesintoasingledissimilaritymatrixwereintroducedbyKaufmanandRousseeuw[KR90].'},
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              'page': 119.0,
              'text': 'HAN09-ch02-039-082-97801238147912011/6/13:15Page82#4482Chapter2GettingtoKnowYourDatastatistics-basedvisualizationofdatausingboxplots,quantileplots,quantile–quantileplots,scatterplots,andloesscurves,seeCleveland[Cle93].PioneeringworkondatavisualizationtechniquesisdescribedinTheVisualDis-playofQuantitativeInformation[Tuf83],EnvisioningInformation[Tuf90],andVisualExplanations:ImagesandQuantities,EvidenceandNarrative[Tuf97],allbyTufte,inadditiontoGraphicsandGraphicInformationProcessingbyBertin[Ber81],VisualizingDatabyCleveland[Cle93],andInformationVisualizationinDataMiningandKnowledgeDiscoveryeditedbyFayyad,Grinstein,andWierse[FGW01].MajorconferencesandsymposiumsonvisualizationincludeACMHumanFactorsinComputingSystems(CHI),Visualization,andtheInternationalSymposiumonInfor-mationVisualization.ResearchonvisualizationisalsopublishedinTransactionsonVisualizationandComputerGraphics,JournalofComputationalandGraphicalStatistics,andIEEEComputerGraphicsandApplications.Manygraphicaluserinterfacesandvisualizationtoolshavebeendevelopedandcanbefoundinvariousdataminingproducts.Severalbooksondatamining(e.g.,DataMiningSolutionsbyWestphalandBlaxton[WB98])presentmanygoodexamplesandvisualsnapshots.Forasurveyofvisualizationtechniques,see“Visualtechniquesforexploringdatabases”byKeim[Kei97].Similarityanddistancemeasuresamongvariousvariableshavebeenintroducedinmanytextbooksthatstudyclusteranalysis,includingHartigan[Har75];JainandDubes[JD88];KaufmanandRousseeuw[KR90];andArabie,Hubert,anddeSoete[AHS96].MethodsforcombiningattributesofdifferenttypesintoasingledissimilaritymatrixwereintroducedbyKaufmanandRousseeuw[KR90].'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page88#688Chapter3DataPreprocessing3.2DataCleaningReal-worlddatatendtobeincomplete,noisy,andinconsistent.Datacleaning(ordatacleansing)routinesattempttofillinmissingvalues,smoothoutnoisewhileidenti-fyingoutliers,andcorrectinconsistenciesinthedata.Inthissection,youwillstudybasicmethodsfordatacleaning.Section3.2.1looksatwaysofhandlingmissingvalues.Section3.2.2explainsdatasmoothingtechniques.Section3.2.3discussesapproachestodatacleaningasaprocess.3.2.1MissingValuesImaginethatyouneedtoanalyzeAllElectronicssalesandcustomerdata.Younotethatmanytupleshavenorecordedvalueforseveralattributessuchascustomerincome.Howcanyougoaboutfillinginthemissingvaluesforthisattribute?Let’slookatthefollowingmethods.1.Ignorethetuple:Thisisusuallydonewhentheclasslabelismissing(assumingtheminingtaskinvolvesclassification).Thismethodisnotveryeffective,unlessthetuplecontainsseveralattributeswithmissingvalues.Itisespeciallypoorwhenthepercent-ageofmissingvaluesperattributevariesconsiderably.Byignoringthetuple,wedonotmakeuseoftheremainingattributes’valuesinthetuple.Suchdatacouldhavebeenusefultothetaskathand.2.Fillinthemissingvaluemanually:Ingeneral,thisapproachistimeconsumingandmaynotbefeasiblegivenalargedatasetwithmanymissingvalues.3.Useaglobalconstanttofillinthemissingvalue:Replaceallmissingattributevaluesbythesameconstantsuchasalabellike“Unknown”or−∞.Ifmissingvaluesarereplacedby,say,“Unknown,”thentheminingprogrammaymistakenlythinkthattheyformaninterestingconcept,sincetheyallhaveavalueincommon—thatof“Unknown.”Hence,althoughthismethodissimple,itisnotfoolproof.4.Useameasureofcentraltendencyfortheattribute(e.g.,themeanormedian)tofillinthemissingvalue:Chapter2discussedmeasuresofcentraltendency,whichindicatethe“middle”valueofadatadistribution.Fornormal(symmetric)datadis-tributions,themeancanbeused,whileskeweddatadistributionshouldemploythemedian(Section2.2).Forexample,supposethatthedatadistributionregard-ingtheincomeofAllElectronicscustomersissymmetricandthatt'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page88#688Chapter3DataPreprocessing3.2DataCleaningReal-worlddatatendtobeincomplete,noisy,andinconsistent.Datacleaning(ordatacleansing)routinesattempttofillinmissingvalues,smoothoutnoisewhileidenti-fyingoutliers,andcorrectinconsistenciesinthedata.Inthissection,youwillstudybasicmethodsfordatacleaning.Section3.2.1looksatwaysofhandlingmissingvalues.Section3.2.2explainsdatasmoothingtechniques.Section3.2.3discussesapproachestodatacleaningasaprocess.3.2.1MissingValuesImaginethatyouneedtoanalyzeAllElectronicssalesandcustomerdata.Younotethatmanytupleshavenorecordedvalueforseveralattributessuchascustomerincome.Howcanyougoaboutfillinginthemissingvaluesforthisattribute?Let’slookatthefollowingmethods.1.Ignorethetuple:Thisisusuallydonewhentheclasslabelismissing(assumingtheminingtaskinvolvesclassification).Thismethodisnotveryeffective,unlessthetuplecontainsseveralattributeswithmissingvalues.Itisespeciallypoorwhenthepercent-ageofmissingvaluesperattributevariesconsiderably.Byignoringthetuple,wedonotmakeuseoftheremainingattributes’valuesinthetuple.Suchdatacouldhavebeenusefultothetaskathand.2.Fillinthemissingvaluemanually:Ingeneral,thisapproachistimeconsumingandmaynotbefeasiblegivenalargedatasetwithmanymissingvalues.3.Useaglobalconstanttofillinthemissingvalue:Replaceallmissingattributevaluesbythesameconstantsuchasalabellike“Unknown”or−∞.Ifmissingvaluesarereplacedby,say,“Unknown,”thentheminingprogrammaymistakenlythinkthattheyformaninterestingconcept,sincetheyallhaveavalueincommon—thatof“Unknown.”Hence,althoughthismethodissimple,itisnotfoolproof.4.Useameasureofcentraltendencyfortheattribute(e.g.,themeanormedian)tofillinthemissingvalue:Chapter2discussedmeasuresofcentraltendency,whichindicatethe“middle”valueofadatadistribution.Fornormal(symmetric)datadis-tributions,themeancanbeused,whileskeweddatadistributionshouldemploythemedian(Section2.2).Forexample,supposethatthedatadistributionregard-ingtheincomeofAllElectronicscustomersissymmetricandthatt'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page89#73.2DataCleaning89induction.Forexample,usingtheothercustomerattributesinyourdataset,youmayconstructadecisiontreetopredictthemissingvaluesforincome.DecisiontreesandBayesianinferencearedescribedindetailinChapters8and9,respectively,whileregressionisintroducedinSection3.4.5.Methods3through6biasthedata—thefilled-invaluemaynotbecorrect.Method6,however,isapopularstrategy.Incomparisontotheothermethods,itusesthemostinformationfromthepresentdatatopredictmissingvalues.Byconsideringtheotherattributes’valuesinitsestimationofthemissingvalueforincome,thereisagreaterchancethattherelationshipsbetweenincomeandtheotherattributesarepreserved.Itisimportanttonotethat,insomecases,amissingvaluemaynotimplyanerrorinthedata!Forexample,whenapplyingforacreditcard,candidatesmaybeaskedtosupplytheirdriver’slicensenumber.Candidateswhodonothaveadriver’slicensemaynaturallyleavethisfieldblank.Formsshouldallowrespondentstospecifyvaluessuchas“notapplicable.”Softwareroutinesmayalsobeusedtouncoverothernullvalues(e.g.,“don’tknow,”“?”or“none”).Ideally,eachattributeshouldhaveoneormorerulesregardingthenullcondition.Therulesmayspecifywhetherornotnullsareallowedand/orhowsuchvaluesshouldbehandledortransformed.Fieldsmayalsobeinten-tionallyleftblankiftheyaretobeprovidedinalaterstepofthebusinessprocess.Hence,althoughwecantryourbesttocleanthedataafteritisseized,gooddatabaseanddataentryproceduredesignshouldhelpminimizethenumberofmissingvaluesorerrorsinthefirstplace.3.2.2NoisyData“Whatisnoise?”Noiseisarandomerrororvarianceinameasuredvariable.InChapter2,wesawhowsomebasicstatisticaldescriptiontechniques(e.g.,boxplotsandscatterplots),andmethodsofdatavisualizationcanbeusedtoidentifyoutliers,whichmayrepresentnoise.Givenanumericattributesuchas,say,price,howcanwe“smooth”outthedatatoremovethenoise?Let’slookatthefollowingdatasmoothingtechniques.Binning:Binningmethodssmoothasorteddatavaluebyconsultingits“neighbor-hood,”thatis,thevaluesaroundit.Thesortedvaluesaredistributedintoa'},
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              'text': 'saredistributedintoanumberof“buckets,”orbins.Becausebinningmethodsconsulttheneighborhoodofvalues,theyperformlocalsmoothing.Figure3.2illustratessomebinningtechniques.Inthisexample,thedataforpricearefirstsortedandthenpartitionedintoequal-frequencybinsofsize3(i.e.,eachbincontainsthreevalues).Insmoothingbybinmeans,eachvalueinabinisreplacedbythemeanvalueofthebin.Forexample,themeanofthevalues4,8,and15inBin1is9.Therefore,eachoriginalvalueinthisbinisreplacedbythevalue9.Similarly,smoothingbybinmedianscanbeemployed,inwhicheachbinvalueisreplacedbythebinmedian.Insmoothingbybinboundaries,theminimumandmaximumvaluesinagivenbinareidentifiedasthebinboundaries.Eachbinvalueisthenreplacedbytheclosestboundaryvalue.Ingeneral,thelargerthewidth,the'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page89#73.2DataCleaning89induction.Forexample,usingtheothercustomerattributesinyourdataset,youmayconstructadecisiontreetopredictthemissingvaluesforincome.DecisiontreesandBayesianinferencearedescribedindetailinChapters8and9,respectively,whileregressionisintroducedinSection3.4.5.Methods3through6biasthedata—thefilled-invaluemaynotbecorrect.Method6,however,isapopularstrategy.Incomparisontotheothermethods,itusesthemostinformationfromthepresentdatatopredictmissingvalues.Byconsideringtheotherattributes’valuesinitsestimationofthemissingvalueforincome,thereisagreaterchancethattherelationshipsbetweenincomeandtheotherattributesarepreserved.Itisimportanttonotethat,insomecases,amissingvaluemaynotimplyanerrorinthedata!Forexample,whenapplyingforacreditcard,candidatesmaybeaskedtosupplytheirdriver’slicensenumber.Candidateswhodonothaveadriver’slicensemaynaturallyleavethisfieldblank.Formsshouldallowrespondentstospecifyvaluessuchas“notapplicable.”Softwareroutinesmayalsobeusedtouncoverothernullvalues(e.g.,“don’tknow,”“?”or“none”).Ideally,eachattributeshouldhaveoneormorerulesregardingthenullcondition.Therulesmayspecifywhetherornotnullsareallowedand/orhowsuchvaluesshouldbehandledortransformed.Fieldsmayalsobeinten-tionallyleftblankiftheyaretobeprovidedinalaterstepofthebusinessprocess.Hence,althoughwecantryourbesttocleanthedataafteritisseized,gooddatabaseanddataentryproceduredesignshouldhelpminimizethenumberofmissingvaluesorerrorsinthefirstplace.3.2.2NoisyData“Whatisnoise?”Noiseisarandomerrororvarianceinameasuredvariable.InChapter2,wesawhowsomebasicstatisticaldescriptiontechniques(e.g.,boxplotsandscatterplots),andmethodsofdatavisualizationcanbeusedtoidentifyoutliers,whichmayrepresentnoise.Givenanumericattributesuchas,say,price,howcanwe“smooth”outthedatatoremovethenoise?Let’slookatthefollowingdatasmoothingtechniques.Binning:Binningmethodssmoothasorteddatavaluebyconsultingits“neighbor-hood,”thatis,thevaluesaroundit.Thesortedvaluesaredistributedintoa'},
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              'page': 126.0,
              'text': 'saredistributedintoanumberof“buckets,”orbins.Becausebinningmethodsconsulttheneighborhoodofvalues,theyperformlocalsmoothing.Figure3.2illustratessomebinningtechniques.Inthisexample,thedataforpricearefirstsortedandthenpartitionedintoequal-frequencybinsofsize3(i.e.,eachbincontainsthreevalues).Insmoothingbybinmeans,eachvalueinabinisreplacedbythemeanvalueofthebin.Forexample,themeanofthevalues4,8,and15inBin1is9.Therefore,eachoriginalvalueinthisbinisreplacedbythevalue9.Similarly,smoothingbybinmedianscanbeemployed,inwhicheachbinvalueisreplacedbythebinmedian.Insmoothingbybinboundaries,theminimumandmaximumvaluesinagivenbinareidentifiedasthebinboundaries.Eachbinvalueisthenreplacedbytheclosestboundaryvalue.Ingeneral,thelargerthewidth,the'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page90#890Chapter3DataPreprocessingSorteddataforprice(indollars):4,8,15,21,21,24,25,28,34Partitioninto(equal-frequency)bins:Bin1:4,8,15Bin2:21,21,24Bin3:25,28,34Smoothingbybinmeans:Bin1:9,9,9Bin2:22,22,22Bin3:29,29,29Smoothingbybinboundaries:Bin1:4,4,15Bin2:21,21,24Bin3:25,25,34Figure3.2Binningmethodsfordatasmoothing.greatertheeffectofthesmoothing.Alternatively,binsmaybeequalwidth,wheretheintervalrangeofvaluesineachbinisconstant.BinningisalsousedasadiscretizationtechniqueandisfurtherdiscussedinSection3.5.Regression:Datasmoothingcanalsobedonebyregression,atechniquethatcon-formsdatavaluestoafunction.Linearregressioninvolvesfindingthe“best”linetofittwoattributes(orvariables)sothatoneattributecanbeusedtopredicttheother.Multiplelinearregressionisanextensionoflinearregression,wheremorethantwoattributesareinvolvedandthedataarefittoamultidimensionalsurface.RegressionisfurtherdescribedinSection3.4.5.Outlieranalysis:Outliersmaybedetectedbyclustering,forexample,wheresimilarvaluesareorganizedintogroups,or“clusters.”Intuitively,valuesthatfalloutsideofthesetofclustersmaybeconsideredoutliers(Figure3.3).Chapter12isdedicatedtothetopicofoutlieranalysis.Manydatasmoothingmethodsarealsousedfordatadiscretization(aformofdatatransformation)anddatareduction.Forexample,thebinningtechniquesdescribedbeforereducethenumberofdistinctvaluesperattribute.Thisactsasaformofdatareductionforlogic-baseddataminingmethods,suchasdecisiontreeinduction,whichrepeatedlymakesvaluecomparisonsonsorteddata.Concepthierarchiesareaformofdatadiscretizationthatcanalsobeusedfordatasmoothing.Aconcepthierarchyforprice,forexample,maymaprealpricevaluesintoinexpensive,moderatelypriced,andexpensive,therebyreducingthenumberofdatavaluestobehandledbythemining'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page90#890Chapter3DataPreprocessingSorteddataforprice(indollars):4,8,15,21,21,24,25,28,34Partitioninto(equal-frequency)bins:Bin1:4,8,15Bin2:21,21,24Bin3:25,28,34Smoothingbybinmeans:Bin1:9,9,9Bin2:22,22,22Bin3:29,29,29Smoothingbybinboundaries:Bin1:4,4,15Bin2:21,21,24Bin3:25,25,34Figure3.2Binningmethodsfordatasmoothing.greatertheeffectofthesmoothing.Alternatively,binsmaybeequalwidth,wheretheintervalrangeofvaluesineachbinisconstant.BinningisalsousedasadiscretizationtechniqueandisfurtherdiscussedinSection3.5.Regression:Datasmoothingcanalsobedonebyregression,atechniquethatcon-formsdatavaluestoafunction.Linearregressioninvolvesfindingthe“best”linetofittwoattributes(orvariables)sothatoneattributecanbeusedtopredicttheother.Multiplelinearregressionisanextensionoflinearregression,wheremorethantwoattributesareinvolvedandthedataarefittoamultidimensionalsurface.RegressionisfurtherdescribedinSection3.4.5.Outlieranalysis:Outliersmaybedetectedbyclustering,forexample,wheresimilarvaluesareorganizedintogroups,or“clusters.”Intuitively,valuesthatfalloutsideofthesetofclustersmaybeconsideredoutliers(Figure3.3).Chapter12isdedicatedtothetopicofoutlieranalysis.Manydatasmoothingmethodsarealsousedfordatadiscretization(aformofdatatransformation)anddatareduction.Forexample,thebinningtechniquesdescribedbeforereducethenumberofdistinctvaluesperattribute.Thisactsasaformofdatareductionforlogic-baseddataminingmethods,suchasdecisiontreeinduction,whichrepeatedlymakesvaluecomparisonsonsorteddata.Concepthierarchiesareaformofdatadiscretizationthatcanalsobeusedfordatasmoothing.Aconcepthierarchyforprice,forexample,maymaprealpricevaluesintoinexpensive,moderatelypriced,andexpensive,therebyreducingthenumberofdatavaluestobehandledbythemining'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page91#93.2DataCleaning91Figure3.3A2-Dcustomerdataplotwithrespecttocustomerlocationsinacity,showingthreedataclusters.Outliersmaybedetectedasvaluesthatfalloutsideoftheclustersets.process.DatadiscretizationisdiscussedinSection3.5.Somemethodsofclassification(e.g.,neuralnetworks)havebuilt-indatasmoothingmechanisms.ClassificationisthetopicofChapters8and9.3.2.3DataCleaningasaProcessMissingvalues,noise,andinconsistenciescontributetoinaccuratedata.Sofar,wehavelookedattechniquesforhandlingmissingdataandforsmoothingdata.“Butdataclean-ingisabigjob.Whataboutdatacleaningasaprocess?Howexactlydoesoneproceedintacklingthistask?Arethereanytoolsouttheretohelp?”Thefirststepindatacleaningasaprocessisdiscrepancydetection.Discrepanciescanbecausedbyseveralfactors,includingpoorlydesigneddataentryformsthathavemanyoptionalfields,humanerrorindataentry,deliberateerrors(e.g.,respondentsnotwant-ingtodivulgeinformationaboutthemselves),anddatadecay(e.g.,outdatedaddresses).Discrepanciesmayalsoarisefrominconsistentdatarepresentationsandinconsistentuseofcodes.Othersourcesofdiscrepanciesincludeerrorsininstrumentationdevicesthatrecorddataandsystemerrors.Errorscanalsooccurwhenthedataare(inadequately)usedforpurposesotherthanoriginallyintended.Theremayalsobeinconsistenciesduetodataintegration(e.g.,whereagivenattributecanhavedifferentnamesindifferentdatabases).22DataintegrationandtheremovalofredundantdatathatcanresultfromsuchintegrationarefurtherdescribedinSection3.3.'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page91#93.2DataCleaning91Figure3.3A2-Dcustomerdataplotwithrespecttocustomerlocationsinacity,showingthreedataclusters.Outliersmaybedetectedasvaluesthatfalloutsideoftheclustersets.process.DatadiscretizationisdiscussedinSection3.5.Somemethodsofclassification(e.g.,neuralnetworks)havebuilt-indatasmoothingmechanisms.ClassificationisthetopicofChapters8and9.3.2.3DataCleaningasaProcessMissingvalues,noise,andinconsistenciescontributetoinaccuratedata.Sofar,wehavelookedattechniquesforhandlingmissingdataandforsmoothingdata.“Butdataclean-ingisabigjob.Whataboutdatacleaningasaprocess?Howexactlydoesoneproceedintacklingthistask?Arethereanytoolsouttheretohelp?”Thefirststepindatacleaningasaprocessisdiscrepancydetection.Discrepanciescanbecausedbyseveralfactors,includingpoorlydesigneddataentryformsthathavemanyoptionalfields,humanerrorindataentry,deliberateerrors(e.g.,respondentsnotwant-ingtodivulgeinformationaboutthemselves),anddatadecay(e.g.,outdatedaddresses).Discrepanciesmayalsoarisefrominconsistentdatarepresentationsandinconsistentuseofcodes.Othersourcesofdiscrepanciesincludeerrorsininstrumentationdevicesthatrecorddataandsystemerrors.Errorscanalsooccurwhenthedataare(inadequately)usedforpurposesotherthanoriginallyintended.Theremayalsobeinconsistenciesduetodataintegration(e.g.,whereagivenattributecanhavedifferentnamesindifferentdatabases).22DataintegrationandtheremovalofredundantdatathatcanresultfromsuchintegrationarefurtherdescribedinSection3.3.'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page95#133.3DataIntegration95χ2CorrelationTestforNominalDataFornominaldata,acorrelationrelationshipbetweentwoattributes,AandB,canbediscoveredbyaχ2(chi-square)test.SupposeAhascdistinctvalues,namelya1,a2,...ac.Bhasrdistinctvalues,namelyb1,b2,...br.ThedatatuplesdescribedbyAandBcanbeshownasacontingencytable,withthecvaluesofAmakingupthecolumnsandthervaluesofBmakinguptherows.Let(Ai,Bj)denotethejointeventthatattributeAtakesonvalueaiandattributeBtakesonvaluebj,thatis,where(A=ai,B=bj).Eachandeverypossible(Ai,Bj)jointeventhasitsowncell(orslot)inthetable.Theχ2value(alsoknownasthePearsonχ2statistic)iscomputedasχ2=c(cid:88)i=1r(cid:88)j=1(oij−eij)2eij,(3.1)whereoijistheobservedfrequency(i.e.,actualcount)ofthejointevent(Ai,Bj)andeijistheexpectedfrequencyof(Ai,Bj),whichcanbecomputedaseij=count(A=ai)×count(B=bj)n,(3.2)wherenisthenumberofdatatuples,count(A=ai)isthenumberoftupleshavingvalueaiforA,andcount(B=bj)isthenumberoftupleshavingvaluebjforB.ThesuminEq.(3.1)iscomputedoverallofther×ccells.Notethatthecellsthatcontributethemosttotheχ2valuearethoseforwhichtheactualcountisverydifferentfromthatexpected.Theχ2statisticteststhehypothesisthatAandBareindependent,thatis,thereisnocorrelationbetweenthem.Thetestisbasedonasignificancelevel,with(r−1)×(c−1)degreesoffreedom.WeillustratetheuseofthisstatisticinExample3.1.Ifthehypothesiscanberejected,thenwesaythatAandBarestatisticallycorrelated.Example3.1Correlationanalysisofnominalattributesusingχ2.Supposethatagroupof1500peoplewassurveyed.Thegenderofeachpersonwasnoted.Eachpersonwaspolledastowhetherhisorherpreferredtypeofreadingmaterialwasfictionornonfiction.Thus,wehavetwoattributes,genderandpreferredreading.Theobservedfrequency(orcount)ofeachpossiblejointeventissummarizedinthecontingencytableshowninTable3.1,wherethenumbersinparenthesesaretheexpectedfrequencies.Theexpectedfrequen-ciesarecalculatedbasedonthedatadistributionforbothattributesusingEq.(3.2).UsingEq.(3.2),wecanverifytheexpectedfrequenciesforeachc'},
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              'text': 'dfrequenciesforeachcell.Forexample,theexpectedfrequencyforthecell(male,fiction)ise11=count(male)×count(fiction)n=300×4501500=90,andsoon.Noticethatinanyrow,thesumoftheexpectedfrequenciesmustequalthetotalobservedfrequencyforthatrow,andthesumoftheexpectedfrequenciesinanycolumnmustalsoequalthetotalobservedfrequencyforthatcolumn.'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page95#133.3DataIntegration95χ2CorrelationTestforNominalDataFornominaldata,acorrelationrelationshipbetweentwoattributes,AandB,canbediscoveredbyaχ2(chi-square)test.SupposeAhascdistinctvalues,namelya1,a2,...ac.Bhasrdistinctvalues,namelyb1,b2,...br.ThedatatuplesdescribedbyAandBcanbeshownasacontingencytable,withthecvaluesofAmakingupthecolumnsandthervaluesofBmakinguptherows.Let(Ai,Bj)denotethejointeventthatattributeAtakesonvalueaiandattributeBtakesonvaluebj,thatis,where(A=ai,B=bj).Eachandeverypossible(Ai,Bj)jointeventhasitsowncell(orslot)inthetable.Theχ2value(alsoknownasthePearsonχ2statistic)iscomputedasχ2=c(cid:88)i=1r(cid:88)j=1(oij−eij)2eij,(3.1)whereoijistheobservedfrequency(i.e.,actualcount)ofthejointevent(Ai,Bj)andeijistheexpectedfrequencyof(Ai,Bj),whichcanbecomputedaseij=count(A=ai)×count(B=bj)n,(3.2)wherenisthenumberofdatatuples,count(A=ai)isthenumberoftupleshavingvalueaiforA,andcount(B=bj)isthenumberoftupleshavingvaluebjforB.ThesuminEq.(3.1)iscomputedoverallofther×ccells.Notethatthecellsthatcontributethemosttotheχ2valuearethoseforwhichtheactualcountisverydifferentfromthatexpected.Theχ2statisticteststhehypothesisthatAandBareindependent,thatis,thereisnocorrelationbetweenthem.Thetestisbasedonasignificancelevel,with(r−1)×(c−1)degreesoffreedom.WeillustratetheuseofthisstatisticinExample3.1.Ifthehypothesiscanberejected,thenwesaythatAandBarestatisticallycorrelated.Example3.1Correlationanalysisofnominalattributesusingχ2.Supposethatagroupof1500peoplewassurveyed.Thegenderofeachpersonwasnoted.Eachpersonwaspolledastowhetherhisorherpreferredtypeofreadingmaterialwasfictionornonfiction.Thus,wehavetwoattributes,genderandpreferredreading.Theobservedfrequency(orcount)ofeachpossiblejointeventissummarizedinthecontingencytableshowninTable3.1,wherethenumbersinparenthesesaretheexpectedfrequencies.Theexpectedfrequen-ciesarecalculatedbasedonthedatadistributionforbothattributesusingEq.(3.2).UsingEq.(3.2),wecanverifytheexpectedfrequenciesforeachc'},
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              'text': 'dfrequenciesforeachcell.Forexample,theexpectedfrequencyforthecell(male,fiction)ise11=count(male)×count(fiction)n=300×4501500=90,andsoon.Noticethatinanyrow,thesumoftheexpectedfrequenciesmustequalthetotalobservedfrequencyforthatrow,andthesumoftheexpectedfrequenciesinanycolumnmustalsoequalthetotalobservedfrequencyforthatcolumn.'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page96#1496Chapter3DataPreprocessingTable3.1Example2.1’s2×2ContingencyTableDatamalefemaleTotalfiction250(90)200(360)450nonfiction50(210)1000(840)1050Total30012001500Note:Aregenderandpreferredreadingcorrelated?UsingEq.(3.1)forχ2computation,wegetχ2=(250−90)290+(50−210)2210+(200−360)2360+(1000−840)2840=284.44+121.90+71.11+30.48=507.93.Forthis2×2table,thedegreesoffreedomare(2−1)(2−1)=1.For1degreeoffree-dom,theχ2valueneededtorejectthehypothesisatthe0.001significancelevelis10.828(takenfromthetableofupperpercentagepointsoftheχ2distribution,typicallyavail-ablefromanytextbookonstatistics).Sinceourcomputedvalueisabovethis,wecanrejectthehypothesisthatgenderandpreferredreadingareindependentandconcludethatthetwoattributesare(strongly)correlatedforthegivengroupofpeople.CorrelationCoefficientforNumericDataFornumericattributes,wecanevaluatethecorrelationbetweentwoattributes,AandB,bycomputingthecorrelationcoefficient(alsoknownasPearson’sproductmomentcoefficient,namedafteritsinventer,KarlPearson).ThisisrA,B=n(cid:88)i=1(ai−¯A)(bi−¯B)nσAσB=n(cid:88)i=1(aibi)−n¯A¯BnσAσB,(3.3)wherenisthenumberoftuples,aiandbiaretherespectivevaluesofAandBintuplei,¯Aand¯BaretherespectivemeanvaluesofAandB,σAandσBaretherespectivestandarddeviationsofAandB(asdefinedinSection2.2.2),and(cid:54)(aibi)isthesumoftheABcross-product(i.e.,foreachtuple,thevalueforAismultipliedbythevalueforBinthattuple).Notethat−1≤rA,B≤+1.IfrA,Bisgreaterthan0,thenAandBarepositivelycorrelated,meaningthatthevaluesofAincreaseasthevaluesofBincrease.Thehigherthevalue,thestrongerthecorrelation(i.e.,themoreeachattributeimpliestheother).Hence,ahighervaluemayindicatethatA(orB)mayberemovedasaredundancy.Iftheresultingvalueisequalto0,thenAandBareindependentandthereisnocorrelationbetweenthem.Iftheresultingvalueislessthan0,thenAandBarenegativelycorrelated,wherethevaluesofoneattributeincreaseasthevaluesoftheotherattributedecrease.Thismeansthateachattributediscouragestheother.Scatterplotscanalsobeusedtovie'},
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              'page': 133.0,
              'text': 'tscanalsobeusedtoviewcorrelationsbetweenattributes(Section2.2.3).Forexample,Figure2.8’s'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page96#1496Chapter3DataPreprocessingTable3.1Example2.1’s2×2ContingencyTableDatamalefemaleTotalfiction250(90)200(360)450nonfiction50(210)1000(840)1050Total30012001500Note:Aregenderandpreferredreadingcorrelated?UsingEq.(3.1)forχ2computation,wegetχ2=(250−90)290+(50−210)2210+(200−360)2360+(1000−840)2840=284.44+121.90+71.11+30.48=507.93.Forthis2×2table,thedegreesoffreedomare(2−1)(2−1)=1.For1degreeoffree-dom,theχ2valueneededtorejectthehypothesisatthe0.001significancelevelis10.828(takenfromthetableofupperpercentagepointsoftheχ2distribution,typicallyavail-ablefromanytextbookonstatistics).Sinceourcomputedvalueisabovethis,wecanrejectthehypothesisthatgenderandpreferredreadingareindependentandconcludethatthetwoattributesare(strongly)correlatedforthegivengroupofpeople.CorrelationCoefficientforNumericDataFornumericattributes,wecanevaluatethecorrelationbetweentwoattributes,AandB,bycomputingthecorrelationcoefficient(alsoknownasPearson’sproductmomentcoefficient,namedafteritsinventer,KarlPearson).ThisisrA,B=n(cid:88)i=1(ai−¯A)(bi−¯B)nσAσB=n(cid:88)i=1(aibi)−n¯A¯BnσAσB,(3.3)wherenisthenumberoftuples,aiandbiaretherespectivevaluesofAandBintuplei,¯Aand¯BaretherespectivemeanvaluesofAandB,σAandσBaretherespectivestandarddeviationsofAandB(asdefinedinSection2.2.2),and(cid:54)(aibi)isthesumoftheABcross-product(i.e.,foreachtuple,thevalueforAismultipliedbythevalueforBinthattuple).Notethat−1≤rA,B≤+1.IfrA,Bisgreaterthan0,thenAandBarepositivelycorrelated,meaningthatthevaluesofAincreaseasthevaluesofBincrease.Thehigherthevalue,thestrongerthecorrelation(i.e.,themoreeachattributeimpliestheother).Hence,ahighervaluemayindicatethatA(orB)mayberemovedasaredundancy.Iftheresultingvalueisequalto0,thenAandBareindependentandthereisnocorrelationbetweenthem.Iftheresultingvalueislessthan0,thenAandBarenegativelycorrelated,wherethevaluesofoneattributeincreaseasthevaluesoftheotherattributedecrease.Thismeansthateachattributediscouragestheother.Scatterplotscanalsobeusedtovie'},
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              'text': 'tscanalsobeusedtoviewcorrelationsbetweenattributes(Section2.2.3).Forexample,Figure2.8’s'},
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              'page': 134.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page97#153.3DataIntegration97scatterplotsrespectivelyshowpositivelycorrelateddataandnegativelycorrelateddata,whileFigure2.9displaysuncorrelateddata.Notethatcorrelationdoesnotimplycausality.Thatis,ifAandBarecorrelated,thisdoesnotnecessarilyimplythatAcausesBorthatBcausesA.Forexample,inanalyzingademographicdatabase,wemayfindthatattributesrepresentingthenumberofhospitalsandthenumberofcartheftsinaregionarecorrelated.Thisdoesnotmeanthatonecausestheother.Bothareactuallycausallylinkedtoathirdattribute,namely,population.CovarianceofNumericDataInprobabilitytheoryandstatistics,correlationandcovariancearetwosimilarmeasuresforassessinghowmuchtwoattributeschangetogether.ConsidertwonumericattributesAandB,andasetofnobservations{(a1,b1),...,(an,bn)}.ThemeanvaluesofAandB,respectively,arealsoknownastheexpectedvaluesonAandB,thatis,E(A)=¯A=(cid:80)ni=1ainandE(B)=¯B=(cid:80)ni=1bin.ThecovariancebetweenAandBisdefinedasCov(A,B)=E((A−¯A)(B−¯B))=(cid:80)ni=1(ai−¯A)(bi−¯B)n.(3.4)IfwecompareEq.(3.3)forrA,B(correlationcoefficient)withEq.(3.4)forcovariance,weseethatrA,B=Cov(A,B)σAσB,(3.5)whereσAandσBarethestandarddeviationsofAandB,respectively.ItcanalsobeshownthatCov(A,B)=E(A·B)−¯A¯B.(3.6)Thisequationmaysimplifycalculations.FortwoattributesAandBthattendtochangetogether,ifAislargerthan¯A(theexpectedvalueofA),thenBislikelytobelargerthan¯B(theexpectedvalueofB).Therefore,thecovariancebetweenAandBispositive.Ontheotherhand,ifoneoftheattributestendstobeaboveitsexpectedvaluewhentheotherattributeisbelowitsexpectedvalue,thenthecovarianceofAandBisnegative.IfAandBareindependent(i.e.,theydonothavecorrelation),thenE(A·B)=E(A)·E(B).Therefore,thecovarianceisCov(A,B)=E(A·B)−¯A¯B=E(A)·E(B)−¯A¯B=0.However,theconverseisnottrue.Somepairsofrandomvariables(attributes)mayhaveacovarianceof0butarenotindependent.Onlyundersomeadditionalassumptions'},
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              'file_name': 'Data%20Mining%20Concepts%20and%20Techniques%20-%20Jiawei%20Han%2C%20Micheline%20Kamber%2C%20Jian%20Pei%20%28PDF%29.pdf',
              'page': 134.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page97#153.3DataIntegration97scatterplotsrespectivelyshowpositivelycorrelateddataandnegativelycorrelateddata,whileFigure2.9displaysuncorrelateddata.Notethatcorrelationdoesnotimplycausality.Thatis,ifAandBarecorrelated,thisdoesnotnecessarilyimplythatAcausesBorthatBcausesA.Forexample,inanalyzingademographicdatabase,wemayfindthatattributesrepresentingthenumberofhospitalsandthenumberofcartheftsinaregionarecorrelated.Thisdoesnotmeanthatonecausestheother.Bothareactuallycausallylinkedtoathirdattribute,namely,population.CovarianceofNumericDataInprobabilitytheoryandstatistics,correlationandcovariancearetwosimilarmeasuresforassessinghowmuchtwoattributeschangetogether.ConsidertwonumericattributesAandB,andasetofnobservations{(a1,b1),...,(an,bn)}.ThemeanvaluesofAandB,respectively,arealsoknownastheexpectedvaluesonAandB,thatis,E(A)=¯A=(cid:80)ni=1ainandE(B)=¯B=(cid:80)ni=1bin.ThecovariancebetweenAandBisdefinedasCov(A,B)=E((A−¯A)(B−¯B))=(cid:80)ni=1(ai−¯A)(bi−¯B)n.(3.4)IfwecompareEq.(3.3)forrA,B(correlationcoefficient)withEq.(3.4)forcovariance,weseethatrA,B=Cov(A,B)σAσB,(3.5)whereσAandσBarethestandarddeviationsofAandB,respectively.ItcanalsobeshownthatCov(A,B)=E(A·B)−¯A¯B.(3.6)Thisequationmaysimplifycalculations.FortwoattributesAandBthattendtochangetogether,ifAislargerthan¯A(theexpectedvalueofA),thenBislikelytobelargerthan¯B(theexpectedvalueofB).Therefore,thecovariancebetweenAandBispositive.Ontheotherhand,ifoneoftheattributestendstobeaboveitsexpectedvaluewhentheotherattributeisbelowitsexpectedvalue,thenthecovarianceofAandBisnegative.IfAandBareindependent(i.e.,theydonothavecorrelation),thenE(A·B)=E(A)·E(B).Therefore,thecovarianceisCov(A,B)=E(A·B)−¯A¯B=E(A)·E(B)−¯A¯B=0.However,theconverseisnottrue.Somepairsofrandomvariables(attributes)mayhaveacovarianceof0butarenotindependent.Onlyundersomeadditionalassumptions'},
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              'page': 135.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page98#1698Chapter3DataPreprocessingTable3.2StockPricesforAllElectronicsandHighTechTimepointAllElectronicsHighTecht1620t2510t3414t435t525(e.g.,thedatafollowmultivariatenormaldistributions)doesacovarianceof0implyindependence.Example3.2Covarianceanalysisofnumericattributes.ConsiderTable3.2,whichpresentsasim-plifiedexampleofstockpricesobservedatfivetimepointsforAllElectronicsandHighTech,ahigh-techcompany.Ifthestocksareaffectedbythesameindustrytrends,willtheirpricesriseorfalltogether?E(AllElectronics)=6+5+4+3+25=205=$4andE(HighTech)=20+10+14+5+55=545=$10.80.Thus,usingEq.(3.4),wecomputeCov(AllElectroncis,HighTech)=6×20+5×10+4×14+3×5+2×55−4×10.80=50.2−43.2=7.Therefore,giventhepositivecovariancewecansaythatstockpricesforbothcompaniesrisetogether.Varianceisaspecialcaseofcovariance,wherethetwoattributesareidentical(i.e.,thecovarianceofanattributewithitself).VariancewasdiscussedinChapter2.3.3.3TupleDuplicationInadditiontodetectingredundanciesbetweenattributes,duplicationshouldalsobedetectedatthetuplelevel(e.g.,wheretherearetwoormoreidenticaltuplesforagivenuniquedataentrycase).Theuseofdenormalizedtables(oftendonetoimproveper-formancebyavoidingjoins)isanothersourceofdataredundancy.Inconsistenciesoftenarisebetweenvariousduplicates,duetoinaccuratedataentryorupdatingsomebutnotalldataoccurrences.Forexample,ifapurchaseorderdatabasecontainsattributesfor'},
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              'file_name': 'Data%20Mining%20Concepts%20and%20Techniques%20-%20Jiawei%20Han%2C%20Micheline%20Kamber%2C%20Jian%20Pei%20%28PDF%29.pdf',
              'page': 135.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page98#1698Chapter3DataPreprocessingTable3.2StockPricesforAllElectronicsandHighTechTimepointAllElectronicsHighTecht1620t2510t3414t435t525(e.g.,thedatafollowmultivariatenormaldistributions)doesacovarianceof0implyindependence.Example3.2Covarianceanalysisofnumericattributes.ConsiderTable3.2,whichpresentsasim-plifiedexampleofstockpricesobservedatfivetimepointsforAllElectronicsandHighTech,ahigh-techcompany.Ifthestocksareaffectedbythesameindustrytrends,willtheirpricesriseorfalltogether?E(AllElectronics)=6+5+4+3+25=205=$4andE(HighTech)=20+10+14+5+55=545=$10.80.Thus,usingEq.(3.4),wecomputeCov(AllElectroncis,HighTech)=6×20+5×10+4×14+3×5+2×55−4×10.80=50.2−43.2=7.Therefore,giventhepositivecovariancewecansaythatstockpricesforbothcompaniesrisetogether.Varianceisaspecialcaseofcovariance,wherethetwoattributesareidentical(i.e.,thecovarianceofanattributewithitself).VariancewasdiscussedinChapter2.3.3.3TupleDuplicationInadditiontodetectingredundanciesbetweenattributes,duplicationshouldalsobedetectedatthetuplelevel(e.g.,wheretherearetwoormoreidenticaltuplesforagivenuniquedataentrycase).Theuseofdenormalizedtables(oftendonetoimproveper-formancebyavoidingjoins)isanothersourceofdataredundancy.Inconsistenciesoftenarisebetweenvariousduplicates,duetoinaccuratedataentryorupdatingsomebutnotalldataoccurrences.Forexample,ifapurchaseorderdatabasecontainsattributesfor'},
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              'page': 138.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page101#193.4DataReduction101cleaningaswell.Givenasetofcoefficients,anapproximationoftheoriginaldatacanbeconstructedbyapplyingtheinverseoftheDWTused.TheDWTiscloselyrelatedtothediscreteFouriertransform(DFT),asignalprocess-ingtechniqueinvolvingsinesandcosines.Ingeneral,however,theDWTachievesbetterlossycompression.Thatis,ifthesamenumberofcoefficientsisretainedforaDWTandaDFTofagivendatavector,theDWTversionwillprovideamoreaccurateapproxima-tionoftheoriginaldata.Hence,foranequivalentapproximation,theDWTrequireslessspacethantheDFT.UnliketheDFT,waveletsarequitelocalizedinspace,contributingtotheconservationoflocaldetail.ThereisonlyoneDFT,yetthereareseveralfamiliesofDWTs.Figure3.4showssomewaveletfamilies.PopularwavelettransformsincludetheHaar-2,Daubechies-4,andDaubechies-6.Thegeneralprocedureforapplyingadiscretewavelettransformusesahierarchicalpyramidalgorithmthathalvesthedataateachiteration,resultinginfastcomputationalspeed.Themethodisasfollows:1.Thelength,L,oftheinputdatavectormustbeanintegerpowerof2.Thisconditioncanbemetbypaddingthedatavectorwithzerosasnecessary(L≥n).2.Eachtransforminvolvesapplyingtwofunctions.Thefirstappliessomedatasmooth-ing,suchasasumorweightedaverage.Thesecondperformsaweighteddifference,whichactstobringoutthedetailedfeaturesofthedata.3.ThetwofunctionsareappliedtopairsofdatapointsinX,thatis,toallpairsofmeasurements(x2i,x2i+1).ThisresultsintwodatasetsoflengthL/2.Ingeneral,theserepresentasmoothedorlow-frequencyversionoftheinputdataandthehigh-frequencycontentofit,respectively.4.Thetwofunctionsarerecursivelyappliedtothedatasetsobtainedinthepreviousloop,untiltheresultingdatasetsobtainedareoflength2.5.Selectedvaluesfromthedatasetsobtainedinthepreviousiterationsaredesignatedthewaveletcoefficientsofthetransformeddata.02460.80.60.40.20.0(cid:2)1.0(cid:2)0.50.00.5(a) '
                      'Haar-2(b) '
                      'Daubechies-41.01.52.00.60.40.20.0Figure3.4Examplesofwaveletfamilies.Thenumbernexttoawaveletnameisthenumberofvanishingmomentsofthewavelet.Thisisasetofmathematicalrelationshipsthatthecoefficientsmustsatisfyandisrelatedtothenumberofcoefficients.'},
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              'page': 138.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page101#193.4DataReduction101cleaningaswell.Givenasetofcoefficients,anapproximationoftheoriginaldatacanbeconstructedbyapplyingtheinverseoftheDWTused.TheDWTiscloselyrelatedtothediscreteFouriertransform(DFT),asignalprocess-ingtechniqueinvolvingsinesandcosines.Ingeneral,however,theDWTachievesbetterlossycompression.Thatis,ifthesamenumberofcoefficientsisretainedforaDWTandaDFTofagivendatavector,theDWTversionwillprovideamoreaccurateapproxima-tionoftheoriginaldata.Hence,foranequivalentapproximation,theDWTrequireslessspacethantheDFT.UnliketheDFT,waveletsarequitelocalizedinspace,contributingtotheconservationoflocaldetail.ThereisonlyoneDFT,yetthereareseveralfamiliesofDWTs.Figure3.4showssomewaveletfamilies.PopularwavelettransformsincludetheHaar-2,Daubechies-4,andDaubechies-6.Thegeneralprocedureforapplyingadiscretewavelettransformusesahierarchicalpyramidalgorithmthathalvesthedataateachiteration,resultinginfastcomputationalspeed.Themethodisasfollows:1.Thelength,L,oftheinputdatavectormustbeanintegerpowerof2.Thisconditioncanbemetbypaddingthedatavectorwithzerosasnecessary(L≥n).2.Eachtransforminvolvesapplyingtwofunctions.Thefirstappliessomedatasmooth-ing,suchasasumorweightedaverage.Thesecondperformsaweighteddifference,whichactstobringoutthedetailedfeaturesofthedata.3.ThetwofunctionsareappliedtopairsofdatapointsinX,thatis,toallpairsofmeasurements(x2i,x2i+1).ThisresultsintwodatasetsoflengthL/2.Ingeneral,theserepresentasmoothedorlow-frequencyversionoftheinputdataandthehigh-frequencycontentofit,respectively.4.Thetwofunctionsarerecursivelyappliedtothedatasetsobtainedinthepreviousloop,untiltheresultingdatasetsobtainedareoflength2.5.Selectedvaluesfromthedatasetsobtainedinthepreviousiterationsaredesignatedthewaveletcoefficientsofthetransformeddata.02460.80.60.40.20.0(cid:2)1.0(cid:2)0.50.00.5(a) '
                      'Haar-2(b) '
                      'Daubechies-41.01.52.00.60.40.20.0Figure3.4Examplesofwaveletfamilies.Thenumbernexttoawaveletnameisthenumberofvanishingmomentsofthewavelet.Thisisasetofmathematicalrelationshipsthatthecoefficientsmustsatisfyandisrelatedtothenumberofcoefficients.'},
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              'page': 139.0,
              'text': 'setofvariables.Theinitialdatacanthenbeprojectedontothissmallerset.PCAoftenrevealsrelationshipsthatwerenotpreviouslysuspectedandtherebyallowsinterpretationsthatwouldnotordinarilyresult.Thebasicprocedureisasfollows:1.Theinputdataarenormalized,sothateachattributefallswithinthesamerange.Thisstephelpsensurethatattributeswithlargedomainswillnotdominateattributeswithsmallerdomains.2.PCAcomputeskorthonormalvectorsthatprovideabasisforthenormalizedinputdata.Theseareunitvectorsthateachpointinadirectionperpendiculartotheothers.Thesevectorsarereferredtoastheprincipalcomponents.Theinputdataarealinearcombinationoftheprincipalcomponents.3.Theprincipalcomponentsaresortedinorderofdecreasing“significance”orstrength.Theprincipalcomponentsessentiallyserveasanewsetofaxesforthedata,'},
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              'page': 139.0,
              'text': 'setofvariables.Theinitialdatacanthenbeprojectedontothissmallerset.PCAoftenrevealsrelationshipsthatwerenotpreviouslysuspectedandtherebyallowsinterpretationsthatwouldnotordinarilyresult.Thebasicprocedureisasfollows:1.Theinputdataarenormalized,sothateachattributefallswithinthesamerange.Thisstephelpsensurethatattributeswithlargedomainswillnotdominateattributeswithsmallerdomains.2.PCAcomputeskorthonormalvectorsthatprovideabasisforthenormalizedinputdata.Theseareunitvectorsthateachpointinadirectionperpendiculartotheothers.Thesevectorsarereferredtoastheprincipalcomponents.Theinputdataarealinearcombinationoftheprincipalcomponents.3.Theprincipalcomponentsaresortedinorderofdecreasing“significance”orstrength.Theprincipalcomponentsessentiallyserveasanewsetofaxesforthedata,'},
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              'page': 143.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page106#24106Chapter3DataPreprocessingspecifytheslopeofthelineandthey-intercept,respectively.Thesecoefficientscanbesolvedforbythemethodofleastsquares,whichminimizestheerrorbetweentheactuallineseparatingthedataandtheestimateoftheline.Multiplelinearregressionisanextensionof(simple)linearregression,whichallowsaresponsevariable,y,tobemodeledasalinearfunctionoftwoormorepredictorvariables.Log-linearmodelsapproximatediscretemultidimensionalprobabilitydistributions.Givenasetoftuplesinndimensions(e.g.,describedbynattributes),wecancon-sidereachtupleasapointinann-dimensionalspace.Log-linearmodelscanbeusedtoestimatetheprobabilityofeachpointinamultidimensionalspaceforasetofdis-cretizedattributes,basedonasmallersubsetofdimensionalcombinations.Thisallowsahigher-dimensionaldataspacetobeconstructedfromlower-dimensionalspaces.Log-linearmodelsarethereforealsousefulfordimensionalityreduction(sincethelower-dimensionalpointstogethertypicallyoccupylessspacethantheoriginaldatapoints)anddatasmoothing(sinceaggregateestimatesinthelower-dimensionalspacearelesssubjecttosamplingvariationsthantheestimatesinthehigher-dimensionalspace).Regressionandlog-linearmodelscanbothbeusedonsparsedata,althoughtheirapplicationmaybelimited.Whilebothmethodscanhandleskeweddata,regressiondoesexceptionallywell.Regressioncanbecomputationallyintensivewhenappliedtohigh-dimensionaldata,whereaslog-linearmodelsshowgoodscalabilityforupto10orsodimensions.Severalsoftwarepackagesexisttosolveregressionproblems.ExamplesincludeSAS(www.sas.com),SPSS(www.spss.com),andS-Plus(www.insightful.com).AnotherusefulresourceisthebookNumericalRecipesinC,byPress,Teukolsky,Vetterling,andFlannery[PTVF07],anditsassociatedsourcecode.3.4.6HistogramsHistogramsusebinningtoapproximatedatadistributionsandareapopularformofdatareduction.HistogramswereintroducedinSection2.2.3.Ahistogramforanattribute,A,partitionsthedatadistributionofAintodisjointsubsets,referredtoasbucketsorbins.Ifeachbucketrepresentsonlyasingl'},
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              'page': 143.0,
              'text': 'representsonlyasingleattribute–value/frequencypair,thebucketsarecalledsingletonbuckets.Often,bucketsinsteadrepresentcontinuousrangesforthegivenattribute.Example3.3Histograms.ThefollowingdataarealistofAllElectronicspricesforcommonlysolditems(roundedtothenearestdollar).Thenumbershavebeensorted:1,1,5,5,5,5,5,8,8,10,10,10,10,12,14,14,14,15,15,15,15,15,15,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,21,21,21,21,25,25,25,25,25,28,28,30,30,30.Figure3.7showsahistogramforthedatausingsingletonbuckets.Tofurtherreducethedata,itiscommontohaveeachbucketdenoteacontinuousvaluerangeforthegivenattribute.InFigure3.8,eachbucketrepresentsadifferent$10rangeforprice.'},
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              'page': 143.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page106#24106Chapter3DataPreprocessingspecifytheslopeofthelineandthey-intercept,respectively.Thesecoefficientscanbesolvedforbythemethodofleastsquares,whichminimizestheerrorbetweentheactuallineseparatingthedataandtheestimateoftheline.Multiplelinearregressionisanextensionof(simple)linearregression,whichallowsaresponsevariable,y,tobemodeledasalinearfunctionoftwoormorepredictorvariables.Log-linearmodelsapproximatediscretemultidimensionalprobabilitydistributions.Givenasetoftuplesinndimensions(e.g.,describedbynattributes),wecancon-sidereachtupleasapointinann-dimensionalspace.Log-linearmodelscanbeusedtoestimatetheprobabilityofeachpointinamultidimensionalspaceforasetofdis-cretizedattributes,basedonasmallersubsetofdimensionalcombinations.Thisallowsahigher-dimensionaldataspacetobeconstructedfromlower-dimensionalspaces.Log-linearmodelsarethereforealsousefulfordimensionalityreduction(sincethelower-dimensionalpointstogethertypicallyoccupylessspacethantheoriginaldatapoints)anddatasmoothing(sinceaggregateestimatesinthelower-dimensionalspacearelesssubjecttosamplingvariationsthantheestimatesinthehigher-dimensionalspace).Regressionandlog-linearmodelscanbothbeusedonsparsedata,althoughtheirapplicationmaybelimited.Whilebothmethodscanhandleskeweddata,regressiondoesexceptionallywell.Regressioncanbecomputationallyintensivewhenappliedtohigh-dimensionaldata,whereaslog-linearmodelsshowgoodscalabilityforupto10orsodimensions.Severalsoftwarepackagesexisttosolveregressionproblems.ExamplesincludeSAS(www.sas.com),SPSS(www.spss.com),andS-Plus(www.insightful.com).AnotherusefulresourceisthebookNumericalRecipesinC,byPress,Teukolsky,Vetterling,andFlannery[PTVF07],anditsassociatedsourcecode.3.4.6HistogramsHistogramsusebinningtoapproximatedatadistributionsandareapopularformofdatareduction.HistogramswereintroducedinSection2.2.3.Ahistogramforanattribute,A,partitionsthedatadistributionofAintodisjointsubsets,referredtoasbucketsorbins.Ifeachbucketrepresentsonlyasingl'},
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              'file_name': 'Data%20Mining%20Concepts%20and%20Techniques%20-%20Jiawei%20Han%2C%20Micheline%20Kamber%2C%20Jian%20Pei%20%28PDF%29.pdf',
              'page': 143.0,
              'text': 'representsonlyasingleattribute–value/frequencypair,thebucketsarecalledsingletonbuckets.Often,bucketsinsteadrepresentcontinuousrangesforthegivenattribute.Example3.3Histograms.ThefollowingdataarealistofAllElectronicspricesforcommonlysolditems(roundedtothenearestdollar).Thenumbershavebeensorted:1,1,5,5,5,5,5,8,8,10,10,10,10,12,14,14,14,15,15,15,15,15,15,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,21,21,21,21,25,25,25,25,25,28,28,30,30,30.Figure3.7showsahistogramforthedatausingsingletonbuckets.Tofurtherreducethedata,itiscommontohaveeachbucketdenoteacontinuousvaluerangeforthegivenattribute.InFigure3.8,eachbucketrepresentsadifferent$10rangeforprice.'},
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              'page': 144.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page107#253.4DataReduction10751010987654321015202530price '
                      '($)countFigure3.7Ahistogramforpriceusingsingletonbuckets—eachbucketrepresentsoneprice–value/frequencypair.1–1011–2021–30price '
                      '($)count2520151050Figure3.8Anequal-widthhistogramforprice,wherevaluesareaggregatedsothateachbuckethasauniformwidthof$10.“Howarethebucketsdeterminedandtheattributevaluespartitioned?”Thereareseveralpartitioningrules,includingthefollowing:Equal-width:Inanequal-widthhistogram,thewidthofeachbucketrangeisuniform(e.g.,thewidthof$10forthebucketsinFigure3.8).Equal-frequency(orequal-depth):Inanequal-frequencyhistogram,thebucketsarecreatedsothat,roughly,thefrequencyofeachbucketisconstant(i.e.,eachbucketcontainsroughlythesamenumberofcontiguousdatasamples).'},
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              'page': 144.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page107#253.4DataReduction10751010987654321015202530price '
                      '($)countFigure3.7Ahistogramforpriceusingsingletonbuckets—eachbucketrepresentsoneprice–value/frequencypair.1–1011–2021–30price '
                      '($)count2520151050Figure3.8Anequal-widthhistogramforprice,wherevaluesareaggregatedsothateachbuckethasauniformwidthof$10.“Howarethebucketsdeterminedandtheattributevaluespartitioned?”Thereareseveralpartitioningrules,includingthefollowing:Equal-width:Inanequal-widthhistogram,thewidthofeachbucketrangeisuniform(e.g.,thewidthof$10forthebucketsinFigure3.8).Equal-frequency(orequal-depth):Inanequal-frequencyhistogram,thebucketsarecreatedsothat,roughly,thefrequencyofeachbucketisconstant(i.e.,eachbucketcontainsroughlythesamenumberofcontiguousdatasamples).'},
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              'page': 145.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page108#26108Chapter3DataPreprocessingHistogramsarehighlyeffectiveatapproximatingbothsparseanddensedata,aswellashighlyskewedanduniformdata.Thehistogramsdescribedbeforeforsingleattributescanbeextendedformultipleattributes.Multidimensionalhistogramscancap-turedependenciesbetweenattributes.Thesehistogramshavebeenfoundeffectiveinapproximatingdatawithuptofiveattributes.Morestudiesareneededregardingtheeffectivenessofmultidimensionalhistogramsforhighdimensionalities.Singletonbucketsareusefulforstoringhigh-frequencyoutliers.3.4.7ClusteringClusteringtechniquesconsiderdatatuplesasobjects.Theypartitiontheobjectsintogroups,orclusters,sothatobjectswithinaclusterare“similar”tooneanotherand“dis-similar”toobjectsinotherclusters.Similarityiscommonlydefinedintermsofhow“close”theobjectsareinspace,basedonadistancefunction.The“quality”ofaclustermayberepresentedbyitsdiameter,themaximumdistancebetweenanytwoobjectsinthecluster.Centroiddistanceisanalternativemeasureofclusterqualityandisdefinedastheaveragedistanceofeachclusterobjectfromtheclustercentroid(denotingthe“averageobject,”oraveragepointinspaceforthecluster).Figure3.3showeda2-Dplotofcustomerdatawithrespecttocustomerlocationsinacity.Threedataclustersarevisible.Indatareduction,theclusterrepresentationsofthedataareusedtoreplacetheactualdata.Theeffectivenessofthistechniquedependsonthedata’snature.Itismuchmoreeffectivefordatathatcanbeorganizedintodistinctclustersthanforsmeareddata.Therearemanymeasuresfordefiningclustersandclusterquality.Clusteringmeth-odsarefurtherdescribedinChapters10and11.3.4.8SamplingSamplingcanbeusedasadatareductiontechniquebecauseitallowsalargedatasettoberepresentedbyamuchsmallerrandomdatasample(orsubset).Supposethatalargedataset,D,containsNtuples.Let’slookatthemostcommonwaysthatwecouldsampleDfordatareduction,asillustratedinFigure3.9.Simplerandomsamplewithoutreplacement(SRSWOR)ofsizes:ThisiscreatedbydrawingsoftheNtuplesfromD(s<N),wheretheprobabilityofdrawinganytupleinDis1/N'},
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              'page': 145.0,
              'text': 'winganytupleinDis1/N,thatis,alltuplesareequallylikelytobesampled.Simplerandomsamplewithreplacement(SRSWR)ofsizes:ThisissimilartoSRSWOR,exceptthateachtimeatupleisdrawnfromD,itisrecordedandthenreplaced.Thatis,afteratupleisdrawn,itisplacedbackinDsothatitmaybedrawnagain.Clustersample:IfthetuplesinDaregroupedintoMmutuallydisjoint“clusters,”thenanSRSofsclusterscanbeobtained,wheres<M.Forexample,tuplesinadatabaseareusuallyretrievedapageatatime,sothateachpagecanbeconsidered'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page108#26108Chapter3DataPreprocessingHistogramsarehighlyeffectiveatapproximatingbothsparseanddensedata,aswellashighlyskewedanduniformdata.Thehistogramsdescribedbeforeforsingleattributescanbeextendedformultipleattributes.Multidimensionalhistogramscancap-turedependenciesbetweenattributes.Thesehistogramshavebeenfoundeffectiveinapproximatingdatawithuptofiveattributes.Morestudiesareneededregardingtheeffectivenessofmultidimensionalhistogramsforhighdimensionalities.Singletonbucketsareusefulforstoringhigh-frequencyoutliers.3.4.7ClusteringClusteringtechniquesconsiderdatatuplesasobjects.Theypartitiontheobjectsintogroups,orclusters,sothatobjectswithinaclusterare“similar”tooneanotherand“dis-similar”toobjectsinotherclusters.Similarityiscommonlydefinedintermsofhow“close”theobjectsareinspace,basedonadistancefunction.The“quality”ofaclustermayberepresentedbyitsdiameter,themaximumdistancebetweenanytwoobjectsinthecluster.Centroiddistanceisanalternativemeasureofclusterqualityandisdefinedastheaveragedistanceofeachclusterobjectfromtheclustercentroid(denotingthe“averageobject,”oraveragepointinspaceforthecluster).Figure3.3showeda2-Dplotofcustomerdatawithrespecttocustomerlocationsinacity.Threedataclustersarevisible.Indatareduction,theclusterrepresentationsofthedataareusedtoreplacetheactualdata.Theeffectivenessofthistechniquedependsonthedata’snature.Itismuchmoreeffectivefordatathatcanbeorganizedintodistinctclustersthanforsmeareddata.Therearemanymeasuresfordefiningclustersandclusterquality.Clusteringmeth-odsarefurtherdescribedinChapters10and11.3.4.8SamplingSamplingcanbeusedasadatareductiontechniquebecauseitallowsalargedatasettoberepresentedbyamuchsmallerrandomdatasample(orsubset).Supposethatalargedataset,D,containsNtuples.Let’slookatthemostcommonwaysthatwecouldsampleDfordatareduction,asillustratedinFigure3.9.Simplerandomsamplewithoutreplacement(SRSWOR)ofsizes:ThisiscreatedbydrawingsoftheNtuplesfromD(s<N),wheretheprobabilityofdrawinganytupleinDis1/N'},
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              'page': 145.0,
              'text': 'winganytupleinDis1/N,thatis,alltuplesareequallylikelytobesampled.Simplerandomsamplewithreplacement(SRSWR)ofsizes:ThisissimilartoSRSWOR,exceptthateachtimeatupleisdrawnfromD,itisrecordedandthenreplaced.Thatis,afteratupleisdrawn,itisplacedbackinDsothatitmaybedrawnagain.Clustersample:IfthetuplesinDaregroupedintoMmutuallydisjoint“clusters,”thenanSRSofsclusterscanbeobtained,wheres<M.Forexample,tuplesinadatabaseareusuallyretrievedapageatatime,sothateachpagecanbeconsidered'},
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              'page': 146.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page109#273.4DataReduction109Cluster '
                      'sampleStartified '
                      'sampleFigure3.9Samplingcanbeusedfordatareduction.acluster.Areduceddatarepresentationcanbeobtainedbyapplying,say,SRSWORtothepages,resultinginaclustersampleofthetuples.Otherclusteringcriteriacon-veyingrichsemanticscanalsobeexplored.Forexample,inaspatialdatabase,wemaychoosetodefineclustersgeographicallybasedonhowcloselydifferentareasarelocated.Stratifiedsample:IfDisdividedintomutuallydisjointpartscalledstrata,astratifiedsampleofDisgeneratedbyobtaininganSRSateachstratum.Thishelpsensurea'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page109#273.4DataReduction109Cluster '
                      'sampleStartified '
                      'sampleFigure3.9Samplingcanbeusedfordatareduction.acluster.Areduceddatarepresentationcanbeobtainedbyapplying,say,SRSWORtothepages,resultinginaclustersampleofthetuples.Otherclusteringcriteriacon-veyingrichsemanticscanalsobeexplored.Forexample,inaspatialdatabase,wemaychoosetodefineclustersgeographicallybasedonhowcloselydifferentareasarelocated.Stratifiedsample:IfDisdividedintomutuallydisjointpartscalledstrata,astratifiedsampleofDisgeneratedbyobtaininganSRSateachstratum.Thishelpsensurea'},
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              'page': 150.0,
              'text': 'ataanalysis.Forexample,changingmea-surementunitsfrommeterstoinchesforheight,orfromkilogramstopoundsforweight,mayleadtoverydifferentresults.Ingeneral,expressinganattributeinsmallerunitswillleadtoalargerrangeforthatattribute,andthustendtogivesuchanattributegreatereffector“weight.”Tohelpavoiddependenceonthechoiceofmeasurementunits,thedatashouldbenormalizedorstandardized.Thisinvolvestransformingthedatatofallwithinasmallerorcommonrangesuchas[−1,1]or[0.0,1.0].(Thetermsstandardizeandnormalizeareusedinterchangeablyindatapreprocessing,althoughinstatistics,thelattertermalsohasotherconnotations.)Normalizingthedataattemptstogiveallattributesanequalweight.Normaliza-tionisparticularlyusefulforclassificationalgorithmsinvolvingneuralnetworksordistancemeasurementssuchasnearest-neighborclassificationandclustering.Ifusingtheneuralnetworkbackpropagationalgorithmforclassificationmining(Chapter9),normalizingtheinputvaluesforeachattributemeasuredinthetrainingtupleswillhelpspeedupthelearningphase.Fordistance-basedmethods,normalizationhelpsprevent'},
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              'page': 150.0,
              'text': 'ataanalysis.Forexample,changingmea-surementunitsfrommeterstoinchesforheight,orfromkilogramstopoundsforweight,mayleadtoverydifferentresults.Ingeneral,expressinganattributeinsmallerunitswillleadtoalargerrangeforthatattribute,andthustendtogivesuchanattributegreatereffector“weight.”Tohelpavoiddependenceonthechoiceofmeasurementunits,thedatashouldbenormalizedorstandardized.Thisinvolvestransformingthedatatofallwithinasmallerorcommonrangesuchas[−1,1]or[0.0,1.0].(Thetermsstandardizeandnormalizeareusedinterchangeablyindatapreprocessing,althoughinstatistics,thelattertermalsohasotherconnotations.)Normalizingthedataattemptstogiveallattributesanequalweight.Normaliza-tionisparticularlyusefulforclassificationalgorithmsinvolvingneuralnetworksordistancemeasurementssuchasnearest-neighborclassificationandclustering.Ifusingtheneuralnetworkbackpropagationalgorithmforclassificationmining(Chapter9),normalizingtheinputvaluesforeachattributemeasuredinthetrainingtupleswillhelpspeedupthelearningphase.Fordistance-basedmethods,normalizationhelpsprevent'},
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              'page': 151.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page114#32114Chapter3DataPreprocessingattributeswithinitiallylargeranges(e.g.,income)fromoutweighingattributeswithinitiallysmallerranges(e.g.,binaryattributes).Itisalsousefulwhengivennopriorknowledgeofthedata.Therearemanymethodsfordatanormalization.Westudymin-maxnormalization,z-scorenormalization,andnormalizationbydecimalscaling.Forourdiscussion,letAbeanumericattributewithnobservedvalues,v1,v2,...,vn.Min-maxnormalizationperformsalineartransformationontheoriginaldata.Sup-posethatminAandmaxAaretheminimumandmaximumvaluesofanattribute,A.Min-maxnormalizationmapsavalue,vi,ofAtov(cid:48)iintherange[newminA,newmaxA]bycomputingv(cid:48)i=vi−minAmaxA−minA(newmaxA−newminA)+newminA.(3.8)Min-maxnormalizationpreservestherelationshipsamongtheoriginaldatavalues.Itwillencounteran“out-of-bounds”errorifafutureinputcasefornormalizationfallsoutsideoftheoriginaldatarangeforA.Example3.4Min-maxnormalization.Supposethattheminimumandmaximumvaluesfortheattributeincomeare$12,000and$98,000,respectively.Wewouldliketomapincometotherange[0.0,1.0].Bymin-maxnormalization,avalueof$73,600forincomeistransformedto73,600−12,00098,000−12,000(1.0−0)+0=0.716.Inz-scorenormalization(orzero-meannormalization),thevaluesforanattribute,A,arenormalizedbasedonthemean(i.e.,average)andstandarddeviationofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi−¯AσA,(3.9)where¯AandσAarethemeanandstandarddeviation,respectively,ofattributeA.ThemeanandstandarddeviationwerediscussedinSection2.2,where¯A=1n(v1+v2+···+vn)andσAiscomputedasthesquarerootofthevarianceofA(seeEq.(2.6)).ThismethodofnormalizationisusefulwhentheactualminimumandmaximumofattributeAareunknown,orwhenthereareoutliersthatdominatethemin-maxnormalization.Example3.5z-scorenormalization.Supposethatthemeanandstandarddeviationofthevaluesfortheattributeincomeare$54,000and$16,000,respectively.Withz-scorenormalization,avalueof$73,600forincomeistransformedto73,600−54,00016,000=1.225.Avariationofthisz-scorenormalizati'},
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              'page': 151.0,
              'text': 'isz-scorenormalizationreplacesthestandarddeviationofEq.(3.9)bythemeanabsolutedeviationofA.ThemeanabsolutedeviationofA,denotedsA,issA=1n(|v1−¯A|+|v2−¯A|+···+|vn−¯A|).(3.10)'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page114#32114Chapter3DataPreprocessingattributeswithinitiallylargeranges(e.g.,income)fromoutweighingattributeswithinitiallysmallerranges(e.g.,binaryattributes).Itisalsousefulwhengivennopriorknowledgeofthedata.Therearemanymethodsfordatanormalization.Westudymin-maxnormalization,z-scorenormalization,andnormalizationbydecimalscaling.Forourdiscussion,letAbeanumericattributewithnobservedvalues,v1,v2,...,vn.Min-maxnormalizationperformsalineartransformationontheoriginaldata.Sup-posethatminAandmaxAaretheminimumandmaximumvaluesofanattribute,A.Min-maxnormalizationmapsavalue,vi,ofAtov(cid:48)iintherange[newminA,newmaxA]bycomputingv(cid:48)i=vi−minAmaxA−minA(newmaxA−newminA)+newminA.(3.8)Min-maxnormalizationpreservestherelationshipsamongtheoriginaldatavalues.Itwillencounteran“out-of-bounds”errorifafutureinputcasefornormalizationfallsoutsideoftheoriginaldatarangeforA.Example3.4Min-maxnormalization.Supposethattheminimumandmaximumvaluesfortheattributeincomeare$12,000and$98,000,respectively.Wewouldliketomapincometotherange[0.0,1.0].Bymin-maxnormalization,avalueof$73,600forincomeistransformedto73,600−12,00098,000−12,000(1.0−0)+0=0.716.Inz-scorenormalization(orzero-meannormalization),thevaluesforanattribute,A,arenormalizedbasedonthemean(i.e.,average)andstandarddeviationofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi−¯AσA,(3.9)where¯AandσAarethemeanandstandarddeviation,respectively,ofattributeA.ThemeanandstandarddeviationwerediscussedinSection2.2,where¯A=1n(v1+v2+···+vn)andσAiscomputedasthesquarerootofthevarianceofA(seeEq.(2.6)).ThismethodofnormalizationisusefulwhentheactualminimumandmaximumofattributeAareunknown,orwhenthereareoutliersthatdominatethemin-maxnormalization.Example3.5z-scorenormalization.Supposethatthemeanandstandarddeviationofthevaluesfortheattributeincomeare$54,000and$16,000,respectively.Withz-scorenormalization,avalueof$73,600forincomeistransformedto73,600−54,00016,000=1.225.Avariationofthisz-scorenormalizati'},
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              'page': 151.0,
              'text': 'isz-scorenormalizationreplacesthestandarddeviationofEq.(3.9)bythemeanabsolutedeviationofA.ThemeanabsolutedeviationofA,denotedsA,issA=1n(|v1−¯A|+|v2−¯A|+···+|vn−¯A|).(3.10)'},
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              'page': 152.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page115#333.5DataTransformationandDataDiscretization115Thus,z-scorenormalizationusingthemeanabsolutedeviationisv(cid:48)i=vi−¯AsA.(3.11)Themeanabsolutedeviation,sA,ismorerobusttooutliersthanthestandarddeviation,σA.Whencomputingthemeanabsolutedeviation,thedeviationsfromthemean(i.e.,|xi−¯x|)arenotsquared;hence,theeffectofoutliersissomewhatreduced.NormalizationbydecimalscalingnormalizesbymovingthedecimalpointofvaluesofattributeA.ThenumberofdecimalpointsmoveddependsonthemaximumabsolutevalueofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi10j,(3.12)wherejisthesmallestintegersuchthatmax(|v(cid:48)i|)<1.Example3.6Decimalscaling.SupposethattherecordedvaluesofArangefrom−986to917.ThemaximumabsolutevalueofAis986.Tonormalizebydecimalscaling,wethereforedivideeachvalueby1000(i.e.,j=3)sothat−986normalizesto−0.986and917normalizesto0.917.Notethatnormalizationcanchangetheoriginaldataquiteabit,especiallywhenusingz-scorenormalizationordecimalscaling.Itisalsonecessarytosavethenormaliza-tionparameters(e.g.,themeanandstandarddeviationifusingz-scorenormalization)sothatfuturedatacanbenormalizedinauniformmanner.3.5.3DiscretizationbyBinningBinningisatop-downsplittingtechniquebasedonaspecifiednumberofbins.Section3.2.2discussedbinningmethodsfordatasmoothing.Thesemethodsarealsousedasdiscretizationmethodsfordatareductionandconcepthierarchygeneration.Forexample,attributevaluescanbediscretizedbyapplyingequal-widthorequal-frequencybinning,andthenreplacingeachbinvaluebythebinmeanormedian,asinsmoothingbybinmeansorsmoothingbybinmedians,respectively.Thesetechniquescanbeappliedrecursivelytotheresultingpartitionstogenerateconcepthierarchies.Binningdoesnotuseclassinformationandisthereforeanunsuperviseddiscretiza-tiontechnique.Itissensitivetotheuser-specifiednumberofbins,aswellasthepresenceofoutliers.3.5.4DiscretizationbyHistogramAnalysisLikebinning,histogramanalysisisanunsuperviseddiscretizationtechniquebecauseitdoesnotuseclassinformation.Histogra'},
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              'page': 152.0,
              'text': 'information.HistogramswereintroducedinSection2.2.3.Ahis-togrampartitionsthevaluesofanattribute,A,intodisjointrangescalledbucketsorbins.'},
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              'file_name': 'Data%20Mining%20Concepts%20and%20Techniques%20-%20Jiawei%20Han%2C%20Micheline%20Kamber%2C%20Jian%20Pei%20%28PDF%29.pdf',
              'page': 152.0,
              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page115#333.5DataTransformationandDataDiscretization115Thus,z-scorenormalizationusingthemeanabsolutedeviationisv(cid:48)i=vi−¯AsA.(3.11)Themeanabsolutedeviation,sA,ismorerobusttooutliersthanthestandarddeviation,σA.Whencomputingthemeanabsolutedeviation,thedeviationsfromthemean(i.e.,|xi−¯x|)arenotsquared;hence,theeffectofoutliersissomewhatreduced.NormalizationbydecimalscalingnormalizesbymovingthedecimalpointofvaluesofattributeA.ThenumberofdecimalpointsmoveddependsonthemaximumabsolutevalueofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi10j,(3.12)wherejisthesmallestintegersuchthatmax(|v(cid:48)i|)<1.Example3.6Decimalscaling.SupposethattherecordedvaluesofArangefrom−986to917.ThemaximumabsolutevalueofAis986.Tonormalizebydecimalscaling,wethereforedivideeachvalueby1000(i.e.,j=3)sothat−986normalizesto−0.986and917normalizesto0.917.Notethatnormalizationcanchangetheoriginaldataquiteabit,especiallywhenusingz-scorenormalizationordecimalscaling.Itisalsonecessarytosavethenormaliza-tionparameters(e.g.,themeanandstandarddeviationifusingz-scorenormalization)sothatfuturedatacanbenormalizedinauniformmanner.3.5.3DiscretizationbyBinningBinningisatop-downsplittingtechniquebasedonaspecifiednumberofbins.Section3.2.2discussedbinningmethodsfordatasmoothing.Thesemethodsarealsousedasdiscretizationmethodsfordatareductionandconcepthierarchygeneration.Forexample,attributevaluescanbediscretizedbyapplyingequal-widthorequal-frequencybinning,andthenreplacingeachbinvaluebythebinmeanormedian,asinsmoothingbybinmeansorsmoothingbybinmedians,respectively.Thesetechniquescanbeappliedrecursivelytotheresultingpartitionstogenerateconcepthierarchies.Binningdoesnotuseclassinformationandisthereforeanunsuperviseddiscretiza-tiontechnique.Itissensitivetotheuser-specifiednumberofbins,aswellasthepresenceofoutliers.3.5.4DiscretizationbyHistogramAnalysisLikebinning,histogramanalysisisanunsuperviseddiscretizationtechniquebecauseitdoesnotuseclassinformation.Histogra'},
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              'page': 152.0,
              'text': 'information.HistogramswereintroducedinSection2.2.3.Ahis-togrampartitionsthevaluesofanattribute,A,intodisjointrangescalledbucketsorbins.'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page116#34116Chapter3DataPreprocessingVariouspartitioningrulescanbeusedtodefinehistograms(Section3.4.6).Inanequal-widthhistogram,forexample,thevaluesarepartitionedintoequal-sizepartitionsorranges(e.g.,earlierinFigure3.8forprice,whereeachbuckethasawidthof$10).Withanequal-frequencyhistogram,thevaluesarepartitionedsothat,ideally,eachpar-titioncontainsthesamenumberofdatatuples.Thehistogramanalysisalgorithmcanbeappliedrecursivelytoeachpartitioninordertoautomaticallygenerateamultilevelcon-cepthierarchy,withtheprocedureterminatingonceaprespecifiednumberofconceptlevelshasbeenreached.Aminimumintervalsizecanalsobeusedperleveltocontroltherecursiveprocedure.Thisspecifiestheminimumwidthofapartition,ortheminimumnumberofvaluesforeachpartitionateachlevel.Histogramscanalsobepartitionedbasedonclusteranalysisofthedatadistribution,asdescribednext.3.5.5DiscretizationbyCluster,DecisionTree,andCorrelationAnalysesClustering,decisiontreeanalysis,andcorrelationanalysiscanbeusedfordatadis-cretization.Webrieflystudyeachoftheseapproaches.Clusteranalysisisapopulardatadiscretizationmethod.Aclusteringalgorithmcanbeappliedtodiscretizeanumericattribute,A,bypartitioningthevaluesofAintoclus-tersorgroups.ClusteringtakesthedistributionofAintoconsideration,aswellastheclosenessofdatapoints,andthereforeisabletoproducehigh-qualitydiscretizationresults.ClusteringcanbeusedtogenerateaconcepthierarchyforAbyfollowingeitheratop-downsplittingstrategyorabottom-upmergingstrategy,whereeachclusterformsanodeoftheconcepthierarchy.Intheformer,eachinitialclusterorpartitionmaybefurtherdecomposedintoseveralsubclusters,formingalowerlevelofthehiera-rchy.Inthelatter,clustersareformedbyrepeatedlygroupingneighboringclustersinordertoformhigher-levelconcepts.ClusteringmethodsfordataminingarestudiedinChapters10and11.Techniquestogeneratedecisiontreesforclassification(Chapter8)canbeappliedtodiscretization.Suchtechniquesemployatop-downsplittingapproach.Unliketheothermethodsmentionedsofar,de'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page121#393.7Exercises1213.7Exercises3.1Dataqualitycanbeassessedintermsofseveralissues,includingaccuracy,completeness,andconsistency.Foreachoftheabovethreeissues,discusshowdataqualityassess-mentcandependontheintendeduseofthedata,givingexamples.Proposetwootherdimensionsofdataquality.3.2Inreal-worlddata,tupleswithmissingvaluesforsomeattributesareacommonoccurrence.Describevariousmethodsforhandlingthisproblem.3.3Exercise2.2gavethefollowingdata(inincreasingorder)fortheattributeage:13,15,16,16,19,20,20,21,22,22,25,25,25,25,30,33,33,35,35,35,35,36,40,45,46,52,70.(a)Usesmoothingbybinmeanstosmooththesedata,usingabindepthof3.Illustrateyoursteps.Commentontheeffectofthistechniqueforthegivendata.(b)Howmightyoudetermineoutliersinthedata?(c)Whatothermethodsaretherefordatasmoothing?3.4Discussissuestoconsiderduringdataintegration.3.5Whatarethevaluerangesofthefollowingnormalizationmethods?(a)min-maxnormalization(b)z-scorenormalization(c)z-scorenormalizationusingthemeanabsolutedeviationinsteadofstandarddevia-tion(d)normalizationbydecimalscaling3.6Usethesemethodstonormalizethefollowinggroupofdata:200,300,400,600,1000(a)min-maxnormalizationbysettingmin=0andmax=1(b)z-scorenormalization(c)z-scorenormalizationusingthemeanabsolutedeviationinsteadofstandarddevia-tion(d)normalizationbydecimalscaling3.7UsingthedataforagegiveninExercise3.3,answerthefollowing:(a)Usemin-maxnormalizationtotransformthevalue35forageontotherange[0.0,1.0].(b)Usez-scorenormalizationtotransformthevalue35forage,wherethestandarddeviationofageis12.94years.(c)Usenormalizationbydecimalscalingtotransformthevalue35forage.(d)Commentonwhichmethodyouwouldprefertouseforthegivendata,givingreasonsastowhy.'},
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              'text': 'HAN10-ch03-083-124-97801238147912011/6/13:16Page122#40122Chapter3DataPreprocessing3.8UsingthedataforageandbodyfatgiveninExercise2.4,answerthefollowing:(a)Normalizethetwoattributesbasedonz-scorenormalization.(b)Calculatethecorrelationcoefficient(Pearson’sproductmomentcoefficient).Arethesetwoattributespositivelyornegativelycorrelated?Computetheircovariance.3.9Supposeagroupof12salespricerecordshasbeensortedasfollows:5,10,11,13,15,35,50,55,72,92,204,215.Partitionthemintothreebinsbyeachofthefollowingmethods:(a)equal-frequency(equal-depth)partitioning(b)equal-widthpartitioning(c)clustering3.10Useaflowcharttosummarizethefollowingproceduresforattributesubsetselection:(a)stepwiseforwardselection(b)stepwisebackwardelimination(c)acombinationofforwardselectionandbackwardelimination3.11UsingthedataforagegiveninExercise3.3,(a)Plotanequal-widthhistogramofwidth10.(b)Sketchexamplesofeachofthefollowingsamplingtechniques:SRSWOR,SRSWR,clustersampling,andstratifiedsampling.Usesamplesofsize5andthestrata“youth,”“middle-aged,”and“senior.”3.12ChiMerge[Ker92]isasupervised,bottom-up(i.e.,merge-based)datadiscretizationmethod.Itreliesonχ2analysis:Adjacentintervalswiththeleastχ2valuesaremergedtogetheruntilthechosenstoppingcriterionsatisfies.(a)BrieflydescribehowChiMergeworks.(b)TaketheIRISdataset,obtainedfromtheUniversityofCalifornia–IrvineMachineLearningDataRepository(www.ics.uci.edu/∼mlearn/MLRepository.html),asadatasettobediscretized.PerformdatadiscretizationforeachofthefournumericattributesusingtheChiMergemethod.(Letthestoppingcriteriabe:max-interval=6).Youneedtowriteasmallprogramtodothistoavoidclumsynumericalcomputation.Submityoursimpleanalysisandyourtestresults:split-points,finalintervals,andthedocumentedsourceprogram.3.13Proposeanalgorithm,inpseudocodeorinyourfavoriteprogramminglanguage,forthefollowing:(a)Theautomaticgenerationofaconcepthierarchyfornominaldatabasedonthenumberofdistinctvaluesofattributesinthegivenschema.(b)Theautomaticgenerationofaconcepthierarchyfornumericdatabasedonth'},
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              'text': 'numericdatabasedontheequal-widthpartitioningrule.'},
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              'page': 303.0,
              'text': 'eislessthan1,thereisanegativecorrelationbetweentheoccur-renceof{game}and{video}.Thenumeratoristhelikelihoodofacustomerpurchasingboth,whilethedenominatoriswhatthelikelihoodwouldhavebeenifthetwopur-chaseswerecompletelyindependent.Suchanegativecorrelationcannotbeidentifiedbyasupport–confidenceframework.Thesecondcorrelationmeasurethatwestudyistheχ2measure,whichwasintro-ducedinChapter3(Eq.3.1).Tocomputetheχ2value,wetakethesquareddifferencebetweentheobservedandexpectedvalueforaslot(AandBpair)inthecontin-gencytable,dividedbytheexpectedvalue.Thisamountissummedforallslotsofthecontingencytable.Let’sperformaχ2analysisofExample6.8.'},
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              'page': 306.0,
              'text': 'rwise,itwouldgenerateunstableresults,asillustratedinD1andD2.'},
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              'page': 315.0,
              'text': 'dacomparativeanalysisofasetofmeasuresforpatternevaluation.'},
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              'page': 330.0,
              'text': '%and(P(B|A)+P(A|B))/2=(0.01+0.01)/2<0.02,againindicatingthatAandBarenegativelycorrelated.Thismatchesourintuition.Themeasuredoesnothavethenull-invarianceproblemofthefirsttwodefinitionsconsidered.Let’sexamineanothercase:Supposethatamong100,000transactions,thestoresold1000needlepackagesofAbutonly10packagesofB;however,everytimepackageBis'},
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              'page': 343.0,
              'text': 'pool.ForeachoftheseKseeds,wefindallthepatternswithinaballofasizespec-ifiedbyτ.Allthepatternsineach“ball”arethenfusedtogethertogenerateasetofsuperpatterns.Thesesuperpatternsformanewpool.IfthepoolcontainsmorethanKpatterns,thenextiterationbeginswiththispoolforthenewroundofrandomdrawing.Asthesupportsetofeverysuperpatternshrinkswitheachnewiteration,theiterationprocessterminates.NotethatPattern-Fusionmergessmallsubpatternsofalargepatterninsteadofincrementally-expandingpatternswithsingleitems.Thisgivesthemethodanadvantagetocircumventmidsizepatternsandprogressonapathleadingtoapotentialcolossalpattern.TheideaisillustratedinFigure7.10.Eachpointshowninthemetricspace'},
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              'page': 346.0,
              'text': 'δ≤1)measuresthetightnessofacluster.ApatternPisδ-coveredbyanotherpatternP(cid:48)ifO(P)⊆O(P(cid:48))andPatDist(P,P(cid:48))≤δ.Asetofpatternsformaδ-clusterifthereexistsarepresentativepatternPrsuchthatforeachpatternPintheset,Pisδ-coveredbyPr.'},
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              'text': 'llyindependentbutoftenclusteredinsmallregions.Thisissomewhatlikefind-ing20populationcentersintheworld,whichmayresultincitiesclusteredinasmallnumberofcountriesratherthanevenlydistributedacrosstheglobe.Instead,mostuserswouldprefertoderivethekmostinterestingpatterns,whicharenotonlysig-nificant,butalsomutuallyindependentandcontaininglittleredundancy.Asmallsetof'},
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              'text': 'ttributebecauseAdaandCathybothpurchasedP1.Clusteringhigh-dimensionaldataisthesearchforclustersandthespaceinwhichtheyexist.Thus,therearetwomajorkindsofmethods:Subspaceclusteringapproachessearchforclustersexistinginsubspacesofthegivenhigh-dimensionaldataspace,whereasubspaceisdefinedusingasubsetofattributesinthefullspace.SubspaceclusteringapproachesarediscussedinSection11.2.2.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page510#14510Chapter11AdvancedClusterAnalysisDimensionalityreductionapproachestrytoconstructamuchlower-dimensionalspaceandsearchforclustersinsuchaspace.Often,amethodmayconstructnewdimensionsbycombiningsomedimensionsfromtheoriginaldata.DimensionalityreductionmethodsarethetopicofSection11.2.4.Ingeneral,clusteringhigh-dimensionaldataraisesseveralnewchallengesinadditiontothoseofconventionalclustering:Amajorissueishowtocreateappropriatemodelsforclustersinhigh-dimensionaldata.Unlikeconventionalclustersinlow-dimensionalspaces,clustershiddeninhigh-dimensionaldataareoftensignificantlysmaller.Forexample,whenclusteringcustomer-purchasedata,wewouldnotexpectmanyuserstohavesimilarpurchasepatterns.Searchingforsuchsmallbutmeaningfulclustersislikefindingneedlesinahaystack.Asshownbefore,theconventionaldistancemeasurescanbeineffective.Instead,weoftenhavetoconsidervariousmoresophisticatedtechniquesthatcanmodelcorrelationsandconsistencyamongobjectsinsubspaces.Therearetypicallyanexponentialnumberofpossiblesubspacesordimensionalityreductionoptions,andthustheoptimalsolutionsareoftencomputationallypro-hibitive.Forexample,iftheoriginaldataspacehas1000dimensions,andwewanttofindclustersofdimensionality10,thenthereare(cid:32)100010(cid:33)=2.63×1023possiblesubspaces.11.2.2SubspaceClusteringMethods“Howcanwefindsubspaceclustersfromhigh-dimensionaldata?”Manymethodshavebeenproposed.Theygenerallycanbecategorizedintothreemajorgroups:subspacesearchmethods,correlation-basedclusteringmethods,andbiclusteringmethods.SubspaceSearchMethodsAsubspacesearchmethodsearchesvarioussubspacesforclusters.Here,aclusterisasubsetofobjectsthataresimilartoeachotherinasubspace.Thesimilarityisoftencap-turedbyconventionalmeasuressuchasdistanceordensity.Forexample,theCLIQUEalgorithmintroducedinSection10.5.2isasubspaceclusteringmethod.Itenumeratessubspacesandtheclustersinthosesubspacesinadimensionality-increasingorder,andappliesantimonotonicitytoprunesubspacesinwhichnoclustermayexist'},
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              'page': 547.0,
              'text': 'ichnoclustermayexist.Amajorchallengethatsubspacesearchmethodsfaceishowtosearchaseriesofsubspaceseffectivelyandefficiently.Generallytherearetwokindsofstrategies:Bottom-upapproachesstartfromlow-dimensionalsubspacesandsearchhigher-dimensionalsubspacesonlywhentheremaybeclustersinthosehigher-dimensional'},
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              'page': 548.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page511#1511.2ClusteringHigh-DimensionalData511subspaces.Variouspruningtechniquesareexploredtoreducethenumberofhigher-dimensionalsubspacesthatneedtobesearched.CLIQUEisanexampleofabottom-upapproach.Top-downapproachesstartfromthefullspaceandsearchsmallerandsmallersub-spacesrecursively.Top-downapproachesareeffectiveonlyifthelocalityassumptionholds,whichrequirethatthesubspaceofaclustercanbedeterminedbythelocalneighborhood.Example11.10PROCLUS,atop-downsubspaceapproach.PROCLUSisak-medoid-likemethodthatfirstgenerateskpotentialclustercentersforahigh-dimensionaldatasetusingasampleofthedataset.Itthenrefinesthesubspaceclustersiteratively.Ineachitera-tion,foreachofthecurrentk-medoids,PROCLUSconsidersthelocalneighborhoodofthemedoidinthewholedataset,andidentifiesasubspacefortheclusterbymini-mizingthestandarddeviationofthedistancesofthepointsintheneighborhoodtothemedoidoneachdimension.Onceallthesubspacesforthemedoidsaredeter-mined,eachpointinthedatasetisassignedtotheclosestmedoidaccordingtothecorrespondingsubspace.Clustersandpossibleoutliersareidentified.Inthenextiter-ation,newmedoidsreplaceexistingonesifdoingsoimprovestheclusteringquality.Correlation-BasedClusteringMethodsWhilesubspacesearchmethodssearchforclusterswithasimilaritythatismeasuredusingconventionalmetricslikedistanceordensity,correlation-basedapproachescanfurtherdiscoverclustersthataredefinedbyadvancedcorrelationmodels.Example11.11Acorrelation-basedapproachusingPCA.Asanexample,aPCA-basedapproachfirstappliesPCA(PrincipalComponentsAnalysis;seeChapter3)toderiveasetofnew,uncorrelateddimensions,andthenmineclustersinthenewspaceoritssubspaces.InadditiontoPCA,otherspacetransformationsmaybeused,suchastheHoughtransformorfractaldimensions.Foradditionaldetailsonsubspacesearchmethodsandcorrelation-basedclusteringmethods,pleaserefertothebibliographicnotes(Section11.7).BiclusteringMethodsInsomeapplications,wewanttoclusterbothobjectsandattributessimultaneously.Theresultingclustersareknownas'},
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              'page': 548.0,
              'text': 'ngclustersareknownasbiclustersandmeetfourrequirements:(1)onlyasmallsetofobjectsparticipateinacluster;(2)aclusteronlyinvolvesasmallnumberofattributes;(3)anobjectmayparticipateinmultipleclusters,ordoesnotparticipateinanycluster;and(4)anattributemaybeinvolvedinmultipleclusters,orisnotinvolvedinanycluster.Section11.2.3discussesbiclusteringindetail.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page512#16512Chapter11AdvancedClusterAnalysis11.2.3BiclusteringIntheclusteranalysisdiscussedsofar,weclusterobjectsaccordingtotheirattributevalues.Objectsandattributesarenottreatedinthesameway.However,insomeapplica-tions,objectsandattributesaredefinedinasymmetricway,wheredataanalysisinvolvessearchingdatamatricesforsubmatricesthatshowuniquepatternsasclusters.Thiskindofclusteringtechniquebelongstothecategoryofbiclustering.Thissectionfirstintroducestwomotivatingapplicationexamplesofbiclustering—geneexpressionandrecommendersystems.Youwillthenlearnaboutthedifferenttypesofbiclusters.Last,wepresentbiclusteringmethods.ApplicationExamplesBiclusteringtechniqueswerefirstproposedtoaddresstheneedsforanalyzinggeneexpressiondata.Ageneisaunitofthepassing-onoftraitsfromalivingorganismtoitsoffspring.Typically,ageneresidesonasegmentofDNA.GenesarecriticalforalllivingthingsbecausetheyspecifyallproteinsandfunctionalRNAchains.Theyholdtheinformationtobuildandmaintainalivingorganism’scellsandpassgenetictraitstooffspring.Synthesisofafunctionalgeneproduct,eitherRNAorprotein,reliesontheprocessofgeneexpression.Agenotypeisthegeneticmakeupofacell,anorganism,oranindividual.Phenotypesareobservablecharacteristicsofanorganism.Geneexpressionisthemostfundamentallevelingeneticsinthatgenotypescausephenotypes.UsingDNAchips(alsoknownasDNAmicroarrays)andotherbiologicalengineer-ingtechniques,wecanmeasuretheexpressionlevelofalargenumber(possiblyall)ofanorganism’sgenes,inanumberofdifferentexperimentalconditions.Suchconditionsmaycorrespondtodifferenttimepointsinanexperimentorsamplesfromdifferentorgans.Roughlyspeaking,thegeneexpressiondataorDNAmicroarraydataareconcep-tuallyagene-sample/conditionmatrix,whereeachrowcorrespondstoonegene,andeachcolumncorrespondstoonesampleorcondition.Eachelementinthematrixisarealnumberandrecordstheexpressionlevelofageneunderaspecificcondition.Figure11.3showsanillustration.Fromtheclusteringviewpoint,aninterestingissueisthatageneexpressiondat'},
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              'page': 549.0,
              'text': 'atageneexpressiondatamatrixcanbeanalyzedintwodimensions—thegenedimensionandthesample/conditiondimension.Whenanalyzinginthegenedimension,wetreateachgeneasanobjectandtreatthesamples/conditionsasattributes.Bymininginthegenedimension,wemayfindpat-ternssharedbymultiplegenes,orclustergenesintogroups.Forexample,wemayfindagroupofgenesthatexpressthemselvessimilarly,whichishighlyinterestinginbioinformatics,suchasinfindingpathways.Whenanalyzinginthesample/conditiondimension,wetreateachsample/conditionasanobjectandtreatthegenesasattributes.Inthisway,wemayfindpatternsofsamples/conditions,orclustersamples/conditionsintogroups.Forexample,wemayfindthedifferencesingeneexpressionbycomparingagroupoftumorsamplesandnontumorsamples.'},
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              'page': 550.0,
              'text': '# 11.2 Clustering High-Dimensional Data\n'
                      '\n'
                      '![Figure 11.3 Microarray data matrix.](path/to/image)\n'
                      '\n'
                      '## Example 11.12 Gene expression\n'
                      '\n'
                      'Gene expression matrices are popular in bioinformatics '
                      'research and development. For example, an important '
                      'task is to classify a new gene using the expression '
                      'data of the gene and that of other genes in known '
                      'classes. Symmetrically, we may classify a new sample '
                      '(e.g., a new patient) using the expression data of the '
                      'sample and that of samples in known classes (e.g., '
                      'tumor and non-tumor). Such tasks are invaluable in '
                      'understanding the mechanisms of diseases and in '
                      'clinical treatment.\n'
                      '\n'
                      'As can be seen, many gene expression data mining '
                      'problems are highly related to cluster analysis. '
                      'However, a challenge here is that, instead of '
                      'clustering in one dimension (e.g., gene or '
                      'sample/condition), in many cases we need to cluster in '
                      'two dimensions simultaneously (e.g., both gene and '
                      'sample/condition). Moreover, unlike the clustering '
                      'models we have discussed so far, a cluster in a gene '
                      'expression data matrix is a *submatrix* and usually has '
                      'the following characteristics:\n'
                      '\n'
                      '- Only a small set of genes participate in the '
                      'cluster.\n'
                      '- The cluster involves only a small subset of '
                      'samples/conditions.\n'
                      '- A gene may participate in multiple clusters, or may '
                      'not participate in any cluster.\n'
                      '- A sample/condition may be involved in multiple '
                      'clusters, or may not be involved in any cluster.\n'
                      '\n'
                      'To find clusters in gene-sample/condition matrices, we '
                      'need new clustering techniques that meet the following '
                      'requirements for biclustering:\n'
                      '\n'
                      '- A cluster of genes is defined using only a subset of '
                      'samples/conditions.\n'
                      '- A cluster of samples/conditions is defined using only '
                      'a subset of genes.'},
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              'page': 551.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page514#18514Chapter11AdvancedClusterAnalysisTheclustersareneitherexclusive(e.g.,whereonegenecanparticipateinmultipleclusters)norexhaustive(e.g.,whereagenemaynotparticipateinanycluster).Biclusteringisusefulnotonlyinbioinformatics,butalsoinotherapplicationsaswell.Considerrecommendersystemsasanexample.Example11.13Usingbiclusteringforarecommendersystem.AllElectronicscollectsdatafromcus-tomers’evaluationsofproductsandusesthedatatorecommendproductstocustomers.Thedatacanbemodeledasacustomer-productmatrix,whereeachrowrepresentsacustomer,andeachcolumnrepresentsaproduct.Eachelementinthematrixrepresentsacustomer’sevaluationofaproduct,whichmaybeascore(e.g.,like,likesomewhat,notlike)orpurchasebehavior(e.g.,buyornot).Figure11.4illustratesthestructure.Thecustomer-productmatrixcanbeanalyzedintwodimensions:thecustomerdimensionandtheproductdimension.Treatingeachcustomerasanobjectandproductsasattributes,AllElectronicscanfindcustomergroupsthathavesimilarpreferencesorpurchasepatterns.Usingproductsasobjectsandcustomersasattributes,AllElectronicscanmineproductgroupsthataresimilarincustomerinterest.Moreover,AllElectronicscanmineclustersinbothcustomersandproductssimulta-neously.Suchaclustercontainsasubsetofcustomersandinvolvesasubsetofproducts.Forexample,AllElectronicsishighlyinterestedinfindingagroupofcustomerswhoalllikethesamegroupofproducts.Suchaclusterisasubmatrixinthecustomer-productmatrix,whereallelementshaveahighvalue.Usingsuchacluster,AllElectronicscanmakerecommendationsintwodirections.First,thecompanycanrecommendproductstonewcustomerswhoaresimilartothecustomersinthecluster.Second,thecompanycanrecommendtocustomersnewproductsthataresimilartothoseinvolvedinthecluster.Aswithbiclustersinageneexpressiondatamatrix,thebiclustersinacustomer-productmatrixusuallyhavethefollowingcharacteristics:Onlyasmallsetofcustomersparticipateinacluster.Aclusterinvolvesonlyasmallsubsetofproducts.Acustomercanparticipateinmultipleclusters,ormaynotparticipateinanycl'},
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              'page': 552.0,
              'text': '# 11.2 Clustering High-Dimensional Data\n'
                      '\n'
                      'A product may be involved in multiple clusters, or may '
                      'not be involved in any cluster.\n'
                      '\n'
                      'Biclustering can be applied to customer-product '
                      'matrices to mine clusters satisfying these '
                      'requirements.\n'
                      '\n'
                      '## Types of Biclusters\n'
                      '\n'
                      "**“How can we model biclusters and mine them?”** Let's "
                      'start with some basic notation. For the sake of '
                      'simplicity, we will use “genes” and “conditions” to '
                      'refer to the two dimensions in our discussion. Our '
                      'discussion can easily be extended to other '
                      'applications. For example, we can simply replace '
                      '“genes” and “conditions” by “customers” and “products” '
                      'to tackle the customer-product biclustering problem. \n'
                      '\n'
                      'Let \\( A = \\{a_1, a_2, \\ldots, a_m\\} \\) be a set '
                      'of genes and \\( B = \\{b_1, b_2, \\ldots, b_n\\} \\) '
                      'be a set of conditions. Let \n'
                      '\n'
                      '\\[ \n'
                      'E = \\{e_{ij}\\} \n'
                      '\\]\n'
                      '\n'
                      'be a gene expression data matrix, that is, a '
                      'gene-condition matrix, where \\( 1 \\leq i \\leq m \\) '
                      'and \\( 1 \\leq j \\leq n \\). A submatrix \\( J '
                      '\\times I \\) is defined by a subset \\( I \\subset A '
                      '\\) of genes and a subset \\( J \\subset B \\) of '
                      'conditions. For example, in the matrix shown in Figure '
                      '11.5, \\( (a_1, a_3, a5) \\) and \\( \\{b_1, b_2, b_3, '
                      'b_4\\} \\) is a submatrix.\n'
                      '\n'
                      'A bicluster is a submatrix where genes and conditions '
                      'follow consistent patterns. We can define different '
                      'types of biclusters based on such patterns.\n'
                      '\n'
                      '- At the simplest case, a submatrix \\( J \\times I \\) '
                      'of \\( E (J \\subset A, I \\subset B) \\) is a '
                      '**bicluster with constant values** if for any \\( i '
                      '\\in I \\) and \\( j \\in J \\), \\( e_{ij} = c \\), '
                      'where \\( c \\) is a constant. For example, the '
                      'submatrix \\( (a_4,a_5,a_3) \\) and \\( '
                      '\\{b_2,b_3,b_4,b_5\\} \\) in Figure 11.5 is a bicluster '
                      'with constant values.\n'
                      '\n'
                      '- A bicluster is interesting if each row has a constant '
                      'value, though different rows may have different values. '
                      'A bicluster with constant values on rows is a submatrix '
                      '\\( J \\times I \\) such that for any \\( i \\in I \\) '
                      'and \\( j \\in J \\), \\( e_{ij} = c + \\beta_j \\), '
                      'where \\( \\beta_j \\) is the adjustment for row \\( j '
                      '\\). For example, Figure 11.6 shows a bicluster with '
                      'constant values on rows.\n'
                      '\n'
                      '- Symmetrically, a bicluster with constant values on '
                      'columns is a submatrix \\( J \\times I \\) such that '
                      'for any \\( i \\in I \\) and \\( j \\in J \\), \\( '
                      'e_{ij} = c + \\beta_i \\), where \\( \\beta_i \\) is '
                      'the adjustment for column \\( i \\).\n'
                      '\n'
                      '|      | b1   | b2   | b3   | ... | bg   |\n'
                      '|------|------|------|------|-----|------|\n'
                      '| a1   | ...  | ...  | ...  | ... | ...  |\n'
                      '| a2   | ...  | ...  | ...  | ... | ...  |\n'
                      '| a3   | ...  | ...  | ...  | ... | ...  |\n'
                      '| ...  | ...  | ...  | ...  | ... | ...  |\n'
                      '| am   | ...  | ...  | ...  | ... | ...  |\n'
                      '\n'
                      'Figure 11.5 Gene-condition matrix, a submatrix, and a '
                      'bicluster.'},
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              'page': 553.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page516#20516Chapter11AdvancedClusterAnalysisMoregenerally,abiclusterisinterestingiftherowschangeinasynchronizedwaywithrespecttothecolumnsandviceversa.Mathematically,abiclusterwithcoherentvalues(alsoknownasapattern-basedcluster)isasubmatrixI×Jsuchthatforanyi∈Iandj∈J,eij=c+αi+βj,whereαiandβjaretheadjustmentforrowiandcolumnj,respectively.Forexample,Figure11.7showsabiclusterwithcoherentvalues.ItcanbeshownthatI×Jisabiclusterwithcoherentvaluesifandonlyifforanyi1,i2∈Iandj1,j2∈J,thenei1j1−ei2j1=ei1j2−ei2j2.Moreover,insteadofusingaddition,wecandefineabiclusterwithcoherentvaluesusingmultiplication,thatis,eij=c·(cid:0)αi·βj(cid:1).Clearly,biclusterswithconstantvaluesonrowsorcolumnsarespecialcasesofbiclusterswithcoherentvalues.Insomeapplications,wemayonlybeinterestedintheup-ordown-regulatedchangesacrossgenesorconditionswithoutconstrainingtheexactvalues.Abiclus-terwithcoherentevolutionsonrowsisasubmatrixI×Jsuchthatforanyi1,i2∈Iandj1,j2∈J,(ei1j1−ei1j2)(ei2j1−ei2j2)≥0.Forexample,Figure11.8showsabiclus-terwithcoherentevolutionsonrows.Symmetrically,wecandefinebiclusterswithcoherentevolutionsoncolumns.Next,westudyhowtominebiclusters.10101010102020202020505050505000000Figure11.6Biclusterwithconstantvaluesonrows.1050307020206040803050907011060040206010Figure11.7Biclusterwithcoherentvalues.105030702020100501000305010090120800802010010Figure11.8Biclusterwithcoherentevolutionsonrows.'},
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              'page': 554.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page517#2111.2ClusteringHigh-DimensionalData517BiclusteringMethodsThepreviousspecificationofthetypesofbiclustersonlyconsidersidealcases.Inrealdatasets,suchperfectbiclustersrarelyexist.Whentheydoexist,theyareusuallyverysmall.Instead,randomnoisecanaffectthereadingsofeijandthuspreventabiclusterinnaturefromappearinginaperfectshape.Therearetwomajortypesofmethodsfordiscoveringbiclustersindatathatmaycomewithnoise.Optimization-basedmethodsconductaniterativesearch.Ateachiteration,thesubmatrixwiththehighestsignificancescoreisidentifiedasabicluster.Theprocessterminateswhenauser-specifiedconditionismet.Duetocostconcernsincomputation,greedysearchisoftenemployedtofindlocaloptimalbiclusters.Enu-merationmethodsuseatolerancethresholdtospecifythedegreeofnoiseallowedinthebiclusterstobemined,andthentriestoenumerateallsubmatricesofbiclustersthatsatisfytherequirements.Weusetheδ-ClusterandMaPlealgorithmsasexamplestoillustratetheseideas.OptimizationUsingtheδ-ClusterAlgorithmForasubmatrix,I×J,themeanoftheithrowiseiJ=1|J|(cid:88)j∈Jeij.(11.16)Symmetrically,themeanofthejthcolumniseIj=1|I|(cid:88)i∈Ieij.(11.17)ThemeanofallelementsinthesubmatrixiseIJ=1|I||J|(cid:88)i∈I,j∈Jeij=1|I|(cid:88)i∈IeiJ=1|J|(cid:88)j∈JeIj.(11.18)Thequalityofthesubmatrixasabiclustercanbemeasuredbythemean-squaredresiduevalueasH(I×J)=1|I||J|(cid:88)i∈I,j∈J(eij−eiJ−eIj+eIJ)2.(11.19)SubmatrixI×Jisaδ-biclusterifH(I×J)≤δ,whereδ≥0isathreshold.Whenδ=0,I×Jisaperfectbiclusterwithcoherentvalues.Bysettingδ>0,ausercanspecifythetoleranceofaveragenoiseperelementagainstaperfectbicluster,becauseinEq.(11.19)theresidueoneachelementisresidue(eij)=eij−eiJ−eIj+eIJ.(11.20)Amaximalδ-biclusterisaδ-biclusterI×Jsuchthattheredoesnotexistanotherδ-biclusterI(cid:48)×J(cid:48),andI⊆I(cid:48),J⊆J(cid:48),andatleastoneinequalityholds.Findingthe'},
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              'page': 555.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page518#22518Chapter11AdvancedClusterAnalysismaximalδ-biclusterofthelargestsizeiscomputationallycostly.Therefore,wecanuseaheuristicgreedysearchmethodtoobtainalocaloptimalcluster.Thealgorithmworksintwophases.Inthedeletionphase,westartfromthewholematrix.Whilethemean-squaredresidueofthematrixisoverδ,weiterativelyremoverowsandcolumns.Ateachiteration,foreachrowi,wecomputethemean-squaredresidueasd(i)=1|J|(cid:88)j∈J(eij−eiJ−eIj+eIJ)2.(11.21)Moreover,foreachcolumnj,wecomputethemean-squaredresidueasd(j)=1|I|(cid:88)i∈I(eij−eiJ−eIj+eIJ)2.(11.22)Weremovetheroworcolumnofthelargestmean-squaredresidue.Attheendofthisphase,weobtainasubmatrixI×Jthatisaδ-bicluster.However,thesubmatrixmaynotbemaximal.Intheadditionphase,weiterativelyexpandtheδ-biclusterI×Jobtainedinthedele-tionphaseaslongastheδ-biclusterrequirementismaintained.Ateachiteration,weconsiderrowsandcolumnsthatarenotinvolvedinthecurrentbiclusterI×Jbycal-culatingtheirmean-squaredresidues.Aroworcolumnofthesmallestmean-squaredresidueisaddedintothecurrentδ-bicluster.Thisgreedyalgorithmcanfindoneδ-biclusteronly.Tofindmultiplebiclustersthatdonothaveheavyoverlaps,wecanrunthealgorithmmultipletimes.Aftereachexecu-tionwhereaδ-biclusterisoutput,wecanreplacetheelementsintheoutputbiclusterbyrandomnumbers.Althoughthegreedyalgorithmmayfindneithertheoptimalbiclustersnorallbiclusters,itisveryfastevenonlargematrices.EnumeratingAllBiclustersUsingMaPleAsmentioned,asubmatrixI×Jisabiclusterwithcoherentvaluesifandonlyifforanyi1,i2∈Iandj1,j2∈J,ei1j1−ei2j1=ei1j2−ei2j2.Forany2×2submatrixofI×J,wecandefineap-scoreasp-score(cid:32)ei1j1ei1j2ei2j1ei2j2(cid:33)=|(ei1j1−ei2j1)−(ei1j2−ei2j2)|.(11.23)AsubmatrixI×Jisaδ-pCluster(forpattern-basedcluster)ifthep-scoreofevery2×2submatrixofI×Jisatmostδ,whereδ≥0isathresholdspecifyingauser’stoleranceofnoiseagainstaperfectbicluster.Here,thep-scorecontrolsthenoiseoneveryelementinabicluster,whilethemean-squaredresiduecapturestheaveragenoise.Aninterestingpropertyofδ-pClust'},
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              'page': 555.0,
              'text': 'ngpropertyofδ-pClusteristhatifI×Jisaδ-pCluster,theneveryx×y(x,y≥2)submatrixofI×Jisalsoaδ-pCluster.Thismonotonicityenables'},
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              'page': 556.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page519#2311.2ClusteringHigh-DimensionalData519ustoobtainasuccinctrepresentationofnonredundantδ-pClusters.Aδ-pClusterismaximalifnomorerowsorcolumnscanbeaddedintotheclusterwhilemaintainingtheδ-pClusterproperty.Toavoidredundancy,insteadoffindingallδ-pClusters,weonlyneedtocomputeallmaximalδ-pClusters.MaPleisanalgorithmthatenumeratesallmaximalδ-pClusters.Itsystematicallyenumerateseverycombinationofconditionsusingasetenumerationtreeandadepth-firstsearch.Thisenumerationframeworkisthesameasthepattern-growthmethodsforfrequentpatternmining(Chapter6).Considergeneexpressiondata.Foreachcon-ditioncombination,J,MaPlefindsthemaximalsubsetsofgenes,I,suchthatI×Jisaδ-pCluster.IfI×Jisnotasubmatrixofanotherδ-pCluster,thenI×Jisamaximalδ-pCluster.Theremaybeahugenumberofconditioncombinations.MaPleprunesmanyunfruitfulcombinationsusingthemonotonicityofδ-pClusters.Foraconditioncom-bination,J,iftheredoesnotexistasetofgenes,I,suchthatI×Jisaδ-pCluster,thenwedonotneedtoconsideranysupersetofJ.Moreover,weshouldconsiderI×Jasacandidateofaδ-pClusteronlyifforevery(|J|−1)-subsetJ(cid:48)ofJ,I×J(cid:48)isaδ-pCluster.MaPlealsoemploysseveralpruningtechniquestospeedupthesearchwhileretainingthecompletenessofreturningallmaximalδ-pClusters.Forexample,whenexaminingacurrentδ-pCluster,I×J,MaPlecollectsallthegenesandconditionsthatmaybeaddedtoexpandthecluster.IfthesecandidategenesandconditionstogetherwithIandJformasubmatrixofaδ-pClusterthathasalreadybeenfound,thenthesearchofI×JandanysupersetofJcanbepruned.InterestedreadersmayrefertothebibliographicnotesforadditionalinformationontheMaPlealgorithm(Section11.7).Aninterestingobservationhereisthatthesearchformaximalδ-pClustersinMaPleissomewhatsimilartominingfrequentcloseditemsets.Consequently,MaPleborrowsthedepth-firstsearchframeworkandideasfromthepruningtechniquesofpattern-growthmethodsforfrequentpatternmining.Thisisanexamplewherefrequentpatternminingandclusteranalysismaysharesimilartechniquesandideas.AnadvantageofMaPleandth'},
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              'page': 556.0,
              'text': 'dvantageofMaPleandtheotheralgorithmsthatenumerateallbiclustersisthattheyguaranteethecompletenessoftheresultsanddonotmissanyoverlappingbiclus-ters.However,achallengeforsuchenumerationalgorithmsisthattheymaybecomeverytimeconsumingifamatrixbecomesverylarge,suchasacustomer-purchasematrixofhundredsofthousandsofcustomersandmillionsofproducts.11.2.4DimensionalityReductionMethodsandSpectralClusteringSubspaceclusteringmethodstrytofindclustersinsubspacesoftheoriginaldataspace.Insomesituations,itismoreeffectivetoconstructanewspaceinsteadofusingsubspacesoftheoriginaldata.Thisisthemotivationbehinddimensionalityreductionmethodsforclusteringhigh-dimensionaldata.Example11.14Clusteringinaderivedspace.ConsiderthethreeclustersofpointsinFigure11.9.Itisnotpossibletoclusterthesepointsinanysubspaceoftheoriginalspace,X×Y,because'},
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              'page': 557.0,
              'text': '# Chapter 11 Advanced Cluster Analysis\n'
                      '\n'
                      '![Figure 11.9](image_url) Clustering in a derived space '
                      'may be more effective.\n'
                      '\n'
                      'all three clusters would end up being projected onto '
                      'overlapping areas in the x and y axes. What if, '
                      'instead, we construct a new dimension, '
                      '\\(-\\frac{x^2}{2} + \\frac{y^2}{2}\\) (shown as a '
                      'dashed line in the figure)? By projecting the points '
                      'onto this new dimension, the three clusters become '
                      'apparent.\n'
                      '\n'
                      'Although Example 11.14 involves only two dimensions, '
                      'the idea of constructing a new space (so that any '
                      'clustering structure that is hidden in the data becomes '
                      'well manifested) can be extended to high-dimensional '
                      'data. Preferably, the newly constructed space should '
                      'have low dimensionality.\n'
                      '\n'
                      'There are many dimensionality reduction methods. A '
                      'straightforward approach is to apply feature selection '
                      'and extraction methods to the data set as those '
                      'discussed in Chapter 3. However, such methods may not '
                      'be able to detect the clustering structure. Therefore, '
                      'methods that combine feature extraction and clustering '
                      'are preferred. In this section, we introduce **spectral '
                      'clustering**, a group of methods that are effective in '
                      'high-dimensional data applications.\n'
                      '\n'
                      '## Figure 11.10\n'
                      '\n'
                      'shows the general framework for spectral clustering '
                      'approaches. The Ng-Jordan-Weiss algorithm is a spectral '
                      'clustering method. Let’s have a look at each step of '
                      'the framework. In doing so, we also note special '
                      'conditions that apply to the Ng-Jordan-Weiss algorithm '
                      'as an example.\n'
                      '\n'
                      'Given a set of objects \\(o_1, o_2, \\ldots, o_n\\), '
                      'the distance between each pair of objects, \\(dist(o_i, '
                      'o_j)\\) \\((1 \\leq i < j \\leq n)\\), and the desired '
                      'number \\(k\\) of clusters, a spectral clustering '
                      'approach works as follows:\n'
                      '\n'
                      '1. Using the distance measure, calculate an affinity '
                      'matrix, \\(W\\), such that\n'
                      '\n'
                      '   \\[\n'
                      '   W_{ij} = e^{-\\frac{(dist(o_i, '
                      'o_j))^2}{\\sigma^2}},\n'
                      '   \\]\n'
                      '\n'
                      '   where \\(\\sigma\\) is a scaling parameter that '
                      'controls how fast the affinity \\(W_{ij}\\) decreases '
                      'as \\(dist(o_i, o_j)\\) increases. In the '
                      'Ng-Jordan-Weiss algorithm, \\(W_{ij}\\) is set to 0.'},
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              'page': 558.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page521#2511.2ClusteringHigh-DimensionalData521[wij]Affinity '
                      'matrixDataA=f(w)Av=λvClustering inthe new spaceCompute '
                      'leadingk eigenvectorsof AProject backto cluster '
                      'theoriginal '
                      'dataFigure11.10Theframeworkofspectralclusteringapproaches.Source:AdaptedfromSlide8athttp://videolectures.net/micued08azranmcl/.2.UsingtheaffinitymatrixW,deriveamatrixA=f(W).Thewayinwhichthisisdonecanvary.TheNg-Jordan-Weissalgorithmdefinesamatrix,D,asadiagonalmatrixsuchthatDiiisthesumoftheithrowofW,thatis,Dii=n(cid:88)j=1Wij.(11.24)AisthensettoA=D−12WD−12.(11.25)3.FindthekleadingeigenvectorsofA.Recallthattheeigenvectorsofasquarematrixarethenonzerovectorsthatremainproportionaltotheoriginalvectorafterbeingmultipliedbythematrix.Mathematically,avectorvisaneigenvectorofmatrixAifAv=λv,whereλiscalledthecorrespondingeigenvalue.ThisstepderivesknewdimensionsfromA,whicharebasedontheaffinitymatrixW.Typically,kshouldbemuchsmallerthanthedimensionalityoftheoriginaldata.TheNg-Jordan-Weissalgorithmcomputesthekeigenvectorswiththelargesteigenvaluesx1,...,xkofA.4.Usingthekleadingeigenvectors,projecttheoriginaldataintothenewspacedefinedbythekleadingeigenvectors,andrunaclusteringalgorithmsuchask-meanstofindkclusters.TheNg-Jordan-WeissalgorithmstackstheklargesteigenvectorsincolumnstoformamatrixX=[x1x2···xk]∈Rn×k.ThealgorithmformsamatrixYbyrenormalizingeachrowinXtohaveunitlength,thatis,Yij=Xij(cid:113)(cid:80)kj=1X2ij.(11.26)ThealgorithmthentreatseachrowinYasapointinthek-dimensionalspaceRk,andrunsk-means(oranyotheralgorithmservingthepartitioningpurpose)toclusterthepointsintokclusters.'},
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              'page': 559.0,
              'text': '# 522\n'
                      '## Chapter 11 Advanced Cluster Analysis\n'
                      '\n'
                      '![Figure '
                      '1.1.11](http://videlectures.net/micub08.azrn.mdl)  \n'
                      'The new dimensions and the clustering results of the '
                      'Ng-Jordan-Weiss algorithm. Source: Adapted from Slide 9 '
                      'at \n'
                      '\n'
                      '5. Assign the original data points to clusters '
                      'according to how the transformed points are assigned in '
                      'the clusters obtained in step 4.  \n'
                      '   In the Ng-Jordan-Weiss algorithm, the original '
                      'object \\( y_i \\) is assigned to the jth cluster if '
                      "and only if matrix \\( Y \\)'s row is assigned to the "
                      'jth cluster as a result of step 4.\n'
                      '\n'
                      'In spectral clustering methods, the dimensionality of '
                      'the new space is set to the desired number of clusters. '
                      'This setting expects that each new dimension should be '
                      'able to manifest a cluster.\n'
                      '\n'
                      '### Example 1.1.15\n'
                      '**The Ng-Jordan-Weiss algorithm.** Consider the set of '
                      'points in Figure 1.1.11. The data set, the affinity '
                      'matrix, the three largest eigenvectors, and the '
                      'normalized vectors are shown. Note that with the three '
                      'new dimensions (formed by the three largest '
                      'eigenvectors), the clusters are easily detected.\n'
                      '\n'
                      'Spectral clustering is effective in high-dimensional '
                      'applications such as image processing. Theoretically, '
                      'it works well when certain conditions apply. '
                      'Scalability, however, is a challenge. Computing '
                      'eigenvectors on a large matrix is costly. Spectral '
                      'clustering can be combined with other clustering '
                      'methods, such as clustering. Additional information on '
                      'other dimensionality reduction clustering methods, such '
                      'as kernel PCA, can be found in the bibliographic notes '
                      '(Section 11.7).\n'
                      '\n'
                      '## 11.3 Clustering Graph and Network Data\n'
                      'Cluster analysis on graph and network data extracts '
                      'valuable knowledge and information. Such data are '
                      'increasingly popular in many applications. We discuss '
                      'applications and challenges of clustering graph and '
                      'network data in Section 11.3.1. Similarity measures for '
                      'this form of clustering are given in Section 11.3.2. '
                      'You will learn about graph clustering methods in '
                      'Section 11.3.3.\n'
                      '\n'
                      'In general, the terms graph and network can be used '
                      'interchangeably. In the rest of this section, we mainly '
                      'use the term **graph**.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page523#2711.3ClusteringGraphandNetworkData52311.3.1ApplicationsandChallengesAsacustomerrelationshipmanageratAllElectronics,younoticethatalotofdatarelatingtocustomersandtheirpurchasebehaviorcanbepreferablymodeledusinggraphs.Example11.16Bipartitegraph.ThecustomerpurchasebehavioratAllElectronicscanberepresentedinabipartitegraph.Inabipartitegraph,verticescanbedividedintotwodisjointsetssothateachedgeconnectsavertexinonesettoavertexintheotherset.FortheAllElectronicscustomerpurchasedata,onesetofverticesrepresentscustomers,withonecustomerpervertex.Theothersetrepresentsproducts,withoneproductpervertex.Anedgeconnectsacustomertoaproduct,representingthepurchaseoftheproductbythecustomer.Figure11.12showsanillustration.“Whatkindofknowledgecanweobtainbyaclusteranalysisofthecustomer-productbipartitegraph?”Byclusteringthecustomerssuchthatthosecustomersbuyingsimilarsetsofproductsareplacedintoonegroup,acustomerrelationshipmanagercanmakeproductrecommendations.Forexample,supposeAdabelongstoacustomerclusterinwhichmostofthecustomerspurchasedadigitalcamerainthelast12months,butAdahasyettopurchaseone.Asmanager,youdecidetorecommendadigitalcameratoher.Alternatively,wecanclusterproductssuchthatthoseproductspurchasedbysimilarsetsofcustomersaregroupedtogether.Thisclusteringinformationcanalsobeusedforproductrecommendations.Forexample,ifadigitalcameraandahigh-speedflashmemorycardbelongtothesameproductcluster,thenwhenacustomerpurchasesadigitalcamera,wecanrecommendthehigh-speedflashmemorycard.Bipartitegraphsarewidelyusedinmanyapplications.Consideranotherexample.Example11.17Websearchengines.Inwebsearchengines,searchlogsarearchivedtorecorduserqueriesandthecorrespondingclick-throughinformation.(Theclick-throughinforma-tiontellsusonwhichpages,givenasaresultofasearch,theuserclicked.)Thequeryandclick-throughinformationcanberepresentedusingabipartitegraph,wherethetwosetsCustomersProductsFigure11.12Bipartitegraphrepresentingcustomer-purchasedata.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page524#28524Chapter11AdvancedClusterAnalysisofverticescorrespondtoqueriesandwebpages,respectively.Anedgelinksaquerytoawebpageifauserclicksthewebpagewhenaskingthequery.Valuableinformationcanbeobtainedbyclusteranalysesonthequery–webpagebipartitegraph.Forinstance,wemayidentifyqueriesposedindifferentlanguages,butthatmeanthesamething,iftheclick-throughinformationforeachqueryissimilar.Asanotherexample,allthewebpagesontheWebformadirectedgraph,alsoknownasthewebgraph,whereeachwebpageisavertex,andeachhyperlinkisanedgepointingfromasourcepagetoadestinationpage.Clusteranalysisonthewebgraphcandisclosecommunities,findhubsandauthoritativewebpages,anddetectwebspams.Inadditiontobipartitegraphs,clusteranalysiscanalsobeappliedtoothertypesofgraphs,includinggeneralgraphs,aselaboratedExample11.18.Example11.18Socialnetwork.Asocialnetworkisasocialstructure.Itcanberepresentedasagraph,wheretheverticesareindividualsororganizations,andthelinksareinterdependenciesbetweenthevertices,representingfriendship,commoninterests,orcollaborativeactivi-ties.AllElectronics’customersformasocialnetwork,whereeachcustomerisavertex,andanedgelinkstwocustomersiftheyknoweachother.Ascustomerrelationshipmanager,youareinterestedinfindingusefulinformationthatcanbederivedfromAllElectronics’socialnetworkthroughclusteranalysis.Youobtainclustersfromthenetwork,wherecustomersinaclusterknoweachotherorhavefriendsincommon.Customerswithinaclustermayinfluenceoneanotherregard-ingpurchasedecisionmaking.Moreover,communicationchannelscanbedesignedtoinformthe“heads”ofclusters(i.e.,the“best”connectedpeopleintheclusters),sothatpromotionalinformationcanbespreadoutquickly.Thus,youmayusecustomerclusteringtopromotesalesatAllElectronics.Asanotherexample,theauthorsofscientificpublicationsformasocialnetwork,wheretheauthorsareverticesandtwoauthorsareconnectedbyanedgeiftheyco-authoredapublication.Thenetworkis,ingeneral,aweightedgraphbecauseanedgebetweentwoauthorscancarryaweightrepresentingthestrength'},
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              'text': 'resentingthestrengthofthecollaborationsuchashowmanypublicationsthetwoauthors(astheendvertices)coauthored.Clus-teringthecoauthornetworkprovidesinsightastocommunitiesofauthorsandpatternsofcollaboration.“Arethereanychallengesspecifictoclusteranalysisongraphandnetworkdata?”Inmostoftheclusteringmethodsdiscussedsofar,objectsarerepresentedusingasetofattributes.Auniquefeatureofgraphandnetworkdataisthatonlyobjects(asvertices)andrelationshipsbetweenthem(asedges)aregiven.Nodimensionsorattributesareexplicitlydefined.Toconductclusteranalysisongraphandnetworkdata,therearetwomajornewchallenges.“Howcanwemeasurethesimilaritybetweentwoobjectsonagraphaccordingly?”Typically,wecannotuseconventionaldistancemeasures,suchasEuclideandis-tance.Instead,weneedtodevelopnewmeasurestoquantifythesimilarity.Such'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page525#2911.3ClusteringGraphandNetworkData525measuresoftenarenotmetric,andthusraisenewchallengesregardingthedevelop-mentofefficientclusteringmethods.SimilaritymeasuresforgraphsarediscussedinSection11.3.2.“Howcanwedesignclusteringmodelsandmethodsthatareeffectiveongraphandnetworkdata?”Graphandnetworkdataareoftencomplicated,carryingtopologicalstructuresthataremoresophisticatedthantraditionalclusteranalysisapplications.Manygraphdatasetsarelarge,suchasthewebgraphcontainingatleasttensofbillionsofwebpagesinthepubliclyindexableWeb.Graphscanalsobesparsewhere,onaverage,avertexisconnectedtoonlyasmallnumberofotherverticesinthegraph.Todiscoveraccurateandusefulknowledgehiddendeepinthedata,agoodclusteringmethodhastoaccommodatethesefactors.ClusteringmethodsforgraphandnetworkdataareintroducedinSection11.3.3.11.3.2SimilarityMeasures“Howcanwemeasurethesimilarityordistancebetweentwoverticesinagraph?”Inourdiscussion,weexaminetwotypesofmeasures:geodesicdistanceanddistancebasedonrandomwalk.GeodesicDistanceAsimplemeasureofthedistancebetweentwoverticesinagraphistheshortestpathbetweenthevertices.Formally,thegeodesicdistancebetweentwoverticesisthelengthintermsofthenumberofedgesoftheshortestpathbetweenthevertices.Fortwoverticesthatarenotconnectedinagraph,thegeodesicdistanceisdefinedasinfinite.Usinggeodesicdistance,wecandefineseveralotherusefulmeasurementsforgraphanalysisandclustering.GivenagraphG=(V,E),whereVisthesetofverticesandEisthesetofedges,wedefinethefollowing:Foravertextv∈V,theeccentricityofv,denotedeccen(v),isthelargestgeodesicdistancebetweenvandanyothervertexu∈V−{v}.Theeccentricityofvcaptureshowfarawayvisfromitsremotestvertexinthegraph.TheradiusofgraphGistheminimumeccentricityofallvertices.Thatis,r=minv∈Veccen(v).(11.27)Theradiuscapturesthedistancebetweenthe“mostcentralpoint”andthe“farthestborder”ofthegraph.ThediameterofgraphGisthemaximumeccentricityofallvertices.Thatis,d=maxv∈Veccen(v).(11.28)Thediameterrepresentsthelargestdistancebetw'},
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              'text': 'elargestdistancebetweenanypairofvertices.Aperipheralvertexisavertexthatachievesthediameter.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page526#30526Chapter11AdvancedClusterAnalysisabcedFigure11.13Agraph,G,whereverticesc,d,andeareperipheral.Example11.19Measurementsbasedongeodesicdistance.ConsidergraphGinFigure11.13.Theeccentricityofais2,thatis,eccen(a)=2,eccen(b)=2,andeccen(c)=eccen(d)=eccen(e)=3.Thus,theradiusofGis2,andthediameteris3.Notethatitisnotnecessarythatd=2×r.Verticesc,d,andeareperipheralvertices.SimRank:SimilarityBasedonRandomWalkandStructuralContextForsomeapplications,geodesicdistancemaybeinappropriateinmeasuringthesimi-laritybetweenverticesinagraph.HereweintroduceSimRank,asimilaritymeasurebasedonrandomwalkandonthestructuralcontextofthegraph.Inmathematics,arandomwalkisatrajectorythatconsistsoftakingsuccessiverandomsteps.Example11.20Similaritybetweenpeopleinasocialnetwork.Let’sconsidermeasuringthesimilaritybetweentwoverticesintheAllElectronicscustomersocialnetworkofExample11.18.Here,similaritycanbeexplainedastheclosenessbetweentwoparticipantsinthenet-work,thatis,howclosetwopeopleareintermsoftherelationshiprepresentedbythesocialnetwork.“Howwellcanthegeodesicdistancemeasuresimilarityandclosenessinsuchanetwork?”SupposeAdaandBobaretwocustomersinthenetwork,andthenetworkisundirected.Thegeodesicdistance(i.e.,thelengthoftheshortestpathbetweenAdaandBob)istheshortestpaththatamessagecanbepassedfromAdatoBobandviceversa.However,thisinformationisnotusefulforAllElectronics’customerrelationshipmanagementbecausethecompanytypicallydoesnotwanttosendaspecificmessagefromonecustomertoanother.Therefore,geodesicdistancedoesnotsuittheapplication.“Whatdoessimilaritymeaninasocialnetwork?”Weconsidertwowaystodefinesimilarity:Twocustomersareconsideredsimilartooneanotheriftheyhavesimilarneighborsinthesocialnetwork.Thisheuristicisintuitivebecause,inpractice,twopeoplereceivingrecommendationsfromagoodnumberofcommonfriendsoftenmakesimilardecisions.Thiskindofsimilarityisbasedonthelocalstructure(i.e.,theneighborhoods)ofthevertices,andthusiscalledstructuralcontext–basedsimilarity.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page527#3111.3ClusteringGraphandNetworkData527SupposeAllElectronicssendspromotionalinformationtobothAdaandBobinthesocialnetwork.AdaandBobmayrandomlyforwardsuchinformationtotheirfriends(orneighbors)inthenetwork.TheclosenessbetweenAdaandBobcanthenbemeasuredbythelikelihoodthatothercustomerssimultaneouslyreceivethepro-motionalinformationthatwasoriginallysenttoAdaandBob.Thiskindofsimilarityisbasedontherandomwalkreachabilityoverthenetwork,andthusisreferredtoassimilaritybasedonrandomwalk.Let’shaveacloserlookatwhatismeantbysimilaritybasedonstructuralcontext,andsimilaritybasedonrandomwalk.Theintuitionbehindsimilaritybasedonstructuralcontextisthattwoverticesinagrapharesimilariftheyareconnectedtosimilarvertices.Tomeasuresuchsimilarity,weneedtodefinethenotionofindividualneighborhood.InadirectedgraphG=(V,E),whereVisthesetofverticesandE⊆V×Visthesetofedges,foravertexv∈V,theindividualin-neighborhoodofvisdefinedasI(v)={u|(u,v)∈E}.(11.29)Symmetrically,wedefinetheindividualout-neighborhoodofvasO(v)={w|(v,w)∈E}.(11.30)FollowingtheintuitionillustratedinExample11.20,wedefineSimRank,astructural-contextsimilarity,withavaluethatisbetween0and1foranypairofver-tices.Foranyvertex,v∈V,thesimilaritybetweenthevertexanditselfiss(v,v)=1becausetheneighborhoodsareidentical.Forverticesu,v∈Vsuchthatu(cid:54)=v,wecandefines(u,v)=C|I(u)||I(v)|(cid:88)x∈I(u)(cid:88)y∈I(v)s(x,y),(11.31)whereCisaconstantbetween0and1.Avertexmaynothaveanyin-neighbors.Thus,wedefineEq.(11.31)tobe0wheneitherI(u)orI(v)is∅.ParameterCspecifiestherateofdecayassimilarityispropagatedacrossedges.“HowcanwecomputeSimRank?”AstraightforwardmethoditerativelyevaluatesEq.(11.31)untilafixedpointisreached.Letsi(u,v)betheSimRankscorecalculatedattheithround.Tobegin,wesets0(u,v)=(cid:40)0ifu(cid:54)=v1ifu=v.(11.32)WeuseEq.(11.31)tocomputesi+1fromsiassi+1(u,v)=C|I(u)||I(v)|(cid:88)x∈I(u)(cid:88)y∈I(v)si(x,y).(11.33)'},
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              'page': 565.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page528#32528Chapter11AdvancedClusterAnalysisItcanbeshownthatlimi→∞si(u,v)=s(u,v).AdditionalmethodsforapproximatingSimRankaregiveninthebibliographicnotes(Section11.7).Now,let’sconsidersimilaritybasedonrandomwalk.Adirectedgraphisstronglyconnectedif,foranytwonodesuandv,thereisapathfromutovandanotherpathfromvtou.Inastronglyconnectedgraph,G=(V,E),foranytwovertices,u,v∈V,wecandefinetheexpecteddistancefromutovasd(u,v)=(cid:88)t:u(cid:32)vP[t]l(t),(11.34)whereu(cid:32)visapathstartingfromuandendingatvthatmaycontaincyclesbutdoesnotreachvuntiltheend.Foratravelingtour,t=w1→w2→···→wk,itslengthisl(t)=k−1.TheprobabilityofthetourisdefinedasP[t]=(cid:40)(cid:81)k−1i=11|O(wi)|ifl(t)>00ifl(t)=0.(11.35)Tomeasuretheprobabilitythatavertexwreceivesamessagethatoriginatedsimulta-neouslyfromuandv,weextendtheexpecteddistancetothenotionofexpectedmeetingdistance,thatis,m(u,v)=(cid:88)t:(u,v)(cid:32)(x,x)P[t]l(t),(11.36)where(u,v)(cid:32)(x,x)isapairoftoursu(cid:32)xandv(cid:32)xofthesamelength.UsingaconstantCbetween0and1,wedefinetheexpectedmeetingprobabilityasp(u,v)=(cid:88)t:(u,v)(cid:32)(x,x)P[t]Cl(t),(11.37)whichisasimilaritymeasurebasedonrandomwalk.Here,theparameterCspecifiestheprobabilityofcontinuingthewalkateachstepofthetrajectory.Ithasbeenshownthats(u,v)=p(u,v)foranytwovertices,uandv.Thatis,SimRankisbasedonbothstructuralcontextandrandomwalk.11.3.3GraphClusteringMethodsLet’sconsiderhowtoconductclusteringonagraph.Wefirstdescribetheintuitionbehindgraphclustering.Wethendiscusstwogeneralcategoriesofgraphclusteringmethods.Tofindclustersinagraph,imaginecuttingthegraphintopieces,eachpiecebeingacluster,suchthattheverticeswithinaclusterarewellconnectedandtheverticesindifferentclustersareconnectedinamuchweakerway.Formally,foragraph,G=(V,E),'},
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              'page': 566.0,
              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page529#3311.3ClusteringGraphandNetworkData529acut,C=(S,T),isapartitioningofthesetofverticesVinG,thatis,V=S∪TandS∩T=∅.Thecutsetofacutisthesetofedges,{(u,v)∈E|u∈S,v∈T}.Thesizeofthecutisthenumberofedgesinthecutset.Forweightedgraphs,thesizeofacutisthesumoftheweightsoftheedgesinthecutset.“Whatkindsofcutsaregoodforderivingclustersingraphs?”Ingraphtheoryandsomenetworkapplications,aminimumcutisofimportance.Acutisminimumifthecut’ssizeisnotgreaterthananyothercut’ssize.Therearepolynomialtimealgorithmstocomputeminimumcutsofgraphs.Canweusethesealgorithmsingraphclustering?Example11.21Cutsandclusters.ConsidergraphGinFigure11.14.Thegraphhastwoclusters:{a,b,c,d,e,f}and{g,h,i,j,k},andoneoutliervertex,l.ConsidercutC1=({a,b,c,d,e,f,g,h,i,j,k},{l}).Onlyoneedge,namely,(e,l),crossesthetwopartitionscreatedbyC1.Therefore,thecutsetofC1is{(e,l)}andthesizeofC1is1.(Notethatthesizeofanycutinaconnectedgraphcannotbesmallerthan1.)Asaminimumcut,C1doesnotleadtoagoodclusteringbecauseitonlyseparatestheoutliervertex,l,fromtherestofthegraph.CutC2=({a,b,c,d,e,f,l},{g,h,i,j,k})leadstoamuchbetterclusteringthanC1.TheedgesinthecutsetofC2arethoseconnectingthetwo“naturalclusters”inthegraph.Specifically,foredges(d,h)and(e,k)thatareinthecutset,mostoftheedgesconnectingd,h,e,andkbelongtoonecluster.Example11.21indicatesthatusingaminimumcutisunlikelytoleadtoagoodclus-tering.Wearebetteroffchoosingacutwhere,foreachvertexuthatisinvolvedinanedgeinthecutset,mostoftheedgesconnectingtoubelongtoonecluster.Formally,letdeg(u)bethedegreeofu,thatis,thenumberofedgesconnectingtou.ThesparsityofacutC=(S,T)isdefinedas(cid:56)=cutsizemin{|S|,|T|}.(11.38)Sparsest '
                      'cut C2Minimum cut '
                      'C1abdefghijklcFigure11.14AgraphGandtwocuts.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page530#34530Chapter11AdvancedClusterAnalysisAcutissparsestifitssparsityisnotgreaterthanthesparsityofanyothercut.Theremaybemorethanonesparsestcut.InExample11.21andFigure11.14,C2isasparsestcut.Usingsparsityastheobjectivefunction,asparsestcuttriestominimizethenumberofedgescrossingthepartitionsandbalancethepartitionsinsize.ConsideraclusteringonagraphG=(V,E)thatpartitionsthegraphintokclusters.ThemodularityofaclusteringassessesthequalityoftheclusteringandisdefinedasQ=k(cid:88)i=1(cid:32)li|E|−(cid:18)di2|E|(cid:19)2(cid:33),(11.39)whereliisthenumberofedgesbetweenverticesintheithcluster,anddiisthesumofthedegreesoftheverticesintheithcluster.Themodularityofaclusteringofagraphisthedifferencebetweenthefractionofalledgesthatfallintoindividualclustersandthefractionthatwoulddosoifthegraphverticeswererandomlyconnected.Theoptimalclusteringofgraphsmaximizesthemodularity.Theoretically,manygraphclusteringproblemscanberegardedasfindinggoodcuts,suchasthesparsestcuts,onthegraph.Inpractice,however,anumberofchallengesexist:Highcomputationalcost:Manygraphcutproblemsarecomputationallyexpen-sive.Thesparsestcutproblem,forexample,isNP-hard.Therefore,findingtheoptimalsolutionsonlargegraphsisoftenimpossible.Agoodtrade-offbetweenefficiency/scalabilityandqualityhastobeachieved.Sophisticatedgraphs:Graphscanbemoresophisticatedthantheonesdescribedhere,involvingweightsand/orcycles.Highdimensionality:Agraphcanhavemanyvertices.Inasimilaritymatrix,avertexisrepresentedasavector(arowinthematrix)withadimensionalitythatisthenumberofverticesinthegraph.Therefore,graphclusteringmethodsmusthandlehighdimensionality.Sparsity:Alargegraphisoftensparse,meaningeachvertexonaverageconnectstoonlyasmallnumberofothervertices.Asimilaritymatrixfromalargesparsegraphcanalsobesparse.Therearetwokindsofmethodsforclusteringgraphdata,whichaddressthesechallenges.Oneusesclusteringmethodsforhigh-dimensionaldata,whiletheotherisdesignedspecificallyforclusteringgraphs.Thefirstgroupofmethodsisba'},
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              'text': 'stgroupofmethodsisbasedongenericclusteringmethodsforhigh-dimensionaldata.TheyextractasimilaritymatrixfromagraphusingasimilaritymeasuresuchasthosediscussedinSection11.3.2.Agenericclusteringmethodcanthenbeappliedonthesimilaritymatrixtodiscoverclusters.Clusteringmethodsfor'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page531#3511.3ClusteringGraphandNetworkData531high-dimensionaldataaretypicallyemployed.Forexample,inmanyscenarios,onceasimilaritymatrixisobtained,spectralclusteringmethods(Section11.2.4)canbeapplied.Spectralclusteringcanapproximateoptimalgraphcutsolutions.Foradditionalinformation,pleaserefertothebibliographicnotes(Section11.7).Thesecondgroupofmethodsisspecifictographs.Theysearchthegraphtofindwell-connectedcomponentsasclusters.Let’slookatamethodcalledSCAN(StructuralClusteringAlgorithmforNetworks)asanexample.Givenanundirectedgraph,G=(V,E),foravertex,u∈V,theneighborhoodofuis(cid:48)(u)={v|(u,v)∈E}∪{u}.Usingtheideaofstructural-contextsimilarity,SCANmeasuresthesimilaritybetweentwovertices,u,v∈V,bythenormalizedcommonneighborhoodsize,thatis,σ(u,v)=|(cid:48)(u)∩(cid:48)(v)|√|(cid:48)(u)||(cid:48)(v)|.(11.40)Thelargerthevaluecomputed,themoresimilarthetwovertices.SCANusesasimilaritythresholdεtodefinetheclustermembership.Foravertex,u∈V,theε-neighborhoodofuisdefinedasNε(u)={v∈(cid:48)(u)|σ(u,v)≥ε}.Theε-neighborhoodofucontainsallneighborsofuwithastructural-contextsimilaritytouthatisatleastε.InSCAN,acorevertexisavertexinsideofacluster.Thatis,u∈Visacorever-texif|Nε(u)|≥µ,whereµisapopularitythreshold.SCANgrowsclustersfromcorevertices.Ifavertexvisintheε-neighborhoodofacoreu,thenvisassignedtothesameclusterasu.Thisprocessofgrowingclusterscontinuesuntilnoclustercanbefurthergrown.Theprocessissimilartothedensity-basedclusteringmethod,DBSCAN(Chapter10).Formally,avertexvcanbedirectlyreachedfromacoreuifv∈Nε(u).Transitively,avertexvcanbereachedfromacoreuifthereexistverticesw1,...,wnsuchthatw1canbereachedfromu,wicanbereachedfromwi−1for1<i≤n,andvcanbereachedfromwn.Moreover,twovertices,u,v∈V,whichmayormaynotbecores,aresaidtobeconnectedifthereexistsacorewsuchthatbothuandvcanbereachedfromw.Allverticesinaclusterareconnected.Aclusterisamaximumsetofverticessuchthateverypairinthesetisconnected.Someverticesmaynotbelongtoanycluster.Suchavertexui'},
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              'text': 'luster.Suchavertexuisahubiftheneighbor-hood(cid:48)(u)ofucontainsverticesfrommorethanonecluster.Ifavertexdoesnotbelongtoanycluster,andisnotahub,itisanoutlier.TheSCANalgorithmisshowninFigure11.15.Thesearchframeworkcloselyresem-blesthecluster-findingprocessinDBSCAN.SCANfindsacutofthegraph,whereeachclusterisasetofverticesthatareconnectedbasedonthetransitivesimilarityinastructuralcontext.AnadvantageofSCANisthatitstimecomplexityislinearwithrespecttothenumberofedges.Inverylargeandsparsegraphs,thenumberofedgesisinthesamescaleofthenumberofvertices.Therefore,SCANisexpectedtohavegoodscalabilityonclusteringlargegraphs.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page532#36532Chapter11AdvancedClusterAnalysisAlgorithm:SCANforclustersongraphdata.Input:agraphG=(V,E),asimilaritythresholdε,andapopulationthresholdµOutput:asetofclustersMethod:setallverticesinVunlabeledforallunlabeledvertexudoifuisacorethengenerateanewcluster-idcinsertallv∈Nε(u)intoaqueueQwhileQ(cid:54)=dow←thefirstvertexinQR←thesetofverticesthatcanbedirectlyreachedfromwforalls∈Rdoifsisnotunlabeledorlabeledasnonmemberthenassignthecurrentcluster-idctosendififsisunlabeledtheninsertsintoqueueQendifendforremovewfromQendwhileelselabeluasnonmemberendifendforforallvertexulabelednonmemberdoif∃x,y∈(cid:48)(u):xandyhavedifferentcluster-idsthenlabeluashubelselabeluasoutlierendifendforFigure11.15SCANalgorithmforclusteranalysisongraphdata.11.4ClusteringwithConstraintsUsersoftenhavebackgroundknowledgethattheywanttointegrateintoclusteranalysis.Theremayalsobeapplication-specificrequirements.Suchinformationcanbemod-eledasclusteringconstraints.Weapproachthetopicofclusteringwithconstraintsintwosteps.Section11.4.1categorizesthetypesofconstraintsforclusteringgraphdata.MethodsforclusteringwithconstraintsareintroducedinSection11.4.2.'},
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              'text': 'HAN18-ch11-497-542-97801238147912011/6/13:24Page533#3711.4ClusteringwithConstraints53311.4.1CategorizationofConstraintsThissectionstudieshowtocategorizetheconstraintsusedinclusteranalysis.Specifi-cally,wecancategorizeconstraintsaccordingtothesubjectsonwhichtheyareset,oronhowstronglytheconstraintsaretobeenforced.AsdiscussedinChapter10,clusteranalysisinvolvesthreeessentialaspects:objectsasinstancesofclusters,clustersasgroupsofobjects,andthesimilarityamongobjects.Therefore,thefirstmethodwediscusscategorizesconstraintsaccordingtowhattheyareappliedto.Wethushavethreetypes:constraintsoninstances,constraintsonclusters,andconstraintsonsimilaritymeasurement.Constraintsoninstances:Aconstraintoninstancesspecifieshowapairorasetofinstancesshouldbegroupedintheclusteranalysis.Twocommontypesofcon-straintsfromthiscategoryinclude:Must-linkconstraints.Ifamust-linkconstraintisspecifiedontwoobjectsxandy,thenxandyshouldbegroupedintooneclusterintheoutputoftheclusteranalysis.Thesemust-linkconstraintsaretransitive.Thatis,ifmust-link(x,y)andmust-link(y,z),thenmust-link(x,z).Cannot-linkconstraints.Cannot-linkconstraintsaretheoppositeofmust-linkconstraints.Ifacannot-linkconstraintisspecifiedontwoobjects,xandy,thenintheoutputoftheclusteranalysis,xandyshouldbelongtodifferentclusters.Cannot-linkconstraintscanbeentailed.Thatis,ifcannot-link(x,y),must-link(x,x(cid:48)),andmust-link(y,y(cid:48)),thencannot-link(x(cid:48),y(cid:48)).Aconstraintoninstancescanbedefinedusingspecificinstances.Alternatively,itcanalsobedefinedusinginstancevariablesorattributesofinstances.Forexample,aconstraint,Constraint(x,y):must-link(x,y)ifdist(x,y)≤(cid:15),usesthedistancebetweenobjectstospecifyamust-linkconstraint.Constraintsonclusters:Aconstraintonclustersspecifiesarequirementontheclusters,possiblyusingattributesoftheclusters.Forexample,aconstraintmayspecifytheminimumnumberofobjectsinacluster,themaximumdiameterofacluster,ortheshapeofacluster(e.g.,aconvex).Thenumberofclustersspecifiedforpartitioningclusteringmethodscan'},
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File: www-fairwaycapecod-com-menu-breakfast-62826.txt

Page: 1

Context: ## Fairway Favorites

**The Mountain**  
grilled homemade biscuit topped with two sausage patties, two scrambled eggs, bacon, cheddar cheese and sausage gravy … served with home fries  
14

**Nanny’s Potato Pancakes**  
two homemade potato pancakes and two eggs … served with toast and choice of bacon, ham or sausage  
13.50

**Corned Beef Hash and Eggs**  
our own recipe, served with two eggs, toast and home fries  
14.50

**Fairway Breakfast**  
three eggs with ham, bacon and sausage… served with toast and home fries  
12.50

**Silver Skillet**  
sauteed onions, peppers, home fries and choice of bacon, ham or sausage, all scrambled with three eggs and topped with a blend of cheddar and mozzarella… served with toast  
12.50

**Country Breakfast**  
grilled homemade biscuit topped with two sausage patties, two eggs and sausage gravy … served with home fries  
12.50

**Two by Two**  
two eggs, short stack of buttermilk pancakes or French toast, and choice of two strips of bacon, two sausage links or slice of ham  
12

**Quinoa Bowl**  
Quinoa mixed with black beans, corn, roasted red peppers and a touch of chipotle pepper…topped with two eggs any style, fresh avocado, pico de gallo and cheddar jack cheese.  
13

**Avocado Toast**  
two slices of toast topped with mashed avocado and poached eggs, finished with our house recipe everything bagel seasoning and fresh micro greens.  
11

## Benedicts

_all served with home fries_

**Traditional**  
English muffin topped with two poached eggs, Canadian bacon, and hollandaise  
12.50

**Eggs Mulligan**  
English muffin topped with poached eggs, our own corned beef hash and hollandaise  
14.50

**Nanny’s**  
homemade potato pancakes topped with poached eggs and hollandaise… served with choice of fresh fruit or home fries  
13

**Garden**  
homemade Italian herbed focaccia toast topped with two poached eggs, spinach, tomatoes, mushrooms and hollandaise  
12.50
####################
File: www-fairwaycapecod-com-menu-breakfast-62826.txt

Page: 1

Context: **Fried Green Tomatoes**  
a bed of spinach topped with two poached eggs, fried green tomatoes, avocado, hollandaise and garnished with basil   
12

**Crab Cakes**  
grilled homemade crab cakes on a bed of fresh spinach, topped with poached eggs and hollandaise  
18

**Portuguese Benedict**  
homemade English muffin bread topped with two poached eggs, grilled linguica, sautéed onions and peppers with hollandaise  
16

## Egg Combos

_served with toast and home fries_

add bacon, ham, chicken sausage or sausage 4 Linguica 6

one egg 6 two eggs 7 three eggs 8

## Omelets

_Sautéed toppings cooked into the eggs, folded over, and served with home fries and toast_ 

**Western**  
sautéed onions, peppers, ham and cheese  
14

**Green Goodness**  
sautéed asparagus, zucchini, broccoli, spinach served with homemade English muffin bread toast  
14

**Athenian**  
spinach, tomatoes, sautéed onions and feta cheese  
14

**Build Your Own**  
plain   
9

**add your own ingredients**

cheese: American, Swiss, cheddar, feta

2 each

ham  
bacon  
sausage  
tomato  
onions  
peppers  
mushrooms  
spinach

2 each

fresh avocado  
linguiça

3 each

hash

4 each

## Pancakes, French Toast and Waffles

_add bacon, ham, sausage, chicken sausage 4 or linguica 6_ 
_100% pure maple syrup 2_ 

**Cinnamon Roll French Toast**  
made with “Hole–In–One” cinnamon roll  
10

**Traditional French Toast**  
made with your choice of white, wheat, or raisin  
two 9 three 10.50

**Belgian Waffle**  
made with our own recipe with malted flour.  
plain 9 warm apples 11

**Buttermilk Pancakes**  
two 7 three 8.50

**Fresh Blueberry Pancakes**  
two 9 three 10.50
####################
File: www-fairwaycapecod-com-menu-breakfast-62826.txt

Page: 1

Context: **Fresh Blueberry Pancakes**  
two 9 three 10.50

**Chocolate Chip Pancakes**  
two 9 three 10.50

**Fresh Blueberry Crunch Pancakes**  
two 10 three 11.50

## Breakfast Sandwiches

**Classic**  
two eggs, cheese and choice of bacon, ham, sausage patty or linguiça +3 … served on a brioche roll  
8

add home fries  
3

substitute flour wrap, buttermilk biscuit or bagel  
1

**Veggie Focaccia**  
scrambled eggs, spinach, mushrooms, tomatoes, sauteed onions and cheddar cheese on focaccia bread… served with home fries  
10

**California Focaccia**  
scrambled eggs, fresh avocado, fresh tomato and cheddar cheese on focaccia bread… served with home fries  
10

## Sides

**Potato Pancakes**  
served with sour cream  
5  
add apple sauce 2

**Biscuit And Gravy**  
8

**Homemade Corned Beef Hash**  
8

**Home Fries**  
3

**Loaded Home Fries**  
loaded with chopped bacon, cheddar cheese, scallions topped with hollandaise sauce  
8

**Crab Cakes** 
15

**Breakfast Protein** 
Bacon, Ham, Sausage, Canadian Bacon or Chicken sausage – 4 Linguiça – 6

**Seasonal Fresh Fruit Cup**  
cup 5

**Oatmeal**  
5  
add Craisins®, raisins, walnuts or chocolate chips 1 each  
add fresh blueberries 1.5

[Fairway Favorites](#favorites) | [Benedicts](#benedicts) | [Egg Combos](#eggcombos) | [Omelets](#omelets) | [Pancakes & French Toast](#pancakes) | [Breakfast Sandwiches](#sandwiches) | [Sides](#sides)

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Breakfast

[Fairway Favorites](#favorites) | [Benedicts](#benedicts) | [Egg Combos](#eggcombos) | [Omelets](#omelets) | [Pancakes & French Toast](#pancakes) | [Breakfast Sandwiches](#sandwiches) | [Sides](#sides)
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Context: inarydissimilarity.Ifobjectsiandjaredescribedbysymmetricbinaryattributes,thentheTable2.3ContingencyTableforBinaryAttributesObjectj10sum1qrq+rObjecti0sts+tsumq+sr+tp
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page71#332.4MeasuringDataSimilarityandDissimilarity71dissimilaritybetweeniandjisd(i,j)=r+sq+r+s+t.(2.13)Forasymmetricbinaryattributes,thetwostatesarenotequallyimportant,suchasthepositive(1)andnegative(0)outcomesofadiseasetest.Giventwoasymmetricbinaryattributes,theagreementoftwo1s(apositivematch)isthenconsideredmoresignifi-cantthanthatoftwo0s(anegativematch).Therefore,suchbinaryattributesareoftenconsidered“monary”(havingonestate).Thedissimilaritybasedontheseattributesiscalledasymmetricbinarydissimilarity,wherethenumberofnegativematches,t,isconsideredunimportantandisthusignoredinthefollowingcomputation:d(i,j)=r+sq+r+s.(2.14)Complementarily,wecanmeasurethedifferencebetweentwobinaryattributesbasedonthenotionofsimilarityinsteadofdissimilarity.Forexample,theasymmetricbinarysimilaritybetweentheobjectsiandjcanbecomputedassim(i,j)=qq+r+s=1−d(i,j).(2.15)Thecoefficientsim(i,j)ofEq.(2.15)iscalledtheJaccardcoefficientandispopularlyreferencedintheliterature.Whenbothsymmetricandasymmetricbinaryattributesoccurinthesamedataset,themixedattributesapproachdescribedinSection2.4.6canbeapplied.Example2.18Dissimilaritybetweenbinaryattributes.Supposethatapatientrecordtable(Table2.4)containstheattributesname,gender,fever,cough,test-1,test-2,test-3,andtest-4,wherenameisanobjectidentifier,genderisasymmetricattribute,andtheremainingattributesareasymmetricbinary.Forasymmetricattributevalues,letthevaluesY(yes)andP(positive)besetto1,andthevalueN(noornegative)besetto0.SupposethatthedistancebetweenobjectsTable2.4RelationalTableWherePatientsAreDescribedbyBinaryAttributesnamegenderfevercoughtest-1test-2test-3test-4JackMYNPNNNJimMYYNNNNMaryFYNPNPN........................
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page72#3472Chapter2GettingtoKnowYourData(patients)iscomputedbasedonlyontheasymmetricattributes.AccordingtoEq.(2.14),thedistancebetweeneachpairofthethreepatients—Jack,Mary,andJim—isd(Jack,Jim)=1+11+1+1=0.67,d(Jack,Mary)=0+12+0+1=0.33,d(Jim,Mary)=1+21+1+2=0.75.ThesemeasurementssuggestthatJimandMaryareunlikelytohaveasimilardiseasebecausetheyhavethehighestdissimilarityvalueamongthethreepairs.Ofthethreepatients,JackandMaryarethemostlikelytohaveasimilardisease.2.4.4DissimilarityofNumericData:MinkowskiDistanceInthissection,wedescribedistancemeasuresthatarecommonlyusedforcomputingthedissimilarityofobjectsdescribedbynumericattributes.ThesemeasuresincludetheEuclidean,Manhattan,andMinkowskidistances.Insomecases,thedataarenormalizedbeforeapplyingdistancecalculations.Thisinvolvestransformingthedatatofallwithinasmallerorcommonrange,suchas[−1,1]or[0.0,1.0].Consideraheightattribute,forexample,whichcouldbemeasuredineithermetersorinches.Ingeneral,expressinganattributeinsmallerunitswillleadtoalargerrangeforthatattribute,andthustendtogivesuchattributesgreatereffector“weight.”Normalizingthedataattemptstogiveallattributesanequalweight.Itmayormaynotbeusefulinaparticularapplication.MethodsfornormalizingdataarediscussedindetailinChapter3ondatapreprocessing.ThemostpopulardistancemeasureisEuclideandistance(i.e.,straightlineor“asthecrowflies”).Leti=(xi1,xi2,...,xip)andj=(xj1,xj2,...,xjp)betwoobjectsdescribedbypnumericattributes.TheEuclideandistancebetweenobjectsiandjisdefinedasd(i,j)=(cid:113)(xi1−xj1)2+(xi2−xj2)2+···+(xip−xjp)2.(2.16)Anotherwell-knownmeasureistheManhattan(orcityblock)distance,namedsobecauseitisthedistanceinblocksbetweenanytwopointsinacity(suchas2blocksdownand3blocksoverforatotalof5blocks).Itisdefinedasd(i,j)=|xi1−xj1|+|xi2−xj2|+···+|xip−xjp|.(2.17)BoththeEuclideanandtheManhattandistancesatisfythefollowingmathematicalproperties:Non-negativity:d(i,j)≥0:Distanceisanon-negativenumber.Identityofindiscernibles:d(i,i)=0:Thedistanceofanob
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page73#352.4MeasuringDataSimilarityandDissimilarity73Symmetry:d(i,j)=d(j,i):Distanceisasymmetricfunction.Triangleinequality:d(i,j)≤d(i,k)+d(k,j):Goingdirectlyfromobjectitoobjectjinspaceisnomorethanmakingadetouroveranyotherobjectk.Ameasurethatsatisfiestheseconditionsisknownasmetric.Pleasenotethatthenon-negativitypropertyisimpliedbytheotherthreeproperties.Example2.19EuclideandistanceandManhattandistance.Letx1=(1,2)andx2=(3,5)repre-senttwoobjectsasshowninFigure2.23.TheEuclideandistancebetweenthetwois√22+32=3.61.TheManhattandistancebetweenthetwois2+3=5.MinkowskidistanceisageneralizationoftheEuclideanandManhattandistances.Itisdefinedasd(i,j)=h(cid:113)|xi1−xj1|h+|xi2−xj2|h+···+|xip−xjp|h,(2.18)wherehisarealnumbersuchthath≥1.(SuchadistanceisalsocalledLpnorminsomeliterature,wherethesymbolpreferstoournotationofh.Wehavekeptpasthenumberofattributestobeconsistentwiththerestofthischapter.)ItrepresentstheManhattandistancewhenh=1(i.e.,L1norm)andEuclideandistancewhenh=2(i.e.,L2norm).Thesupremumdistance(alsoreferredtoasLmax,L∞normandastheChebyshevdistance)isageneralizationoftheMinkowskidistanceforh→∞.Tocomputeit,wefindtheattributefthatgivesthemaximumdifferenceinvaluesbetweenthetwoobjects.Thisdifferenceisthesupremumdistance,definedmoreformallyas:d(i,j)=limh→∞p(cid:88)f=1|xif−xjf|h1h=pmaxf|xif−xjf|.(2.19)TheL∞normisalsoknownastheuniformnorm.1223543321x2 = (3,5)x1 = (1,2)Euclidean distance= (22 + 32)1/2 = 3.61Manhattan distance= 2 + 3 = 5Supremum distance= 5 – 2 = 3Figure2.23Euclidean,Manhattan,andsupremumdistancesbetweentwoobjects.
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page74#3674Chapter2GettingtoKnowYourDataExample2.20Supremumdistance.Let’susethesametwoobjects,x1=(1,2)andx2=(3,5),asinFigure2.23.Thesecondattributegivesthegreatestdifferencebetweenvaluesfortheobjects,whichis5−2=3.Thisisthesupremumdistancebetweenbothobjects.Ifeachattributeisassignedaweightaccordingtoitsperceivedimportance,theweightedEuclideandistancecanbecomputedasd(i,j)=(cid:113)w1|xi1−xj1|2+w2|xi2−xj2|2+···+wm|xip−xjp|2.(2.20)Weightingcanalsobeappliedtootherdistancemeasuresaswell.2.4.5ProximityMeasuresforOrdinalAttributesThevaluesofanordinalattributehaveameaningfulorderorrankingaboutthem,yetthemagnitudebetweensuccessivevaluesisunknown(Section2.1.4).Anexam-pleincludesthesequencesmall,medium,largeforasizeattribute.Ordinalattributesmayalsobeobtainedfromthediscretizationofnumericattributesbysplittingthevaluerangeintoafinitenumberofcategories.Thesecategoriesareorganizedintoranks.Thatis,therangeofanumericattributecanbemappedtoanordinalattributefhavingMfstates.Forexample,therangeoftheinterval-scaledattributetemperature(inCelsius)canbeorganizedintothefollowingstates:−30to−10,−10to10,10to30,repre-sentingthecategoriescoldtemperature,moderatetemperature,andwarmtemperature,respectively.LetMrepresentthenumberofpossiblestatesthatanordinalattributecanhave.Theseorderedstatesdefinetheranking1,...,Mf.“Howareordinalattributeshandled?”Thetreatmentofordinalattributesisquitesimilartothatofnumericattributeswhencomputingdissimilaritybetweenobjects.Supposethatfisanattributefromasetofordinalattributesdescribingnobjects.Thedissimilaritycomputationwithrespecttofinvolvesthefollowingsteps:1.Thevalueofffortheithobjectisxif,andfhasMforderedstates,representingtheranking1,...,Mf.Replaceeachxifbyitscorrespondingrank,rif∈{1,...,Mf}.2.Sinceeachordinalattributecanhaveadifferentnumberofstates,itisoftennecessarytomaptherangeofeachattributeonto[0.0,1.0]sothateachattributehasequalweight.Weperformsuchdatanormalizationbyreplacingtherankrifoftheithobjectinthefthatt
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Context: ithobjectinthefthattributebyzif=rif−1Mf−1.(2.21)3.DissimilaritycanthenbecomputedusinganyofthedistancemeasuresdescribedinSection2.4.4fornumericattributes,usingziftorepresentthefvaluefortheithobject.
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page75#372.4MeasuringDataSimilarityandDissimilarity75Example2.21Dissimilaritybetweenordinalattributes.SupposethatwehavethesampledatashownearlierinTable2.2,exceptthatthistimeonlytheobject-identifierandthecontinuousordinalattribute,test-2,areavailable.Therearethreestatesfortest-2:fair,good,andexcellent,thatis,Mf=3.Forstep1,ifwereplaceeachvaluefortest-2byitsrank,thefourobjectsareassignedtheranks3,1,2,and3,respectively.Step2normalizestherankingbymappingrank1to0.0,rank2to0.5,andrank3to1.0.Forstep3,wecanuse,say,theEuclideandistance(Eq.2.16),whichresultsinthefollowingdissimilaritymatrix:01.000.50.5001.00.50.Therefore,objects1and2arethemostdissimilar,asareobjects2and4(i.e.,d(2,1)=1.0andd(4,2)=1.0).Thismakesintuitivesensesinceobjects1and4arebothexcellent.Object2isfair,whichisattheoppositeendoftherangeofvaluesfortest-2.Similarityvaluesforordinalattributescanbeinterpretedfromdissimilarityassim(i,j)=1−d(i,j).2.4.6DissimilarityforAttributesofMixedTypesSections2.4.2through2.4.5discussedhowtocomputethedissimilaritybetweenobjectsdescribedbyattributesofthesametype,wherethesetypesmaybeeithernominal,sym-metricbinary,asymmetricbinary,numeric,orordinal.However,inmanyrealdatabases,objectsaredescribedbyamixtureofattributetypes.Ingeneral,adatabasecancontainalloftheseattributetypes.“So,howcanwecomputethedissimilaritybetweenobjectsofmixedattributetypes?”Oneapproachistogroupeachtypeofattributetogether,performingseparatedatamining(e.g.,clustering)analysisforeachtype.Thisisfeasibleiftheseanalysesderivecompatibleresults.However,inrealapplications,itisunlikelythataseparateanalysisperattributetypewillgeneratecompatibleresults.Amorepreferableapproachistoprocessallattributetypestogether,performingasingleanalysis.Onesuchtechniquecombinesthedifferentattributesintoasingledis-similaritymatrix,bringingallofthemeaningfulattributesontoacommonscaleoftheinterval[0.0,1.0].Supposethatthedatasetcontainspattributesofmixedtype.Thedissimilar
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Context: edtype.Thedissimilarityd(i,j)betweenobjectsiandjisdefinedasd(i,j)=(cid:80)pf=1δ(f)ijd(f)ij(cid:80)pf=1δ(f)ij,(2.22)
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page76#3876Chapter2GettingtoKnowYourDatawheretheindicatorδ(f)ij=0ifeither(1)xiforxjfismissing(i.e.,thereisnomea-surementofattributefforobjectiorobjectj),or(2)xif=xjf=0andattributefisasymmetricbinary;otherwise,δ(f)ij=1.Thecontributionofattributeftothedissimilaritybetweeniandj(i.e.,d(f)ij)iscomputeddependentonitstype:Iffisnumeric:d(f)ij=|xif−xjf|maxhxhf−minhxhf,wherehrunsoverallnonmissingobjectsforattributef.Iffisnominalorbinary:d(f)ij=0ifxif=xjf;otherwise,d(f)ij=1.Iffisordinal:computetheranksrifandzif=rif−1Mf−1,andtreatzifasnumeric.Thesestepsareidenticaltowhatwehavealreadyseenforeachoftheindividualattributetypes.Theonlydifferenceisfornumericattributes,wherewenormalizesothatthevaluesmaptotheinterval[0.0,1.0].Thus,thedissimilaritybetweenobjectscanbecomputedevenwhentheattributesdescribingtheobjectsareofdifferenttypes.Example2.22Dissimilaritybetweenattributesofmixedtype.Let’scomputeadissimilaritymatrixfortheobjectsinTable2.2.Nowwewillconsideralloftheattributes,whichareofdifferenttypes.InExamples2.17and2.21,weworkedoutthedissimilaritymatricesforeachoftheindividualattributes.Theprocedureswefollowedfortest-1(whichisnominal)andtest-2(whichisordinal)arethesameasoutlinedearlierforprocessingattributesofmixedtypes.Therefore,wecanusethedissimilaritymatricesobtainedfortest-1andtest-2laterwhenwecomputeEq.(2.22).First,however,weneedtocomputethedissimilaritymatrixforthethirdattribute,test-3(whichisnumeric).Thatis,wemustcomputed(3)ij.Followingthecasefornumericattributes,weletmaxhxh=64andminhxh=22.ThedifferencebetweenthetwoisusedinEq.(2.22)tonormalizethevaluesofthedissimilaritymatrix.Theresultingdissimilaritymatrixfortest-3is00.5500.451.0000.400.140.860.WecannowusethedissimilaritymatricesforthethreeattributesinourcomputationofEq.(2.22).Theindicatorδ(f)ij=1foreachofthethreeattributes,f.Weget,forexample,d(3,1)=1(1)+1(0.50)+1(0.45)3=0.65.Theresultingdissimilaritymatrixobtainedforthe
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page77#392.4MeasuringDataSimilarityandDissimilarity77datadescribedbythethreeattributesofmixedtypesis:00.8500.650.8300.130.710.790.FromTable2.2,wecanintuitivelyguessthatobjects1and4arethemostsimilar,basedontheirvaluesfortest-1andtest-2.Thisisconfirmedbythedissimilaritymatrix,whered(4,1)isthelowestvalueforanypairofdifferentobjects.Similarly,thematrixindicatesthatobjects1and2aretheleastsimilar.2.4.7CosineSimilarityAdocumentcanberepresentedbythousandsofattributes,eachrecordingthefrequencyofaparticularword(suchasakeyword)orphraseinthedocument.Thus,eachdocu-mentisanobjectrepresentedbywhatiscalledaterm-frequencyvector.Forexample,inTable2.5,weseethatDocument1containsfiveinstancesofthewordteam,whilehockeyoccursthreetimes.Thewordcoachisabsentfromtheentiredocument,asindicatedbyacountvalueof0.Suchdatacanbehighlyasymmetric.Term-frequencyvectorsaretypicallyverylongandsparse(i.e.,theyhavemany0val-ues).Applicationsusingsuchstructuresincludeinformationretrieval,textdocumentclustering,biologicaltaxonomy,andgenefeaturemapping.Thetraditionaldistancemeasuresthatwehavestudiedinthischapterdonotworkwellforsuchsparsenumericdata.Forexample,twoterm-frequencyvectorsmayhavemany0valuesincommon,meaningthatthecorrespondingdocumentsdonotsharemanywords,butthisdoesnotmakethemsimilar.Weneedameasurethatwillfocusonthewordsthatthetwodocu-mentsdohaveincommon,andtheoccurrencefrequencyofsuchwords.Inotherwords,weneedameasurefornumericdatathatignoreszero-matches.Cosinesimilarityisameasureofsimilaritythatcanbeusedtocomparedocu-mentsor,say,givearankingofdocumentswithrespecttoagivenvectorofquerywords.Letxandybetwovectorsforcomparison.UsingthecosinemeasureasaTable2.5DocumentVectororTerm-FrequencyVectorDocumentteamcoachhockeybaseballsoccerpenaltyscorewinlossseasonDocument15030200200Document23020110101Document30702100300Document40100122030
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page78#4078Chapter2GettingtoKnowYourDatasimilarityfunction,wehavesim(x,y)=x·y||x||||y||,(2.23)where||x||istheEuclideannormofvectorx=(x1,x2,...,xp),definedas(cid:113)x21+x22+···+x2p.Conceptually,itisthelengthofthevector.Similarly,||y||istheEuclideannormofvectory.Themeasurecomputesthecosineoftheanglebetweenvec-torsxandy.Acosinevalueof0meansthatthetwovectorsareat90degreestoeachother(orthogonal)andhavenomatch.Thecloserthecosinevalueto1,thesmallertheangleandthegreaterthematchbetweenvectors.NotethatbecausethecosinesimilaritymeasuredoesnotobeyallofthepropertiesofSection2.4.4definingmetricmeasures,itisreferredtoasanonmetricmeasure.Example2.23Cosinesimilaritybetweentwoterm-frequencyvectors.Supposethatxandyarethefirsttwoterm-frequencyvectorsinTable2.5.Thatis,x=(5,0,3,0,2,0,0,2,0,0)andy=(3,0,2,0,1,1,0,1,0,1).Howsimilararexandy?UsingEq.(2.23)tocomputethecosinesimilaritybetweenthetwovectors,weget:xt·y=5×3+0×0+3×2+0×0+2×1+0×1+0×0+2×1+0×0+0×1=25||x||=(cid:112)52+02+32+02+22+02+02+22+02+02=6.48||y||=(cid:112)32+02+22+02+12+12+02+12+02+12=4.12sim(x,y)=0.94Therefore,ifwewereusingthecosinesimilaritymeasuretocomparethesedocuments,theywouldbeconsideredquitesimilar.Whenattributesarebinary-valued,thecosinesimilarityfunctioncanbeinterpretedintermsofsharedfeaturesorattributes.Supposeanobjectxpossessestheithattributeifxi=1.Thenxt·yisthenumberofattributespossessed(i.e.,shared)bybothxandy,and|x||y|isthegeometricmeanofthenumberofattributespossessedbyxandthenumberpossessedbyy.Thus,sim(x,y)isameasureofrelativepossessionofcommonattributes.Asimplevariationofcosinesimilarityfortheprecedingscenarioissim(x,y)=x·yx·x+y·y−x·y,(2.24)whichistheratioofthenumberofattributessharedbyxandytothenumberofattributespossessedbyxory.Thisfunction,knownastheTanimotocoefficientorTanimotodistance,isfrequentlyusedininformationretrievalandbiologytaxonomy.
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page79#412.6Exercises792.5SummaryDatasetsaremadeupofdataobjects.Adataobjectrepresentsanentity.Dataobjectsaredescribedbyattributes.Attributescanbenominal,binary,ordinal,ornumeric.Thevaluesofanominal(orcategorical)attributearesymbolsornamesofthings,whereeachvaluerepresentssomekindofcategory,code,orstate.Binaryattributesarenominalattributeswithonlytwopossiblestates(suchas1and0ortrueandfalse).Ifthetwostatesareequallyimportant,theattributeissymmetric;otherwiseitisasymmetric.Anordinalattributeisanattributewithpossiblevaluesthathaveameaningfulorderorrankingamongthem,butthemagnitudebetweensuccessivevaluesisnotknown.Anumericattributeisquantitative(i.e.,itisameasurablequantity)representedinintegerorrealvalues.Numericattributetypescanbeinterval-scaledorratio-scaled.Thevaluesofaninterval-scaledattributearemeasuredinfixedandequalunits.Ratio-scaledattributesarenumericattributeswithaninherentzero-point.Measurementsareratio-scaledinthatwecanspeakofvaluesasbeinganorderofmagnitudelargerthantheunitofmeasurement.Basicstatisticaldescriptionsprovidetheanalyticalfoundationfordatapreprocess-ing.Thebasicstatisticalmeasuresfordatasummarizationincludemean,weightedmean,median,andmodeformeasuringthecentraltendencyofdata;andrange,quan-tiles,quartiles,interquartilerange,variance,andstandarddeviationformeasuringthedispersionofdata.Graphicalrepresentations(e.g.,boxplots,quantileplots,quantile–quantileplots,histograms,andscatterplots)facilitatevisualinspectionofthedataandarethususefulfordatapreprocessingandmining.Datavisualizationtechniquesmaybepixel-oriented,geometric-based,icon-based,orhierarchical.Thesemethodsapplytomultidimensionalrelationaldata.Additionaltechniqueshavebeenproposedforthevisualizationofcomplexdata,suchastextandsocialnetworks.Measuresofobjectsimilarityanddissimilarityareusedindataminingapplicationssuchasclustering,outlieranalysis,andnearest-neighborclassification.Suchmea-suresofproximitycanbecomputedforeachattributetypestudiedinthischapt
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Context: pestudiedinthischapter,orforcombinationsofsuchattributes.ExamplesincludetheJaccardcoefficientforasymmetricbinaryattributesandEuclidean,Manhattan,Minkowski,andsupremumdistancesfornumericattributes.Forapplicationsinvolvingsparsenumericdatavec-tors,suchasterm-frequencyvectors,thecosinemeasureandtheTanimotocoefficientareoftenusedintheassessmentofsimilarity.2.6Exercises2.1Givethreeadditionalcommonlyusedstatisticalmeasuresthatarenotalreadyillus-tratedinthischapterforthecharacterizationofdatadispersion.Discusshowtheycanbecomputedefficientlyinlargedatabases.
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Context: pestudiedinthischapter,orforcombinationsofsuchattributes.ExamplesincludetheJaccardcoefficientforasymmetricbinaryattributesandEuclidean,Manhattan,Minkowski,andsupremumdistancesfornumericattributes.Forapplicationsinvolvingsparsenumericdatavec-tors,suchasterm-frequencyvectors,thecosinemeasureandtheTanimotocoefficientareoftenusedintheassessmentofsimilarity.2.6Exercises2.1Givethreeadditionalcommonlyusedstatisticalmeasuresthatarenotalreadyillus-tratedinthischapterforthecharacterizationofdatadispersion.Discusshowtheycanbecomputedefficientlyinlargedatabases.
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Context: # Chapter 2 Getting to Know Your Data

## 2.2 Data Analysis

Suppose that the data for analysis includes the attribute `age`. The age values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20, 21, 22, 22, 25, 25, 30, 33, 33, 35, 35, 35, 36, 40, 45, 46, 52, 70.

1. **What is the mean of the data?**
2. **What is the median?**
3. **What is the mode of the data?** Comment on the data's modality (i.e., bimodal, trimodal, etc.).
4. **What is the midrange of the data?**
5. **Can you find (roughly) the first quartile (Q1) and the third quartile (Q3) of the data?**
6. **Give the five-number summary of the data.**
7. **Show a boxplot of the data.**
8. **How is a quantile–quantile plot different from a quantile plot?**

## 2.3 Frequency Distribution

Suppose that the values for a given set of data are grouped into intervals. The intervals and corresponding frequencies are as follows:

| Age   | Frequency |
|-------|-----------|
| 1–5   | 200       |
| 6–15  | 450       |
| 16–20 | 300       |
| 21–50 | 1500      |
| 51–80 | 700       |
| 81–110| 44        |

**Compute an approximate median value for the data.**

## 2.4 Age and Body Fat Data

Suppose that a hospital tested the age and body fat data for 18 randomly selected adults with the following results:

| Age |    |    |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|-----|----|----|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|     | 23 | 23 | 27| 27| 39| 41| 47| 49| 50|   |   |   |   |   |   |   |   |
| %fat| 9.5| 26.5| 7.8| 17.8| 31.4| 25.9| 27.4| 27.2| 31.2|   |   |   |   |   |   |   |   |

| Age |    |    |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|-----|----|----|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|     | 52 | 54 | 54| 56| 57| 58| 58| 60| 61|   |   |   |   |   |   |   |   |
| %fat| 34.6| 42.5| 28.8| 33.4| 30.2| 34.1| 32.9| 41.2| 35.7|   |   |   |   |   |   |   |   |

1. **Calculate the mean, median, and standard deviation of age and %fat.**
2. **Draw the boxplots for age and %fat.**
3. **Draw a scatter plot and a q-q plot based on these two variables.**

## 2.5 Dissimilarity of Objects

Briefly outline how to compute the dissimilarity between objects described by the following:

- **Nominal attributes**
- **Asymmetric binary attributes**
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Context: # Chapter 2: Getting to Know Your Data

## 2.2 

Suppose that the data for analysis includes the attribute **age**. The age values for the data tuples are (in increasing order) 13, 15, 16, 16, 19, 20, 21, 22, 22, 25, 25, 30, 33, 33, 35, 35, 35, 36, 40, 45, 46, 52, 70.

(a) What is the **mean** of the data? What is the **median**?  
(b) What is the **mode** of the data? Comment on the data's modality (i.e., bimodal, trimodal, etc.).  
(c) What is the **range** of the data?  
(d) Can you find (roughly) the first quartile (Q1) and the third quartile (Q3) of the data?  
(e) Give the **five-number summary** of the data.  
(f) Show a **boxplot** of the data.  
(g) How is a **quantile–quantile plot** different from a **quantile plot**?

## 2.3 

Suppose that the values for a given set of data are grouped into intervals. The intervals and corresponding frequencies are as follows:

| age      | frequency |
|----------|-----------|
| 1–5      | 200       |
| 6–15     | 450       |
| 16–20    | 300       |
| 21–50    | 1500      |
| 51–80    | 700       |
| 81–110   | 44        |

Compute an **approximate median** value for the data.

## 2.4 

Suppose that a hospital tested the age and body fat data for 18 randomly selected adults with the following results:

| age | 23 | 23 | 27 | 27 | 39 | 41 | 47 | 49 | 50 |
|-----|----|----|----|----|----|----|----|----|----|
| %fat | 9.5 | 26.5 | 7.8 | 17.8 | 31.4 | 25.9 | 27.4 | 27.2 | 31.2 |

| age | 52 | 54 | 54 | 56 | 57 | 58 | 58 | 60 | 61 |
|-----|----|----|----|----|----|----|----|----|----|
| %fat | 34.6 | 42.5 | 28.8 | 33.4 | 30.2 | 34.1 | 32.9 | 41.2 | 35.7 |

(a) Calculate the mean, median, and standard deviation of age and %fat.  
(b) Draw the **boxplots** for age and %fat.  
(c) Draw a **scatter plot** and a **q-q plot** based on these two variables.

## 2.5 

Briefly outline how to compute the dissimilarity between objects described by the following:  
(a) **Nominal attributes**  
(b) **Asymmetric binary attributes**
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Context: ## 2.7 Bibliographic Notes

### 2.6

Given two objects represented by the tuples (22, 1, 42, 10) and (20, 0, 36, 8):

1. Compute the **Euclidean distance** between the two objects.
2. Compute the **Manhattan distance** between the two objects.
3. Compute the **Minkowski distance** between the two objects, using \( q = 3 \).
4. Compute the **supremum distance** between the two objects.

### 2.7

The median is one of the most important holistic measures in data analysis. Propose several methods for median approximation. Analyze their respective complexity under different parameter settings and decide to what extent the real value can be approximated. Moreover, suggest a heuristic strategy to balance between accuracy and complexity and then apply it to all methods you have given.

### 2.8

It is important to define or select similarity measures in data analysis. However, there is no commonly accepted subjective similarity measure. Results can vary depending on the similarity measures used. Nonetheless, seemingly different similarity measures may be equivalent after some transformation.

Suppose we have the following 2-D data set:

|   A₁ |   A₂ |
|------|------|
| x₁   | 1.5  | 1.7  |
| x₂   | 2.2  | 1.9  |
| x₃   | 1.6  | 1.8  |
| x₄   | 1.2  | 1.5  |
| x₅   | 1.5  | 1.0  |

1. Consider the data as 2-D data points. Given a new data point, \( x = (1.4, 1.6) \) as a query, rank the database points based on similarity with the query using Euclidean distance, Manhattan distance, supremum distance, and cosine similarity.
2. Normalize the data set to make the norm of each data point equal to 1. Use Euclidean distance on the transformed data to rank the data points.

### 2.7 Bibliographic Notes

Methods for descriptive data summarization have been studied in the statistics literature long before the onset of computers. Good summaries of statistical descriptive data mining methods include Freedman, Pisani, and Purves [FP97] and Devore [Dev95].
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Context: # 2.7 Bibliographic Notes

## 2.6
Given two objects represented by the tuples (22, 1, 42, 10) and (20, 0, 36, 8):

1. Compute the Euclidean distance between the two objects.
2. Compute the Manhattan distance between the two objects.
3. Compute the Minkowski distance between the two objects, using \( q = 3 \).
4. Compute the supremum distance between the two objects.

## 2.7
The median is one of the most important holistic measures in data analysis. Propose several methods for median approximation. Analyze their respective complexity under different parameter settings and decide to what extent the real value can be approximated. Moreover, suggest a heuristic strategy to balance between accuracy and complexity and then apply it to all methods you have given.

## 2.8
It is important to define or select similarity measures in data analysis. However, there is no commonly accepted subjective similarity measure. Results can vary depending on the similarity measures used. Nonetheless, seemingly different similarity measures may be equivalent after some transformation.

Suppose we have the following 2-D data set:

|     |  A₁   |  A₂   |
|-----|-------|-------|
| x₁  |  1.5  |  1.7  |
| x₂  |  2.2  |  1.9  |
| x₃  |  1.6  |  1.8  |
| x₄  |  1.2  |  1.5  |
| x₅  |  1.5  |  1.0  |

1. Consider the data as 2-D data points. Given a new data point, \( x = (1.4, 1.6) \) as a query, rank the database points based on similarity with the query using Euclidean distance, Manhattan distance, supremum distance, and cosine similarity.
2. Normalize the data set to make the norm of each data point equal to 1. Use Euclidean distance on the transformed data to rank the data points.
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page82#4482Chapter2GettingtoKnowYourDatastatistics-basedvisualizationofdatausingboxplots,quantileplots,quantile–quantileplots,scatterplots,andloesscurves,seeCleveland[Cle93].PioneeringworkondatavisualizationtechniquesisdescribedinTheVisualDis-playofQuantitativeInformation[Tuf83],EnvisioningInformation[Tuf90],andVisualExplanations:ImagesandQuantities,EvidenceandNarrative[Tuf97],allbyTufte,inadditiontoGraphicsandGraphicInformationProcessingbyBertin[Ber81],VisualizingDatabyCleveland[Cle93],andInformationVisualizationinDataMiningandKnowledgeDiscoveryeditedbyFayyad,Grinstein,andWierse[FGW01].MajorconferencesandsymposiumsonvisualizationincludeACMHumanFactorsinComputingSystems(CHI),Visualization,andtheInternationalSymposiumonInfor-mationVisualization.ResearchonvisualizationisalsopublishedinTransactionsonVisualizationandComputerGraphics,JournalofComputationalandGraphicalStatistics,andIEEEComputerGraphicsandApplications.Manygraphicaluserinterfacesandvisualizationtoolshavebeendevelopedandcanbefoundinvariousdataminingproducts.Severalbooksondatamining(e.g.,DataMiningSolutionsbyWestphalandBlaxton[WB98])presentmanygoodexamplesandvisualsnapshots.Forasurveyofvisualizationtechniques,see“Visualtechniquesforexploringdatabases”byKeim[Kei97].Similarityanddistancemeasuresamongvariousvariableshavebeenintroducedinmanytextbooksthatstudyclusteranalysis,includingHartigan[Har75];JainandDubes[JD88];KaufmanandRousseeuw[KR90];andArabie,Hubert,anddeSoete[AHS96].MethodsforcombiningattributesofdifferenttypesintoasingledissimilaritymatrixwereintroducedbyKaufmanandRousseeuw[KR90].
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Context: HAN09-ch02-039-082-97801238147912011/6/13:15Page82#4482Chapter2GettingtoKnowYourDatastatistics-basedvisualizationofdatausingboxplots,quantileplots,quantile–quantileplots,scatterplots,andloesscurves,seeCleveland[Cle93].PioneeringworkondatavisualizationtechniquesisdescribedinTheVisualDis-playofQuantitativeInformation[Tuf83],EnvisioningInformation[Tuf90],andVisualExplanations:ImagesandQuantities,EvidenceandNarrative[Tuf97],allbyTufte,inadditiontoGraphicsandGraphicInformationProcessingbyBertin[Ber81],VisualizingDatabyCleveland[Cle93],andInformationVisualizationinDataMiningandKnowledgeDiscoveryeditedbyFayyad,Grinstein,andWierse[FGW01].MajorconferencesandsymposiumsonvisualizationincludeACMHumanFactorsinComputingSystems(CHI),Visualization,andtheInternationalSymposiumonInfor-mationVisualization.ResearchonvisualizationisalsopublishedinTransactionsonVisualizationandComputerGraphics,JournalofComputationalandGraphicalStatistics,andIEEEComputerGraphicsandApplications.Manygraphicaluserinterfacesandvisualizationtoolshavebeendevelopedandcanbefoundinvariousdataminingproducts.Severalbooksondatamining(e.g.,DataMiningSolutionsbyWestphalandBlaxton[WB98])presentmanygoodexamplesandvisualsnapshots.Forasurveyofvisualizationtechniques,see“Visualtechniquesforexploringdatabases”byKeim[Kei97].Similarityanddistancemeasuresamongvariousvariableshavebeenintroducedinmanytextbooksthatstudyclusteranalysis,includingHartigan[Har75];JainandDubes[JD88];KaufmanandRousseeuw[KR90];andArabie,Hubert,anddeSoete[AHS96].MethodsforcombiningattributesofdifferenttypesintoasingledissimilaritymatrixwereintroducedbyKaufmanandRousseeuw[KR90].
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page88#688Chapter3DataPreprocessing3.2DataCleaningReal-worlddatatendtobeincomplete,noisy,andinconsistent.Datacleaning(ordatacleansing)routinesattempttofillinmissingvalues,smoothoutnoisewhileidenti-fyingoutliers,andcorrectinconsistenciesinthedata.Inthissection,youwillstudybasicmethodsfordatacleaning.Section3.2.1looksatwaysofhandlingmissingvalues.Section3.2.2explainsdatasmoothingtechniques.Section3.2.3discussesapproachestodatacleaningasaprocess.3.2.1MissingValuesImaginethatyouneedtoanalyzeAllElectronicssalesandcustomerdata.Younotethatmanytupleshavenorecordedvalueforseveralattributessuchascustomerincome.Howcanyougoaboutfillinginthemissingvaluesforthisattribute?Let’slookatthefollowingmethods.1.Ignorethetuple:Thisisusuallydonewhentheclasslabelismissing(assumingtheminingtaskinvolvesclassification).Thismethodisnotveryeffective,unlessthetuplecontainsseveralattributeswithmissingvalues.Itisespeciallypoorwhenthepercent-ageofmissingvaluesperattributevariesconsiderably.Byignoringthetuple,wedonotmakeuseoftheremainingattributes’valuesinthetuple.Suchdatacouldhavebeenusefultothetaskathand.2.Fillinthemissingvaluemanually:Ingeneral,thisapproachistimeconsumingandmaynotbefeasiblegivenalargedatasetwithmanymissingvalues.3.Useaglobalconstanttofillinthemissingvalue:Replaceallmissingattributevaluesbythesameconstantsuchasalabellike“Unknown”or−∞.Ifmissingvaluesarereplacedby,say,“Unknown,”thentheminingprogrammaymistakenlythinkthattheyformaninterestingconcept,sincetheyallhaveavalueincommon—thatof“Unknown.”Hence,althoughthismethodissimple,itisnotfoolproof.4.Useameasureofcentraltendencyfortheattribute(e.g.,themeanormedian)tofillinthemissingvalue:Chapter2discussedmeasuresofcentraltendency,whichindicatethe“middle”valueofadatadistribution.Fornormal(symmetric)datadis-tributions,themeancanbeused,whileskeweddatadistributionshouldemploythemedian(Section2.2).Forexample,supposethatthedatadistributionregard-ingtheincomeofAllElectronicscustomersissymmetricandthatt
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page88#688Chapter3DataPreprocessing3.2DataCleaningReal-worlddatatendtobeincomplete,noisy,andinconsistent.Datacleaning(ordatacleansing)routinesattempttofillinmissingvalues,smoothoutnoisewhileidenti-fyingoutliers,andcorrectinconsistenciesinthedata.Inthissection,youwillstudybasicmethodsfordatacleaning.Section3.2.1looksatwaysofhandlingmissingvalues.Section3.2.2explainsdatasmoothingtechniques.Section3.2.3discussesapproachestodatacleaningasaprocess.3.2.1MissingValuesImaginethatyouneedtoanalyzeAllElectronicssalesandcustomerdata.Younotethatmanytupleshavenorecordedvalueforseveralattributessuchascustomerincome.Howcanyougoaboutfillinginthemissingvaluesforthisattribute?Let’slookatthefollowingmethods.1.Ignorethetuple:Thisisusuallydonewhentheclasslabelismissing(assumingtheminingtaskinvolvesclassification).Thismethodisnotveryeffective,unlessthetuplecontainsseveralattributeswithmissingvalues.Itisespeciallypoorwhenthepercent-ageofmissingvaluesperattributevariesconsiderably.Byignoringthetuple,wedonotmakeuseoftheremainingattributes’valuesinthetuple.Suchdatacouldhavebeenusefultothetaskathand.2.Fillinthemissingvaluemanually:Ingeneral,thisapproachistimeconsumingandmaynotbefeasiblegivenalargedatasetwithmanymissingvalues.3.Useaglobalconstanttofillinthemissingvalue:Replaceallmissingattributevaluesbythesameconstantsuchasalabellike“Unknown”or−∞.Ifmissingvaluesarereplacedby,say,“Unknown,”thentheminingprogrammaymistakenlythinkthattheyformaninterestingconcept,sincetheyallhaveavalueincommon—thatof“Unknown.”Hence,althoughthismethodissimple,itisnotfoolproof.4.Useameasureofcentraltendencyfortheattribute(e.g.,themeanormedian)tofillinthemissingvalue:Chapter2discussedmeasuresofcentraltendency,whichindicatethe“middle”valueofadatadistribution.Fornormal(symmetric)datadis-tributions,themeancanbeused,whileskeweddatadistributionshouldemploythemedian(Section2.2).Forexample,supposethatthedatadistributionregard-ingtheincomeofAllElectronicscustomersissymmetricandthatt
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page89#73.2DataCleaning89induction.Forexample,usingtheothercustomerattributesinyourdataset,youmayconstructadecisiontreetopredictthemissingvaluesforincome.DecisiontreesandBayesianinferencearedescribedindetailinChapters8and9,respectively,whileregressionisintroducedinSection3.4.5.Methods3through6biasthedata—thefilled-invaluemaynotbecorrect.Method6,however,isapopularstrategy.Incomparisontotheothermethods,itusesthemostinformationfromthepresentdatatopredictmissingvalues.Byconsideringtheotherattributes’valuesinitsestimationofthemissingvalueforincome,thereisagreaterchancethattherelationshipsbetweenincomeandtheotherattributesarepreserved.Itisimportanttonotethat,insomecases,amissingvaluemaynotimplyanerrorinthedata!Forexample,whenapplyingforacreditcard,candidatesmaybeaskedtosupplytheirdriver’slicensenumber.Candidateswhodonothaveadriver’slicensemaynaturallyleavethisfieldblank.Formsshouldallowrespondentstospecifyvaluessuchas“notapplicable.”Softwareroutinesmayalsobeusedtouncoverothernullvalues(e.g.,“don’tknow,”“?”or“none”).Ideally,eachattributeshouldhaveoneormorerulesregardingthenullcondition.Therulesmayspecifywhetherornotnullsareallowedand/orhowsuchvaluesshouldbehandledortransformed.Fieldsmayalsobeinten-tionallyleftblankiftheyaretobeprovidedinalaterstepofthebusinessprocess.Hence,althoughwecantryourbesttocleanthedataafteritisseized,gooddatabaseanddataentryproceduredesignshouldhelpminimizethenumberofmissingvaluesorerrorsinthefirstplace.3.2.2NoisyData“Whatisnoise?”Noiseisarandomerrororvarianceinameasuredvariable.InChapter2,wesawhowsomebasicstatisticaldescriptiontechniques(e.g.,boxplotsandscatterplots),andmethodsofdatavisualizationcanbeusedtoidentifyoutliers,whichmayrepresentnoise.Givenanumericattributesuchas,say,price,howcanwe“smooth”outthedatatoremovethenoise?Let’slookatthefollowingdatasmoothingtechniques.Binning:Binningmethodssmoothasorteddatavaluebyconsultingits“neighbor-hood,”thatis,thevaluesaroundit.Thesortedvaluesaredistributedintoa
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Context: saredistributedintoanumberof“buckets,”orbins.Becausebinningmethodsconsulttheneighborhoodofvalues,theyperformlocalsmoothing.Figure3.2illustratessomebinningtechniques.Inthisexample,thedataforpricearefirstsortedandthenpartitionedintoequal-frequencybinsofsize3(i.e.,eachbincontainsthreevalues).Insmoothingbybinmeans,eachvalueinabinisreplacedbythemeanvalueofthebin.Forexample,themeanofthevalues4,8,and15inBin1is9.Therefore,eachoriginalvalueinthisbinisreplacedbythevalue9.Similarly,smoothingbybinmedianscanbeemployed,inwhicheachbinvalueisreplacedbythebinmedian.Insmoothingbybinboundaries,theminimumandmaximumvaluesinagivenbinareidentifiedasthebinboundaries.Eachbinvalueisthenreplacedbytheclosestboundaryvalue.Ingeneral,thelargerthewidth,the
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page89#73.2DataCleaning89induction.Forexample,usingtheothercustomerattributesinyourdataset,youmayconstructadecisiontreetopredictthemissingvaluesforincome.DecisiontreesandBayesianinferencearedescribedindetailinChapters8and9,respectively,whileregressionisintroducedinSection3.4.5.Methods3through6biasthedata—thefilled-invaluemaynotbecorrect.Method6,however,isapopularstrategy.Incomparisontotheothermethods,itusesthemostinformationfromthepresentdatatopredictmissingvalues.Byconsideringtheotherattributes’valuesinitsestimationofthemissingvalueforincome,thereisagreaterchancethattherelationshipsbetweenincomeandtheotherattributesarepreserved.Itisimportanttonotethat,insomecases,amissingvaluemaynotimplyanerrorinthedata!Forexample,whenapplyingforacreditcard,candidatesmaybeaskedtosupplytheirdriver’slicensenumber.Candidateswhodonothaveadriver’slicensemaynaturallyleavethisfieldblank.Formsshouldallowrespondentstospecifyvaluessuchas“notapplicable.”Softwareroutinesmayalsobeusedtouncoverothernullvalues(e.g.,“don’tknow,”“?”or“none”).Ideally,eachattributeshouldhaveoneormorerulesregardingthenullcondition.Therulesmayspecifywhetherornotnullsareallowedand/orhowsuchvaluesshouldbehandledortransformed.Fieldsmayalsobeinten-tionallyleftblankiftheyaretobeprovidedinalaterstepofthebusinessprocess.Hence,althoughwecantryourbesttocleanthedataafteritisseized,gooddatabaseanddataentryproceduredesignshouldhelpminimizethenumberofmissingvaluesorerrorsinthefirstplace.3.2.2NoisyData“Whatisnoise?”Noiseisarandomerrororvarianceinameasuredvariable.InChapter2,wesawhowsomebasicstatisticaldescriptiontechniques(e.g.,boxplotsandscatterplots),andmethodsofdatavisualizationcanbeusedtoidentifyoutliers,whichmayrepresentnoise.Givenanumericattributesuchas,say,price,howcanwe“smooth”outthedatatoremovethenoise?Let’slookatthefollowingdatasmoothingtechniques.Binning:Binningmethodssmoothasorteddatavaluebyconsultingits“neighbor-hood,”thatis,thevaluesaroundit.Thesortedvaluesaredistributedintoa
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Context: saredistributedintoanumberof“buckets,”orbins.Becausebinningmethodsconsulttheneighborhoodofvalues,theyperformlocalsmoothing.Figure3.2illustratessomebinningtechniques.Inthisexample,thedataforpricearefirstsortedandthenpartitionedintoequal-frequencybinsofsize3(i.e.,eachbincontainsthreevalues).Insmoothingbybinmeans,eachvalueinabinisreplacedbythemeanvalueofthebin.Forexample,themeanofthevalues4,8,and15inBin1is9.Therefore,eachoriginalvalueinthisbinisreplacedbythevalue9.Similarly,smoothingbybinmedianscanbeemployed,inwhicheachbinvalueisreplacedbythebinmedian.Insmoothingbybinboundaries,theminimumandmaximumvaluesinagivenbinareidentifiedasthebinboundaries.Eachbinvalueisthenreplacedbytheclosestboundaryvalue.Ingeneral,thelargerthewidth,the
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page90#890Chapter3DataPreprocessingSorteddataforprice(indollars):4,8,15,21,21,24,25,28,34Partitioninto(equal-frequency)bins:Bin1:4,8,15Bin2:21,21,24Bin3:25,28,34Smoothingbybinmeans:Bin1:9,9,9Bin2:22,22,22Bin3:29,29,29Smoothingbybinboundaries:Bin1:4,4,15Bin2:21,21,24Bin3:25,25,34Figure3.2Binningmethodsfordatasmoothing.greatertheeffectofthesmoothing.Alternatively,binsmaybeequalwidth,wheretheintervalrangeofvaluesineachbinisconstant.BinningisalsousedasadiscretizationtechniqueandisfurtherdiscussedinSection3.5.Regression:Datasmoothingcanalsobedonebyregression,atechniquethatcon-formsdatavaluestoafunction.Linearregressioninvolvesfindingthe“best”linetofittwoattributes(orvariables)sothatoneattributecanbeusedtopredicttheother.Multiplelinearregressionisanextensionoflinearregression,wheremorethantwoattributesareinvolvedandthedataarefittoamultidimensionalsurface.RegressionisfurtherdescribedinSection3.4.5.Outlieranalysis:Outliersmaybedetectedbyclustering,forexample,wheresimilarvaluesareorganizedintogroups,or“clusters.”Intuitively,valuesthatfalloutsideofthesetofclustersmaybeconsideredoutliers(Figure3.3).Chapter12isdedicatedtothetopicofoutlieranalysis.Manydatasmoothingmethodsarealsousedfordatadiscretization(aformofdatatransformation)anddatareduction.Forexample,thebinningtechniquesdescribedbeforereducethenumberofdistinctvaluesperattribute.Thisactsasaformofdatareductionforlogic-baseddataminingmethods,suchasdecisiontreeinduction,whichrepeatedlymakesvaluecomparisonsonsorteddata.Concepthierarchiesareaformofdatadiscretizationthatcanalsobeusedfordatasmoothing.Aconcepthierarchyforprice,forexample,maymaprealpricevaluesintoinexpensive,moderatelypriced,andexpensive,therebyreducingthenumberofdatavaluestobehandledbythemining
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page90#890Chapter3DataPreprocessingSorteddataforprice(indollars):4,8,15,21,21,24,25,28,34Partitioninto(equal-frequency)bins:Bin1:4,8,15Bin2:21,21,24Bin3:25,28,34Smoothingbybinmeans:Bin1:9,9,9Bin2:22,22,22Bin3:29,29,29Smoothingbybinboundaries:Bin1:4,4,15Bin2:21,21,24Bin3:25,25,34Figure3.2Binningmethodsfordatasmoothing.greatertheeffectofthesmoothing.Alternatively,binsmaybeequalwidth,wheretheintervalrangeofvaluesineachbinisconstant.BinningisalsousedasadiscretizationtechniqueandisfurtherdiscussedinSection3.5.Regression:Datasmoothingcanalsobedonebyregression,atechniquethatcon-formsdatavaluestoafunction.Linearregressioninvolvesfindingthe“best”linetofittwoattributes(orvariables)sothatoneattributecanbeusedtopredicttheother.Multiplelinearregressionisanextensionoflinearregression,wheremorethantwoattributesareinvolvedandthedataarefittoamultidimensionalsurface.RegressionisfurtherdescribedinSection3.4.5.Outlieranalysis:Outliersmaybedetectedbyclustering,forexample,wheresimilarvaluesareorganizedintogroups,or“clusters.”Intuitively,valuesthatfalloutsideofthesetofclustersmaybeconsideredoutliers(Figure3.3).Chapter12isdedicatedtothetopicofoutlieranalysis.Manydatasmoothingmethodsarealsousedfordatadiscretization(aformofdatatransformation)anddatareduction.Forexample,thebinningtechniquesdescribedbeforereducethenumberofdistinctvaluesperattribute.Thisactsasaformofdatareductionforlogic-baseddataminingmethods,suchasdecisiontreeinduction,whichrepeatedlymakesvaluecomparisonsonsorteddata.Concepthierarchiesareaformofdatadiscretizationthatcanalsobeusedfordatasmoothing.Aconcepthierarchyforprice,forexample,maymaprealpricevaluesintoinexpensive,moderatelypriced,andexpensive,therebyreducingthenumberofdatavaluestobehandledbythemining
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page91#93.2DataCleaning91Figure3.3A2-Dcustomerdataplotwithrespecttocustomerlocationsinacity,showingthreedataclusters.Outliersmaybedetectedasvaluesthatfalloutsideoftheclustersets.process.DatadiscretizationisdiscussedinSection3.5.Somemethodsofclassification(e.g.,neuralnetworks)havebuilt-indatasmoothingmechanisms.ClassificationisthetopicofChapters8and9.3.2.3DataCleaningasaProcessMissingvalues,noise,andinconsistenciescontributetoinaccuratedata.Sofar,wehavelookedattechniquesforhandlingmissingdataandforsmoothingdata.“Butdataclean-ingisabigjob.Whataboutdatacleaningasaprocess?Howexactlydoesoneproceedintacklingthistask?Arethereanytoolsouttheretohelp?”Thefirststepindatacleaningasaprocessisdiscrepancydetection.Discrepanciescanbecausedbyseveralfactors,includingpoorlydesigneddataentryformsthathavemanyoptionalfields,humanerrorindataentry,deliberateerrors(e.g.,respondentsnotwant-ingtodivulgeinformationaboutthemselves),anddatadecay(e.g.,outdatedaddresses).Discrepanciesmayalsoarisefrominconsistentdatarepresentationsandinconsistentuseofcodes.Othersourcesofdiscrepanciesincludeerrorsininstrumentationdevicesthatrecorddataandsystemerrors.Errorscanalsooccurwhenthedataare(inadequately)usedforpurposesotherthanoriginallyintended.Theremayalsobeinconsistenciesduetodataintegration(e.g.,whereagivenattributecanhavedifferentnamesindifferentdatabases).22DataintegrationandtheremovalofredundantdatathatcanresultfromsuchintegrationarefurtherdescribedinSection3.3.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page91#93.2DataCleaning91Figure3.3A2-Dcustomerdataplotwithrespecttocustomerlocationsinacity,showingthreedataclusters.Outliersmaybedetectedasvaluesthatfalloutsideoftheclustersets.process.DatadiscretizationisdiscussedinSection3.5.Somemethodsofclassification(e.g.,neuralnetworks)havebuilt-indatasmoothingmechanisms.ClassificationisthetopicofChapters8and9.3.2.3DataCleaningasaProcessMissingvalues,noise,andinconsistenciescontributetoinaccuratedata.Sofar,wehavelookedattechniquesforhandlingmissingdataandforsmoothingdata.“Butdataclean-ingisabigjob.Whataboutdatacleaningasaprocess?Howexactlydoesoneproceedintacklingthistask?Arethereanytoolsouttheretohelp?”Thefirststepindatacleaningasaprocessisdiscrepancydetection.Discrepanciescanbecausedbyseveralfactors,includingpoorlydesigneddataentryformsthathavemanyoptionalfields,humanerrorindataentry,deliberateerrors(e.g.,respondentsnotwant-ingtodivulgeinformationaboutthemselves),anddatadecay(e.g.,outdatedaddresses).Discrepanciesmayalsoarisefrominconsistentdatarepresentationsandinconsistentuseofcodes.Othersourcesofdiscrepanciesincludeerrorsininstrumentationdevicesthatrecorddataandsystemerrors.Errorscanalsooccurwhenthedataare(inadequately)usedforpurposesotherthanoriginallyintended.Theremayalsobeinconsistenciesduetodataintegration(e.g.,whereagivenattributecanhavedifferentnamesindifferentdatabases).22DataintegrationandtheremovalofredundantdatathatcanresultfromsuchintegrationarefurtherdescribedinSection3.3.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page95#133.3DataIntegration95χ2CorrelationTestforNominalDataFornominaldata,acorrelationrelationshipbetweentwoattributes,AandB,canbediscoveredbyaχ2(chi-square)test.SupposeAhascdistinctvalues,namelya1,a2,...ac.Bhasrdistinctvalues,namelyb1,b2,...br.ThedatatuplesdescribedbyAandBcanbeshownasacontingencytable,withthecvaluesofAmakingupthecolumnsandthervaluesofBmakinguptherows.Let(Ai,Bj)denotethejointeventthatattributeAtakesonvalueaiandattributeBtakesonvaluebj,thatis,where(A=ai,B=bj).Eachandeverypossible(Ai,Bj)jointeventhasitsowncell(orslot)inthetable.Theχ2value(alsoknownasthePearsonχ2statistic)iscomputedasχ2=c(cid:88)i=1r(cid:88)j=1(oij−eij)2eij,(3.1)whereoijistheobservedfrequency(i.e.,actualcount)ofthejointevent(Ai,Bj)andeijistheexpectedfrequencyof(Ai,Bj),whichcanbecomputedaseij=count(A=ai)×count(B=bj)n,(3.2)wherenisthenumberofdatatuples,count(A=ai)isthenumberoftupleshavingvalueaiforA,andcount(B=bj)isthenumberoftupleshavingvaluebjforB.ThesuminEq.(3.1)iscomputedoverallofther×ccells.Notethatthecellsthatcontributethemosttotheχ2valuearethoseforwhichtheactualcountisverydifferentfromthatexpected.Theχ2statisticteststhehypothesisthatAandBareindependent,thatis,thereisnocorrelationbetweenthem.Thetestisbasedonasignificancelevel,with(r−1)×(c−1)degreesoffreedom.WeillustratetheuseofthisstatisticinExample3.1.Ifthehypothesiscanberejected,thenwesaythatAandBarestatisticallycorrelated.Example3.1Correlationanalysisofnominalattributesusingχ2.Supposethatagroupof1500peoplewassurveyed.Thegenderofeachpersonwasnoted.Eachpersonwaspolledastowhetherhisorherpreferredtypeofreadingmaterialwasfictionornonfiction.Thus,wehavetwoattributes,genderandpreferredreading.Theobservedfrequency(orcount)ofeachpossiblejointeventissummarizedinthecontingencytableshowninTable3.1,wherethenumbersinparenthesesaretheexpectedfrequencies.Theexpectedfrequen-ciesarecalculatedbasedonthedatadistributionforbothattributesusingEq.(3.2).UsingEq.(3.2),wecanverifytheexpectedfrequenciesforeachc
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Context: dfrequenciesforeachcell.Forexample,theexpectedfrequencyforthecell(male,fiction)ise11=count(male)×count(fiction)n=300×4501500=90,andsoon.Noticethatinanyrow,thesumoftheexpectedfrequenciesmustequalthetotalobservedfrequencyforthatrow,andthesumoftheexpectedfrequenciesinanycolumnmustalsoequalthetotalobservedfrequencyforthatcolumn.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page95#133.3DataIntegration95χ2CorrelationTestforNominalDataFornominaldata,acorrelationrelationshipbetweentwoattributes,AandB,canbediscoveredbyaχ2(chi-square)test.SupposeAhascdistinctvalues,namelya1,a2,...ac.Bhasrdistinctvalues,namelyb1,b2,...br.ThedatatuplesdescribedbyAandBcanbeshownasacontingencytable,withthecvaluesofAmakingupthecolumnsandthervaluesofBmakinguptherows.Let(Ai,Bj)denotethejointeventthatattributeAtakesonvalueaiandattributeBtakesonvaluebj,thatis,where(A=ai,B=bj).Eachandeverypossible(Ai,Bj)jointeventhasitsowncell(orslot)inthetable.Theχ2value(alsoknownasthePearsonχ2statistic)iscomputedasχ2=c(cid:88)i=1r(cid:88)j=1(oij−eij)2eij,(3.1)whereoijistheobservedfrequency(i.e.,actualcount)ofthejointevent(Ai,Bj)andeijistheexpectedfrequencyof(Ai,Bj),whichcanbecomputedaseij=count(A=ai)×count(B=bj)n,(3.2)wherenisthenumberofdatatuples,count(A=ai)isthenumberoftupleshavingvalueaiforA,andcount(B=bj)isthenumberoftupleshavingvaluebjforB.ThesuminEq.(3.1)iscomputedoverallofther×ccells.Notethatthecellsthatcontributethemosttotheχ2valuearethoseforwhichtheactualcountisverydifferentfromthatexpected.Theχ2statisticteststhehypothesisthatAandBareindependent,thatis,thereisnocorrelationbetweenthem.Thetestisbasedonasignificancelevel,with(r−1)×(c−1)degreesoffreedom.WeillustratetheuseofthisstatisticinExample3.1.Ifthehypothesiscanberejected,thenwesaythatAandBarestatisticallycorrelated.Example3.1Correlationanalysisofnominalattributesusingχ2.Supposethatagroupof1500peoplewassurveyed.Thegenderofeachpersonwasnoted.Eachpersonwaspolledastowhetherhisorherpreferredtypeofreadingmaterialwasfictionornonfiction.Thus,wehavetwoattributes,genderandpreferredreading.Theobservedfrequency(orcount)ofeachpossiblejointeventissummarizedinthecontingencytableshowninTable3.1,wherethenumbersinparenthesesaretheexpectedfrequencies.Theexpectedfrequen-ciesarecalculatedbasedonthedatadistributionforbothattributesusingEq.(3.2).UsingEq.(3.2),wecanverifytheexpectedfrequenciesforeachc
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Context: dfrequenciesforeachcell.Forexample,theexpectedfrequencyforthecell(male,fiction)ise11=count(male)×count(fiction)n=300×4501500=90,andsoon.Noticethatinanyrow,thesumoftheexpectedfrequenciesmustequalthetotalobservedfrequencyforthatrow,andthesumoftheexpectedfrequenciesinanycolumnmustalsoequalthetotalobservedfrequencyforthatcolumn.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page96#1496Chapter3DataPreprocessingTable3.1Example2.1’s2×2ContingencyTableDatamalefemaleTotalfiction250(90)200(360)450nonfiction50(210)1000(840)1050Total30012001500Note:Aregenderandpreferredreadingcorrelated?UsingEq.(3.1)forχ2computation,wegetχ2=(250−90)290+(50−210)2210+(200−360)2360+(1000−840)2840=284.44+121.90+71.11+30.48=507.93.Forthis2×2table,thedegreesoffreedomare(2−1)(2−1)=1.For1degreeoffree-dom,theχ2valueneededtorejectthehypothesisatthe0.001significancelevelis10.828(takenfromthetableofupperpercentagepointsoftheχ2distribution,typicallyavail-ablefromanytextbookonstatistics).Sinceourcomputedvalueisabovethis,wecanrejectthehypothesisthatgenderandpreferredreadingareindependentandconcludethatthetwoattributesare(strongly)correlatedforthegivengroupofpeople.CorrelationCoefficientforNumericDataFornumericattributes,wecanevaluatethecorrelationbetweentwoattributes,AandB,bycomputingthecorrelationcoefficient(alsoknownasPearson’sproductmomentcoefficient,namedafteritsinventer,KarlPearson).ThisisrA,B=n(cid:88)i=1(ai−¯A)(bi−¯B)nσAσB=n(cid:88)i=1(aibi)−n¯A¯BnσAσB,(3.3)wherenisthenumberoftuples,aiandbiaretherespectivevaluesofAandBintuplei,¯Aand¯BaretherespectivemeanvaluesofAandB,σAandσBaretherespectivestandarddeviationsofAandB(asdefinedinSection2.2.2),and(cid:54)(aibi)isthesumoftheABcross-product(i.e.,foreachtuple,thevalueforAismultipliedbythevalueforBinthattuple).Notethat−1≤rA,B≤+1.IfrA,Bisgreaterthan0,thenAandBarepositivelycorrelated,meaningthatthevaluesofAincreaseasthevaluesofBincrease.Thehigherthevalue,thestrongerthecorrelation(i.e.,themoreeachattributeimpliestheother).Hence,ahighervaluemayindicatethatA(orB)mayberemovedasaredundancy.Iftheresultingvalueisequalto0,thenAandBareindependentandthereisnocorrelationbetweenthem.Iftheresultingvalueislessthan0,thenAandBarenegativelycorrelated,wherethevaluesofoneattributeincreaseasthevaluesoftheotherattributedecrease.Thismeansthateachattributediscouragestheother.Scatterplotscanalsobeusedtovie
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Context: tscanalsobeusedtoviewcorrelationsbetweenattributes(Section2.2.3).Forexample,Figure2.8’s
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page96#1496Chapter3DataPreprocessingTable3.1Example2.1’s2×2ContingencyTableDatamalefemaleTotalfiction250(90)200(360)450nonfiction50(210)1000(840)1050Total30012001500Note:Aregenderandpreferredreadingcorrelated?UsingEq.(3.1)forχ2computation,wegetχ2=(250−90)290+(50−210)2210+(200−360)2360+(1000−840)2840=284.44+121.90+71.11+30.48=507.93.Forthis2×2table,thedegreesoffreedomare(2−1)(2−1)=1.For1degreeoffree-dom,theχ2valueneededtorejectthehypothesisatthe0.001significancelevelis10.828(takenfromthetableofupperpercentagepointsoftheχ2distribution,typicallyavail-ablefromanytextbookonstatistics).Sinceourcomputedvalueisabovethis,wecanrejectthehypothesisthatgenderandpreferredreadingareindependentandconcludethatthetwoattributesare(strongly)correlatedforthegivengroupofpeople.CorrelationCoefficientforNumericDataFornumericattributes,wecanevaluatethecorrelationbetweentwoattributes,AandB,bycomputingthecorrelationcoefficient(alsoknownasPearson’sproductmomentcoefficient,namedafteritsinventer,KarlPearson).ThisisrA,B=n(cid:88)i=1(ai−¯A)(bi−¯B)nσAσB=n(cid:88)i=1(aibi)−n¯A¯BnσAσB,(3.3)wherenisthenumberoftuples,aiandbiaretherespectivevaluesofAandBintuplei,¯Aand¯BaretherespectivemeanvaluesofAandB,σAandσBaretherespectivestandarddeviationsofAandB(asdefinedinSection2.2.2),and(cid:54)(aibi)isthesumoftheABcross-product(i.e.,foreachtuple,thevalueforAismultipliedbythevalueforBinthattuple).Notethat−1≤rA,B≤+1.IfrA,Bisgreaterthan0,thenAandBarepositivelycorrelated,meaningthatthevaluesofAincreaseasthevaluesofBincrease.Thehigherthevalue,thestrongerthecorrelation(i.e.,themoreeachattributeimpliestheother).Hence,ahighervaluemayindicatethatA(orB)mayberemovedasaredundancy.Iftheresultingvalueisequalto0,thenAandBareindependentandthereisnocorrelationbetweenthem.Iftheresultingvalueislessthan0,thenAandBarenegativelycorrelated,wherethevaluesofoneattributeincreaseasthevaluesoftheotherattributedecrease.Thismeansthateachattributediscouragestheother.Scatterplotscanalsobeusedtovie
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Context: tscanalsobeusedtoviewcorrelationsbetweenattributes(Section2.2.3).Forexample,Figure2.8’s
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page97#153.3DataIntegration97scatterplotsrespectivelyshowpositivelycorrelateddataandnegativelycorrelateddata,whileFigure2.9displaysuncorrelateddata.Notethatcorrelationdoesnotimplycausality.Thatis,ifAandBarecorrelated,thisdoesnotnecessarilyimplythatAcausesBorthatBcausesA.Forexample,inanalyzingademographicdatabase,wemayfindthatattributesrepresentingthenumberofhospitalsandthenumberofcartheftsinaregionarecorrelated.Thisdoesnotmeanthatonecausestheother.Bothareactuallycausallylinkedtoathirdattribute,namely,population.CovarianceofNumericDataInprobabilitytheoryandstatistics,correlationandcovariancearetwosimilarmeasuresforassessinghowmuchtwoattributeschangetogether.ConsidertwonumericattributesAandB,andasetofnobservations{(a1,b1),...,(an,bn)}.ThemeanvaluesofAandB,respectively,arealsoknownastheexpectedvaluesonAandB,thatis,E(A)=¯A=(cid:80)ni=1ainandE(B)=¯B=(cid:80)ni=1bin.ThecovariancebetweenAandBisdefinedasCov(A,B)=E((A−¯A)(B−¯B))=(cid:80)ni=1(ai−¯A)(bi−¯B)n.(3.4)IfwecompareEq.(3.3)forrA,B(correlationcoefficient)withEq.(3.4)forcovariance,weseethatrA,B=Cov(A,B)σAσB,(3.5)whereσAandσBarethestandarddeviationsofAandB,respectively.ItcanalsobeshownthatCov(A,B)=E(A·B)−¯A¯B.(3.6)Thisequationmaysimplifycalculations.FortwoattributesAandBthattendtochangetogether,ifAislargerthan¯A(theexpectedvalueofA),thenBislikelytobelargerthan¯B(theexpectedvalueofB).Therefore,thecovariancebetweenAandBispositive.Ontheotherhand,ifoneoftheattributestendstobeaboveitsexpectedvaluewhentheotherattributeisbelowitsexpectedvalue,thenthecovarianceofAandBisnegative.IfAandBareindependent(i.e.,theydonothavecorrelation),thenE(A·B)=E(A)·E(B).Therefore,thecovarianceisCov(A,B)=E(A·B)−¯A¯B=E(A)·E(B)−¯A¯B=0.However,theconverseisnottrue.Somepairsofrandomvariables(attributes)mayhaveacovarianceof0butarenotindependent.Onlyundersomeadditionalassumptions
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page97#153.3DataIntegration97scatterplotsrespectivelyshowpositivelycorrelateddataandnegativelycorrelateddata,whileFigure2.9displaysuncorrelateddata.Notethatcorrelationdoesnotimplycausality.Thatis,ifAandBarecorrelated,thisdoesnotnecessarilyimplythatAcausesBorthatBcausesA.Forexample,inanalyzingademographicdatabase,wemayfindthatattributesrepresentingthenumberofhospitalsandthenumberofcartheftsinaregionarecorrelated.Thisdoesnotmeanthatonecausestheother.Bothareactuallycausallylinkedtoathirdattribute,namely,population.CovarianceofNumericDataInprobabilitytheoryandstatistics,correlationandcovariancearetwosimilarmeasuresforassessinghowmuchtwoattributeschangetogether.ConsidertwonumericattributesAandB,andasetofnobservations{(a1,b1),...,(an,bn)}.ThemeanvaluesofAandB,respectively,arealsoknownastheexpectedvaluesonAandB,thatis,E(A)=¯A=(cid:80)ni=1ainandE(B)=¯B=(cid:80)ni=1bin.ThecovariancebetweenAandBisdefinedasCov(A,B)=E((A−¯A)(B−¯B))=(cid:80)ni=1(ai−¯A)(bi−¯B)n.(3.4)IfwecompareEq.(3.3)forrA,B(correlationcoefficient)withEq.(3.4)forcovariance,weseethatrA,B=Cov(A,B)σAσB,(3.5)whereσAandσBarethestandarddeviationsofAandB,respectively.ItcanalsobeshownthatCov(A,B)=E(A·B)−¯A¯B.(3.6)Thisequationmaysimplifycalculations.FortwoattributesAandBthattendtochangetogether,ifAislargerthan¯A(theexpectedvalueofA),thenBislikelytobelargerthan¯B(theexpectedvalueofB).Therefore,thecovariancebetweenAandBispositive.Ontheotherhand,ifoneoftheattributestendstobeaboveitsexpectedvaluewhentheotherattributeisbelowitsexpectedvalue,thenthecovarianceofAandBisnegative.IfAandBareindependent(i.e.,theydonothavecorrelation),thenE(A·B)=E(A)·E(B).Therefore,thecovarianceisCov(A,B)=E(A·B)−¯A¯B=E(A)·E(B)−¯A¯B=0.However,theconverseisnottrue.Somepairsofrandomvariables(attributes)mayhaveacovarianceof0butarenotindependent.Onlyundersomeadditionalassumptions
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page98#1698Chapter3DataPreprocessingTable3.2StockPricesforAllElectronicsandHighTechTimepointAllElectronicsHighTecht1620t2510t3414t435t525(e.g.,thedatafollowmultivariatenormaldistributions)doesacovarianceof0implyindependence.Example3.2Covarianceanalysisofnumericattributes.ConsiderTable3.2,whichpresentsasim-plifiedexampleofstockpricesobservedatfivetimepointsforAllElectronicsandHighTech,ahigh-techcompany.Ifthestocksareaffectedbythesameindustrytrends,willtheirpricesriseorfalltogether?E(AllElectronics)=6+5+4+3+25=205=$4andE(HighTech)=20+10+14+5+55=545=$10.80.Thus,usingEq.(3.4),wecomputeCov(AllElectroncis,HighTech)=6×20+5×10+4×14+3×5+2×55−4×10.80=50.2−43.2=7.Therefore,giventhepositivecovariancewecansaythatstockpricesforbothcompaniesrisetogether.Varianceisaspecialcaseofcovariance,wherethetwoattributesareidentical(i.e.,thecovarianceofanattributewithitself).VariancewasdiscussedinChapter2.3.3.3TupleDuplicationInadditiontodetectingredundanciesbetweenattributes,duplicationshouldalsobedetectedatthetuplelevel(e.g.,wheretherearetwoormoreidenticaltuplesforagivenuniquedataentrycase).Theuseofdenormalizedtables(oftendonetoimproveper-formancebyavoidingjoins)isanothersourceofdataredundancy.Inconsistenciesoftenarisebetweenvariousduplicates,duetoinaccuratedataentryorupdatingsomebutnotalldataoccurrences.Forexample,ifapurchaseorderdatabasecontainsattributesfor
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page98#1698Chapter3DataPreprocessingTable3.2StockPricesforAllElectronicsandHighTechTimepointAllElectronicsHighTecht1620t2510t3414t435t525(e.g.,thedatafollowmultivariatenormaldistributions)doesacovarianceof0implyindependence.Example3.2Covarianceanalysisofnumericattributes.ConsiderTable3.2,whichpresentsasim-plifiedexampleofstockpricesobservedatfivetimepointsforAllElectronicsandHighTech,ahigh-techcompany.Ifthestocksareaffectedbythesameindustrytrends,willtheirpricesriseorfalltogether?E(AllElectronics)=6+5+4+3+25=205=$4andE(HighTech)=20+10+14+5+55=545=$10.80.Thus,usingEq.(3.4),wecomputeCov(AllElectroncis,HighTech)=6×20+5×10+4×14+3×5+2×55−4×10.80=50.2−43.2=7.Therefore,giventhepositivecovariancewecansaythatstockpricesforbothcompaniesrisetogether.Varianceisaspecialcaseofcovariance,wherethetwoattributesareidentical(i.e.,thecovarianceofanattributewithitself).VariancewasdiscussedinChapter2.3.3.3TupleDuplicationInadditiontodetectingredundanciesbetweenattributes,duplicationshouldalsobedetectedatthetuplelevel(e.g.,wheretherearetwoormoreidenticaltuplesforagivenuniquedataentrycase).Theuseofdenormalizedtables(oftendonetoimproveper-formancebyavoidingjoins)isanothersourceofdataredundancy.Inconsistenciesoftenarisebetweenvariousduplicates,duetoinaccuratedataentryorupdatingsomebutnotalldataoccurrences.Forexample,ifapurchaseorderdatabasecontainsattributesfor
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page101#193.4DataReduction101cleaningaswell.Givenasetofcoefficients,anapproximationoftheoriginaldatacanbeconstructedbyapplyingtheinverseoftheDWTused.TheDWTiscloselyrelatedtothediscreteFouriertransform(DFT),asignalprocess-ingtechniqueinvolvingsinesandcosines.Ingeneral,however,theDWTachievesbetterlossycompression.Thatis,ifthesamenumberofcoefficientsisretainedforaDWTandaDFTofagivendatavector,theDWTversionwillprovideamoreaccurateapproxima-tionoftheoriginaldata.Hence,foranequivalentapproximation,theDWTrequireslessspacethantheDFT.UnliketheDFT,waveletsarequitelocalizedinspace,contributingtotheconservationoflocaldetail.ThereisonlyoneDFT,yetthereareseveralfamiliesofDWTs.Figure3.4showssomewaveletfamilies.PopularwavelettransformsincludetheHaar-2,Daubechies-4,andDaubechies-6.Thegeneralprocedureforapplyingadiscretewavelettransformusesahierarchicalpyramidalgorithmthathalvesthedataateachiteration,resultinginfastcomputationalspeed.Themethodisasfollows:1.Thelength,L,oftheinputdatavectormustbeanintegerpowerof2.Thisconditioncanbemetbypaddingthedatavectorwithzerosasnecessary(L≥n).2.Eachtransforminvolvesapplyingtwofunctions.Thefirstappliessomedatasmooth-ing,suchasasumorweightedaverage.Thesecondperformsaweighteddifference,whichactstobringoutthedetailedfeaturesofthedata.3.ThetwofunctionsareappliedtopairsofdatapointsinX,thatis,toallpairsofmeasurements(x2i,x2i+1).ThisresultsintwodatasetsoflengthL/2.Ingeneral,theserepresentasmoothedorlow-frequencyversionoftheinputdataandthehigh-frequencycontentofit,respectively.4.Thetwofunctionsarerecursivelyappliedtothedatasetsobtainedinthepreviousloop,untiltheresultingdatasetsobtainedareoflength2.5.Selectedvaluesfromthedatasetsobtainedinthepreviousiterationsaredesignatedthewaveletcoefficientsofthetransformeddata.02460.80.60.40.20.0(cid:2)1.0(cid:2)0.50.00.5(a) Haar-2(b) Daubechies-41.01.52.00.60.40.20.0Figure3.4Examplesofwaveletfamilies.Thenumbernexttoawaveletnameisthenumberofvanishingmomentsofthewavelet.Thisisasetofmathematicalrelationshipsthatthecoefficientsmustsatisfyandisrelatedtothenumberofcoefficients.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page101#193.4DataReduction101cleaningaswell.Givenasetofcoefficients,anapproximationoftheoriginaldatacanbeconstructedbyapplyingtheinverseoftheDWTused.TheDWTiscloselyrelatedtothediscreteFouriertransform(DFT),asignalprocess-ingtechniqueinvolvingsinesandcosines.Ingeneral,however,theDWTachievesbetterlossycompression.Thatis,ifthesamenumberofcoefficientsisretainedforaDWTandaDFTofagivendatavector,theDWTversionwillprovideamoreaccurateapproxima-tionoftheoriginaldata.Hence,foranequivalentapproximation,theDWTrequireslessspacethantheDFT.UnliketheDFT,waveletsarequitelocalizedinspace,contributingtotheconservationoflocaldetail.ThereisonlyoneDFT,yetthereareseveralfamiliesofDWTs.Figure3.4showssomewaveletfamilies.PopularwavelettransformsincludetheHaar-2,Daubechies-4,andDaubechies-6.Thegeneralprocedureforapplyingadiscretewavelettransformusesahierarchicalpyramidalgorithmthathalvesthedataateachiteration,resultinginfastcomputationalspeed.Themethodisasfollows:1.Thelength,L,oftheinputdatavectormustbeanintegerpowerof2.Thisconditioncanbemetbypaddingthedatavectorwithzerosasnecessary(L≥n).2.Eachtransforminvolvesapplyingtwofunctions.Thefirstappliessomedatasmooth-ing,suchasasumorweightedaverage.Thesecondperformsaweighteddifference,whichactstobringoutthedetailedfeaturesofthedata.3.ThetwofunctionsareappliedtopairsofdatapointsinX,thatis,toallpairsofmeasurements(x2i,x2i+1).ThisresultsintwodatasetsoflengthL/2.Ingeneral,theserepresentasmoothedorlow-frequencyversionoftheinputdataandthehigh-frequencycontentofit,respectively.4.Thetwofunctionsarerecursivelyappliedtothedatasetsobtainedinthepreviousloop,untiltheresultingdatasetsobtainedareoflength2.5.Selectedvaluesfromthedatasetsobtainedinthepreviousiterationsaredesignatedthewaveletcoefficientsofthetransformeddata.02460.80.60.40.20.0(cid:2)1.0(cid:2)0.50.00.5(a) Haar-2(b) Daubechies-41.01.52.00.60.40.20.0Figure3.4Examplesofwaveletfamilies.Thenumbernexttoawaveletnameisthenumberofvanishingmomentsofthewavelet.Thisisasetofmathematicalrelationshipsthatthecoefficientsmustsatisfyandisrelatedtothenumberofcoefficients.
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Context: setofvariables.Theinitialdatacanthenbeprojectedontothissmallerset.PCAoftenrevealsrelationshipsthatwerenotpreviouslysuspectedandtherebyallowsinterpretationsthatwouldnotordinarilyresult.Thebasicprocedureisasfollows:1.Theinputdataarenormalized,sothateachattributefallswithinthesamerange.Thisstephelpsensurethatattributeswithlargedomainswillnotdominateattributeswithsmallerdomains.2.PCAcomputeskorthonormalvectorsthatprovideabasisforthenormalizedinputdata.Theseareunitvectorsthateachpointinadirectionperpendiculartotheothers.Thesevectorsarereferredtoastheprincipalcomponents.Theinputdataarealinearcombinationoftheprincipalcomponents.3.Theprincipalcomponentsaresortedinorderofdecreasing“significance”orstrength.Theprincipalcomponentsessentiallyserveasanewsetofaxesforthedata,
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Context: setofvariables.Theinitialdatacanthenbeprojectedontothissmallerset.PCAoftenrevealsrelationshipsthatwerenotpreviouslysuspectedandtherebyallowsinterpretationsthatwouldnotordinarilyresult.Thebasicprocedureisasfollows:1.Theinputdataarenormalized,sothateachattributefallswithinthesamerange.Thisstephelpsensurethatattributeswithlargedomainswillnotdominateattributeswithsmallerdomains.2.PCAcomputeskorthonormalvectorsthatprovideabasisforthenormalizedinputdata.Theseareunitvectorsthateachpointinadirectionperpendiculartotheothers.Thesevectorsarereferredtoastheprincipalcomponents.Theinputdataarealinearcombinationoftheprincipalcomponents.3.Theprincipalcomponentsaresortedinorderofdecreasing“significance”orstrength.Theprincipalcomponentsessentiallyserveasanewsetofaxesforthedata,
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page106#24106Chapter3DataPreprocessingspecifytheslopeofthelineandthey-intercept,respectively.Thesecoefficientscanbesolvedforbythemethodofleastsquares,whichminimizestheerrorbetweentheactuallineseparatingthedataandtheestimateoftheline.Multiplelinearregressionisanextensionof(simple)linearregression,whichallowsaresponsevariable,y,tobemodeledasalinearfunctionoftwoormorepredictorvariables.Log-linearmodelsapproximatediscretemultidimensionalprobabilitydistributions.Givenasetoftuplesinndimensions(e.g.,describedbynattributes),wecancon-sidereachtupleasapointinann-dimensionalspace.Log-linearmodelscanbeusedtoestimatetheprobabilityofeachpointinamultidimensionalspaceforasetofdis-cretizedattributes,basedonasmallersubsetofdimensionalcombinations.Thisallowsahigher-dimensionaldataspacetobeconstructedfromlower-dimensionalspaces.Log-linearmodelsarethereforealsousefulfordimensionalityreduction(sincethelower-dimensionalpointstogethertypicallyoccupylessspacethantheoriginaldatapoints)anddatasmoothing(sinceaggregateestimatesinthelower-dimensionalspacearelesssubjecttosamplingvariationsthantheestimatesinthehigher-dimensionalspace).Regressionandlog-linearmodelscanbothbeusedonsparsedata,althoughtheirapplicationmaybelimited.Whilebothmethodscanhandleskeweddata,regressiondoesexceptionallywell.Regressioncanbecomputationallyintensivewhenappliedtohigh-dimensionaldata,whereaslog-linearmodelsshowgoodscalabilityforupto10orsodimensions.Severalsoftwarepackagesexisttosolveregressionproblems.ExamplesincludeSAS(www.sas.com),SPSS(www.spss.com),andS-Plus(www.insightful.com).AnotherusefulresourceisthebookNumericalRecipesinC,byPress,Teukolsky,Vetterling,andFlannery[PTVF07],anditsassociatedsourcecode.3.4.6HistogramsHistogramsusebinningtoapproximatedatadistributionsandareapopularformofdatareduction.HistogramswereintroducedinSection2.2.3.Ahistogramforanattribute,A,partitionsthedatadistributionofAintodisjointsubsets,referredtoasbucketsorbins.Ifeachbucketrepresentsonlyasingl
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Context: representsonlyasingleattribute–value/frequencypair,thebucketsarecalledsingletonbuckets.Often,bucketsinsteadrepresentcontinuousrangesforthegivenattribute.Example3.3Histograms.ThefollowingdataarealistofAllElectronicspricesforcommonlysolditems(roundedtothenearestdollar).Thenumbershavebeensorted:1,1,5,5,5,5,5,8,8,10,10,10,10,12,14,14,14,15,15,15,15,15,15,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,21,21,21,21,25,25,25,25,25,28,28,30,30,30.Figure3.7showsahistogramforthedatausingsingletonbuckets.Tofurtherreducethedata,itiscommontohaveeachbucketdenoteacontinuousvaluerangeforthegivenattribute.InFigure3.8,eachbucketrepresentsadifferent$10rangeforprice.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page106#24106Chapter3DataPreprocessingspecifytheslopeofthelineandthey-intercept,respectively.Thesecoefficientscanbesolvedforbythemethodofleastsquares,whichminimizestheerrorbetweentheactuallineseparatingthedataandtheestimateoftheline.Multiplelinearregressionisanextensionof(simple)linearregression,whichallowsaresponsevariable,y,tobemodeledasalinearfunctionoftwoormorepredictorvariables.Log-linearmodelsapproximatediscretemultidimensionalprobabilitydistributions.Givenasetoftuplesinndimensions(e.g.,describedbynattributes),wecancon-sidereachtupleasapointinann-dimensionalspace.Log-linearmodelscanbeusedtoestimatetheprobabilityofeachpointinamultidimensionalspaceforasetofdis-cretizedattributes,basedonasmallersubsetofdimensionalcombinations.Thisallowsahigher-dimensionaldataspacetobeconstructedfromlower-dimensionalspaces.Log-linearmodelsarethereforealsousefulfordimensionalityreduction(sincethelower-dimensionalpointstogethertypicallyoccupylessspacethantheoriginaldatapoints)anddatasmoothing(sinceaggregateestimatesinthelower-dimensionalspacearelesssubjecttosamplingvariationsthantheestimatesinthehigher-dimensionalspace).Regressionandlog-linearmodelscanbothbeusedonsparsedata,althoughtheirapplicationmaybelimited.Whilebothmethodscanhandleskeweddata,regressiondoesexceptionallywell.Regressioncanbecomputationallyintensivewhenappliedtohigh-dimensionaldata,whereaslog-linearmodelsshowgoodscalabilityforupto10orsodimensions.Severalsoftwarepackagesexisttosolveregressionproblems.ExamplesincludeSAS(www.sas.com),SPSS(www.spss.com),andS-Plus(www.insightful.com).AnotherusefulresourceisthebookNumericalRecipesinC,byPress,Teukolsky,Vetterling,andFlannery[PTVF07],anditsassociatedsourcecode.3.4.6HistogramsHistogramsusebinningtoapproximatedatadistributionsandareapopularformofdatareduction.HistogramswereintroducedinSection2.2.3.Ahistogramforanattribute,A,partitionsthedatadistributionofAintodisjointsubsets,referredtoasbucketsorbins.Ifeachbucketrepresentsonlyasingl
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Context: representsonlyasingleattribute–value/frequencypair,thebucketsarecalledsingletonbuckets.Often,bucketsinsteadrepresentcontinuousrangesforthegivenattribute.Example3.3Histograms.ThefollowingdataarealistofAllElectronicspricesforcommonlysolditems(roundedtothenearestdollar).Thenumbershavebeensorted:1,1,5,5,5,5,5,8,8,10,10,10,10,12,14,14,14,15,15,15,15,15,15,18,18,18,18,18,18,18,18,20,20,20,20,20,20,20,21,21,21,21,25,25,25,25,25,28,28,30,30,30.Figure3.7showsahistogramforthedatausingsingletonbuckets.Tofurtherreducethedata,itiscommontohaveeachbucketdenoteacontinuousvaluerangeforthegivenattribute.InFigure3.8,eachbucketrepresentsadifferent$10rangeforprice.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page107#253.4DataReduction10751010987654321015202530price ($)countFigure3.7Ahistogramforpriceusingsingletonbuckets—eachbucketrepresentsoneprice–value/frequencypair.1–1011–2021–30price ($)count2520151050Figure3.8Anequal-widthhistogramforprice,wherevaluesareaggregatedsothateachbuckethasauniformwidthof$10.“Howarethebucketsdeterminedandtheattributevaluespartitioned?”Thereareseveralpartitioningrules,includingthefollowing:Equal-width:Inanequal-widthhistogram,thewidthofeachbucketrangeisuniform(e.g.,thewidthof$10forthebucketsinFigure3.8).Equal-frequency(orequal-depth):Inanequal-frequencyhistogram,thebucketsarecreatedsothat,roughly,thefrequencyofeachbucketisconstant(i.e.,eachbucketcontainsroughlythesamenumberofcontiguousdatasamples).
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page107#253.4DataReduction10751010987654321015202530price ($)countFigure3.7Ahistogramforpriceusingsingletonbuckets—eachbucketrepresentsoneprice–value/frequencypair.1–1011–2021–30price ($)count2520151050Figure3.8Anequal-widthhistogramforprice,wherevaluesareaggregatedsothateachbuckethasauniformwidthof$10.“Howarethebucketsdeterminedandtheattributevaluespartitioned?”Thereareseveralpartitioningrules,includingthefollowing:Equal-width:Inanequal-widthhistogram,thewidthofeachbucketrangeisuniform(e.g.,thewidthof$10forthebucketsinFigure3.8).Equal-frequency(orequal-depth):Inanequal-frequencyhistogram,thebucketsarecreatedsothat,roughly,thefrequencyofeachbucketisconstant(i.e.,eachbucketcontainsroughlythesamenumberofcontiguousdatasamples).
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page108#26108Chapter3DataPreprocessingHistogramsarehighlyeffectiveatapproximatingbothsparseanddensedata,aswellashighlyskewedanduniformdata.Thehistogramsdescribedbeforeforsingleattributescanbeextendedformultipleattributes.Multidimensionalhistogramscancap-turedependenciesbetweenattributes.Thesehistogramshavebeenfoundeffectiveinapproximatingdatawithuptofiveattributes.Morestudiesareneededregardingtheeffectivenessofmultidimensionalhistogramsforhighdimensionalities.Singletonbucketsareusefulforstoringhigh-frequencyoutliers.3.4.7ClusteringClusteringtechniquesconsiderdatatuplesasobjects.Theypartitiontheobjectsintogroups,orclusters,sothatobjectswithinaclusterare“similar”tooneanotherand“dis-similar”toobjectsinotherclusters.Similarityiscommonlydefinedintermsofhow“close”theobjectsareinspace,basedonadistancefunction.The“quality”ofaclustermayberepresentedbyitsdiameter,themaximumdistancebetweenanytwoobjectsinthecluster.Centroiddistanceisanalternativemeasureofclusterqualityandisdefinedastheaveragedistanceofeachclusterobjectfromtheclustercentroid(denotingthe“averageobject,”oraveragepointinspaceforthecluster).Figure3.3showeda2-Dplotofcustomerdatawithrespecttocustomerlocationsinacity.Threedataclustersarevisible.Indatareduction,theclusterrepresentationsofthedataareusedtoreplacetheactualdata.Theeffectivenessofthistechniquedependsonthedata’snature.Itismuchmoreeffectivefordatathatcanbeorganizedintodistinctclustersthanforsmeareddata.Therearemanymeasuresfordefiningclustersandclusterquality.Clusteringmeth-odsarefurtherdescribedinChapters10and11.3.4.8SamplingSamplingcanbeusedasadatareductiontechniquebecauseitallowsalargedatasettoberepresentedbyamuchsmallerrandomdatasample(orsubset).Supposethatalargedataset,D,containsNtuples.Let’slookatthemostcommonwaysthatwecouldsampleDfordatareduction,asillustratedinFigure3.9.Simplerandomsamplewithoutreplacement(SRSWOR)ofsizes:ThisiscreatedbydrawingsoftheNtuplesfromD(s<N),wheretheprobabilityofdrawinganytupleinDis1/N
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Context: winganytupleinDis1/N,thatis,alltuplesareequallylikelytobesampled.Simplerandomsamplewithreplacement(SRSWR)ofsizes:ThisissimilartoSRSWOR,exceptthateachtimeatupleisdrawnfromD,itisrecordedandthenreplaced.Thatis,afteratupleisdrawn,itisplacedbackinDsothatitmaybedrawnagain.Clustersample:IfthetuplesinDaregroupedintoMmutuallydisjoint“clusters,”thenanSRSofsclusterscanbeobtained,wheres<M.Forexample,tuplesinadatabaseareusuallyretrievedapageatatime,sothateachpagecanbeconsidered
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page108#26108Chapter3DataPreprocessingHistogramsarehighlyeffectiveatapproximatingbothsparseanddensedata,aswellashighlyskewedanduniformdata.Thehistogramsdescribedbeforeforsingleattributescanbeextendedformultipleattributes.Multidimensionalhistogramscancap-turedependenciesbetweenattributes.Thesehistogramshavebeenfoundeffectiveinapproximatingdatawithuptofiveattributes.Morestudiesareneededregardingtheeffectivenessofmultidimensionalhistogramsforhighdimensionalities.Singletonbucketsareusefulforstoringhigh-frequencyoutliers.3.4.7ClusteringClusteringtechniquesconsiderdatatuplesasobjects.Theypartitiontheobjectsintogroups,orclusters,sothatobjectswithinaclusterare“similar”tooneanotherand“dis-similar”toobjectsinotherclusters.Similarityiscommonlydefinedintermsofhow“close”theobjectsareinspace,basedonadistancefunction.The“quality”ofaclustermayberepresentedbyitsdiameter,themaximumdistancebetweenanytwoobjectsinthecluster.Centroiddistanceisanalternativemeasureofclusterqualityandisdefinedastheaveragedistanceofeachclusterobjectfromtheclustercentroid(denotingthe“averageobject,”oraveragepointinspaceforthecluster).Figure3.3showeda2-Dplotofcustomerdatawithrespecttocustomerlocationsinacity.Threedataclustersarevisible.Indatareduction,theclusterrepresentationsofthedataareusedtoreplacetheactualdata.Theeffectivenessofthistechniquedependsonthedata’snature.Itismuchmoreeffectivefordatathatcanbeorganizedintodistinctclustersthanforsmeareddata.Therearemanymeasuresfordefiningclustersandclusterquality.Clusteringmeth-odsarefurtherdescribedinChapters10and11.3.4.8SamplingSamplingcanbeusedasadatareductiontechniquebecauseitallowsalargedatasettoberepresentedbyamuchsmallerrandomdatasample(orsubset).Supposethatalargedataset,D,containsNtuples.Let’slookatthemostcommonwaysthatwecouldsampleDfordatareduction,asillustratedinFigure3.9.Simplerandomsamplewithoutreplacement(SRSWOR)ofsizes:ThisiscreatedbydrawingsoftheNtuplesfromD(s<N),wheretheprobabilityofdrawinganytupleinDis1/N
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Context: winganytupleinDis1/N,thatis,alltuplesareequallylikelytobesampled.Simplerandomsamplewithreplacement(SRSWR)ofsizes:ThisissimilartoSRSWOR,exceptthateachtimeatupleisdrawnfromD,itisrecordedandthenreplaced.Thatis,afteratupleisdrawn,itisplacedbackinDsothatitmaybedrawnagain.Clustersample:IfthetuplesinDaregroupedintoMmutuallydisjoint“clusters,”thenanSRSofsclusterscanbeobtained,wheres<M.Forexample,tuplesinadatabaseareusuallyretrievedapageatatime,sothateachpagecanbeconsidered
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page109#273.4DataReduction109Cluster sampleStartified sampleFigure3.9Samplingcanbeusedfordatareduction.acluster.Areduceddatarepresentationcanbeobtainedbyapplying,say,SRSWORtothepages,resultinginaclustersampleofthetuples.Otherclusteringcriteriacon-veyingrichsemanticscanalsobeexplored.Forexample,inaspatialdatabase,wemaychoosetodefineclustersgeographicallybasedonhowcloselydifferentareasarelocated.Stratifiedsample:IfDisdividedintomutuallydisjointpartscalledstrata,astratifiedsampleofDisgeneratedbyobtaininganSRSateachstratum.Thishelpsensurea
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page109#273.4DataReduction109Cluster sampleStartified sampleFigure3.9Samplingcanbeusedfordatareduction.acluster.Areduceddatarepresentationcanbeobtainedbyapplying,say,SRSWORtothepages,resultinginaclustersampleofthetuples.Otherclusteringcriteriacon-veyingrichsemanticscanalsobeexplored.Forexample,inaspatialdatabase,wemaychoosetodefineclustersgeographicallybasedonhowcloselydifferentareasarelocated.Stratifiedsample:IfDisdividedintomutuallydisjointpartscalledstrata,astratifiedsampleofDisgeneratedbyobtaininganSRSateachstratum.Thishelpsensurea
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Context: ataanalysis.Forexample,changingmea-surementunitsfrommeterstoinchesforheight,orfromkilogramstopoundsforweight,mayleadtoverydifferentresults.Ingeneral,expressinganattributeinsmallerunitswillleadtoalargerrangeforthatattribute,andthustendtogivesuchanattributegreatereffector“weight.”Tohelpavoiddependenceonthechoiceofmeasurementunits,thedatashouldbenormalizedorstandardized.Thisinvolvestransformingthedatatofallwithinasmallerorcommonrangesuchas[−1,1]or[0.0,1.0].(Thetermsstandardizeandnormalizeareusedinterchangeablyindatapreprocessing,althoughinstatistics,thelattertermalsohasotherconnotations.)Normalizingthedataattemptstogiveallattributesanequalweight.Normaliza-tionisparticularlyusefulforclassificationalgorithmsinvolvingneuralnetworksordistancemeasurementssuchasnearest-neighborclassificationandclustering.Ifusingtheneuralnetworkbackpropagationalgorithmforclassificationmining(Chapter9),normalizingtheinputvaluesforeachattributemeasuredinthetrainingtupleswillhelpspeedupthelearningphase.Fordistance-basedmethods,normalizationhelpsprevent
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Context: ataanalysis.Forexample,changingmea-surementunitsfrommeterstoinchesforheight,orfromkilogramstopoundsforweight,mayleadtoverydifferentresults.Ingeneral,expressinganattributeinsmallerunitswillleadtoalargerrangeforthatattribute,andthustendtogivesuchanattributegreatereffector“weight.”Tohelpavoiddependenceonthechoiceofmeasurementunits,thedatashouldbenormalizedorstandardized.Thisinvolvestransformingthedatatofallwithinasmallerorcommonrangesuchas[−1,1]or[0.0,1.0].(Thetermsstandardizeandnormalizeareusedinterchangeablyindatapreprocessing,althoughinstatistics,thelattertermalsohasotherconnotations.)Normalizingthedataattemptstogiveallattributesanequalweight.Normaliza-tionisparticularlyusefulforclassificationalgorithmsinvolvingneuralnetworksordistancemeasurementssuchasnearest-neighborclassificationandclustering.Ifusingtheneuralnetworkbackpropagationalgorithmforclassificationmining(Chapter9),normalizingtheinputvaluesforeachattributemeasuredinthetrainingtupleswillhelpspeedupthelearningphase.Fordistance-basedmethods,normalizationhelpsprevent
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page114#32114Chapter3DataPreprocessingattributeswithinitiallylargeranges(e.g.,income)fromoutweighingattributeswithinitiallysmallerranges(e.g.,binaryattributes).Itisalsousefulwhengivennopriorknowledgeofthedata.Therearemanymethodsfordatanormalization.Westudymin-maxnormalization,z-scorenormalization,andnormalizationbydecimalscaling.Forourdiscussion,letAbeanumericattributewithnobservedvalues,v1,v2,...,vn.Min-maxnormalizationperformsalineartransformationontheoriginaldata.Sup-posethatminAandmaxAaretheminimumandmaximumvaluesofanattribute,A.Min-maxnormalizationmapsavalue,vi,ofAtov(cid:48)iintherange[newminA,newmaxA]bycomputingv(cid:48)i=vi−minAmaxA−minA(newmaxA−newminA)+newminA.(3.8)Min-maxnormalizationpreservestherelationshipsamongtheoriginaldatavalues.Itwillencounteran“out-of-bounds”errorifafutureinputcasefornormalizationfallsoutsideoftheoriginaldatarangeforA.Example3.4Min-maxnormalization.Supposethattheminimumandmaximumvaluesfortheattributeincomeare$12,000and$98,000,respectively.Wewouldliketomapincometotherange[0.0,1.0].Bymin-maxnormalization,avalueof$73,600forincomeistransformedto73,600−12,00098,000−12,000(1.0−0)+0=0.716.Inz-scorenormalization(orzero-meannormalization),thevaluesforanattribute,A,arenormalizedbasedonthemean(i.e.,average)andstandarddeviationofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi−¯AσA,(3.9)where¯AandσAarethemeanandstandarddeviation,respectively,ofattributeA.ThemeanandstandarddeviationwerediscussedinSection2.2,where¯A=1n(v1+v2+···+vn)andσAiscomputedasthesquarerootofthevarianceofA(seeEq.(2.6)).ThismethodofnormalizationisusefulwhentheactualminimumandmaximumofattributeAareunknown,orwhenthereareoutliersthatdominatethemin-maxnormalization.Example3.5z-scorenormalization.Supposethatthemeanandstandarddeviationofthevaluesfortheattributeincomeare$54,000and$16,000,respectively.Withz-scorenormalization,avalueof$73,600forincomeistransformedto73,600−54,00016,000=1.225.Avariationofthisz-scorenormalizati
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Context: isz-scorenormalizationreplacesthestandarddeviationofEq.(3.9)bythemeanabsolutedeviationofA.ThemeanabsolutedeviationofA,denotedsA,issA=1n(|v1−¯A|+|v2−¯A|+···+|vn−¯A|).(3.10)
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page114#32114Chapter3DataPreprocessingattributeswithinitiallylargeranges(e.g.,income)fromoutweighingattributeswithinitiallysmallerranges(e.g.,binaryattributes).Itisalsousefulwhengivennopriorknowledgeofthedata.Therearemanymethodsfordatanormalization.Westudymin-maxnormalization,z-scorenormalization,andnormalizationbydecimalscaling.Forourdiscussion,letAbeanumericattributewithnobservedvalues,v1,v2,...,vn.Min-maxnormalizationperformsalineartransformationontheoriginaldata.Sup-posethatminAandmaxAaretheminimumandmaximumvaluesofanattribute,A.Min-maxnormalizationmapsavalue,vi,ofAtov(cid:48)iintherange[newminA,newmaxA]bycomputingv(cid:48)i=vi−minAmaxA−minA(newmaxA−newminA)+newminA.(3.8)Min-maxnormalizationpreservestherelationshipsamongtheoriginaldatavalues.Itwillencounteran“out-of-bounds”errorifafutureinputcasefornormalizationfallsoutsideoftheoriginaldatarangeforA.Example3.4Min-maxnormalization.Supposethattheminimumandmaximumvaluesfortheattributeincomeare$12,000and$98,000,respectively.Wewouldliketomapincometotherange[0.0,1.0].Bymin-maxnormalization,avalueof$73,600forincomeistransformedto73,600−12,00098,000−12,000(1.0−0)+0=0.716.Inz-scorenormalization(orzero-meannormalization),thevaluesforanattribute,A,arenormalizedbasedonthemean(i.e.,average)andstandarddeviationofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi−¯AσA,(3.9)where¯AandσAarethemeanandstandarddeviation,respectively,ofattributeA.ThemeanandstandarddeviationwerediscussedinSection2.2,where¯A=1n(v1+v2+···+vn)andσAiscomputedasthesquarerootofthevarianceofA(seeEq.(2.6)).ThismethodofnormalizationisusefulwhentheactualminimumandmaximumofattributeAareunknown,orwhenthereareoutliersthatdominatethemin-maxnormalization.Example3.5z-scorenormalization.Supposethatthemeanandstandarddeviationofthevaluesfortheattributeincomeare$54,000and$16,000,respectively.Withz-scorenormalization,avalueof$73,600forincomeistransformedto73,600−54,00016,000=1.225.Avariationofthisz-scorenormalizati
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Context: isz-scorenormalizationreplacesthestandarddeviationofEq.(3.9)bythemeanabsolutedeviationofA.ThemeanabsolutedeviationofA,denotedsA,issA=1n(|v1−¯A|+|v2−¯A|+···+|vn−¯A|).(3.10)
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page115#333.5DataTransformationandDataDiscretization115Thus,z-scorenormalizationusingthemeanabsolutedeviationisv(cid:48)i=vi−¯AsA.(3.11)Themeanabsolutedeviation,sA,ismorerobusttooutliersthanthestandarddeviation,σA.Whencomputingthemeanabsolutedeviation,thedeviationsfromthemean(i.e.,|xi−¯x|)arenotsquared;hence,theeffectofoutliersissomewhatreduced.NormalizationbydecimalscalingnormalizesbymovingthedecimalpointofvaluesofattributeA.ThenumberofdecimalpointsmoveddependsonthemaximumabsolutevalueofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi10j,(3.12)wherejisthesmallestintegersuchthatmax(|v(cid:48)i|)<1.Example3.6Decimalscaling.SupposethattherecordedvaluesofArangefrom−986to917.ThemaximumabsolutevalueofAis986.Tonormalizebydecimalscaling,wethereforedivideeachvalueby1000(i.e.,j=3)sothat−986normalizesto−0.986and917normalizesto0.917.Notethatnormalizationcanchangetheoriginaldataquiteabit,especiallywhenusingz-scorenormalizationordecimalscaling.Itisalsonecessarytosavethenormaliza-tionparameters(e.g.,themeanandstandarddeviationifusingz-scorenormalization)sothatfuturedatacanbenormalizedinauniformmanner.3.5.3DiscretizationbyBinningBinningisatop-downsplittingtechniquebasedonaspecifiednumberofbins.Section3.2.2discussedbinningmethodsfordatasmoothing.Thesemethodsarealsousedasdiscretizationmethodsfordatareductionandconcepthierarchygeneration.Forexample,attributevaluescanbediscretizedbyapplyingequal-widthorequal-frequencybinning,andthenreplacingeachbinvaluebythebinmeanormedian,asinsmoothingbybinmeansorsmoothingbybinmedians,respectively.Thesetechniquescanbeappliedrecursivelytotheresultingpartitionstogenerateconcepthierarchies.Binningdoesnotuseclassinformationandisthereforeanunsuperviseddiscretiza-tiontechnique.Itissensitivetotheuser-specifiednumberofbins,aswellasthepresenceofoutliers.3.5.4DiscretizationbyHistogramAnalysisLikebinning,histogramanalysisisanunsuperviseddiscretizationtechniquebecauseitdoesnotuseclassinformation.Histogra
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Context: information.HistogramswereintroducedinSection2.2.3.Ahis-togrampartitionsthevaluesofanattribute,A,intodisjointrangescalledbucketsorbins.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page115#333.5DataTransformationandDataDiscretization115Thus,z-scorenormalizationusingthemeanabsolutedeviationisv(cid:48)i=vi−¯AsA.(3.11)Themeanabsolutedeviation,sA,ismorerobusttooutliersthanthestandarddeviation,σA.Whencomputingthemeanabsolutedeviation,thedeviationsfromthemean(i.e.,|xi−¯x|)arenotsquared;hence,theeffectofoutliersissomewhatreduced.NormalizationbydecimalscalingnormalizesbymovingthedecimalpointofvaluesofattributeA.ThenumberofdecimalpointsmoveddependsonthemaximumabsolutevalueofA.Avalue,vi,ofAisnormalizedtov(cid:48)ibycomputingv(cid:48)i=vi10j,(3.12)wherejisthesmallestintegersuchthatmax(|v(cid:48)i|)<1.Example3.6Decimalscaling.SupposethattherecordedvaluesofArangefrom−986to917.ThemaximumabsolutevalueofAis986.Tonormalizebydecimalscaling,wethereforedivideeachvalueby1000(i.e.,j=3)sothat−986normalizesto−0.986and917normalizesto0.917.Notethatnormalizationcanchangetheoriginaldataquiteabit,especiallywhenusingz-scorenormalizationordecimalscaling.Itisalsonecessarytosavethenormaliza-tionparameters(e.g.,themeanandstandarddeviationifusingz-scorenormalization)sothatfuturedatacanbenormalizedinauniformmanner.3.5.3DiscretizationbyBinningBinningisatop-downsplittingtechniquebasedonaspecifiednumberofbins.Section3.2.2discussedbinningmethodsfordatasmoothing.Thesemethodsarealsousedasdiscretizationmethodsfordatareductionandconcepthierarchygeneration.Forexample,attributevaluescanbediscretizedbyapplyingequal-widthorequal-frequencybinning,andthenreplacingeachbinvaluebythebinmeanormedian,asinsmoothingbybinmeansorsmoothingbybinmedians,respectively.Thesetechniquescanbeappliedrecursivelytotheresultingpartitionstogenerateconcepthierarchies.Binningdoesnotuseclassinformationandisthereforeanunsuperviseddiscretiza-tiontechnique.Itissensitivetotheuser-specifiednumberofbins,aswellasthepresenceofoutliers.3.5.4DiscretizationbyHistogramAnalysisLikebinning,histogramanalysisisanunsuperviseddiscretizationtechniquebecauseitdoesnotuseclassinformation.Histogra
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Context: information.HistogramswereintroducedinSection2.2.3.Ahis-togrampartitionsthevaluesofanattribute,A,intodisjointrangescalledbucketsorbins.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page116#34116Chapter3DataPreprocessingVariouspartitioningrulescanbeusedtodefinehistograms(Section3.4.6).Inanequal-widthhistogram,forexample,thevaluesarepartitionedintoequal-sizepartitionsorranges(e.g.,earlierinFigure3.8forprice,whereeachbuckethasawidthof$10).Withanequal-frequencyhistogram,thevaluesarepartitionedsothat,ideally,eachpar-titioncontainsthesamenumberofdatatuples.Thehistogramanalysisalgorithmcanbeappliedrecursivelytoeachpartitioninordertoautomaticallygenerateamultilevelcon-cepthierarchy,withtheprocedureterminatingonceaprespecifiednumberofconceptlevelshasbeenreached.Aminimumintervalsizecanalsobeusedperleveltocontroltherecursiveprocedure.Thisspecifiestheminimumwidthofapartition,ortheminimumnumberofvaluesforeachpartitionateachlevel.Histogramscanalsobepartitionedbasedonclusteranalysisofthedatadistribution,asdescribednext.3.5.5DiscretizationbyCluster,DecisionTree,andCorrelationAnalysesClustering,decisiontreeanalysis,andcorrelationanalysiscanbeusedfordatadis-cretization.Webrieflystudyeachoftheseapproaches.Clusteranalysisisapopulardatadiscretizationmethod.Aclusteringalgorithmcanbeappliedtodiscretizeanumericattribute,A,bypartitioningthevaluesofAintoclus-tersorgroups.ClusteringtakesthedistributionofAintoconsideration,aswellastheclosenessofdatapoints,andthereforeisabletoproducehigh-qualitydiscretizationresults.ClusteringcanbeusedtogenerateaconcepthierarchyforAbyfollowingeitheratop-downsplittingstrategyorabottom-upmergingstrategy,whereeachclusterformsanodeoftheconcepthierarchy.Intheformer,eachinitialclusterorpartitionmaybefurtherdecomposedintoseveralsubclusters,formingalowerlevelofthehiera-rchy.Inthelatter,clustersareformedbyrepeatedlygroupingneighboringclustersinordertoformhigher-levelconcepts.ClusteringmethodsfordataminingarestudiedinChapters10and11.Techniquestogeneratedecisiontreesforclassification(Chapter8)canbeappliedtodiscretization.Suchtechniquesemployatop-downsplittingapproach.Unliketheothermethodsmentionedsofar,de
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page121#393.7Exercises1213.7Exercises3.1Dataqualitycanbeassessedintermsofseveralissues,includingaccuracy,completeness,andconsistency.Foreachoftheabovethreeissues,discusshowdataqualityassess-mentcandependontheintendeduseofthedata,givingexamples.Proposetwootherdimensionsofdataquality.3.2Inreal-worlddata,tupleswithmissingvaluesforsomeattributesareacommonoccurrence.Describevariousmethodsforhandlingthisproblem.3.3Exercise2.2gavethefollowingdata(inincreasingorder)fortheattributeage:13,15,16,16,19,20,20,21,22,22,25,25,25,25,30,33,33,35,35,35,35,36,40,45,46,52,70.(a)Usesmoothingbybinmeanstosmooththesedata,usingabindepthof3.Illustrateyoursteps.Commentontheeffectofthistechniqueforthegivendata.(b)Howmightyoudetermineoutliersinthedata?(c)Whatothermethodsaretherefordatasmoothing?3.4Discussissuestoconsiderduringdataintegration.3.5Whatarethevaluerangesofthefollowingnormalizationmethods?(a)min-maxnormalization(b)z-scorenormalization(c)z-scorenormalizationusingthemeanabsolutedeviationinsteadofstandarddevia-tion(d)normalizationbydecimalscaling3.6Usethesemethodstonormalizethefollowinggroupofdata:200,300,400,600,1000(a)min-maxnormalizationbysettingmin=0andmax=1(b)z-scorenormalization(c)z-scorenormalizationusingthemeanabsolutedeviationinsteadofstandarddevia-tion(d)normalizationbydecimalscaling3.7UsingthedataforagegiveninExercise3.3,answerthefollowing:(a)Usemin-maxnormalizationtotransformthevalue35forageontotherange[0.0,1.0].(b)Usez-scorenormalizationtotransformthevalue35forage,wherethestandarddeviationofageis12.94years.(c)Usenormalizationbydecimalscalingtotransformthevalue35forage.(d)Commentonwhichmethodyouwouldprefertouseforthegivendata,givingreasonsastowhy.
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Context: HAN10-ch03-083-124-97801238147912011/6/13:16Page122#40122Chapter3DataPreprocessing3.8UsingthedataforageandbodyfatgiveninExercise2.4,answerthefollowing:(a)Normalizethetwoattributesbasedonz-scorenormalization.(b)Calculatethecorrelationcoefficient(Pearson’sproductmomentcoefficient).Arethesetwoattributespositivelyornegativelycorrelated?Computetheircovariance.3.9Supposeagroupof12salespricerecordshasbeensortedasfollows:5,10,11,13,15,35,50,55,72,92,204,215.Partitionthemintothreebinsbyeachofthefollowingmethods:(a)equal-frequency(equal-depth)partitioning(b)equal-widthpartitioning(c)clustering3.10Useaflowcharttosummarizethefollowingproceduresforattributesubsetselection:(a)stepwiseforwardselection(b)stepwisebackwardelimination(c)acombinationofforwardselectionandbackwardelimination3.11UsingthedataforagegiveninExercise3.3,(a)Plotanequal-widthhistogramofwidth10.(b)Sketchexamplesofeachofthefollowingsamplingtechniques:SRSWOR,SRSWR,clustersampling,andstratifiedsampling.Usesamplesofsize5andthestrata“youth,”“middle-aged,”and“senior.”3.12ChiMerge[Ker92]isasupervised,bottom-up(i.e.,merge-based)datadiscretizationmethod.Itreliesonχ2analysis:Adjacentintervalswiththeleastχ2valuesaremergedtogetheruntilthechosenstoppingcriterionsatisfies.(a)BrieflydescribehowChiMergeworks.(b)TaketheIRISdataset,obtainedfromtheUniversityofCalifornia–IrvineMachineLearningDataRepository(www.ics.uci.edu/∼mlearn/MLRepository.html),asadatasettobediscretized.PerformdatadiscretizationforeachofthefournumericattributesusingtheChiMergemethod.(Letthestoppingcriteriabe:max-interval=6).Youneedtowriteasmallprogramtodothistoavoidclumsynumericalcomputation.Submityoursimpleanalysisandyourtestresults:split-points,finalintervals,andthedocumentedsourceprogram.3.13Proposeanalgorithm,inpseudocodeorinyourfavoriteprogramminglanguage,forthefollowing:(a)Theautomaticgenerationofaconcepthierarchyfornominaldatabasedonthenumberofdistinctvaluesofattributesinthegivenschema.(b)Theautomaticgenerationofaconcepthierarchyfornumericdatabasedonth
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Context: numericdatabasedontheequal-widthpartitioningrule.
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Context: eislessthan1,thereisanegativecorrelationbetweentheoccur-renceof{game}and{video}.Thenumeratoristhelikelihoodofacustomerpurchasingboth,whilethedenominatoriswhatthelikelihoodwouldhavebeenifthetwopur-chaseswerecompletelyindependent.Suchanegativecorrelationcannotbeidentifiedbyasupport–confidenceframework.Thesecondcorrelationmeasurethatwestudyistheχ2measure,whichwasintro-ducedinChapter3(Eq.3.1).Tocomputetheχ2value,wetakethesquareddifferencebetweentheobservedandexpectedvalueforaslot(AandBpair)inthecontin-gencytable,dividedbytheexpectedvalue.Thisamountissummedforallslotsofthecontingencytable.Let’sperformaχ2analysisofExample6.8.
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Context: rwise,itwouldgenerateunstableresults,asillustratedinD1andD2.
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Context: dacomparativeanalysisofasetofmeasuresforpatternevaluation.
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Context: %and(P(B|A)+P(A|B))/2=(0.01+0.01)/2<0.02,againindicatingthatAandBarenegativelycorrelated.Thismatchesourintuition.Themeasuredoesnothavethenull-invarianceproblemofthefirsttwodefinitionsconsidered.Let’sexamineanothercase:Supposethatamong100,000transactions,thestoresold1000needlepackagesofAbutonly10packagesofB;however,everytimepackageBis
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Context: pool.ForeachoftheseKseeds,wefindallthepatternswithinaballofasizespec-ifiedbyτ.Allthepatternsineach“ball”arethenfusedtogethertogenerateasetofsuperpatterns.Thesesuperpatternsformanewpool.IfthepoolcontainsmorethanKpatterns,thenextiterationbeginswiththispoolforthenewroundofrandomdrawing.Asthesupportsetofeverysuperpatternshrinkswitheachnewiteration,theiterationprocessterminates.NotethatPattern-Fusionmergessmallsubpatternsofalargepatterninsteadofincrementally-expandingpatternswithsingleitems.Thisgivesthemethodanadvantagetocircumventmidsizepatternsandprogressonapathleadingtoapotentialcolossalpattern.TheideaisillustratedinFigure7.10.Eachpointshowninthemetricspace
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Context: δ≤1)measuresthetightnessofacluster.ApatternPisδ-coveredbyanotherpatternP(cid:48)ifO(P)⊆O(P(cid:48))andPatDist(P,P(cid:48))≤δ.Asetofpatternsformaδ-clusterifthereexistsarepresentativepatternPrsuchthatforeachpatternPintheset,Pisδ-coveredbyPr.
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Context: llyindependentbutoftenclusteredinsmallregions.Thisissomewhatlikefind-ing20populationcentersintheworld,whichmayresultincitiesclusteredinasmallnumberofcountriesratherthanevenlydistributedacrosstheglobe.Instead,mostuserswouldprefertoderivethekmostinterestingpatterns,whicharenotonlysig-nificant,butalsomutuallyindependentandcontaininglittleredundancy.Asmallsetof
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Context: ttributebecauseAdaandCathybothpurchasedP1.Clusteringhigh-dimensionaldataisthesearchforclustersandthespaceinwhichtheyexist.Thus,therearetwomajorkindsofmethods:Subspaceclusteringapproachessearchforclustersexistinginsubspacesofthegivenhigh-dimensionaldataspace,whereasubspaceisdefinedusingasubsetofattributesinthefullspace.SubspaceclusteringapproachesarediscussedinSection11.2.2.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page510#14510Chapter11AdvancedClusterAnalysisDimensionalityreductionapproachestrytoconstructamuchlower-dimensionalspaceandsearchforclustersinsuchaspace.Often,amethodmayconstructnewdimensionsbycombiningsomedimensionsfromtheoriginaldata.DimensionalityreductionmethodsarethetopicofSection11.2.4.Ingeneral,clusteringhigh-dimensionaldataraisesseveralnewchallengesinadditiontothoseofconventionalclustering:Amajorissueishowtocreateappropriatemodelsforclustersinhigh-dimensionaldata.Unlikeconventionalclustersinlow-dimensionalspaces,clustershiddeninhigh-dimensionaldataareoftensignificantlysmaller.Forexample,whenclusteringcustomer-purchasedata,wewouldnotexpectmanyuserstohavesimilarpurchasepatterns.Searchingforsuchsmallbutmeaningfulclustersislikefindingneedlesinahaystack.Asshownbefore,theconventionaldistancemeasurescanbeineffective.Instead,weoftenhavetoconsidervariousmoresophisticatedtechniquesthatcanmodelcorrelationsandconsistencyamongobjectsinsubspaces.Therearetypicallyanexponentialnumberofpossiblesubspacesordimensionalityreductionoptions,andthustheoptimalsolutionsareoftencomputationallypro-hibitive.Forexample,iftheoriginaldataspacehas1000dimensions,andwewanttofindclustersofdimensionality10,thenthereare(cid:32)100010(cid:33)=2.63×1023possiblesubspaces.11.2.2SubspaceClusteringMethods“Howcanwefindsubspaceclustersfromhigh-dimensionaldata?”Manymethodshavebeenproposed.Theygenerallycanbecategorizedintothreemajorgroups:subspacesearchmethods,correlation-basedclusteringmethods,andbiclusteringmethods.SubspaceSearchMethodsAsubspacesearchmethodsearchesvarioussubspacesforclusters.Here,aclusterisasubsetofobjectsthataresimilartoeachotherinasubspace.Thesimilarityisoftencap-turedbyconventionalmeasuressuchasdistanceordensity.Forexample,theCLIQUEalgorithmintroducedinSection10.5.2isasubspaceclusteringmethod.Itenumeratessubspacesandtheclustersinthosesubspacesinadimensionality-increasingorder,andappliesantimonotonicitytoprunesubspacesinwhichnoclustermayexist
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Context: ichnoclustermayexist.Amajorchallengethatsubspacesearchmethodsfaceishowtosearchaseriesofsubspaceseffectivelyandefficiently.Generallytherearetwokindsofstrategies:Bottom-upapproachesstartfromlow-dimensionalsubspacesandsearchhigher-dimensionalsubspacesonlywhentheremaybeclustersinthosehigher-dimensional
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page511#1511.2ClusteringHigh-DimensionalData511subspaces.Variouspruningtechniquesareexploredtoreducethenumberofhigher-dimensionalsubspacesthatneedtobesearched.CLIQUEisanexampleofabottom-upapproach.Top-downapproachesstartfromthefullspaceandsearchsmallerandsmallersub-spacesrecursively.Top-downapproachesareeffectiveonlyifthelocalityassumptionholds,whichrequirethatthesubspaceofaclustercanbedeterminedbythelocalneighborhood.Example11.10PROCLUS,atop-downsubspaceapproach.PROCLUSisak-medoid-likemethodthatfirstgenerateskpotentialclustercentersforahigh-dimensionaldatasetusingasampleofthedataset.Itthenrefinesthesubspaceclustersiteratively.Ineachitera-tion,foreachofthecurrentk-medoids,PROCLUSconsidersthelocalneighborhoodofthemedoidinthewholedataset,andidentifiesasubspacefortheclusterbymini-mizingthestandarddeviationofthedistancesofthepointsintheneighborhoodtothemedoidoneachdimension.Onceallthesubspacesforthemedoidsaredeter-mined,eachpointinthedatasetisassignedtotheclosestmedoidaccordingtothecorrespondingsubspace.Clustersandpossibleoutliersareidentified.Inthenextiter-ation,newmedoidsreplaceexistingonesifdoingsoimprovestheclusteringquality.Correlation-BasedClusteringMethodsWhilesubspacesearchmethodssearchforclusterswithasimilaritythatismeasuredusingconventionalmetricslikedistanceordensity,correlation-basedapproachescanfurtherdiscoverclustersthataredefinedbyadvancedcorrelationmodels.Example11.11Acorrelation-basedapproachusingPCA.Asanexample,aPCA-basedapproachfirstappliesPCA(PrincipalComponentsAnalysis;seeChapter3)toderiveasetofnew,uncorrelateddimensions,andthenmineclustersinthenewspaceoritssubspaces.InadditiontoPCA,otherspacetransformationsmaybeused,suchastheHoughtransformorfractaldimensions.Foradditionaldetailsonsubspacesearchmethodsandcorrelation-basedclusteringmethods,pleaserefertothebibliographicnotes(Section11.7).BiclusteringMethodsInsomeapplications,wewanttoclusterbothobjectsandattributessimultaneously.Theresultingclustersareknownas
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Context: ngclustersareknownasbiclustersandmeetfourrequirements:(1)onlyasmallsetofobjectsparticipateinacluster;(2)aclusteronlyinvolvesasmallnumberofattributes;(3)anobjectmayparticipateinmultipleclusters,ordoesnotparticipateinanycluster;and(4)anattributemaybeinvolvedinmultipleclusters,orisnotinvolvedinanycluster.Section11.2.3discussesbiclusteringindetail.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page512#16512Chapter11AdvancedClusterAnalysis11.2.3BiclusteringIntheclusteranalysisdiscussedsofar,weclusterobjectsaccordingtotheirattributevalues.Objectsandattributesarenottreatedinthesameway.However,insomeapplica-tions,objectsandattributesaredefinedinasymmetricway,wheredataanalysisinvolvessearchingdatamatricesforsubmatricesthatshowuniquepatternsasclusters.Thiskindofclusteringtechniquebelongstothecategoryofbiclustering.Thissectionfirstintroducestwomotivatingapplicationexamplesofbiclustering—geneexpressionandrecommendersystems.Youwillthenlearnaboutthedifferenttypesofbiclusters.Last,wepresentbiclusteringmethods.ApplicationExamplesBiclusteringtechniqueswerefirstproposedtoaddresstheneedsforanalyzinggeneexpressiondata.Ageneisaunitofthepassing-onoftraitsfromalivingorganismtoitsoffspring.Typically,ageneresidesonasegmentofDNA.GenesarecriticalforalllivingthingsbecausetheyspecifyallproteinsandfunctionalRNAchains.Theyholdtheinformationtobuildandmaintainalivingorganism’scellsandpassgenetictraitstooffspring.Synthesisofafunctionalgeneproduct,eitherRNAorprotein,reliesontheprocessofgeneexpression.Agenotypeisthegeneticmakeupofacell,anorganism,oranindividual.Phenotypesareobservablecharacteristicsofanorganism.Geneexpressionisthemostfundamentallevelingeneticsinthatgenotypescausephenotypes.UsingDNAchips(alsoknownasDNAmicroarrays)andotherbiologicalengineer-ingtechniques,wecanmeasuretheexpressionlevelofalargenumber(possiblyall)ofanorganism’sgenes,inanumberofdifferentexperimentalconditions.Suchconditionsmaycorrespondtodifferenttimepointsinanexperimentorsamplesfromdifferentorgans.Roughlyspeaking,thegeneexpressiondataorDNAmicroarraydataareconcep-tuallyagene-sample/conditionmatrix,whereeachrowcorrespondstoonegene,andeachcolumncorrespondstoonesampleorcondition.Eachelementinthematrixisarealnumberandrecordstheexpressionlevelofageneunderaspecificcondition.Figure11.3showsanillustration.Fromtheclusteringviewpoint,aninterestingissueisthatageneexpressiondat
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Context: atageneexpressiondatamatrixcanbeanalyzedintwodimensions—thegenedimensionandthesample/conditiondimension.Whenanalyzinginthegenedimension,wetreateachgeneasanobjectandtreatthesamples/conditionsasattributes.Bymininginthegenedimension,wemayfindpat-ternssharedbymultiplegenes,orclustergenesintogroups.Forexample,wemayfindagroupofgenesthatexpressthemselvessimilarly,whichishighlyinterestinginbioinformatics,suchasinfindingpathways.Whenanalyzinginthesample/conditiondimension,wetreateachsample/conditionasanobjectandtreatthegenesasattributes.Inthisway,wemayfindpatternsofsamples/conditions,orclustersamples/conditionsintogroups.Forexample,wemayfindthedifferencesingeneexpressionbycomparingagroupoftumorsamplesandnontumorsamples.
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Context: # 11.2 Clustering High-Dimensional Data

![Figure 11.3 Microarray data matrix.](path/to/image)

## Example 11.12 Gene expression

Gene expression matrices are popular in bioinformatics research and development. For example, an important task is to classify a new gene using the expression data of the gene and that of other genes in known classes. Symmetrically, we may classify a new sample (e.g., a new patient) using the expression data of the sample and that of samples in known classes (e.g., tumor and non-tumor). Such tasks are invaluable in understanding the mechanisms of diseases and in clinical treatment.

As can be seen, many gene expression data mining problems are highly related to cluster analysis. However, a challenge here is that, instead of clustering in one dimension (e.g., gene or sample/condition), in many cases we need to cluster in two dimensions simultaneously (e.g., both gene and sample/condition). Moreover, unlike the clustering models we have discussed so far, a cluster in a gene expression data matrix is a *submatrix* and usually has the following characteristics:

- Only a small set of genes participate in the cluster.
- The cluster involves only a small subset of samples/conditions.
- A gene may participate in multiple clusters, or may not participate in any cluster.
- A sample/condition may be involved in multiple clusters, or may not be involved in any cluster.

To find clusters in gene-sample/condition matrices, we need new clustering techniques that meet the following requirements for biclustering:

- A cluster of genes is defined using only a subset of samples/conditions.
- A cluster of samples/conditions is defined using only a subset of genes.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page514#18514Chapter11AdvancedClusterAnalysisTheclustersareneitherexclusive(e.g.,whereonegenecanparticipateinmultipleclusters)norexhaustive(e.g.,whereagenemaynotparticipateinanycluster).Biclusteringisusefulnotonlyinbioinformatics,butalsoinotherapplicationsaswell.Considerrecommendersystemsasanexample.Example11.13Usingbiclusteringforarecommendersystem.AllElectronicscollectsdatafromcus-tomers’evaluationsofproductsandusesthedatatorecommendproductstocustomers.Thedatacanbemodeledasacustomer-productmatrix,whereeachrowrepresentsacustomer,andeachcolumnrepresentsaproduct.Eachelementinthematrixrepresentsacustomer’sevaluationofaproduct,whichmaybeascore(e.g.,like,likesomewhat,notlike)orpurchasebehavior(e.g.,buyornot).Figure11.4illustratesthestructure.Thecustomer-productmatrixcanbeanalyzedintwodimensions:thecustomerdimensionandtheproductdimension.Treatingeachcustomerasanobjectandproductsasattributes,AllElectronicscanfindcustomergroupsthathavesimilarpreferencesorpurchasepatterns.Usingproductsasobjectsandcustomersasattributes,AllElectronicscanmineproductgroupsthataresimilarincustomerinterest.Moreover,AllElectronicscanmineclustersinbothcustomersandproductssimulta-neously.Suchaclustercontainsasubsetofcustomersandinvolvesasubsetofproducts.Forexample,AllElectronicsishighlyinterestedinfindingagroupofcustomerswhoalllikethesamegroupofproducts.Suchaclusterisasubmatrixinthecustomer-productmatrix,whereallelementshaveahighvalue.Usingsuchacluster,AllElectronicscanmakerecommendationsintwodirections.First,thecompanycanrecommendproductstonewcustomerswhoaresimilartothecustomersinthecluster.Second,thecompanycanrecommendtocustomersnewproductsthataresimilartothoseinvolvedinthecluster.Aswithbiclustersinageneexpressiondatamatrix,thebiclustersinacustomer-productmatrixusuallyhavethefollowingcharacteristics:Onlyasmallsetofcustomersparticipateinacluster.Aclusterinvolvesonlyasmallsubsetofproducts.Acustomercanparticipateinmultipleclusters,ormaynotparticipateinanycl
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Context: # 11.2 Clustering High-Dimensional Data

A product may be involved in multiple clusters, or may not be involved in any cluster.

Biclustering can be applied to customer-product matrices to mine clusters satisfying these requirements.

## Types of Biclusters

**“How can we model biclusters and mine them?”** Let's start with some basic notation. For the sake of simplicity, we will use “genes” and “conditions” to refer to the two dimensions in our discussion. Our discussion can easily be extended to other applications. For example, we can simply replace “genes” and “conditions” by “customers” and “products” to tackle the customer-product biclustering problem. 

Let \( A = \{a_1, a_2, \ldots, a_m\} \) be a set of genes and \( B = \{b_1, b_2, \ldots, b_n\} \) be a set of conditions. Let 

\[ 
E = \{e_{ij}\} 
\]

be a gene expression data matrix, that is, a gene-condition matrix, where \( 1 \leq i \leq m \) and \( 1 \leq j \leq n \). A submatrix \( J \times I \) is defined by a subset \( I \subset A \) of genes and a subset \( J \subset B \) of conditions. For example, in the matrix shown in Figure 11.5, \( (a_1, a_3, a5) \) and \( \{b_1, b_2, b_3, b_4\} \) is a submatrix.

A bicluster is a submatrix where genes and conditions follow consistent patterns. We can define different types of biclusters based on such patterns.

- At the simplest case, a submatrix \( J \times I \) of \( E (J \subset A, I \subset B) \) is a **bicluster with constant values** if for any \( i \in I \) and \( j \in J \), \( e_{ij} = c \), where \( c \) is a constant. For example, the submatrix \( (a_4,a_5,a_3) \) and \( \{b_2,b_3,b_4,b_5\} \) in Figure 11.5 is a bicluster with constant values.

- A bicluster is interesting if each row has a constant value, though different rows may have different values. A bicluster with constant values on rows is a submatrix \( J \times I \) such that for any \( i \in I \) and \( j \in J \), \( e_{ij} = c + \beta_j \), where \( \beta_j \) is the adjustment for row \( j \). For example, Figure 11.6 shows a bicluster with constant values on rows.

- Symmetrically, a bicluster with constant values on columns is a submatrix \( J \times I \) such that for any \( i \in I \) and \( j \in J \), \( e_{ij} = c + \beta_i \), where \( \beta_i \) is the adjustment for column \( i \).

|      | b1   | b2   | b3   | ... | bg   |
|------|------|------|------|-----|------|
| a1   | ...  | ...  | ...  | ... | ...  |
| a2   | ...  | ...  | ...  | ... | ...  |
| a3   | ...  | ...  | ...  | ... | ...  |
| ...  | ...  | ...  | ...  | ... | ...  |
| am   | ...  | ...  | ...  | ... | ...  |

Figure 11.5 Gene-condition matrix, a submatrix, and a bicluster.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page516#20516Chapter11AdvancedClusterAnalysisMoregenerally,abiclusterisinterestingiftherowschangeinasynchronizedwaywithrespecttothecolumnsandviceversa.Mathematically,abiclusterwithcoherentvalues(alsoknownasapattern-basedcluster)isasubmatrixI×Jsuchthatforanyi∈Iandj∈J,eij=c+αi+βj,whereαiandβjaretheadjustmentforrowiandcolumnj,respectively.Forexample,Figure11.7showsabiclusterwithcoherentvalues.ItcanbeshownthatI×Jisabiclusterwithcoherentvaluesifandonlyifforanyi1,i2∈Iandj1,j2∈J,thenei1j1−ei2j1=ei1j2−ei2j2.Moreover,insteadofusingaddition,wecandefineabiclusterwithcoherentvaluesusingmultiplication,thatis,eij=c·(cid:0)αi·βj(cid:1).Clearly,biclusterswithconstantvaluesonrowsorcolumnsarespecialcasesofbiclusterswithcoherentvalues.Insomeapplications,wemayonlybeinterestedintheup-ordown-regulatedchangesacrossgenesorconditionswithoutconstrainingtheexactvalues.Abiclus-terwithcoherentevolutionsonrowsisasubmatrixI×Jsuchthatforanyi1,i2∈Iandj1,j2∈J,(ei1j1−ei1j2)(ei2j1−ei2j2)≥0.Forexample,Figure11.8showsabiclus-terwithcoherentevolutionsonrows.Symmetrically,wecandefinebiclusterswithcoherentevolutionsoncolumns.Next,westudyhowtominebiclusters.10101010102020202020505050505000000Figure11.6Biclusterwithconstantvaluesonrows.1050307020206040803050907011060040206010Figure11.7Biclusterwithcoherentvalues.105030702020100501000305010090120800802010010Figure11.8Biclusterwithcoherentevolutionsonrows.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page517#2111.2ClusteringHigh-DimensionalData517BiclusteringMethodsThepreviousspecificationofthetypesofbiclustersonlyconsidersidealcases.Inrealdatasets,suchperfectbiclustersrarelyexist.Whentheydoexist,theyareusuallyverysmall.Instead,randomnoisecanaffectthereadingsofeijandthuspreventabiclusterinnaturefromappearinginaperfectshape.Therearetwomajortypesofmethodsfordiscoveringbiclustersindatathatmaycomewithnoise.Optimization-basedmethodsconductaniterativesearch.Ateachiteration,thesubmatrixwiththehighestsignificancescoreisidentifiedasabicluster.Theprocessterminateswhenauser-specifiedconditionismet.Duetocostconcernsincomputation,greedysearchisoftenemployedtofindlocaloptimalbiclusters.Enu-merationmethodsuseatolerancethresholdtospecifythedegreeofnoiseallowedinthebiclusterstobemined,andthentriestoenumerateallsubmatricesofbiclustersthatsatisfytherequirements.Weusetheδ-ClusterandMaPlealgorithmsasexamplestoillustratetheseideas.OptimizationUsingtheδ-ClusterAlgorithmForasubmatrix,I×J,themeanoftheithrowiseiJ=1|J|(cid:88)j∈Jeij.(11.16)Symmetrically,themeanofthejthcolumniseIj=1|I|(cid:88)i∈Ieij.(11.17)ThemeanofallelementsinthesubmatrixiseIJ=1|I||J|(cid:88)i∈I,j∈Jeij=1|I|(cid:88)i∈IeiJ=1|J|(cid:88)j∈JeIj.(11.18)Thequalityofthesubmatrixasabiclustercanbemeasuredbythemean-squaredresiduevalueasH(I×J)=1|I||J|(cid:88)i∈I,j∈J(eij−eiJ−eIj+eIJ)2.(11.19)SubmatrixI×Jisaδ-biclusterifH(I×J)≤δ,whereδ≥0isathreshold.Whenδ=0,I×Jisaperfectbiclusterwithcoherentvalues.Bysettingδ>0,ausercanspecifythetoleranceofaveragenoiseperelementagainstaperfectbicluster,becauseinEq.(11.19)theresidueoneachelementisresidue(eij)=eij−eiJ−eIj+eIJ.(11.20)Amaximalδ-biclusterisaδ-biclusterI×Jsuchthattheredoesnotexistanotherδ-biclusterI(cid:48)×J(cid:48),andI⊆I(cid:48),J⊆J(cid:48),andatleastoneinequalityholds.Findingthe
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page518#22518Chapter11AdvancedClusterAnalysismaximalδ-biclusterofthelargestsizeiscomputationallycostly.Therefore,wecanuseaheuristicgreedysearchmethodtoobtainalocaloptimalcluster.Thealgorithmworksintwophases.Inthedeletionphase,westartfromthewholematrix.Whilethemean-squaredresidueofthematrixisoverδ,weiterativelyremoverowsandcolumns.Ateachiteration,foreachrowi,wecomputethemean-squaredresidueasd(i)=1|J|(cid:88)j∈J(eij−eiJ−eIj+eIJ)2.(11.21)Moreover,foreachcolumnj,wecomputethemean-squaredresidueasd(j)=1|I|(cid:88)i∈I(eij−eiJ−eIj+eIJ)2.(11.22)Weremovetheroworcolumnofthelargestmean-squaredresidue.Attheendofthisphase,weobtainasubmatrixI×Jthatisaδ-bicluster.However,thesubmatrixmaynotbemaximal.Intheadditionphase,weiterativelyexpandtheδ-biclusterI×Jobtainedinthedele-tionphaseaslongastheδ-biclusterrequirementismaintained.Ateachiteration,weconsiderrowsandcolumnsthatarenotinvolvedinthecurrentbiclusterI×Jbycal-culatingtheirmean-squaredresidues.Aroworcolumnofthesmallestmean-squaredresidueisaddedintothecurrentδ-bicluster.Thisgreedyalgorithmcanfindoneδ-biclusteronly.Tofindmultiplebiclustersthatdonothaveheavyoverlaps,wecanrunthealgorithmmultipletimes.Aftereachexecu-tionwhereaδ-biclusterisoutput,wecanreplacetheelementsintheoutputbiclusterbyrandomnumbers.Althoughthegreedyalgorithmmayfindneithertheoptimalbiclustersnorallbiclusters,itisveryfastevenonlargematrices.EnumeratingAllBiclustersUsingMaPleAsmentioned,asubmatrixI×Jisabiclusterwithcoherentvaluesifandonlyifforanyi1,i2∈Iandj1,j2∈J,ei1j1−ei2j1=ei1j2−ei2j2.Forany2×2submatrixofI×J,wecandefineap-scoreasp-score(cid:32)ei1j1ei1j2ei2j1ei2j2(cid:33)=|(ei1j1−ei2j1)−(ei1j2−ei2j2)|.(11.23)AsubmatrixI×Jisaδ-pCluster(forpattern-basedcluster)ifthep-scoreofevery2×2submatrixofI×Jisatmostδ,whereδ≥0isathresholdspecifyingauser’stoleranceofnoiseagainstaperfectbicluster.Here,thep-scorecontrolsthenoiseoneveryelementinabicluster,whilethemean-squaredresiduecapturestheaveragenoise.Aninterestingpropertyofδ-pClust
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Context: ngpropertyofδ-pClusteristhatifI×Jisaδ-pCluster,theneveryx×y(x,y≥2)submatrixofI×Jisalsoaδ-pCluster.Thismonotonicityenables
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page519#2311.2ClusteringHigh-DimensionalData519ustoobtainasuccinctrepresentationofnonredundantδ-pClusters.Aδ-pClusterismaximalifnomorerowsorcolumnscanbeaddedintotheclusterwhilemaintainingtheδ-pClusterproperty.Toavoidredundancy,insteadoffindingallδ-pClusters,weonlyneedtocomputeallmaximalδ-pClusters.MaPleisanalgorithmthatenumeratesallmaximalδ-pClusters.Itsystematicallyenumerateseverycombinationofconditionsusingasetenumerationtreeandadepth-firstsearch.Thisenumerationframeworkisthesameasthepattern-growthmethodsforfrequentpatternmining(Chapter6).Considergeneexpressiondata.Foreachcon-ditioncombination,J,MaPlefindsthemaximalsubsetsofgenes,I,suchthatI×Jisaδ-pCluster.IfI×Jisnotasubmatrixofanotherδ-pCluster,thenI×Jisamaximalδ-pCluster.Theremaybeahugenumberofconditioncombinations.MaPleprunesmanyunfruitfulcombinationsusingthemonotonicityofδ-pClusters.Foraconditioncom-bination,J,iftheredoesnotexistasetofgenes,I,suchthatI×Jisaδ-pCluster,thenwedonotneedtoconsideranysupersetofJ.Moreover,weshouldconsiderI×Jasacandidateofaδ-pClusteronlyifforevery(|J|−1)-subsetJ(cid:48)ofJ,I×J(cid:48)isaδ-pCluster.MaPlealsoemploysseveralpruningtechniquestospeedupthesearchwhileretainingthecompletenessofreturningallmaximalδ-pClusters.Forexample,whenexaminingacurrentδ-pCluster,I×J,MaPlecollectsallthegenesandconditionsthatmaybeaddedtoexpandthecluster.IfthesecandidategenesandconditionstogetherwithIandJformasubmatrixofaδ-pClusterthathasalreadybeenfound,thenthesearchofI×JandanysupersetofJcanbepruned.InterestedreadersmayrefertothebibliographicnotesforadditionalinformationontheMaPlealgorithm(Section11.7).Aninterestingobservationhereisthatthesearchformaximalδ-pClustersinMaPleissomewhatsimilartominingfrequentcloseditemsets.Consequently,MaPleborrowsthedepth-firstsearchframeworkandideasfromthepruningtechniquesofpattern-growthmethodsforfrequentpatternmining.Thisisanexamplewherefrequentpatternminingandclusteranalysismaysharesimilartechniquesandideas.AnadvantageofMaPleandth
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Context: dvantageofMaPleandtheotheralgorithmsthatenumerateallbiclustersisthattheyguaranteethecompletenessoftheresultsanddonotmissanyoverlappingbiclus-ters.However,achallengeforsuchenumerationalgorithmsisthattheymaybecomeverytimeconsumingifamatrixbecomesverylarge,suchasacustomer-purchasematrixofhundredsofthousandsofcustomersandmillionsofproducts.11.2.4DimensionalityReductionMethodsandSpectralClusteringSubspaceclusteringmethodstrytofindclustersinsubspacesoftheoriginaldataspace.Insomesituations,itismoreeffectivetoconstructanewspaceinsteadofusingsubspacesoftheoriginaldata.Thisisthemotivationbehinddimensionalityreductionmethodsforclusteringhigh-dimensionaldata.Example11.14Clusteringinaderivedspace.ConsiderthethreeclustersofpointsinFigure11.9.Itisnotpossibletoclusterthesepointsinanysubspaceoftheoriginalspace,X×Y,because
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Context: # Chapter 11 Advanced Cluster Analysis

![Figure 11.9](image_url) Clustering in a derived space may be more effective.

all three clusters would end up being projected onto overlapping areas in the x and y axes. What if, instead, we construct a new dimension, \(-\frac{x^2}{2} + \frac{y^2}{2}\) (shown as a dashed line in the figure)? By projecting the points onto this new dimension, the three clusters become apparent.

Although Example 11.14 involves only two dimensions, the idea of constructing a new space (so that any clustering structure that is hidden in the data becomes well manifested) can be extended to high-dimensional data. Preferably, the newly constructed space should have low dimensionality.

There are many dimensionality reduction methods. A straightforward approach is to apply feature selection and extraction methods to the data set as those discussed in Chapter 3. However, such methods may not be able to detect the clustering structure. Therefore, methods that combine feature extraction and clustering are preferred. In this section, we introduce **spectral clustering**, a group of methods that are effective in high-dimensional data applications.

## Figure 11.10

shows the general framework for spectral clustering approaches. The Ng-Jordan-Weiss algorithm is a spectral clustering method. Let’s have a look at each step of the framework. In doing so, we also note special conditions that apply to the Ng-Jordan-Weiss algorithm as an example.

Given a set of objects \(o_1, o_2, \ldots, o_n\), the distance between each pair of objects, \(dist(o_i, o_j)\) \((1 \leq i < j \leq n)\), and the desired number \(k\) of clusters, a spectral clustering approach works as follows:

1. Using the distance measure, calculate an affinity matrix, \(W\), such that

   \[
   W_{ij} = e^{-\frac{(dist(o_i, o_j))^2}{\sigma^2}},
   \]

   where \(\sigma\) is a scaling parameter that controls how fast the affinity \(W_{ij}\) decreases as \(dist(o_i, o_j)\) increases. In the Ng-Jordan-Weiss algorithm, \(W_{ij}\) is set to 0.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page521#2511.2ClusteringHigh-DimensionalData521[wij]Affinity matrixDataA=f(w)Av=λvClustering inthe new spaceCompute leadingk eigenvectorsof AProject backto cluster theoriginal dataFigure11.10Theframeworkofspectralclusteringapproaches.Source:AdaptedfromSlide8athttp://videolectures.net/micued08azranmcl/.2.UsingtheaffinitymatrixW,deriveamatrixA=f(W).Thewayinwhichthisisdonecanvary.TheNg-Jordan-Weissalgorithmdefinesamatrix,D,asadiagonalmatrixsuchthatDiiisthesumoftheithrowofW,thatis,Dii=n(cid:88)j=1Wij.(11.24)AisthensettoA=D−12WD−12.(11.25)3.FindthekleadingeigenvectorsofA.Recallthattheeigenvectorsofasquarematrixarethenonzerovectorsthatremainproportionaltotheoriginalvectorafterbeingmultipliedbythematrix.Mathematically,avectorvisaneigenvectorofmatrixAifAv=λv,whereλiscalledthecorrespondingeigenvalue.ThisstepderivesknewdimensionsfromA,whicharebasedontheaffinitymatrixW.Typically,kshouldbemuchsmallerthanthedimensionalityoftheoriginaldata.TheNg-Jordan-Weissalgorithmcomputesthekeigenvectorswiththelargesteigenvaluesx1,...,xkofA.4.Usingthekleadingeigenvectors,projecttheoriginaldataintothenewspacedefinedbythekleadingeigenvectors,andrunaclusteringalgorithmsuchask-meanstofindkclusters.TheNg-Jordan-WeissalgorithmstackstheklargesteigenvectorsincolumnstoformamatrixX=[x1x2···xk]∈Rn×k.ThealgorithmformsamatrixYbyrenormalizingeachrowinXtohaveunitlength,thatis,Yij=Xij(cid:113)(cid:80)kj=1X2ij.(11.26)ThealgorithmthentreatseachrowinYasapointinthek-dimensionalspaceRk,andrunsk-means(oranyotheralgorithmservingthepartitioningpurpose)toclusterthepointsintokclusters.
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Context: # 522
## Chapter 11 Advanced Cluster Analysis

![Figure 1.1.11](http://videlectures.net/micub08.azrn.mdl)  
The new dimensions and the clustering results of the Ng-Jordan-Weiss algorithm. Source: Adapted from Slide 9 at 

5. Assign the original data points to clusters according to how the transformed points are assigned in the clusters obtained in step 4.  
   In the Ng-Jordan-Weiss algorithm, the original object \( y_i \) is assigned to the jth cluster if and only if matrix \( Y \)'s row is assigned to the jth cluster as a result of step 4.

In spectral clustering methods, the dimensionality of the new space is set to the desired number of clusters. This setting expects that each new dimension should be able to manifest a cluster.

### Example 1.1.15
**The Ng-Jordan-Weiss algorithm.** Consider the set of points in Figure 1.1.11. The data set, the affinity matrix, the three largest eigenvectors, and the normalized vectors are shown. Note that with the three new dimensions (formed by the three largest eigenvectors), the clusters are easily detected.

Spectral clustering is effective in high-dimensional applications such as image processing. Theoretically, it works well when certain conditions apply. Scalability, however, is a challenge. Computing eigenvectors on a large matrix is costly. Spectral clustering can be combined with other clustering methods, such as clustering. Additional information on other dimensionality reduction clustering methods, such as kernel PCA, can be found in the bibliographic notes (Section 11.7).

## 11.3 Clustering Graph and Network Data
Cluster analysis on graph and network data extracts valuable knowledge and information. Such data are increasingly popular in many applications. We discuss applications and challenges of clustering graph and network data in Section 11.3.1. Similarity measures for this form of clustering are given in Section 11.3.2. You will learn about graph clustering methods in Section 11.3.3.

In general, the terms graph and network can be used interchangeably. In the rest of this section, we mainly use the term **graph**.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page523#2711.3ClusteringGraphandNetworkData52311.3.1ApplicationsandChallengesAsacustomerrelationshipmanageratAllElectronics,younoticethatalotofdatarelatingtocustomersandtheirpurchasebehaviorcanbepreferablymodeledusinggraphs.Example11.16Bipartitegraph.ThecustomerpurchasebehavioratAllElectronicscanberepresentedinabipartitegraph.Inabipartitegraph,verticescanbedividedintotwodisjointsetssothateachedgeconnectsavertexinonesettoavertexintheotherset.FortheAllElectronicscustomerpurchasedata,onesetofverticesrepresentscustomers,withonecustomerpervertex.Theothersetrepresentsproducts,withoneproductpervertex.Anedgeconnectsacustomertoaproduct,representingthepurchaseoftheproductbythecustomer.Figure11.12showsanillustration.“Whatkindofknowledgecanweobtainbyaclusteranalysisofthecustomer-productbipartitegraph?”Byclusteringthecustomerssuchthatthosecustomersbuyingsimilarsetsofproductsareplacedintoonegroup,acustomerrelationshipmanagercanmakeproductrecommendations.Forexample,supposeAdabelongstoacustomerclusterinwhichmostofthecustomerspurchasedadigitalcamerainthelast12months,butAdahasyettopurchaseone.Asmanager,youdecidetorecommendadigitalcameratoher.Alternatively,wecanclusterproductssuchthatthoseproductspurchasedbysimilarsetsofcustomersaregroupedtogether.Thisclusteringinformationcanalsobeusedforproductrecommendations.Forexample,ifadigitalcameraandahigh-speedflashmemorycardbelongtothesameproductcluster,thenwhenacustomerpurchasesadigitalcamera,wecanrecommendthehigh-speedflashmemorycard.Bipartitegraphsarewidelyusedinmanyapplications.Consideranotherexample.Example11.17Websearchengines.Inwebsearchengines,searchlogsarearchivedtorecorduserqueriesandthecorrespondingclick-throughinformation.(Theclick-throughinforma-tiontellsusonwhichpages,givenasaresultofasearch,theuserclicked.)Thequeryandclick-throughinformationcanberepresentedusingabipartitegraph,wherethetwosetsCustomersProductsFigure11.12Bipartitegraphrepresentingcustomer-purchasedata.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page524#28524Chapter11AdvancedClusterAnalysisofverticescorrespondtoqueriesandwebpages,respectively.Anedgelinksaquerytoawebpageifauserclicksthewebpagewhenaskingthequery.Valuableinformationcanbeobtainedbyclusteranalysesonthequery–webpagebipartitegraph.Forinstance,wemayidentifyqueriesposedindifferentlanguages,butthatmeanthesamething,iftheclick-throughinformationforeachqueryissimilar.Asanotherexample,allthewebpagesontheWebformadirectedgraph,alsoknownasthewebgraph,whereeachwebpageisavertex,andeachhyperlinkisanedgepointingfromasourcepagetoadestinationpage.Clusteranalysisonthewebgraphcandisclosecommunities,findhubsandauthoritativewebpages,anddetectwebspams.Inadditiontobipartitegraphs,clusteranalysiscanalsobeappliedtoothertypesofgraphs,includinggeneralgraphs,aselaboratedExample11.18.Example11.18Socialnetwork.Asocialnetworkisasocialstructure.Itcanberepresentedasagraph,wheretheverticesareindividualsororganizations,andthelinksareinterdependenciesbetweenthevertices,representingfriendship,commoninterests,orcollaborativeactivi-ties.AllElectronics’customersformasocialnetwork,whereeachcustomerisavertex,andanedgelinkstwocustomersiftheyknoweachother.Ascustomerrelationshipmanager,youareinterestedinfindingusefulinformationthatcanbederivedfromAllElectronics’socialnetworkthroughclusteranalysis.Youobtainclustersfromthenetwork,wherecustomersinaclusterknoweachotherorhavefriendsincommon.Customerswithinaclustermayinfluenceoneanotherregard-ingpurchasedecisionmaking.Moreover,communicationchannelscanbedesignedtoinformthe“heads”ofclusters(i.e.,the“best”connectedpeopleintheclusters),sothatpromotionalinformationcanbespreadoutquickly.Thus,youmayusecustomerclusteringtopromotesalesatAllElectronics.Asanotherexample,theauthorsofscientificpublicationsformasocialnetwork,wheretheauthorsareverticesandtwoauthorsareconnectedbyanedgeiftheyco-authoredapublication.Thenetworkis,ingeneral,aweightedgraphbecauseanedgebetweentwoauthorscancarryaweightrepresentingthestrength
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Context: resentingthestrengthofthecollaborationsuchashowmanypublicationsthetwoauthors(astheendvertices)coauthored.Clus-teringthecoauthornetworkprovidesinsightastocommunitiesofauthorsandpatternsofcollaboration.“Arethereanychallengesspecifictoclusteranalysisongraphandnetworkdata?”Inmostoftheclusteringmethodsdiscussedsofar,objectsarerepresentedusingasetofattributes.Auniquefeatureofgraphandnetworkdataisthatonlyobjects(asvertices)andrelationshipsbetweenthem(asedges)aregiven.Nodimensionsorattributesareexplicitlydefined.Toconductclusteranalysisongraphandnetworkdata,therearetwomajornewchallenges.“Howcanwemeasurethesimilaritybetweentwoobjectsonagraphaccordingly?”Typically,wecannotuseconventionaldistancemeasures,suchasEuclideandis-tance.Instead,weneedtodevelopnewmeasurestoquantifythesimilarity.Such
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page525#2911.3ClusteringGraphandNetworkData525measuresoftenarenotmetric,andthusraisenewchallengesregardingthedevelop-mentofefficientclusteringmethods.SimilaritymeasuresforgraphsarediscussedinSection11.3.2.“Howcanwedesignclusteringmodelsandmethodsthatareeffectiveongraphandnetworkdata?”Graphandnetworkdataareoftencomplicated,carryingtopologicalstructuresthataremoresophisticatedthantraditionalclusteranalysisapplications.Manygraphdatasetsarelarge,suchasthewebgraphcontainingatleasttensofbillionsofwebpagesinthepubliclyindexableWeb.Graphscanalsobesparsewhere,onaverage,avertexisconnectedtoonlyasmallnumberofotherverticesinthegraph.Todiscoveraccurateandusefulknowledgehiddendeepinthedata,agoodclusteringmethodhastoaccommodatethesefactors.ClusteringmethodsforgraphandnetworkdataareintroducedinSection11.3.3.11.3.2SimilarityMeasures“Howcanwemeasurethesimilarityordistancebetweentwoverticesinagraph?”Inourdiscussion,weexaminetwotypesofmeasures:geodesicdistanceanddistancebasedonrandomwalk.GeodesicDistanceAsimplemeasureofthedistancebetweentwoverticesinagraphistheshortestpathbetweenthevertices.Formally,thegeodesicdistancebetweentwoverticesisthelengthintermsofthenumberofedgesoftheshortestpathbetweenthevertices.Fortwoverticesthatarenotconnectedinagraph,thegeodesicdistanceisdefinedasinfinite.Usinggeodesicdistance,wecandefineseveralotherusefulmeasurementsforgraphanalysisandclustering.GivenagraphG=(V,E),whereVisthesetofverticesandEisthesetofedges,wedefinethefollowing:Foravertextv∈V,theeccentricityofv,denotedeccen(v),isthelargestgeodesicdistancebetweenvandanyothervertexu∈V−{v}.Theeccentricityofvcaptureshowfarawayvisfromitsremotestvertexinthegraph.TheradiusofgraphGistheminimumeccentricityofallvertices.Thatis,r=minv∈Veccen(v).(11.27)Theradiuscapturesthedistancebetweenthe“mostcentralpoint”andthe“farthestborder”ofthegraph.ThediameterofgraphGisthemaximumeccentricityofallvertices.Thatis,d=maxv∈Veccen(v).(11.28)Thediameterrepresentsthelargestdistancebetw
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Context: elargestdistancebetweenanypairofvertices.Aperipheralvertexisavertexthatachievesthediameter.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page526#30526Chapter11AdvancedClusterAnalysisabcedFigure11.13Agraph,G,whereverticesc,d,andeareperipheral.Example11.19Measurementsbasedongeodesicdistance.ConsidergraphGinFigure11.13.Theeccentricityofais2,thatis,eccen(a)=2,eccen(b)=2,andeccen(c)=eccen(d)=eccen(e)=3.Thus,theradiusofGis2,andthediameteris3.Notethatitisnotnecessarythatd=2×r.Verticesc,d,andeareperipheralvertices.SimRank:SimilarityBasedonRandomWalkandStructuralContextForsomeapplications,geodesicdistancemaybeinappropriateinmeasuringthesimi-laritybetweenverticesinagraph.HereweintroduceSimRank,asimilaritymeasurebasedonrandomwalkandonthestructuralcontextofthegraph.Inmathematics,arandomwalkisatrajectorythatconsistsoftakingsuccessiverandomsteps.Example11.20Similaritybetweenpeopleinasocialnetwork.Let’sconsidermeasuringthesimilaritybetweentwoverticesintheAllElectronicscustomersocialnetworkofExample11.18.Here,similaritycanbeexplainedastheclosenessbetweentwoparticipantsinthenet-work,thatis,howclosetwopeopleareintermsoftherelationshiprepresentedbythesocialnetwork.“Howwellcanthegeodesicdistancemeasuresimilarityandclosenessinsuchanetwork?”SupposeAdaandBobaretwocustomersinthenetwork,andthenetworkisundirected.Thegeodesicdistance(i.e.,thelengthoftheshortestpathbetweenAdaandBob)istheshortestpaththatamessagecanbepassedfromAdatoBobandviceversa.However,thisinformationisnotusefulforAllElectronics’customerrelationshipmanagementbecausethecompanytypicallydoesnotwanttosendaspecificmessagefromonecustomertoanother.Therefore,geodesicdistancedoesnotsuittheapplication.“Whatdoessimilaritymeaninasocialnetwork?”Weconsidertwowaystodefinesimilarity:Twocustomersareconsideredsimilartooneanotheriftheyhavesimilarneighborsinthesocialnetwork.Thisheuristicisintuitivebecause,inpractice,twopeoplereceivingrecommendationsfromagoodnumberofcommonfriendsoftenmakesimilardecisions.Thiskindofsimilarityisbasedonthelocalstructure(i.e.,theneighborhoods)ofthevertices,andthusiscalledstructuralcontext–basedsimilarity.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page527#3111.3ClusteringGraphandNetworkData527SupposeAllElectronicssendspromotionalinformationtobothAdaandBobinthesocialnetwork.AdaandBobmayrandomlyforwardsuchinformationtotheirfriends(orneighbors)inthenetwork.TheclosenessbetweenAdaandBobcanthenbemeasuredbythelikelihoodthatothercustomerssimultaneouslyreceivethepro-motionalinformationthatwasoriginallysenttoAdaandBob.Thiskindofsimilarityisbasedontherandomwalkreachabilityoverthenetwork,andthusisreferredtoassimilaritybasedonrandomwalk.Let’shaveacloserlookatwhatismeantbysimilaritybasedonstructuralcontext,andsimilaritybasedonrandomwalk.Theintuitionbehindsimilaritybasedonstructuralcontextisthattwoverticesinagrapharesimilariftheyareconnectedtosimilarvertices.Tomeasuresuchsimilarity,weneedtodefinethenotionofindividualneighborhood.InadirectedgraphG=(V,E),whereVisthesetofverticesandE⊆V×Visthesetofedges,foravertexv∈V,theindividualin-neighborhoodofvisdefinedasI(v)={u|(u,v)∈E}.(11.29)Symmetrically,wedefinetheindividualout-neighborhoodofvasO(v)={w|(v,w)∈E}.(11.30)FollowingtheintuitionillustratedinExample11.20,wedefineSimRank,astructural-contextsimilarity,withavaluethatisbetween0and1foranypairofver-tices.Foranyvertex,v∈V,thesimilaritybetweenthevertexanditselfiss(v,v)=1becausetheneighborhoodsareidentical.Forverticesu,v∈Vsuchthatu(cid:54)=v,wecandefines(u,v)=C|I(u)||I(v)|(cid:88)x∈I(u)(cid:88)y∈I(v)s(x,y),(11.31)whereCisaconstantbetween0and1.Avertexmaynothaveanyin-neighbors.Thus,wedefineEq.(11.31)tobe0wheneitherI(u)orI(v)is∅.ParameterCspecifiestherateofdecayassimilarityispropagatedacrossedges.“HowcanwecomputeSimRank?”AstraightforwardmethoditerativelyevaluatesEq.(11.31)untilafixedpointisreached.Letsi(u,v)betheSimRankscorecalculatedattheithround.Tobegin,wesets0(u,v)=(cid:40)0ifu(cid:54)=v1ifu=v.(11.32)WeuseEq.(11.31)tocomputesi+1fromsiassi+1(u,v)=C|I(u)||I(v)|(cid:88)x∈I(u)(cid:88)y∈I(v)si(x,y).(11.33)
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page528#32528Chapter11AdvancedClusterAnalysisItcanbeshownthatlimi→∞si(u,v)=s(u,v).AdditionalmethodsforapproximatingSimRankaregiveninthebibliographicnotes(Section11.7).Now,let’sconsidersimilaritybasedonrandomwalk.Adirectedgraphisstronglyconnectedif,foranytwonodesuandv,thereisapathfromutovandanotherpathfromvtou.Inastronglyconnectedgraph,G=(V,E),foranytwovertices,u,v∈V,wecandefinetheexpecteddistancefromutovasd(u,v)=(cid:88)t:u(cid:32)vP[t]l(t),(11.34)whereu(cid:32)visapathstartingfromuandendingatvthatmaycontaincyclesbutdoesnotreachvuntiltheend.Foratravelingtour,t=w1→w2→···→wk,itslengthisl(t)=k−1.TheprobabilityofthetourisdefinedasP[t]=(cid:40)(cid:81)k−1i=11|O(wi)|ifl(t)>00ifl(t)=0.(11.35)Tomeasuretheprobabilitythatavertexwreceivesamessagethatoriginatedsimulta-neouslyfromuandv,weextendtheexpecteddistancetothenotionofexpectedmeetingdistance,thatis,m(u,v)=(cid:88)t:(u,v)(cid:32)(x,x)P[t]l(t),(11.36)where(u,v)(cid:32)(x,x)isapairoftoursu(cid:32)xandv(cid:32)xofthesamelength.UsingaconstantCbetween0and1,wedefinetheexpectedmeetingprobabilityasp(u,v)=(cid:88)t:(u,v)(cid:32)(x,x)P[t]Cl(t),(11.37)whichisasimilaritymeasurebasedonrandomwalk.Here,theparameterCspecifiestheprobabilityofcontinuingthewalkateachstepofthetrajectory.Ithasbeenshownthats(u,v)=p(u,v)foranytwovertices,uandv.Thatis,SimRankisbasedonbothstructuralcontextandrandomwalk.11.3.3GraphClusteringMethodsLet’sconsiderhowtoconductclusteringonagraph.Wefirstdescribetheintuitionbehindgraphclustering.Wethendiscusstwogeneralcategoriesofgraphclusteringmethods.Tofindclustersinagraph,imaginecuttingthegraphintopieces,eachpiecebeingacluster,suchthattheverticeswithinaclusterarewellconnectedandtheverticesindifferentclustersareconnectedinamuchweakerway.Formally,foragraph,G=(V,E),
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page529#3311.3ClusteringGraphandNetworkData529acut,C=(S,T),isapartitioningofthesetofverticesVinG,thatis,V=S∪TandS∩T=∅.Thecutsetofacutisthesetofedges,{(u,v)∈E|u∈S,v∈T}.Thesizeofthecutisthenumberofedgesinthecutset.Forweightedgraphs,thesizeofacutisthesumoftheweightsoftheedgesinthecutset.“Whatkindsofcutsaregoodforderivingclustersingraphs?”Ingraphtheoryandsomenetworkapplications,aminimumcutisofimportance.Acutisminimumifthecut’ssizeisnotgreaterthananyothercut’ssize.Therearepolynomialtimealgorithmstocomputeminimumcutsofgraphs.Canweusethesealgorithmsingraphclustering?Example11.21Cutsandclusters.ConsidergraphGinFigure11.14.Thegraphhastwoclusters:{a,b,c,d,e,f}and{g,h,i,j,k},andoneoutliervertex,l.ConsidercutC1=({a,b,c,d,e,f,g,h,i,j,k},{l}).Onlyoneedge,namely,(e,l),crossesthetwopartitionscreatedbyC1.Therefore,thecutsetofC1is{(e,l)}andthesizeofC1is1.(Notethatthesizeofanycutinaconnectedgraphcannotbesmallerthan1.)Asaminimumcut,C1doesnotleadtoagoodclusteringbecauseitonlyseparatestheoutliervertex,l,fromtherestofthegraph.CutC2=({a,b,c,d,e,f,l},{g,h,i,j,k})leadstoamuchbetterclusteringthanC1.TheedgesinthecutsetofC2arethoseconnectingthetwo“naturalclusters”inthegraph.Specifically,foredges(d,h)and(e,k)thatareinthecutset,mostoftheedgesconnectingd,h,e,andkbelongtoonecluster.Example11.21indicatesthatusingaminimumcutisunlikelytoleadtoagoodclus-tering.Wearebetteroffchoosingacutwhere,foreachvertexuthatisinvolvedinanedgeinthecutset,mostoftheedgesconnectingtoubelongtoonecluster.Formally,letdeg(u)bethedegreeofu,thatis,thenumberofedgesconnectingtou.ThesparsityofacutC=(S,T)isdefinedas(cid:56)=cutsizemin{|S|,|T|}.(11.38)Sparsest cut C2Minimum cut C1abdefghijklcFigure11.14AgraphGandtwocuts.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page530#34530Chapter11AdvancedClusterAnalysisAcutissparsestifitssparsityisnotgreaterthanthesparsityofanyothercut.Theremaybemorethanonesparsestcut.InExample11.21andFigure11.14,C2isasparsestcut.Usingsparsityastheobjectivefunction,asparsestcuttriestominimizethenumberofedgescrossingthepartitionsandbalancethepartitionsinsize.ConsideraclusteringonagraphG=(V,E)thatpartitionsthegraphintokclusters.ThemodularityofaclusteringassessesthequalityoftheclusteringandisdefinedasQ=k(cid:88)i=1(cid:32)li|E|−(cid:18)di2|E|(cid:19)2(cid:33),(11.39)whereliisthenumberofedgesbetweenverticesintheithcluster,anddiisthesumofthedegreesoftheverticesintheithcluster.Themodularityofaclusteringofagraphisthedifferencebetweenthefractionofalledgesthatfallintoindividualclustersandthefractionthatwoulddosoifthegraphverticeswererandomlyconnected.Theoptimalclusteringofgraphsmaximizesthemodularity.Theoretically,manygraphclusteringproblemscanberegardedasfindinggoodcuts,suchasthesparsestcuts,onthegraph.Inpractice,however,anumberofchallengesexist:Highcomputationalcost:Manygraphcutproblemsarecomputationallyexpen-sive.Thesparsestcutproblem,forexample,isNP-hard.Therefore,findingtheoptimalsolutionsonlargegraphsisoftenimpossible.Agoodtrade-offbetweenefficiency/scalabilityandqualityhastobeachieved.Sophisticatedgraphs:Graphscanbemoresophisticatedthantheonesdescribedhere,involvingweightsand/orcycles.Highdimensionality:Agraphcanhavemanyvertices.Inasimilaritymatrix,avertexisrepresentedasavector(arowinthematrix)withadimensionalitythatisthenumberofverticesinthegraph.Therefore,graphclusteringmethodsmusthandlehighdimensionality.Sparsity:Alargegraphisoftensparse,meaningeachvertexonaverageconnectstoonlyasmallnumberofothervertices.Asimilaritymatrixfromalargesparsegraphcanalsobesparse.Therearetwokindsofmethodsforclusteringgraphdata,whichaddressthesechallenges.Oneusesclusteringmethodsforhigh-dimensionaldata,whiletheotherisdesignedspecificallyforclusteringgraphs.Thefirstgroupofmethodsisba
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Context: stgroupofmethodsisbasedongenericclusteringmethodsforhigh-dimensionaldata.TheyextractasimilaritymatrixfromagraphusingasimilaritymeasuresuchasthosediscussedinSection11.3.2.Agenericclusteringmethodcanthenbeappliedonthesimilaritymatrixtodiscoverclusters.Clusteringmethodsfor
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page531#3511.3ClusteringGraphandNetworkData531high-dimensionaldataaretypicallyemployed.Forexample,inmanyscenarios,onceasimilaritymatrixisobtained,spectralclusteringmethods(Section11.2.4)canbeapplied.Spectralclusteringcanapproximateoptimalgraphcutsolutions.Foradditionalinformation,pleaserefertothebibliographicnotes(Section11.7).Thesecondgroupofmethodsisspecifictographs.Theysearchthegraphtofindwell-connectedcomponentsasclusters.Let’slookatamethodcalledSCAN(StructuralClusteringAlgorithmforNetworks)asanexample.Givenanundirectedgraph,G=(V,E),foravertex,u∈V,theneighborhoodofuis(cid:48)(u)={v|(u,v)∈E}∪{u}.Usingtheideaofstructural-contextsimilarity,SCANmeasuresthesimilaritybetweentwovertices,u,v∈V,bythenormalizedcommonneighborhoodsize,thatis,σ(u,v)=|(cid:48)(u)∩(cid:48)(v)|√|(cid:48)(u)||(cid:48)(v)|.(11.40)Thelargerthevaluecomputed,themoresimilarthetwovertices.SCANusesasimilaritythresholdεtodefinetheclustermembership.Foravertex,u∈V,theε-neighborhoodofuisdefinedasNε(u)={v∈(cid:48)(u)|σ(u,v)≥ε}.Theε-neighborhoodofucontainsallneighborsofuwithastructural-contextsimilaritytouthatisatleastε.InSCAN,acorevertexisavertexinsideofacluster.Thatis,u∈Visacorever-texif|Nε(u)|≥µ,whereµisapopularitythreshold.SCANgrowsclustersfromcorevertices.Ifavertexvisintheε-neighborhoodofacoreu,thenvisassignedtothesameclusterasu.Thisprocessofgrowingclusterscontinuesuntilnoclustercanbefurthergrown.Theprocessissimilartothedensity-basedclusteringmethod,DBSCAN(Chapter10).Formally,avertexvcanbedirectlyreachedfromacoreuifv∈Nε(u).Transitively,avertexvcanbereachedfromacoreuifthereexistverticesw1,...,wnsuchthatw1canbereachedfromu,wicanbereachedfromwi−1for1<i≤n,andvcanbereachedfromwn.Moreover,twovertices,u,v∈V,whichmayormaynotbecores,aresaidtobeconnectedifthereexistsacorewsuchthatbothuandvcanbereachedfromw.Allverticesinaclusterareconnected.Aclusterisamaximumsetofverticessuchthateverypairinthesetisconnected.Someverticesmaynotbelongtoanycluster.Suchavertexui
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Context: luster.Suchavertexuisahubiftheneighbor-hood(cid:48)(u)ofucontainsverticesfrommorethanonecluster.Ifavertexdoesnotbelongtoanycluster,andisnotahub,itisanoutlier.TheSCANalgorithmisshowninFigure11.15.Thesearchframeworkcloselyresem-blesthecluster-findingprocessinDBSCAN.SCANfindsacutofthegraph,whereeachclusterisasetofverticesthatareconnectedbasedonthetransitivesimilarityinastructuralcontext.AnadvantageofSCANisthatitstimecomplexityislinearwithrespecttothenumberofedges.Inverylargeandsparsegraphs,thenumberofedgesisinthesamescaleofthenumberofvertices.Therefore,SCANisexpectedtohavegoodscalabilityonclusteringlargegraphs.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page532#36532Chapter11AdvancedClusterAnalysisAlgorithm:SCANforclustersongraphdata.Input:agraphG=(V,E),asimilaritythresholdε,andapopulationthresholdµOutput:asetofclustersMethod:setallverticesinVunlabeledforallunlabeledvertexudoifuisacorethengenerateanewcluster-idcinsertallv∈Nε(u)intoaqueueQwhileQ(cid:54)=dow←thefirstvertexinQR←thesetofverticesthatcanbedirectlyreachedfromwforalls∈Rdoifsisnotunlabeledorlabeledasnonmemberthenassignthecurrentcluster-idctosendififsisunlabeledtheninsertsintoqueueQendifendforremovewfromQendwhileelselabeluasnonmemberendifendforforallvertexulabelednonmemberdoif∃x,y∈(cid:48)(u):xandyhavedifferentcluster-idsthenlabeluashubelselabeluasoutlierendifendforFigure11.15SCANalgorithmforclusteranalysisongraphdata.11.4ClusteringwithConstraintsUsersoftenhavebackgroundknowledgethattheywanttointegrateintoclusteranalysis.Theremayalsobeapplication-specificrequirements.Suchinformationcanbemod-eledasclusteringconstraints.Weapproachthetopicofclusteringwithconstraintsintwosteps.Section11.4.1categorizesthetypesofconstraintsforclusteringgraphdata.MethodsforclusteringwithconstraintsareintroducedinSection11.4.2.
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Context: HAN18-ch11-497-542-97801238147912011/6/13:24Page533#3711.4ClusteringwithConstraints53311.4.1CategorizationofConstraintsThissectionstudieshowtocategorizetheconstraintsusedinclusteranalysis.Specifi-cally,wecancategorizeconstraintsaccordingtothesubjectsonwhichtheyareset,oronhowstronglytheconstraintsaretobeenforced.AsdiscussedinChapter10,clusteranalysisinvolvesthreeessentialaspects:objectsasinstancesofclusters,clustersasgroupsofobjects,andthesimilarityamongobjects.Therefore,thefirstmethodwediscusscategorizesconstraintsaccordingtowhattheyareappliedto.Wethushavethreetypes:constraintsoninstances,constraintsonclusters,andconstraintsonsimilaritymeasurement.Constraintsoninstances:Aconstraintoninstancesspecifieshowapairorasetofinstancesshouldbegroupedintheclusteranalysis.Twocommontypesofcon-straintsfromthiscategoryinclude:Must-linkconstraints.Ifamust-linkconstraintisspecifiedontwoobjectsxandy,thenxandyshouldbegroupedintooneclusterintheoutputoftheclusteranalysis.Thesemust-linkconstraintsaretransitive.Thatis,ifmust-link(x,y)andmust-link(y,z),thenmust-link(x,z).Cannot-linkconstraints.Cannot-linkconstraintsaretheoppositeofmust-linkconstraints.Ifacannot-linkconstraintisspecifiedontwoobjects,xandy,thenintheoutputoftheclusteranalysis,xandyshouldbelongtodifferentclusters.Cannot-linkconstraintscanbeentailed.Thatis,ifcannot-link(x,y),must-link(x,x(cid:48)),andmust-link(y,y(cid:48)),thencannot-link(x(cid:48),y(cid:48)).Aconstraintoninstancescanbedefinedusingspecificinstances.Alternatively,itcanalsobedefinedusinginstancevariablesorattributesofinstances.Forexample,aconstraint,Constraint(x,y):must-link(x,y)ifdist(x,y)≤(cid:15),usesthedistancebetweenobjectstospecifyamust-linkconstraint.Constraintsonclusters:Aconstraintonclustersspecifiesarequirementontheclusters,possiblyusingattributesoftheclusters.Forexample,aconstraintmayspecifytheminimumnumberofobjectsinacluster,themaximumdiameterofacluster,ortheshapeofacluster(e.g.,aconvex).Thenumberofclustersspecifiedforpartitioningclusteringmethodscan
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FINAL ANSWER
Final Answer: Dieser Text handelt von verschiedenen Aspekten des Data Mining und der Datenanalyse, insbesondere aus dem Buch "Data Mining Concepts and Techniques" von Jiawei Han, Micheline Kamber und Jian Pei.

Die spezifischen Themen umfassen:

1. Datenvorverarbeitung
- Umgang mit fehlenden Werten
- Bereinigung von Daten
- Normalisierung von Daten
- Diskretisierung von Attributen

2. Ähnlichkeits- und Distanzmaße
- Berechnung von Ähnlichkeit zwischen Objekten
- Verschiedene Distanzmetriken (Euklidische Distanz, Manhattan-Distanz)
- Umgang mit Attributen unterschiedlicher Typen

3. Clustering-Techniken
- Clustering in hochdimensionalen Daten
- Clustering von Graphen und Netzwerkdaten
- Spektrales Clustering
- Biclustering

4. Statistische Methoden
- Berechnung von Mittelwerten, Median, Varianz
- Korrelationsanalyse
- Kovarianz

Der Text ist sehr technisch und mathematisch orientiert und dient als Lehrbuch und Referenz für fortgeschrittene Techniken im Data Mining.
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