2006 - Thèse - Contextualisation, visualisation et évaluation en apprentiss

2006 - Thèse - Contextualisation, visualisation et évaluation en apprentiss

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†† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† –† p†‘–†‘˜r€i–” ——––™™ ‘‘ ‘‘„„ ’ †† †† $$ ! †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †—˜†“†˜m y† —†™€† —†– ˆ† ‘† „†˜“—˜–‘mm––—™™€€ ƒ—–™ ‘o‘† „†’$ † $ o  † † † † † † † † † † † † † † † † † † † † † † † † † † & %  % – j e ˜ € „ lm # † † $ !Y † † † † † † † † † † † † † † † † † † † † † † † † † E D G  – j e˜€„ lm # ‘† † $  X$ Y2¦ † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † —–mm–Wp¨˜€©“G—©–¨™D‘R„ ’F I 0†&$ X YY †† †† †† †† †† †† †† †† †† †† †† †† n† q† 1† †—˜†‘–† j† y†p „† ˆ† –†ˆ †–™†—†•€† –†d y† — † ‘†„†˜“†p † —†—“† m i† † ‘o“† m†x o† Y Y † † † † † † † † † † † † † † † † † † † n q 1 — ˜ ‘ – j y p „ ˆ – ˆ l ‘ € – ˜ — ym i † † ˆ !!## †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† † n†q—1–™€—y˜‘˜p–yj€y˜—p„ j–——– ˆ—–‘™„‘‘˜„“ ˜ˆ‘–u—y™“€ d—–‘g„— ‘–†˜u†‡ † X$ #) † † † † † † † † † † † † † † † † † † † † † † † † † † † ™W—D •€–Vd©yF— Ry“W—D‘„W—¨‘–˜uW ¨‡D © 0†& $ Y"## †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† †† † —˜† “†˜m y† —†™€† —†– ˆ† ‘† „†˜†“ ˜†‘–‘—“™m€ x $!††  ƒ $#!! 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H D W D  $  ¦¤ ¢ ¡ ¡¢ ¥ ¦¤ ¥ ¡¢ ©¨ ¡¨ ¡¤ ¦ ¢  ¡¤  © † – € “  ˆ ™ j € – ˜‘ – — — “ m p “ m ˜ ‘ – ˜ € “ d d “ – € y ˜ „ • –mm – • y „ ‘ “ m – y y – € ˆ ‘ „ d ™ € ˜ – € –m „ d “ € ˜ u– ˜ “ € € y „ d ‘ „ f — –m s “ ˜ € „ … ‘ „ p ˜ ‘ „ — — –mm – — ym d f — – y l ‘ „m ˜ ‘ „ — — – € y ˜ „ • — –m — ym d – y y ˜‘ “• € – — s „ f –m d j – u– € “ ƒ † ™ € ˜‘ „ p ‘ – € –m d j – u– m – • y „ ‘ ˜ y „ ˜ – ™  € d „ € d d “ — ym d “ m – — — “ m p “ m € – j ˜ — – ˆ – m s “ d “ p – € ˜ ˆ — € „ m “ ˜ — – … ˜ p – s „ n † ˜ ‘ –  ˜ € “ d d “ m – m m – y y “ m – € y ˜  „ • – ˆ – — — “ m p “ m – ™  p „ — — “ ˜ — – m “  ˜  ‘  – m d j – u – – y y “ ep © † ™ —  • € – d y — – l “ — —  ˜ ‘ – € d d “ m – ˆ – € ˆ “ p – m — ‘ “ ˆ † y “ – m s “ ˜ y ˆ ‘ „  ˜ “ ˜ d “ ˆ “ m  — ‘ “ – € ˜ ‘ „ j o† y “ – m s “ ˜ – n † — – ™ € ˜ ‘ „ p ‘ – € ™ ˆ — – € y ˜ „ • — – € ˜ y “ ˆ – y‘ ‘ „ p – — — “ m p “ m € y— ˜ – f — – y y ˜— € ™ ˜ p “ € “ p — – — € y— ˜‘ “ — “ s – — ‘ – f — –m s “ ˜ € „ … ‘ „ p — – € y ˜ „ • — – ˆ y „ — – € “  ˆ ™ j € – ˜‘ — – € y ˜ „ • — – ˆ f — – ‘ ˆ “ ˜ p — – ˆ – — — “ m p “ m – ˆ – ˜ € “ d ˜ “ … – ™ € ˜ ‘ „ p ‘ – € – € y ˜ „ • –mm – • y „ ‘ – ‘ y — € – ‘ j € – ˜ ™ ˆ – ˆ –m d j – u– € “ d € l “ — ˜ y – d m f — – € y ˜ „ • — –m ˜‘ “ ‘ € – p ‘ „ p –m d j – u– – € ˜ „ ‘ – ˆ — “ p –m — ‘ “ ’ † — ˜‘–˜“ d — – €˜ y“ — – ˆ — – m m – p p – •“ — – —  m “ ‘ “ — – — – ˆ ™ ˜  € “ m  j  — “ m – ˆ ‘ „  ˜ p ‘ „ … ‘ – ˜ ‘ –  ˜ “ d y “ – • y „ ‘ – p – ˆ –  ˆ “ m “ j – ˆ – y y — € –m € – ym “• ™ ˆ –m s — — „ d ˜— – m f — – ™ ‘ – j ™ ˜ ™ ˜‘ „ — – — m “ ‘ “ — – j j — –m —m – y y — –m € y „ d — ˜ ‘ – ˜ “ d — – € ˜ y “ ˆ ˜ “ ˜ ™ m – ˆ – p ‘ “ — — “ ‘ ‘ „ p “ m ˜ – f ˜‘ – ˜ “ d ‘ y ˆ – — m “ ‘ “ ˆ — ˜ “ ˜m y — ™ € — –m — ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ ™ ‚ – ‘ p – ˆ ™ j “ m – ‘ € p ‘ „ p ™ — • € – d y— – l “ — — ˜‘ – € d d “ m – ˆ ‘ „ ˜ “ pm d d “ ˆ –m d j – u– ‘ ” † – ™ ˜ d “ ˆ “ — ym d “ m – — — “ m p “ — ™ € ˜ ‘ „ p ‘ – € –m d j – u– m – • y „ ‘ ˜ y „ ˜ € – p „ — — “ ˆ – ˜ y — ‘ – – ˜ ˜ – j € – d  y y –m k ˆ „ j ‘ y – € ˆ ‘ – € d d “ € y „ d — ™ — — “ m p ™ ˆ — ‘ € y „ … — –m d j – u– — –m € – —m ˜ y ˆ — € „m “ ˜ l “ — m f — “ p – p — ‘ “ ’ †¤  £  £  ¥¢  ! ¤  ¨  £  ˆ y „    ¥ ¤ ¡  ¤ §     £ ¤ ¥   ˆ – m € “ d ‘ „ f – — — “ m p € y – m – ˆ – ˜ ˜ – y y  ˜ ™ m p – •“ —  ‘ € y „ … ˜ ‘ „ — — –m d j – u– — –m — ˜ – — – y‘ ‘ „ p ˜‘ „ — — –m s— — „ d — – — — “ m p — –m  z † – — — “ m p – ‘ y ˜‘ “  p „ — — “ € y –m ‘ – — – ™ ‘ ‘ „ ˆ — – ˆ € – ˜ – y y  ˜ ™ ˆ – m s “ d “ p – € ˜ ˆ ˜ — – £  ¢ ¨    ¨ ¢ “ m – ˆ m “ € ™ ‘ ™ l … ˜ p – s „ n  ©  © © © ©©  ©   © © ©  § © © ©   ©© ©© © © H © © B C A  C ¦ @  H ED C B ¨ ¨¨ © @ 9 ¥  † —–€ s€“ — – ˆ € “ d — – ™ ˜ ‘ – — ™ € d – € — – ™ ‘ ‘ „ ˆ — – ˆ — “ p y “ ™ — — – € ™ ˜‘ — – j j „ — — y „ ‘ — y „ ‘ f — d j – ˜ – j k u y – ˆ ‘ y — ‘ “ ’ † –mm – € „ l ™ ˜ “ p y „ – y y € ™ j y ‘ – € y ˜ “ ‘ – ˆ – € ˜ ˜ ‘ – • y – d — ˜ y s € ˜ ˜ “ — –m ¤ „ ˜ – € y –m “• ˜ y s € ˜ ˜ “ – j € „ … — y „ — — – ™ ˜ ‘ – — ™ € d – € — – ™ ‘ ‘ „ ˆ — – ˆ  ym – p ‚ ™ € ˜‘ „ p ‘ – € ˜ ‘ – • y „ — — ym d –m — “ p y “ ™ — — – € ™ ˜ ‘  ˜ ‘ – j – m “ d  p ‘  € d — – j j „ — — y „ ‘ — y „ ‘ f — ™ ‘ – j — ‘ „ • “ — y „ ‘ – y y u y “ •“ € ˜ — – m — ‘ “ ’ † – ™ — m ˜ y ˜ — –  y y ¤¥ !¥ ¥¤  ¤¥  ¡ ! ¤ ¡   § ¨ “ m ˜ ‘ – • y „ — ˜ — – p — € „ m “ f — – m y p ™ m „ j — – m – € € p ™ ˆ € y „ d –  j  ep ‘ – y „ u y “  € ™ ˜ “ j — – m – €  € p ™ ˆ € y „ d – y y  —  e d ‘ – – j j „ p f — ˜ ‘ “ — „ d j „ p — ˜ ‘ – € ™  ˆ – € ˜ ‘ – — – u – m d j „ p — ‘ „  ˜ p “ € – ˜ ‘ — – ˆ € – ˜ ‘ – — ™ € d – € – ˜ “ e y „ — ‘ „ m – y y — € „ m f ‘ ‘ ‡ † ¤  ¦¥ ¢  ! ¤ £ £ ! u y “ ‘ „  ˜ — „ d d „ € “ d f ¤ ¡ £  £ ¢ ¤ £ £ ! – ˆ — “ p – p — ‘ “ ˆ – m € “ d ‘ w † — – € y — – j — – ‘ “ ˜ € – p – ˆ – mm – € „ d j – ˜ ‘ „  ˜ ym „ • ™ m – m d j – u – € “ d ˜ ‘ “ y y  ˆ ‘ f ¤ ¥ ¡ ¢ — – ˆ € “ d — – ™ ˜ ‘ – — ™ € d – € ˜ ‘ –  „ — — – mm – y y – m s — — „ d ˜ ‘ – j – m “ l ™ ˜ — – m r † ¤ £ ¥ § – ˆ – j € „ … — y „ — — –  ‘ € y „ … ˜ ‘ „ — — – ™ ‘ ‘ „ ˆ — – ‘ “ ˜ € – i † ˜‘– j– y y u“ ˜‘ — — ™ ˜– y y˜™ — ˜‘– j yp „ ˆ – ˆ y „ — ™ € y˜p y€˜—  j–— — ˜‘– j yp „ ˆ – ˆ — y d€ „ p – ˆ – m d j – u – € “ d — “ p – m ˜ — – i † ¤¥ ¥  — – ˆ € “ d — – ™ ˜ ‘ – — ™ € d – € – € ˜ ˜ ‘ – • y – d — – ™ ‘ ‘ „ ˆ — – ‘ “ ˜ € – p f — € y –mm “ € “ ƒ † — ™ € ™ ˆ— ‘ „ p — ˜ y s € ˜ ˜ “ — –m € y — — ™ ˜m s “ s „ € d – ˆ — ‘ „ ˜ y s € ˜— ˆ — – ˆ € “ d — ˜ – s „ — – ˆ — € y –m “• — –m € – ˜ ‘ – — ™ € d – € ˜ y – d ‘ „ f –m “ € ™ ‘ ™ l — ym d – € k ‘ “ j – ’ † m – € „ l ™ ˜ “ p ˜ y s € ˜ ˜ “ ‘ y € y— — –m s — — „ d — ™ ˜m “ ˆ „ j – ˆ –m s j – — ‘ – ‘ y ‘ – s y „ f – y y € ™ j y‘ ˜ y s € ˜ ˜ “ ‘ y € y— „ ˜ ‘ ™ ˆ – ˆ — – m m “ • € – ˜ ‘  — – ˆ – € ˜ ˜ ‘ – • y – d — ˜ – s „ u y “ — – ™  p „ — — “ — € y – m “ • — – m — € „ m “ f ¢ f “ ˆ  ’ ˆ ‘ “ p „ x¡ ¤ ¡  ¨   ˜ ‘ „ — — – ™ ‘ ‘ „ ˆ — – m – y y — € „ m f – m d j – u – € “ ƒ † — – ™ € ˜ ‘ „ p ‘ – € – € ˜ ˜ ‘ – j – m “ l ™ ˜ ‘ – • y – d — – u–m d j „ p — ym d — ‘ „ ˜ “ ˜‘ – — ™ € d – € — – € ˜ y “ ˆ — “ j f — – ™ ‘ ‘ „ ˆ – ˆ –m s j – — ‘ – ‘ y – € € p ™ ˆ    © ©  ¡¢ © © ©  © ©   © !      Y Y Y  © ! ©¨ ¤ £ ¥ ¢ ¨ ¡ © † § £ ¥¤   ¡¨ ¢ y „ f   ¥¤ ¡  £ £ ¤ §   £¤¥   ˆ – m € “ d ‘ „ f — y ‘ ‘ „ p ‘ ˜ ‘ „ — – € s j „ ‘ € y – m ˜ – — – — — “ m p — –m  — ˜ – f — –m s ‘ „ d— ˆ ˜‘ „ — – ˜ ˜ – y y ˜ ™ — ‘ “ — — –m d j – u– — – ˆ —m y – — — f „ € “ € ˜‘ „ p #  H § B C A  D  D C ¦ @  H ED C B ¨ ©© © ¨¨ @ 9 8   † – € y ˜ „ • – ˜ ˜ – p – ˆ u € d – m u y –  j y “ € – ep „ € d d “ ˆ —  “ j f – ™ € ˜ ‘ „ p ‘ – € – € y ˜ „ • – mm – • y „ ‘ “ m — – ™ ‘ j € – ˜ ™ ˆ ™ € d — – — — “ m p — „ € ˜ — – ˆ – ‘ y m € – p „ — — “ ˆ ˜‘ – j –m d j— — ym d — € „m “ ˜— – ‘ – € ˆ “ p – p — ‘ “ ˆ … ˜ p – s „ n † — “ p – p — ‘ “ ˆ — – € y ˜ „ • — – m € y — – m d j – u – – € ˜ „ ‘ – ˜ d “ ˆ “ † y “ – m s “ ˜ – n † £   ¤¥ § ¥ – ˆ –m € “ d ‘ „ f – y y € ™ j y ‘ — “ j –mm – € „ l ™ ˜ “ p — “ d ˜— – ‘ — –m d j – u– — – ˆ – — — “ m p “ m – y y — € „m f ‘ ‘ ‡ † – ˜ —  m  s “ s „ € d ™ —  • € – d y — – l “ — —  ˜ ‘ – € d d “ ‘ – — – ™ —  m  ˜ y — – ™ ‘ ‘ „ ’ † ¦ ¥ ¤£ © ©  ¥ 7 ( 4) ( ( %) ( % ( %) ( ( &) ( © $ ¨  7 &) ( %) ( 0) ( &) ( ( %) ( &) ( 4) ( 0) ( § ¥ ¥ ¦ # "  § © ¨¥ © ! 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YYYYYY YYYYY YYYY! YYYY YYYYY!  ¦ ¥ ¤£ Y# $! "  ¤ m–— – ˆ – € “  ˆ ™ j € – ˜ ‘ –p ‘–——– ˜€„…‘„p m–— – ˆ – ‘ ˆ “ ˜ p m„ ‘“ e˜™ – € “  ˆ ™ j € – ˜ ‘ –p ‘–——– ˜€„…‘„p –p ‘–——– – ‘ ˆ “ ˜ p m–— – ˆ – € “  ˆ ™ j € – ˜ ‘ m–— – ˆ o –p ‘–——– – € “  ˆ ™ j € – ˜ ‘ ˜€„…‘„p mdl –——“mp – p ‘ “ — — y d € y – y l ‘ „m — – € ˆ ‘m  p ˜‘ “ € y s € “ p ˜‘ “  ˜‘ – ˆ # " " "" !"  ¢ ¦ ©£ ¤ ¡£ ¦ ¡¢ ¡ ¤ ¢ ¡ ¦ ¦ ©¨ ¡¤  ¡ ¡¡¡ – ˆ  — ‘ “ ˜ – j € – d — – d y „ € l — m – ˜ – ˆ – ˜ € – • y „ p ™ ˆ “ n † u € d € y – m – ˆ ˜ – € y – y l ‘ „ m € y – m – ˆ ‘ „  ˜ p ‘ „ … ‘ – — – € y ˜ „ • – ˆ –m s j – — ‘ – ‘ y € – d y „ € l – € – ˆ ‘ „ “ … – ‘ y –m d j – u– € “ d – € ˜‘ „ j †  – € y l  “ n  ¦ ¥ ¤£ ¡ † ™ — • € – d y— ‘ „ ‘ – l “ — — ˜‘ – € d d “ ‘ – — – ™ —m ˜ y — – ™ ‘ ‘ „ ’ † Y Y  Y YYYYYYYY!$! YYYYY## YYYYY YYYYYY# YYYYYY YYYYY YYYY! YYYY YYYYY! ¥ Y $ !!! Y!$ # Y$ !$ Y # $! 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˜ p – m – m € y „ d ¨  £ §  ! £ ¤ – € y ˜ p – m – ˆ – u “ – j k  —  „ € ˜ ‘ y €  € „ € y „ d ‘ „  ˜ p – — – y y “ ep – ˆ ‘ “ m ™ ˜ ‘ – — ™ € d ˜ ‘ – j – m “ l ™ ˜ — – ‘ “ m  s ‘ ” † — – ˜ ‘ “ ˜ —  u – — – y y  ‘ ep – ˜ — – m — – ˜ y „ ˜ – ˆ — m  “ ˜ ™ ˆ — – m — ‘ “ ˆ € – € ˜ ‘ – € — “ d ˜ ‘ “ ˜ “ e y „ — – ‘ € y – ˜ p –m –m € y „ d – — k e ˜ “ m – ˆ – ˆ y — ym d – € y ˜ p –m – ‘ y  — ‘ “ ˜‘ “ ˜ ˜ – j € – d f — ™ y y „ • ™ — ˜ ‘ „ d — € y –— ym d ˜‘ – j –m “ ‘ ˜ ‘ –mm “ ˜ ™ ˆ – ˜ € “ d ‘ – — – u– ‘ ‘ “ — € y –— ym ƒ † r –  ˜ € “ d “ m — ‘ “ ˆ — – ™ ˜ ‘ – — ™ € d ˜ ‘ „ — u y “•“ € ˜ — „ ‘ € “ d — – ˜ € – • y „ — – • ˜ p – d — € – d ˜ – — ‘ „ — ym p ‘ „ p — –m f ‘ ‘ ‡ † – € ˆ “ p – p — ‘ “ ˆ – ˜‘ “ ˜ € „ d j – ˜ € – • y „ – y y ˜ “ j ™m s „ € d – ‘ y – y ˜ ˜— ‘ „ p  y y f — – j e ˜ € „ lm “ —m – ˜ – ˆ ‘ „ — “ € “ d j „ p “ m – ˆ y „ f l ‘ € – ˜— ym p – ˆ — – j e ˜ € „ lm “ ˆ — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ym “• ™ m – ˆ – ‘ “ j „ ˆ –m — ‘ “ ˆ — ‘ „ ˜ y s € ˜ ‘ „ p — „ ‘ – ™ € p “ — ‘ „ p ˜— – r r r – ˜ € “ d “ m — y ƒ † r r –  ˜ € “ d “ m — ‘ “ ˆ — – ™ ˜ ‘ – — ™ € d – ˜  y — ‘ – ˜ ‘ „ — f – € ˆ “ p – p — ‘ “ ˆ — – ™ ‘ – j — ‘ „ •“ — y „ ‘ – y y — ‘ „  ˜ “ ˜ ‘ – j € ™ d u– — –m – y y  — ‘ “ f – ‘ “ j „ ˆ – p — ‘ “ ˆ — – y y j e ˜ € „ lm “ — ‘ „ ˜ y s € ˜ ‘ „ p — „ † r – ˜ € “ d “ m — ‘ “ ˆ l ‘ € – ˜— ym p ‘ – ˜ € “ m – ˆ ˜ “ ˜ ™ m € – ˜‘ – — ™ € d € “ d ˜ ‘ – j –mm – € y ˜ “ ‘ p ‘ „ ˆ — ‘ „ ¡ ‘ – j j „ p — y „ † ‘ „  ˜ “ ym “ • ™ ‘ „ — € y „ d £   ¨  ¡© ! ¤  ¤  £ ¢ £   £  ! § £  ¥ ¤   ¡ ¨ ¢ ¤ ¨ ¥ ¤¢  ¨ ¤ ¥ – ˆ ˜ € ™ ˜ ‘  m ˜ ‘ – j – m “ l ™ — ‘ „ € € – • — y „ ‘ ˜ ‡ † € y – ˜ “ —  m  ˜ y m – m s  — ‘ – e ™ € d j „ p – € k  ‘ “ j – ˆ — ˜ “ ˜ m y — ™ € — – m € – ˜ ‘ – — ™ € d – ˆ ‘ “ — ‘ „  — ‘ – j  ˆ u y – ˆ ‘ – £   ¢¤ ¥  € y – m ˜ – ¤ ¨ §¦¥ — – ˆ € – — m ˜ y ˆ ˜ € ™ ˜ ‘ m — ‘ „ € € – • — y „ † € y – ˜ “ — m ˜ y m – ˆ ˜ € “ d “ m – ˆ  € „  € d “ – p ‘ “ — —  “ ‘ ‘ „ p – ˆ — “ d ˜ ‘ “ € ™  y y – € – ‘ ˜ – f — – m d j – u – — – m ˜ ‘ “ •  € p ™ ˆ — ˜ y s  € ˜ ˜ “ ˆ — – d ˜ — ˜ ‘ – € ™  ˆ ˜ – — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ ˜‘ – — ™ € d ˜ y € s y “ f — – ˜ ‘ “ y y ‘ “ j — – ™ ‘ ‘ „ ˆ u y “ – p “ … – € “ … – ˆ –m s “ d “ p – ˆ „ e ˜ ™ j – ‘ y ˆ – € • y ¤ ‘ – – — j “ m – ˆ  ym – p y y — ‘ “ ‘ „ ˜ “ —m “ y ˜ u– ˜ ‘ „ p “ m – ˆ – j km s „ € d y “ – p “ … – € “ … € y „ d ¤  ¨   ¥  ¤ ¨¦ !  — – ˆ € – — m ˜ y ˆ ˜ € ™ ˜ ‘ m — € „ m “ — ‘ „ € € – • — y „   ©    © © © © ©  ! ©  ©  © ©© $ ©©   © !  ©©  ¤ ¡¤ ¤ ¡¤ ¤  ¡ £ ¤ ¥ ¤  ¥ ¡ ¥ ¦¥ £ ¡ §¨ £ ¤ # ¦ ¡ § £ ¡¤ £ ¡ ¤ § ¡ § £ §¤ ¥ ¦ ¢ £ § ¦ ¢¤ ¡ § ¦ † # — – u– ‘ ‘ “ ‘ – – ˜ y— “ m € “ d — ‘ € y „ … ˜ ‘ „ — —m “ ˜ ™ ˆ — –m d j “ — ym d – ’ † — – y y  ˜ —  € ™ ˜ p “ € “ p — – — – ˆ – m s j – — ‘ –© m – ˆ ‘ „  ˜ p ‘ „ … ‘ – – ™  € d „ € d d “ — ym d “ m – ˆ „ e ˜ ™ j “ m € – ep € – ep – € “ € € y „ d € y – ˜ “ — m ˜ y© m f l ‘ € – ˜ — ym p – m € y „ d – ™ l “ —  • ‘ – ‘ „  ˜ “ p m d d “© m ‘ „ m – — f  — ‘ # † — – — — – m s “ … — – — – ˆ ˜ – — – p € „ … — – — – ˆ f — – m “ d p ‘ € d — – y y  ˜ —  € ™ ˜ p “ € “ p — – — – ˆ m “ € ™ ‘ ™ l y € – d “ ‘ y — ‘ „ ‘ ‘ „ ˆ — y „ ‘ f — – ˆ „ e ˜ ™ j — – p – ˆ – ‘ y p “ ep € y „ ƒ † — – y y  — — “ m p — ym d — – m l ‘ € – ˜ — ym p – ˆ — – ˆ „ e ˜ ™ j — – ˆ ‘ „  ˜ “ ˜ ‘ – — ™ € d “ m  ™ € p “ — ‘ „ p ˜ — – o – € ˜ d “ ep – m —  y ƒ † – j k m s „ € d y ˆ – m “ € ™ ‘ ™ l – y • – ‘ y  – € ˜ d “ ep – m — ‘ “ ˆ ˆ € „ s “© ˆ ˜ y „ ˜ — ‘ „ ˜ ‘ – — ™ € d — y „  † – p “ … – € “ …  € y „ d — – ™ —m ˜ y — ym d — –m — – ˆ „ e ˜ ™ j — – ˆ – y y  — ‘ “ f – € ˆ “ p – p — ‘ “ ˆ — – ™ € ˜ ‘ „ p ‘ – € — – y y  ˜ “ j ™ m s „ € d — – ˜ ‘ – € ™ ! ˆ — – ˆ m “ € ™ ‘ ™ l y € – d “ ‘ y € – ‘ ‘ „ ˆ – ˆ ˜ " ˜ ym d — “ j f l ‘ € – ˜— ym p y ˆ – € ˆ “ p –m — ‘ “ ˆ ˜ ‘ “ ˜— u– — – ˆ „ e ˜ ™ j ˜ – — ‘ „ ˜ „ ‘ — – ˆ – m s j – — ‘ –© m – ˆ – • ˜ — y “ e u – – ˜ — m – ‘ y € ‘ € y „ … — “ d — ‘ „ ˆ ‘ – ˜ ™ € d – ‘ — y „ ‘ f –  ˜ € “ d – ˜ ˜ – p — ‘ “ ’ † — – j km s „ € d — ‘ “ ˜ € – p  – p “ … ˜ – ‘ „  ˜ “ p m d d “© ˆ — d j “ ep — ‘ “ ˜ € – p — ‘ “ ˆ — – € ˜ y “© ˆ – y y — ™  € d „ € d d “ — ym d ˜ ‘ „ — — – ˆ „ e ˜ ™ j — – ‘ “ ˜ € – p ˜ – — – € k ˜ € p — ‘ “ ˜ € – p f – € ˜‘ „ p € “ ƒ † — –m s “ – l “ — •‘ – — – j km s „ € d — – ˆ ˜ – — – y y ˜ “ j ™m s „ € d — – ˆ – m s j – — ‘ –© m € y — — – € ˜ y “© ˆ – y y — € y – mm – j ˜ ‘ – j – y y k — ‘ € ˜ ‘ ˜ ‘ „ — – ‘ – ˆ „ e ˜ ™ j – ‘ y p y “  ‘ – € k ˜ € p ‘ y p y “© y y l ‘ € – ˜ — ym p – m € y — ˜ ‘ “ mm “ •“ € ˜ ™ ˜ y “ ‘ y j j „ p “ m — ‘ “ ˆ —  j ˆ “  — ‘ “ ˜ — – m r † — ™  ˜ ‘ – ˆ  — € – ˜ — ym p — – ˆ – m s — ‘ – e ™ € d j „ p ‘ „  ˜ “ ˜ ‘ – — ™ € d – € – ‘ y € ‘ € y „ … – ˆ ˜ — – … ˜ p –  s „© m — — – p “ p £– — ym d ˜ ‘ „ € – — — – € k ‘ € – ˆ — – p – y y — € „m “ f – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p — – ˆ € –m s p „ €™yjd——––n€˜† y—˜“©yˆˆ–„y€ dy ——–€–™˜˜—dy“mˆp“—–uˆy–™˜j— ‘m“–€ˆ™ ‘–™ˆl‘‘„–˜—„€‘„m–“‘˜y‘„€“— d—–l€‘k˜€€–p˜——ym–mp˜ –‘ˆy©€ˆy—˜—€–™™˜—‘“©sm —€– ‘ˆ„™eˆ˜ – ˆ – m s  — — „ d ˜ ‘ – j – m “ l ™ ˜ — – m  f ‘  ‘ ‡ † ˜ ‘ “ — — – € ™ ˜ ‘  — y m d € – m ™ • ™ € – — ˜ y – d ˜ ‘ – ep y “ • – ep – —  y y — € – ˜ — ym p — – ˆ € – € ™ ˆ  — ‘ „ p f – € “ € ˜ ‘ „ p y “ — “ p — – € ˜ y “© ˆ — ‘ “ ’ † — ™ j € „ … — – d y „ € l — – ˆ – ˜ ‘ “ ˜ € „ d j — ‘™„!˜ˆ“€—“˜d‘™–—j––‘˜y€„€dyj™„—p“usy˜“‘–—˜j‘––m“mpd–pˆ‘—€–dd˜y—„–€˜l“—˜–m yˆ—™€€– l‘€y–©jˆ ™™˜–€m““y… y– ˆ– ˆ˜ y–€sk˜–m€p— ‘–“m ˆ—€—„˜m‘“–f—mp˜‘––ˆ€ — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — – ˆ € y— ™ —m ˜ y ˜ — – l ‘ € – ˜— ym p –m  z † — – ™ ‘ ‘ „ ˆ — – ˆ ‘ „— — – € d j „ p –mm – ˜ – ‘ y  –™m ‘„˜“ j€„…‘© ˆ –˜€– d “m –— j ‘ j –mm–© yy—€„m –m“ j˜ d„ –™ l y  ˜—– ‘„˜ ym„— – ‘ y —€„m“ f – p “ p £ – — ym d ˜ ‘ – j – ˜ “ € ˜ ‘ y € y „ d — – ™ ‘ ‘ „ ˆ – ˆ y –  ‘ y© ˆ – mm “ ˜ “ m – € y ˆ ™ € – ˆ ˜ — – … ˜ p –  s „© m – y y — € „ m f – m d j – u – € “ ƒ † € –  € “ • ˜ y – d ‘ „  ˜ ym „ — – ‘ y© ˆ ˜  € ™ ˜ ‘© m – ˆ – j  j ‘ „  ˜ ‘  ™ ˆ “ m ˜ – f — – j  j — – m — “ d ˜ ‘ „ — – ‘ — ‘ „  ˜ “ € ™ ˆ  — ‘ „ p — – m f l ‘ € – ˜ — ym p – m € y „ d – ™ l “ —  • ‘ – ‘ „  ˜ “ p m d d “© m ‘ „ m – z † — ˜ p ‘ ˜ —  ˆ — € – ˜ — ym p — – ˆ  — ™  p „ — — “ ˜ ‘ „ — — ˜ ‘ – € ™ ! ˆ – j j „ p — ™ € ™ ˆ  — ‘ „ p ˜ ‘ „ —  y y u y – p – y y — € „m “ € – ˜— ym p – j  j y “ — ™ p „ — — “ — ‘ “ ˜‘ „ — — – € “ m j— – j j „ p — ™ € ™ ˆ— ‘ „ p ˜‘ „ —  y y — ˜ –  s „ — – n † ™ ˜ € “ m j — – ˆ ‘ „  ˜ „ ‘ – ‘ “ ˜ € – p – ‘ y© ˆ ‘ „  ˜ p ‘ „ … ‘ –   ¥¤   ¡¨ ¢ y „ — – d y „ € l — ˜ ‘ – € ™ ! ˆ ‘ – — ˜ –  s „© ˆ – m s j – — ‘ – ‘ y € – € “ d ™ —  – ˜ —  — ‘ „ p f m “ € ™ ‘ ™ l — k € ˜ f … ˜ p –  s „ ‘ „ z † p ˜– f — – y y˜—˜“ ˜— f ‘‘„“˜—–‘—ly„–p€–s€j‘„€‘–˜–˜“ˆd—f yl ‘d–‘ˆ j–™“ˆ˜“yˆ˜™f l–‘ep‘€€–“e–pm––€‘–ˆep–“yjy‚˜“—™j˜ y™m“s‘„y€j j‘„yp —˜—––˜‘l–‘€™€!–˜ˆ— y—m‘p“ –ˆn—–™ ‘ d– ¡ ¡   © o  ‘ „  ˜ d € p — – ˆ “ m € – j € „ …— ‘ “ € ˜ – € ˆ ˜ — – p f — – ™ ‘ ‘ „ ˆ – ˆ – — “ s – ‘ y ¥¤  ¥ ¢  ! € y „ d – ™ — m ˜ y — „ …€ “ d ˜— – ‘ „ ˜ “ ˜ ‘ – j l – — “ m f ‘ ‘ ‡ 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d y „ € l – € – ˆ ˜ — – l ‘ € – ˜ — ym p y ˆ … ˜ p – s „ n    ©    ¥ ©   © EH B @ I H H  C G  D  H E  ¤ £   ©  ¦¦ † p ˜ – f – y y ˜ ™ ‘ ™ l – — m “ ‘ “ ‘ – f – y y ˜ “ j € „ … ‘ „ s ‘ – f — – ˜ ‘ – • – ˆ – — m “ ‘ “ ‘ – f — –mm – y ˜ u– ˜ — – ™ ‘ ‘ „ ˆ — – ˆ – —  m “ ‘ “ ‘ – f s – ‰ y ˆ – —  m “ ‘ “ ‘ – – ™ —  m  ˜ y – € ˜ ˜ y – d – y y  ‘ ep – ˜ – ˜ ˜ – p f – j j – ’ † –  ˆ “ m “ j “ m € y „ d — –m s— — „ d — – — y “ p — – ˜ ‘ – € ™  ˆ — ‘ “ ˜‘ “ ˜ ™ – € f – ™ ‘ ‘ „ ˆ – ˆ “ m “ j – ‘ y ˆ — ˜‘ – ˜ ˜ “ — ˜‘ – ˜ “ d – ˆ — – d y „ € l — ˜‘ – € ™  ˆ –m d j – u– € “ d ˜‘ “ m s p ‘ – f — ˜‘ – ˜ “ d – ˆ — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — – ˆ € y— u y “ p  ˆ ™ j — p  ˜ — „ ‘ l “  ˆ — – ˆ € – y ˜ p – – € y „ d ™ € ™ •“ – m d j – u – € “ d ˜ — – — – y y  ‘ ep – ˜ — – m m – ˜ – ˆ ˜ € ™ ˜ ‘ n † — – ™ ‘ ‘ „ ˆ — – p — ˜‘ – p “ — y „ — — –m k ˆ „ j – ˆ ‘ „ ˜— – l l y— “ m ˜ – f — – ™ ‘ ‘ „ ˆ — –m ˜ ‘ “ — € ™ ˜ p “ € “ p — – m l k € – ˆ – p ‘ – € ™ … ‘ m € “ d f ¤  ¢  ! ¥  £ 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€ “ p ˜ y s € ˜ ˜ “ – y y ‘ y ‘ y € “ d — ˜ – s „ — – ˆ – u–m d j „ p  © © ©    !!  !    ©© © ©   ©   ©   ©    ©  © ©   " ¤ § £¢ ¢  ¡ £ ¥ ¤ £ ¡ ¢ ¡¢ ¢ ¤ £  Y ×Z  ©! ©  © ©   ©  !  ©  ©  ‘ „— ‘ – j ˆ “ m € y— — ˜ – s „ — – p – ˆ — € y –m “• — –m € “ p ˜— – — –m l ‘“€˜ — – ˆ €“ d — ™ ˜‘–— ™ € d– € — ˜ – s „ ˆ – d y „ € l y ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d— – — y „ — –m f – j j – ’ † p ‘„ ˆ ˜—– – d y„€l –p – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d — – — y „ — – n † — ˜ – s „ — – € ˜ y “ — – ˆ € – p ‘ – € ™  ˆ — –m — “ d – ˆ “ ‘ ‘ „— ‘ – j ˆ – ˜ ˜ – p f – ˆ ‘ „ ˜ ‘ ™ ˆ – ˆ –mm “• € – ˜ ‘ m – ˆ –m s j – — ‘ – m ˜‘ “ € • y „ p ‘ „— ‘ – j ˆ “ m € y— — ˜ – s „ — – p – ˆ — –m s— — „ d — € y –m “• — –m f – € ˜ ‘ „ p € “ d ‘ „— ‘ – j ˆ “ m € y— — – ™ • –m ™ — € y –m “• — – ˆ ˜ – ‘ „ — ‘ – j ˆ “ m € y— — € y –m “• — –m s “ … – ˆ € “ d ™ — € ™ ˜ p “ € “ p ˜— – m y y – y y € “ j – € ‘ „ f — ˆ ‘ „ € — – ˆ € “ d — ™ ˜‘–— ™ € d– € — ˜– s „ — – ˆ –m s j–— ‘– m €“ d ™ j€ „… – d y „ € l 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— u– ˜ y – d  y y ˜ y € s –m € – € ™ l – ˆ –m s “ d “ p – € ˜ ˆ – ˆ „ e ˜ ™ j – ‘ y € y „ d –m ˜ y – € ˜ ˜‘ – j –m “ l ™ ˜ y – d m m – ‘ ‘ „ ˜ ˆ “ € ˜ l ‘ € – ˜ — ym p – ˆ — – ˆ „ e ˜ ™ j — – ˆ l ‘ € – ˜— ym p – p “ d— s y — – ˆ — – ˆ „ e ˜ ™ j — –m y “ – • ‘ – p – y l ‘ ˜— ˆ ‘ „ f ˜ ˆ ˜‘– j– €˜ y“ ˜‘– j j– ˆ ™ p ™ € d – ™ y y „•™ – ˜ u– ˜‘ „ p y ˆ – y y˜“ j™m s „ € d “m ‘ „ ‘ y „ – ˜ d j „ p ‘ – – € ˆ ‘ – € d — – ˆ „ e ˜ ™ j — – ˆ ™ ˜ p “ d “ p “ m – ‘ € – p ‘ „ p –m s— — „ d ‘ „ ˜ p ‘ ˜— ˆ – € ˜ y “ – ‘ y – ˜ € „ … — ym d “ m ˜— – – p ‘ “ ‘ – ˜ € “ d d “ ˆ – € y — – j “ m ˜‘ „ ˆ € – ˜— ym p y “ ˜ – s „ – y y “ ep ˜ ‘ “ ‘ l  — — “ ‘ – !¥  £ l ‘  € – ˜ — y m p ‘ –  ˜ € – • ‘ „ p – € ˜ ˜ y – d   l ‘  € – ˜ — y m p ‘ y y y ˜ ‘ – — ™ € d — k ˆ — € y – m m  “ ˆ — ‘ „ ˜ „ ‘ — ™ j € „ … — € – ˜ — y m p — – ˆ ‘ y p “ ep – p ‘ “ ‘ – ˜ € “ d d “ ˆ – m s “  € “ • ™ € l – ˆ ‘ y ˜ – s „ – y y “ ep ˜ ‘ –  p „ — — “  y y   — – ˆ „ e ˜ ™ j — – ˆ f € – ˜ — y m p – y y  ‘ y 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– ™  € “• — ™ ˜ — ‘ – ˆ – ˆ ˜ – — – mm “ ˜ – ˆ — € – ˜ — ym p — – ˆ – y y  ˜ “ j ™ m s „ € ƒ o† ¥ ¦ ¤ § £¢ ¢  ¡ £ ¥ ¤ £ ¡ ¢ ¡¢ ¢ ¤ £ Yo  ©©  © ©© – ‘ y ˜‘ „ — – € k ˜ € p —m – ˜ – ˆ — “ q †¤ £ ¥¤  £ £   ¡¨   ˆ — “ p – p — ‘ “ ˆ –m € “ d ‘ w † — € – ˜ — ym p € – ˜ ‘ ‘ „ ˜ “ € “ d ™ — “ m y „ € – ˜ — ym p “ € ˜‘ – ˜ € – ‘ m —m – ˜ f — – y y € ™ j y‘ — – € k ˜ € p — – ˆ € – —m ˜ ” † o † — ™ € d „ € d d “ — ym d – € ˜ ˜‘ “• y „ d — ˜ ‘ – j – d y „ € l – € — – € ˜ y “ ˆ f – ™ — • € – d y — ‘ „ ‘ – ˆ „ e ˜ ™ j – ‘ y € y „ d € –  ˜ ‘ – ˆ ˆ ˜ ‘ – ‘ ˜ € – d ˜— – m y y — – d y „ € l u y “ ˜ ‘ – j – € “ — — – p ™ ‘ — “ d ˜ ‘ – ˆ ‘ „ d — – € € „ p – ‘ ™ — • € – d y— – j km s „ € d ‘ y ˆ — – — — “ m p — –m — “ q † — –m “  ˜ ‘ — – — — “ m p — –m – • y „ € ˜ – € – ˆ „ e ˜ ™ j “ m — € – • € – — s „ ˜ – — ™ ˜ – y y ˜ ™ — – ™ ‘ ‘ „ ˆ – ˆ u y – — – ˆ € – —m ˜ ” † † — –mm – ™ € — – ™ ‘ ‘ „ ˆ u y “ — ™ —m “ € ™ ‘ ™ l – € ˜ — “ d ˜‘ – • y – d – ‘ — –mm – p  ˜ € “ — – ™ ‘ ‘ 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y l ‘ € – ˜ — ym p – p “ d — s y — – ˆ — – ˆ „ e ˜ ™ j — – m € y „ d f ‘ ‘ ‡ † ˜– “ € y “ ‘ „ f † – € y l “ m – ˆ € – ˜— ym p –m € y „ d f –m d j – u– €“ ƒ † € y „ d — € – ˜— ym p — – ˆ –m s j – — ‘ – m – ˆ – ™ — „ d j „ p – ™ ™ € p ‘ „ ˜ ˜ € “ d “ m ˜ – f — ™ p „ — — “ — ˜ – s „ — – p – ˆ – € s j „ ‘ –m f € – ˜— ym p y “ — ™ p „ — — “ — ˜ – s „ — – ˆ — – p ˆ ‘ — – ˆ –m s j – — ‘ – m f — € – ˜— ym p — – p – ˆ –m f — ™  ˜‘ – ˆ — € – ˜— ym p – ˆ – € s j „ ‘ –m “ € – ˜ „ ‘ ‘ w † – — “ s “ m – ˆ –m s j – — ‘ – m € y— — ™ ˜m “ ˆ „ j — – p – ˆ — – p ‘ – y y ™ € … — –m ˜– f ‘ „— ‘ – j ˆ – ˜ ˜ – p € y— — –m s— — „ d — ™ ˜m “ ˆ „ j — – ˆ –m s j – — ‘ – m “ € – ˜ „ ‘ ‘ „ f –mm – € „ l ™ ˜ “ p ‘ „ — ‘ – j ˆ – ‘ y € y „ ƒ † –m d j – u– € “ d — –m “  ˜ “ d— — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s – ˆ ‘ „ ˜ “ ˜ ‘ – j l – — “ m – ˆ – € ˆ “ p –m — ‘ “ ˆ ‘ – ˆm p y – – p “ d— – m ‚ – y y € ˜ ™ j „ ™ l — ‘ – — ‘ y “ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d— – m ¤ „ ‘ „ ˜ “ pm d d “ – ‘ y ˆ – € ˆ “ p –m — ‘ “ ˆ – p “ m d – — ‘ „ — – ™ ˜ • ™ ˜ ‘ “ ˆ ‘ – d – p “ € – — – — “ e d – ˜ ˜ – i † ˜ y ˆ ™ € — ym d –mm “• € – ˜‘ ‘ y € y — – ‘ ™ ˆ ‘ „— ‘ – j ˆ – ‘ y y y – p ‘ – y ‘ ˆ — ym d ˜ “ ‘ – l € “ m — ym d m “  ˜ ‘ –mm “• € – ˜‘ ‘ y € y — – ‘ ™ ˆ ‘ „— ‘ – j ˆ – ‘ y y y – ˜ • ™ ‘ „ f — ‘ # † ˜ „ — — ˜ – s „ — – ˆ ‘ „ ˜ ‘ ™ ˆ – ˆ –mm “• € – ˜ ‘ m – y y –mm – ˜ – ™ —m “ j € „ ‘ ˜— – – y y € ™ j y‘ ‘ „— ‘ – j ˆ – ˜ y „ ˜ – y y “ € – — „ d d y— ‘ w † ‘ „— ‘ – j ˆ “ m € y— € y –m “• “ — ˜‘ “ ˆ ‘ „ d— – € € „ p f – — “ s “ m – ˆ ˜ – s „ –m f €y„ d f “ € – ˜ „ ‘ ‘ w † — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d— – m ˜ ‘ “ j € „ … — ‘ „  — ‘ – j  ˆ – ˆ – € s j „ ‘ – m ˜ – – — “ s “ m – ˆ — ˜ – s „ ˆ – € s j „ ‘ – m p – •“ f –mm “ ˜ – ˆ – p  € ˜ “ j – ‘ y ˜ — – l ‘ € – ˜ — ym p – ˆ – j e ˜ € „ lm “ ‘ y ˆ – ™ € ˜ ‘ – m – y y u y “•“ € ˜ — „ ‘ – ˆ –  ˜ € “ d – € y – “ j “ m — ‘ “ ˆ — ‘ „ € ™ ˆ  — ‘ „ p — y „ ‘ f † ‘ „  ˜ p – — “ m — ‘ “ ˆ ™ •  ˜ „ j — ‘ „ •“ m — y „ ‘ – j j „ i k ∈ [1, K ] N1 = 4  d Dk Ck Sk ©    © keme © ©  S Ck D1 = {1, 2, 3, 4} Ck C k Nk ©  © © ©©  © © © M −i = (xi1 , ..., xiM ) → x © C1  ©  N i ∈ [1, N ]  P d M odalitesd F requencesd K [0, 1] S xid ©  ©  © ieme N ×M D  D  H E @ E  ¤  © © ¦ † – € ˜ y “ m ‘ „ ˜ “ pm d d “ – ‘ y ˆ € – € “• ˜‘ – • y – d l ‘ € – ˜— ym p y ˆ ˜ € ™ ˜‘ ˆ — – € k ˜ € p — –m ˜ – — – ™ ‘ ‘ „ ˆ — – ˆ — ™ ˜ ™ € d „ € d — –m f ˜ ‘ – j j – ˆ ™ p ™ € d ™ y y „ • ™ — ‘ „ •“ m — y „ ‘ – j j „ p f € “ p — m – ™ € — ‘ „  ˜ “ p m d d “ ˆ — “ p — – € ˜ y “ ˆ — ‘ “ ˆ – ˜ ‘ “ € “ l — “ d ˜— – ‘ ™ ˜ p “ p £– ‘ „ — f — – d “ ˜ ™ u y – ˆ — – p – ˆ — € „m — ˜‘ “ — “ …— ˜ “ — — ˜ “ ˜m y— ™ € — – ˆ ˜‘ – ˜ s „ – ˆ „ e ˜ ™ j – ‘ y  — – j j f ˜ ‘ “ ˆ ‘ – d – i † —  ‘ € y „ … — ˜ “ ˜ m y — ™ € — – ˆ – p ‘ – ‘  ˜ € – d “ m ˜ ‘ “ •“ ‘ – – ˜ ˜ – j m y y € y „ d ™ ˆ y ˜ ™ – ‘ “ j „ ˆ y ˆ ˜ € – d u– ‘ y ˆ € – — „ d— ˆ — € „m “ ˜ y “ … mr † —m – ™ € — ‘ „ ˜ “ pm d d “ ˆ — “ p — – ˆ – ˆ „ e ˜ ™ j “ m € – ˜ ‘ „ €… ‘ „ p – ˆ – € “ — — – p ™ ‘ – ˜ y— ‘ – ˜— – m f – € “ ‘ jm ™ € d – — “ e d –mm – ˜ – ‘ y — k € d # † — ™ ˜ ™ € d „ € d — – — € –  € ™ • – ˆ p ‘ „ ˆ ˜ – f — – ™m " € ˜‘ „ p — ‘ „ ˜ “ y ˜— — – ˜‘ – € ™  ˆ — ‘ “ ˆ ™ ˜ p “ p £– ‘ „ — € – • € – — s „ ˆ ˜ – j € – d “ m – p € “ p f —m – p  ˜ € “ — – ™ ‘ ‘ „ ˆ – ˆ u y – — – ˆ – p “ … – ™ ym “• ™ l ‘ € – ˜— ym p – ˆ – ˆ „ e˜™ j “m – ˆ ˜‘– j– ˜€ „ d j „ p –m €– ˜— – ˜ €“ d ˜‘–• y „— – p ‘– j j „ p ‘ „ f – y y˜“ € d ‘ ‡ † — – ™ ‘ ‘ „ ˆ – ˆ u y – – ˆ — – d ˜ — ˜ ‘ – € ™  ˆ ™ — m “ € ™ ‘ ™ l – € ˜ ˜ y – d – ‘ – ˆ „ e ˜ ™ j “ m – ˆ ˜ € ™ ˜‘ m f — ym d – ’ † – € ˜ y “ ‘ y y y € y –mm – j ˜ — – ˜‘ – j – d y „ € l – € ‘ y — – € ˆ – ˆ y „ ˜ € ™ ˜‘ ‘ „ — € –  ˜‘ “ y y – ˆ – y y ˜ “ j ™m s „ € d — ym d d y „ p y “ – s ˜ — – m f — ‘ – — y ˆ “ ™ ‘ ‘ „ ˆ ˜ ‘ – j – d y „ € l – € ‘ y  — – € ˆ – ˆ ˜ € – d u– ‘ y –m s— — „ d ˜— – m — — “ q † – y y  p ™ d— ‘ „ ˜ “ pm d d “ ˆ d j “ ep ‘ y — ‘ “ ˆ ™ ‘ ‘ „ ˆ l ‘  € – ˜ — y m p ‘ y ˆ — ‘ – — – m € – y m “ • ™ € y „ d ˜ € – d u – ‘ y € – —  m  ˜ ” † † — ˜‘ – ‘ ˜ € – d — y m d – € ˜ ˜ ‘ – • y – d ˜ ‘ – ep y “ • – ep – —  y y — € – ˜ — y m p — – ˆ f —  „ … € “ d € “ p € – —  m  ˜ y – € k ˜  € p ‘ s ‘ y — € y „ y „ ˜ — “ d ˜— – ‘ — € – ˜ — ym p — – ˆ ‘ „ ˜ “ € “ d ™ — “ m f –m d j – u– € “ ƒ † l ‘ € – ˜— ym p ‘ „ s ‘ y ˜— – y y – p – ˆ — – ‘ ™ ˆ ™ € d — ‘ „ ˜ „ ‘ — – ˆ ˜‘ – —m ˜ y —m € “ p ™ ˜ • ˜ p – s y — – ˆ – ˜‘ “ ˜ € „ d j ˜ € “ d © © © © ©  © ©  © !  ! ©  © ©  ©  ©  © © © "   o ¤ § £¢ ¢  ¡ £ ¥ ¤ £ ¡ ¢ ¡¢ ¢ ¤ £ € y– y l ‘ „m ¦ † — ‘ „˜“ ˜ „ ‘ – ˆ –m d j– u ‡ † Y # o ¥ $ 32 1  − → − x4 → x3 − → − x2 → x1 C1 ¡ ¡ ¡ ¢ ¡ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ YYY Y Y Y o YYYY u € d oo ¦ ¡ ¢ ©¨ ¡¨ ¢ © — ‘ „  ˜ „ ‘ u y – ˆ € y — — – ™ — “ s ˜ ‘ „ — — – € y — – j — – i † ¤   ¥ £ m ˜ – ¤¥ ¡ ¤  “ m ‚ — – ™ — m ˜ y ˜ ‘ – j – y y  — — “ m p ˜ ‘ „ — l ‘  € – ˜ — y m p y ˆ ¤ £ ¥ ¤  ¤   ¨  ¡  “ m – ˆ — – € y — – j u y – ˆ f – ™ ‘ ‘ „ ˆ – y y “ e p – ˆ — – y ‘ ‘ „ p — – — — “ m p — – ˆ ‘ „ ˜ p ‘ „ … ‘ – l ‘ € – ˜— ym p ‘ y ˆ ™ ˜m “ y y “ m € – ym “• ™ – ˜ “ e y „ — ‘ „ m — f ‘ ‘ ‡  © ¢ ¡ ©  K  k =1 i∈Dk E 2 (P ) = →µ dist(−i , −k ) x→ ‚ — – ˜ ‘ – ‘ ˜ € – d — ym d — –m – j j „ p — – ™ € ™ ˆ — ‘ „ p — € „m “ ˜ ‘ “ ˜ ™ – € y— – j – ˜ ˜ – p ˜ ‘ – — j ‘ j  y y — ‘ „ ˜ ym „ — — –m f – y y ˜ “ € ˆ “ y y € y – € € – m ˜— – l ‘ € – ˜— ym p ‘ y ˆ ¤ £ ¥¤  £ ¨  ¡  “ m € – ym “• ™ € y „ d ˜ ‘ – j – y y — — “ m p – ™ —m ˜ y – € y— – j – ‘ y f – d p ‘ € d – j j –m € y— – ™ — “ x © 1 Nk  ©  Radius(Ck ) = i∈Dk →µ dist(−i , −k ) x→ © ‚ € – ˜— ym p y ˆ € y – € ™ ˜‘ m — ‘ „ ˜ “• € – — s „ — – ˆ ™ ˜ € “ m j— “ m € – —  j u “ j € y „ d € – —  j ‘ j f – ˆ  „ € ˜ ‘ – p ‘ „ — ˜ – € – ˜ — ym p y ˆ — – € s j – j – € ˜ ‘ – – ‘ ‘ – „ j – p ‘ “ ˜— ˆ f € – ˜— ym p y ˆ — y ˆ “ € –m ˜ — – – ™ —m ˜ y ˜ ‘ – • y „ — — ym d “ m – € y — – j “ m f € – ˜— ym p ‘ y ˆ ¤ £ ¥¤  £ “ m € – — € ™ ˜ p “ € “ p € y „ d — – ™ —m ˜ y – € ˜ ˜‘ – • y – d — – € y— – j — – ˜‘ – € ™  ˆ  — f – j j – ’   £   £ ¢ (xid − xjd )2 M  © ! d=1 →→ dist(−i , −j ) = xx  † † † # — – u – ‘ ‘ “ ‘ – – l “ ˜ ‘ “ •“ ˆ — ‘ „ ˜‘–— ™ € d ‘– — y „ † — – ™ l“ — •‘– – €˜ ˜‘– j–m“ l™ ˜‘–• y– d — – € y— – j — – €˜ y“ ˆ —“ q † — k € d“ p – ™ ˜ ‘ – — ™ € d f – ‘ ‘ – ˆm p y – – p ‘ “ ˜ — ˆ “ m ˜ ‘ – j –m s “ ˜ s y ˆ ‘ ˜— – – ™ —m ˜ y — ym d “ m – € y— – j “ n † — ˜ – s „ – € ˜ ‘ – ™ ˜ € “ m j— “ m € ‘ ™ ˆ € y „ d — – ™ —m ˜ y – € ˜ ˜ ‘ – • y – d – p ‘ “ ˜— ˆ – ˆ — – € y— – j — – ˜‘ – € ™  ’ † – p ‘ “  € “ •„ p – ˆ – p  € ˜ “ j “ — ˜ – – ˆ  „ € ˜ ‘ – p ‘ „ — ˜ ‘ – j – y y  d ˜ ‚ ˜ ‘ – j – y y  ˜ —  ˜ “ ˜ — y „ † ˜– ˜– ‚ € – ˜— ym p y ˆ — – € s j – j — – ˆ –m s j – — ‘ – m ˜ ‘ “ ‘ – ˜‘ „ p m “ j ‘ j –m l ‘ “ ˜ p – € € – d  e f € – ˜— ym p y “ ˜ ‘ “ ‘ – ˜ € “ d d “ — ˜ – s „ — – ˆ ¨ ¥ £ § ¤¥ ! £   – m ˜ ‘ – j – y y  d ˜ ‚ – • ˜ p ‘ „ ‘ „ p – y y  l „ m ‘ „  — — – € d u – – ‘ y †   maxd = max{i∈Dk } xid }     mind = min{i∈Dk } xid  ©  ! ©  {[mind , maxd ] | d ∈ S  " {i ∈ Dk | ∃ d ∈ S ∀ j ∈ Dk , xid ≥ xjd } ∪ {i ∈ Dk | ∃ d ∈ S ∀ j ∈ Dk , xid ≤ xjd} ‚ € – ˜— ym p y ˆ — – € k ˜‘ „ € … u y “ ˜ ‘ – • y „ € ˜ – —  y y u y – p ˜ ‘ – j – y y  d ˜ ‚ € – ˜ — y m p y “ ˜ ‘ “ ‘ – ˜ € “ d d “ — ˜ ‘ “ ˜ —  ˆ — ˜ – s „ ˆ – m s j – — ‘ – ‘ y † o ™ m – d d “ f ™ ˜  •“ € l – ˆ – € ˜ ‘ – p ‘ „ — – ˆ – ep „ € d f € – ˜ — y m p y ˆ — – € s j – j — – ˆ ‘ y m † © ¤ ! !  © −= 1 → µk Nk i∈Dk −i → x  ‚ € – ˜ — y m p y ˆ ™ ˜  • “ € l – ˆ – € ˜ ‘ – p f ¤ ! ¥  £ ¤¢ ‘ „ — † ‚ — – ˜‘ “• y— — –m ˜ ‘ “ ˜ ™ ˜‘ – j – y y— — “ m p — – ™ —m ˜ y — ym d — –m — ‘ „ ˜ “ ˜ ‘ – — ™ € d – € — –m f — € – ˜— ym p — –m € – ˜ ‘ – — ™ € d – € – ˆ — ‘ „ ¡ “ … — – ˜‘ – € ™  ˆ – ˜— u– mr −k → µ Ck ! "o ¤ § £¢ ¢  ¡ £ ¥ ¤ £ ¡ ¢ ¡¢ ¢ ¤ £ † – y ˆ ‘ – ˜ ˜ “ – ‘ € – ˜ u– ‘ „ ˜ ym „ — “ m u y – j –m ˆ ‘ „ d— – € € „ p  y y –mm – p ˜— – p € “ p – ˜ ‘ – ‘ ˜ € – d — ym d “ m – j j „ p – ™ € ™ ˆ— ‘ „ p — € „m “ ˜— – f – ™ p „ — — “ – d „ € ˜‘ ‡ m – — j ‘ j y „ f – ™ p „ — — “ – € y— – j v “ m – — j u“ j  y y ‘ „ ˜ ˜ € “ d “ n © ©  K  P k =1 E (P ) = Nk × Ek N ‚ – m m  “ ˜ € y – m € “ d – ™ € ™ ˆ ‘ „ d f € – ˜ — y m p – y y “ ep – ˆ — – d „ € ˜ ‘ ‡ — – ˆ – j j „ — “ m € “ d – ™ ym “• ™ ˜— – ‘ „ ˜ ˜ € “ d – ‘ y ˆ – d „ € ˜‘ ‡ m f ˜‘ – j –m “ ‘ ˜ ‡ © i P recision(i, k) × log P recision(i, k)  ‚ ˜ y— – j j „ p – ™m y pm “ p –mm – ˜ ‘ “ y y ˜— – Ck  ©  Ek = − € – ˜— ym p y ˆ Ek 2 × Rappel(i, k) × P recision(i, k) Rappel(i, k) + P recision(i, k) © – d „ €˜‘ ‡ n F (i, k) =   ‚ – p ‘ “ ˜ € „ d j – j j “ m ˜‘ – “ ‘ „— p ™ € ƒ “ m ˜ – m – d d “ g –m – y y – p ‘ „ ¡ “ … – ˆ ™ ‘ ‘ „ ˜ — „ d ˜ ‘ – j – y y  d ˜ f ‘ „  —  p ™ € ƒ “ m € y — m – d d “ g y ˆ – p ‘ “ ˜ € „ d j m € – m " € ˜ ‘ „ p € y „ d ™ — m ˜ y ˜ — – ¤ „ © 32 1 β F (i, k) = (1 + β ) × Rappel(i, k) × P recision(i, k) β × Rappel(i, k) + P recision(i, k) i F (P ) = N Ci M axK F (i, k) k =1 N  ‚ ˜ y — – j j „ p – ™m y pm “ p ˜— – – € y— – j v “ n † € – ˜— ym p –m — ‘ “ ˆ – — — “ m p “ m – ˆ — – € s j – j – ˆ – € s j „ ‘ –m ˜ – f € – ˜— ym p y ˆ — – € s j – j – ˆ – € s j „ ‘ –m f – — — “ m p “ m – ˆ — – € s j – j – ˆ f £   ¢ ¥ “ m ˜ – ¨¤   ¢ – m ‚ † – € y l “ m € “ d — – ™ € ˜ — ymm ˜ ‘ „ —  y y – € s j „ ‘ –m ˜‘“ ˜ „ ‘™ ˆ Ck i Nk i N Ci Ck Nik   † ‘ „— p ™ € ƒ – ˆ ˜ – m – d d “ g – ˆ — ‘ „ ˜ „ ‘ — – ˆ ‘ „ ˜ “ € ˜— ymmr † ¥ ¦ Rappel(i, k) = Nik N Ci P recision(i, k) = Ck Cik Nik Nk Ci o ¦ ¡ ¢ ©¨ ¡¨ ¢ † — – ˜ ‘ “ ˜ —  u – — – ˆ „ e ˜ ™ j u y “ ˜ € „ d d “ € € “ d — – l “ ˜ ‘ “•“ — ‘ “ ˜ € – p – ˆ k — — „ d — ‘ „ — „ d „ € d — y „ ‘ – y y – ˆ „ e ˜ ™ j –mm – • y „ ‘ “ m – y y — ‘ „ € € – • — y „ ‘ f – € „ p ‘ – n † l ‘ € – ˜— ym p y ˆ – € ˆ “ p –m — ‘ “ ˆ ‘ „ ˜ “ ym “• ™ m – ˆ – ˜ ‘ “ ˜ € „ d j – ˜ € – • y „ – y y ˜ “ j ™m s „ € d “ m – ˆ € „ s “ – ˜‘ “• y— r r r – ˜ € “ d “ m f ‘ ‘ ‡ † € y – ˜ “ —m ˜ y m € “ d –m s “ ˜ ™ € d € – ˜ ‘ ˜‘ – j –m p “ … ˜ “ j € „ … ‘ y — y „ — — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d “ m ˜ – f € y – ˜ “ —m ˜ y m – ˆ ˜ € “ d “ m – ˆ — – — y y – €  € „ € d “ — – p ‘ “ — — “ ‘ ‘ „ p – ˆ – € s j „ ‘ y ˆ ‘ „ ˜ “ — j ‘ j “ m – € km y p ˜ € “ d ‘ „ ˜‘ – ˜ ˜ “ – ‘ y p ‘ „ ˆ — ‘ „ ˜ € „ d — y „ † — – p ‘ “ — — “ ‘ ‘ „ p – ˆ ‘ „ ˜ p “ € ˜ u– m – ‘ € – p ‘ „ p — ‘ „ — • — y „ ‘ – y y ‘ „ ˜ “ pm d d “ n † — – ˜‘“ ˜— u– — – ˆ „ e ˜ ™ j u y “ ˜ € „ d d “ € € “ d — ˜ y „ ˜ “ — ‘  “ ˜ € – p ˜ ‘ – ˜ ‘ – — ™ € d  y y ˜ – f – ™ y y „ • ™ — ‘ „ •“ — y „ ‘ – y y – ˜ u– ˜‘ „ p y ˆ – ˜‘ “ ˜ € „ d j ˜ – – ˜‘ – p ™ € – y y ˜ “ j ™m s „ € d “ m – ˜ d j „ p ‘ – ˜‘ – ‘ ‘ – € d  y y f l ‘ € – ˜ — ym p – ˆ — – ˆ „ e ˜ ™ j — –mm – • y „ ‘ u y – ˆ – ˜ y — ‘ – — ‘ „ — „ d „ € d — y „ ‘ f ˜ y—  y y r r – ˜ € “ d “ m — ‘ “ ’ † # — – u– ‘ ‘ “ ‘ – — –m s ‘ „ d— ˆ ˜‘ – j –m “ l ™ ˜ ‘ “ ˜ ™ — m “ ˜ ™ ˆ – ˆ – l “ ˜ ‘ “•“ ˆ f l ‘ € – ˜ — ym p ‘ – — – y y  — — “ m p — ym d — – m — – ˜ ‘ “ ˜ —  u – — – ˆ „ e ˜ ™ j — – ˆ – ™ m m  “ ˜ ™ ˆ — y m d ‘ „  ˜ “ ˜ ‘ – — ™ € d “ m ™ € p “ — ‘ „ p ˜ — – ˜ ‘ “ •  y — – € ˜  d “ ep – n † — – ˜  j  m — – — ˜ – — ˜ y „ ˜ “ — – — – ‘ y p “ e p p – •“ — – ˜ ‘ – € ™  ˆ l ‘  € – ˜ — y m p – ˆ — – ˆ „ e ˜ ™ j — – — y – € s j „ ‘ – ˆ  — — y “ – ˜ —  u – m r † ™ € ™ ˆ— ‘ „ p – j km s „ € d y ˆ ‘ „ ˜ p ‘ „ … ‘ – –m s “ m “ ™ € d y “ ™ y ˜ p – – – € ˜ ˜ „ ˆ — ˜ – s „ – € ˜‘ – ™ ˜  € “ m  j  — “ m € – y m “ • ™ € y „ d – ™ —  m  ˜ y – p ‘ “ ˜ —  ˆ – ˆ ‘ „  ˜ p ‘ „ … “ m – ˆ ˜ ‘ “ ˜ € „ d j  u  „ ep – m f – € „ p ‘ – n † † # — – u – ‘ ‘ “ ‘ – — – m s ‘ „ d —  ˆ ˜ ‘ “ ˜ ™ ˜ ‘ „ d – p € y — — m “ ˜ ™ ˆ – ˆ – l “ ˜ ‘ “•“ ˆ f — ˜ – s „ – € ˜ ‘ – ™ ˜  € “ m  j  — “ m – ˆ — – m s  — — „ d — ‘ „  ˜ „ ‘ — – ˜ ‘ – € ™  ˆ – ˜ —  u – m  y y y • ˜ ‘ – j – m “ l ™ — ‘ „ •“ — y „ † — ‘ „ — – s — – — € –  ˜‘ – ˆ ˆ ‘ “ – ™ — • ‘ „ ˜ “ pm d d “ m € –  ˜ ‘ – ˆ ˆ – € ˜ ˜ ‘ – y y ™ — ‘ „ p € “ d ˜ „ ˆ l ‘ € – ˜ — ym p – ˆ – ˆ „ e ˜ ™ j – ‘ y ˆ ‘ „ ˜ “ —m ˜ y m – ˆ y „ – € • y ¤ ‘ – – — j “ m – ˆ – d “ ˜ ™ – € k j – € d “ n † ˜‘ – € ™  ˆ – € ˜ ˜ y – d  y y l ‘ € – ˜ — y m p y ˆ ™ ˜  m “ y y “ m – ˆ – € k ˜  € p ‘ y p ‘ „ ˆ ˜ – f ˜ ‘ – € ™  ˆ …  ˜ p – s „ ‘ y ˜ ‘ “ “ – ‘ y p “ e p f l ‘  € – ˜ — y m p y “ — ‘ „  ˜ “ p  m d d “ — – ˜ ‘ – € ™  ˆ – ˜ —  u – m  y y y • p ‘ „ ˆ — ‘ „ •“ — y „ ‘ f – € ˜  d “ ep – p — ‘ “ ’ D @ I H  ©    ©   © © !   © ©  © ! ! © ! © ! !  ©     ©© ¦ o ¤ § £¢ ¢  ¡ £ ¥ ¤ £ ¡ ¢ ¡¢ ¢ ¤ £ $o ˜ y „ ˜ ‘ „  ˜ y m „ — – ‘ y p – •“ ˜ ‘ – € € “ j ™ ˆ — – ˜ ‘ “ ˆ ‘ – p — “ — – ˆ „ e ˜ ™ j — – m ‚ ‘ „  ˜ “ —  m “  p ™ d — ˜ – ‘ „  ˜ “ —  m “ € ™ ‘ ™ l – € ˜‘ – – l “ — — ˜ ‘ – € d d “ ‘ – – y y— — “ m p – p ‘ – € ™  ˆ “ m y “ – • ‘ – p p ‘ „ ˆ – • y „ € ˜ – € ‘ w † – — “ s “ m — ‘ “ ˆ — ˜ – s „ ˆ – y y — € – ˜ — ym p – ˆ ˜ ‘ “ ˜ y “ –  ep € “ € ™  e “ m – ˆ — – mm y – … u y “ € ‘ – ˜ s „ y y — y – p ˜ – f –m s— — „ d — ˜ ‘ – € ™  ˆ — ym d — –m ˜ ‘ – „ — — ˜ ‘ “ ˜m y— ™ € — € – ˜— ym p — –m – y y – p – € k ‘ j – ˆ — € – ˜— ym p — –m ˜ ‘ – j – •— — – p p y— ˜‘ “ — • ˆ — y d f – — “ s “ m – ˆ — ˜ – s „ — – ˆ – m s j – — ‘ – m ˜ ‘ “ ‘ – ˜ ‘ „ p € – ˜ — y m p – y y  ‘ y ‘ y p – •“ ˜ ‘ “ € € “ j ™ ˆ f – ˜ ‘ “ ˆ ‘ – p — – ˆ – ˆ „ e ˜ ™ j “ m ˜ – † – ™ j € „ … –  ep € “ € ™  e “ m – ˆ – ‘ p “ € “ m ™ p „ ˜ — € – ˜— ym p – y y ‘ y ‘ y — ‘ “ ˆ — ‘ y ™ € ˜‘ – „ — — ˜ – s „ — –m — y „ ˜ – y y – p y y — y f — – € “ m j— — ym d — –m – j j „ p — ™ € ™ ˆ— ‘ „ p — € – ˜ — ym p — –m ˜‘ – j – •— — – p p y— ˜ ‘ “ ‘ ‘ „— y … — y d f – — “ s “ m — ‘ “ ˆ u y “  ˜ ‘ — ˜ – s „ ˆ – y y — € – ˜ — ym p – ˆ ˜ ‘ “ ˜ y “ p – •“ ˜ ‘ “ € € “ j ™ ˆ f – ˜ ‘ “ ˆ ‘ – p — “ – ˆ „ e ˜ ™ j “ m † ‚ — – m “ d  p ‘  € d — – ˆ „ e ˜ ™ j u y – ˆ – ˜ —  u – m  f — € – ˜ — y m p – ˆ –  ep € “ € ™  e – m m – ˜ – ‘ y € – j € „ … – ˆ ‘ # † — – € y ˜ „ • — –m ˜‘ “ ‘ € – p ‘ „ p –m d j – u– – € ˜ „ ‘ – ˆ — “ p –m — ‘ “ ˆ –  ep € “ € ™  e – m m – ˜ – ‘ y – € ˜ ‘ „ j † o – € y l “ n † — – €  “ m  j  — – j j „ p — ™ € ™ ˆ  — ‘ „ p — ˜ – s „ ˆ – d y „ € l ‘ “ ˜ € – p ‘ y — – y y  p ™ d— ˜‘ „ — — € – ˜— ym p — –m — ym d f – € s € “ m — ‘ “ ˆ ˆ ‘ – p — – ˆ ‘ „ — ym d — y d f – — “ s “ m – ˆ — ˜ – s „ — – ˆ –m s j – — ‘ – m ˜‘ “ ‘ – ˜ ‘ „ p € – ˜— ym p – y y ‘ y ‘ y ™ p „ — — “ ˜— – – € s € “ m – ˆ – ‘ p “ € “ m © † — € – ˜ — y m p – ˆ –  e p € “ € ™  e – ‘ y € – ™ € p – ˆ ˜ — – – y y  ep € “ € ™  e l ‘  € – ˜ — y m p y ˆ ˜ ‘ – j – ˆ ‘ „ … – n C A ¢H £ B @ B H £ ¦ DH B C E AI F  ¥   © ©  © ! !   ©  ©    ©  ©   © ©   † ˜ ‘ – ˆ ™ p ™ € d – € ˜ d “ ep – m — ‘ “ ˆ ™ — „ d „ € d — ‘ „ •“ — y „ ‘ – y y l ‘ € – ˜ — ym p – ˆ — – ˆ „ e ˜ ™ j u y “ — – ™  p „ — — “ — – y y  ˜ —  € ™ ˜ p “ € “ p — – ˆ y “ – m s “ ˜ y ˆ – ˆ “ m ˜‘ – j –m “ ‘ — ‘ „ j y — ™ € — y „ ‘ – y y f — – ˜ jm — – — ˜ – — ˜ y „ ˜ “ — – — – ˜ y— ‘ – — ‘ „mm “ ˜ ™ ˆ — y „ † – € ˜ d “ ep y ˆ † – € y l “ m — ‘ “ ˆ ™ ˜ ‘ – — ™ € d — – € y ˜ „ • — – m ˜ ‘ “ ‘ € – p ‘ „ p – m d j – u – m € y — ‘ „  ˜ “ p  m d d “ ‘ „ — — ‘ „ € ˜ ‘ „ j — y „ ‘ —  y d f — – m s  — — „ d — – •  ˜ “ ‘ € – ˜ m “ — – ˜ ‘ – € ™  ˆ ˜ ‘ – j – m m – y ˜ ‘ – • ™ p – •“ f m “ € ™ ‘ ™ l ˜ ‘ – j – ‘ ‘ „  ˜ p ‘ „ … ‘ „ — ˆ € „ s “ ˆ — ‘ „ ˜ ‘ – — ™ € d — y „ ‘ f — – ˆ „ e ˜ ™ j — – ˆ – ‘ y p “ ep € y „ ƒ † # — – u– ‘ ‘ “ ‘ – — ™ — „ d „ € d – ˜ y— ‘ – ˜‘ „ — —m “ ˜ ™ ˆ – ˆ — ym ƒ † — – y y— — “ m p — ym d — –m — – ˆ „ e ˜ ™ j — – ˆ ‘ „  € „ e ˆ € y „ ˜ ‘ y € – ˜ ‘ – — ™ € d – ˆ  —  „ ep p ‘ „ ˆ — ‘ „ • “ — y „ ‘ f – € ˜  d “ ep – p — ‘ “ ’ † € y – ˜ p – m – m € y „ d € – € l ™ ˜ ‘ – m p £ ˆ — k € ˜ ˜ ‘ – j – m s “ s „ € d ˜ “ € – — ˜ – — ‘ „  ˜ “ l ˜ — – • ‘ ˆ — ‘ “ —  „ € ˜ – ˆ — ym d ˜ ‘ – j – € — ˜ “ € – ˆ ‘ “ j – ˆ – • ˜— y “ e u– – ˜—m – ‘ y € ‘ € y „ … ‘ – f — – ™ — „ d „ € d ™ ˜ ™ ˜ ‘ „  y y — – • ˜ “ ‘ € – ˜m “ ˆ ˜ – — – ™ d d „ m – • ™ ˆ ™ ˜ ™ ˜ ‘ „  y y l ‘  € – ˜ — y m p – ˆ — – ˆ „ e ˜ ™ j – ˆ ˜ ‘ “ ˜ € „ d j  – € s j „ ‘ y “ – p “v   ©     © ! ©  © © ¥ ¥ ¦¥ £ ¡ §¨ £ ¤ © ¡ ¤ ¨ ¡ ¤  ¡ §¢ § ¦ ¥¤ £ ¢ ¡ © † – m d j – u – m € y — – y y  ep € “ € ™  e l ‘  € – ˜ — y m i ¥ Y  #  !  $     " o      Y  # ! $ † o ¦ 32 1 " o   € – ˜ — ym i  € – ˜— ym i o € – ˜— ym i !o ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ −k = 1 → µ Nk i∈Dk −i → x ‚ — ™p „— — “ ˜ ‘ „ — € y –m  y y — ˜ – s „ — – ˆ ‘ „ ˜ p ‘ „ … ‘ – — € – ˜ — ym p — – ˆ — – ˆ  „ € ˜‘ – p — –m € y „ – € ˜ ˜ – j ‚ – € ˜ y “ m ‘ „  ˜ “ € ™ ˜ – ‘ y ˆ € – ˜ — ym p – ˆ – l ‘ “ ep ˜ – s „ ‘ y — ‘ „ j y “ y y ˜ ‘ “ ˜ —  y d † o – ep „ € d — y m d – m ˜ — – – ˆ  „ € ˜ ‘ – p – m ˜ ‘ „ ˆ € – ˜ — y m p y “ ˜ – s „ – y y “ ep € – ‘ l  — — “ † — ™ ep € – e p – € — € – ˜ — y m p — – m ˜ ‘ “ ˜ ‘ – — ™ € d – € u y “  ˜  ‘  — – ˆ  „ € ˜ ‘ – p — – ˆ – m s j – — ‘ – m ˜ ‘ „ € – j € „ …  y y – — “ s “ m – ˆ — ˜ – s „ ˜ ‘ – j – € „ ˜ “ ™ m “ € —  „ ep † ‚ – ˜ ‘ “ • y — “ m ˜ — – f † o – € y l “ m € “ d – ™ € ˜ — ymm f ¢ s f† m “ ˜ – “ ˆ  ’¡ — ‘ “ – j l ‘  € – ˜ — y m p – ˆ – ˆ „ e ˜ ™ j “ m f — ™ ep € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ DH B C E AI F  ¥ © ©  ©  K     K [email protected] £¤ ©  ¥   T  ! 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©   ©  © !  ©  ©©  ! © ©     !  ©    ! !   #o T   ¦ ¡ ¡ ¤ § ¢ ¡ © ¤ ¨ ¦ £ £ † – ‘ € – ˜ ‘ ™ ˜m “ y y – ˆ – € k ˜ € p ‘ “ ˜ € – p ‘ y ˜‘ “ — j ˜ d „ ˜ “ ˜m y— ™ € –m € – ‘ ‘ „ ˜ p –m ™ — – ˆ — y d f — – ˆ  „ € ˜ ‘ – p — – ˆ –m “  ˜ ‘ ‘ „ ˜ p –m ™ — “ m – € „ ˜ “ ™m “ ‘ „ ¡ “ … – ˆ € – € “• ˜ ‘ “ — “ … ‘ – – ˆ „ e ˜ ™ j “ m € – € ™ ˜ ˆ ˜— – ‘ „ ˜ “ ˜ jm – ˜ ˜ – p € –mm “ d € y „ d –  —  „ ep ‘ „  ˜ ym „ — “ m f — “ p — – ˆ ˜ € “ d ym d “ m — ‘ “ ’ † — – m “ p „ m — – m “ j ˜ d „ — ‘ „  ˜ ym „ — — – ˆ — € – • – l € – • ‘ „ p ˜ – — – ˆ  „ € ˜‘ – p — – ˆ –m “  ˜ ‘ ‘ „ ˜ p –m ™ — “ m –m s — ‘ – — — k € ˜ — — y “ ˜— – – ˆ „ e ˜ ™ j – ˜ ˜ – i † — – y y  p ™ d— — ym d p ‘ „ ˆ ˜ ‘ „ —  y y f — € – ˜— ym p – ˆ ˜‘ “ ˜ € „ d j — ym d – € s j „ ‘ ‘ y € y— — ™ — “ s u y – p — – u–m d j „ p — ym d — –m k ˆ „ j — –m – —m “ ‘ ™ d  y y – j € – ˜ – € ˜ y “ ‘ y ˆ ˜ – l ‘ € – ˜— ym p y ˆ – ‘ € – ˜ ‘ ™ ˜m “ y y “ m ˜ ‘ “ ˜ ™ – € – j € – ˜ ‘ y ˆ ™ — „ d j „ p ˜ ‘ – j – y y— — “ m p ˜— – – € k ˜ € p m – ˜ ‘ y y y — y d f  —  „ ep – m k ˆ „ j y ˆ ‘ „  ˜ “ — m “ € ™ ‘ ™ l ˜ – ™ ˜ p  p ™ d — – € ˜ ‘ – – y y  — — “ m p —  j „ € d j „ p – m – • y „ € ˜ – € ‘ „ f – y y  ˜ —  ˜ “ ˜ — ˜ — – ˜ – p – ˆ ‘ „  ˜ p y € ˜ — ‘ „ p “ m — ‘ “ ’ † ¢ o f ‘ “ m ‡ ˆ ‘ “ m € – j “ ¡ — – ™ ‘ ‘ „ ˆ u y “ ™ € d „ € d d “ — ym d –m ˜‘ – j – ‘ ‘ „ ˜ ˜ € “ d –m € – ‘ ‘ „ ˜ p –m ™ — – ˆ ˜‘ “ ˜ ˜ – j € – d – y y ˜— ˜ “ ˜— ˜— – ˜ ‘ y € – —  m  ˜ y ˆ —  y d f — ™ ep € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m ˜ ‘ “  ˆ „ j ‘ – — ‘ „  ˜  ˜ € “ d — € y –  — y m d € – € ™ ‘ ™ l – ˆ ˜ — – – j k m s „ € d – p – € ˆ y „ — ™ € € y „ d –  —  „ ep ‘ „  ˜ y m „ — “ m f ˜ ‘ – j – y y  d  t † – y ‘ ‘ „ p — “ d ˜ — – ‘ ‘ „  ˜ “ j € „ … ‘  – ˜ ˜ – p f — “ p u y – € s j „ ‘ – ˆ — ‘ “ ˆ f ˜ ‘ – j – — y – € y – e m “ q † — ™ ep € – e p – € — € – ˜ — y m p – ˆ – € s j „ ‘ y ˆ ‘ „ ˜ “ p  p ™ d— “ m – ‘ € – p ‘ „ p – € ˆ “ p – p — ‘ “ ˆ – ˜‘ “ ˜ € „ d j – y y ˜ “ j ™m s „ € d – ‘ ” † — € – ˜— ym p u y “ — ˜ – s „ — – ˆ — ‘ „ ˜ “ ‘ l— — “ — –m — – ˜ y „ ˜ — k € d “ ˜‘ – j ™ ‘ “ ˜m y j— € y „ — j ˜‘ „ — — – ˆ  „ € ˜‘ – p — – m — y „ ˜ ˜  „ — f € y „ —  j ˜ ‘ „ — ˜ ‘ – j – l ‘ “ ep – m € “ d — ™ ‘ € – p ‘ „ p — € – ˜ — y m p — – ˆ — – ˆ  „ € ˜ ‘ – p u y – ˆ — – m f ™ € ˜ ‘ „ p ‘ – € ˜ — – ˜ – s „ ‘ y ˆ ‘ „  ˜ “ ‘ l  — — “ ˆ ˜ ‘ – j – l ‘ “ ep ‘ y y y —  „ … – y y “ ep ˜  „ — ‚ ‘ „ ‘ y „ –m “ ˜‘ – j ™ € p ‘ – € k ‘ “ j – ˆ – ™ y ˜ p – – – € ˜ ˜ y – d – € y ˆ ™ p „ € d – ˜ ˜ – p – ˆ – d “ ˜ ™ – j k — „ € ˜ “ n ˜– ©   ©  ©   © ! ©  © YY  ©  ©  ©  Dk = {i ∈ D| ∀ j ∈ [1, K ] →µ →µ j = k, dist(−i , −k ) < dist(−i , −j )} x→ x→ j  u„€ d € y–m – ˆ ‘„˜p ‘„… ‘– —€–˜— ymp uy“ —˜– s„ —–‚ˆ ——–‘ˆ„„˜€“˜‘‘l–p——u“y“—––m•€yy„„‘ uy–“€˜™˜˜– j  † – y y  ep € “ € ™  e l ‘  € – ˜ — y m p y “ — – ™  p „ — — “ — – y y  ˜ —  € ™ ˜ p “ € “ i y„ y„ y„ y„ ‘„ ‘ ‘„ ‘ ‘„ ‘ y„ ‘„ ‘ ‘„ ‘ y„ O(M × N 2 ) –  ep € “ € ™  e —m y – — y „ — €–˜—DyH Gmp" – ˆ –€ s j„ ‘ ¥ ¥  † o ¦ ¥ ¤£ — – y y € ˜‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– –—€ d ˆ€“ e – j ˜ ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜— ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d !©   €„ D€d(“ ©—W– p‘F“©——G“G‘‘„p Y" ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ § 0 ' ¤  ¨ § ¦ ¦¥!¨¨"© ¤' ¦'¦£0¦ ¨§ ¨§ ¨§ ¨§ ¨§ ¨ ¨§ ¨ ¨§ " ¥ ' ¨ ¤  ¨  ¤ ' £ ¥ ¤£ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢¡ ¢ ¢¡ ©  † –m d j – u– m € y — — ‘ “ – j l ‘ € – ˜ — ym i Y # € y– y l ‘ „m Y # € y– y l ‘ „m o ¥ $ † o ¦ 32 1  YYY Y Y Y o  — – ˆ „ €˜‘– p uy“–• y„ o $ YYYY u € d  YYY Y Y Y o u y “  ˜ ‘ —– ˆ„€˜‘– i YYYY u € d " ¦ ¡ ¡ ¤ § ¢ ¡ © ¤ ¨ ¦ £ £ ˜ ‘ – j – y y  — — “ m p – — „ d d y — ‘ „ f – y y  € ™ j y ‘ – d ˜ – ˆ ˜ ‘ „ — — – ™ ‘ ‘ „ ˆ — – m – y y — € „ m f – m d j – u – € “ ƒ † – l ‘“m™ j y ˆ — – ˜‘“ — „ d j „ p — – ˜‘– €™  ˆ u y“ — ˜– s „ — – ˆ — ‘ „˜“ ˜p – “ u y“ ˜‘“ ˆ ‘ „ d— – €€ „ p –mk ˆ „ j y ˆ — ™ ep “ p — – € ˜ k j “ € “ d — – m ˜ – f – l ‘ “ m ™ j y ˆ — ™ ˜m s “ s „ € d — – m – y y  — ‘ “ f — ‘ „  ˜ y s € ˜ —  ˆ — – p – ˆ — – €˜k j“ €“ d — –m f — – ™•€–— s „ — – ™ ‘ ‘ „ ˆ — – ˆ ‘ „˜p ‘ „ … ‘– f €–• y „ €˜ – ˆ — € „m“ ˜‘“ ˜™ – y y˜“ j™m s „ € d “ m f — ™ ˜m s “ s „ € d – ˆ — ‘ „ ˜ y s € ˜— ˆ – ˆ – l ‘ “ m ™ j ‘ y € “ d — – ™ € ™ ‘ ™ l ™ ˜ ™ ˜‘ „ — – ™ ‘ ‘ „ ˆ — –m – y y – € k ˆ— ‘ „ p ‘ w † — y ‘ ‘ „ p ‘ — – € ˜ k j “ € “ d – ˆ — “ j f – y ‘ ‘ „ p – y y € ˜ ™ j “ € “ d  „m – ‘ y ‘ „m – — — – ™ € ™ ‘ ™ l ™ ˜ ™ ˜ ‘ „ — – ™ ‘ ‘ „ ˆ — – m – y y € – — „ d d y — – ˜ —  — ‘ „ p l ‘  € – ˜ — y m p y ˆ – y y  ˜ —  ˜ “ ˜ — – ep „ € d d “ n C A ¢ H E  H E @ E  ¦ D H B C E  A I F ¥ ¥ !  !  ©  † — ‘ “ – j l ‘ € – ˜— ym p y “ — – ™ p „ — — “ — – y y ˜— € ™ ˜ p “ € “ i ‘„ ‘ ‘„ ‘ y„ ‘„ ‘ y„ ‘„ ‘ ‘„ ‘ y„ y„ y„ ‘„ ‘ ¥ † o ¦ ¥ ¤£ — – y y € ˜ ‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– – —€ d ˆ€“ e – j ˜  ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜— ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y — ™ € — – ˆ ‘ „ ˜ “ ˜ ‘ – — ™ € d !©   K O(M × N × K ) K — – ˆ „ €˜‘– p — €– ˜— ympDH–Gˆ"–€ s j„ ‘ €„ D€d(“ ©—W– p ‘F“©——G“G‘‘„p © !  #© # † – ep „ € d d “ – ˜ ˜ – p – ˆ — – y y  ˜ —  € ™ ˜ p “ € “ p — – ˜ ‘ – € ™  ˆ — – m – j y — ™ € † o y “ –m s “ ˜ –m f ˜‘ – j –m “ ‘ v † – j e ˜ € „ lm “ m – ˆ – ™ € ˜‘ – ‘ – — ‘ € y „ … — ˜ – s „ ˆ – € s j „ ‘ y “ ˜ € „ d d “ € € “ d – €  “ ™ ‘  m ™ ˜  u – m d j „ p – ‘ y €  „ •“ ˆ ˜ ‘ “ ˜ € „ d j  – l “ ˜ ‘ “ • “ m “ – ˆ „ e ˜ ™ j – ˜ ˜ – p f – € ˜ ‘ „ p € “ ƒ † ¢ f† m “ ˜ – “ ˆ  ’¡ — ‘ “ – j – ˆ „ e ˜ ™ j “ m – — m “ € ™ ‘ ™ l  y y – ˆ „ e ˜ ™ j – ‘ y ˜ ‘ – j – m “ l ™ ˜ ‘ – y ˜  ˜ — ‘ „ p  ¤ ¡     £ !  ¤ ¡ £ — – n † ¢ f ˜ € – “ • „ h ˆ ‘ “ u y – m – i¡ — ‘ “ – j l ‘ € – ˜ — ym p –m – —m “ € ™ ‘ ™ l – y y ˜— ˜ “ ˜ — l ‘ € – ˜— ym p –m – y y – ˜ ‘ “• y — ‘ „ ˜ p – — “ m — ‘ “ ˆ — ‘ „ € € – • — y „ † — € – ˜ — ym p — – ˆ ‘ „  ˜ “ ˜ ‘ – — ™ € d – € “ m € y „ d – ep  € — ym d – m k ˆ „ j ‘ y € – — m ˜ y – ˜ —  — ‘ „ p ‘ „  ˜ ym „ — – € ˜ y “ – ‘ ” † ™  p ™ d — ™ € d m y – — – € ˜ y “ ‘ y ˆ — y „ — — – ˆ ‘ – ˜— – — – ˆ  „ € ˜ ‘ – p — € y –m – € ˜ ‘ – – p ‘ “ ˜— ˆ “ m – y y — € „m — ™ ‘ ‘ „— y … ˜ ‘ „ — — € – ˜— ym p u y – ˆ ˜ – f ™  p ™ d— ™ € d m y – — ‘ “ ˜ € – p ‘ y ˆ — y— — – ˆ y “ ˜— – – p ‘ “  € “• “ — ˆ ‘ “ y y ™ — • ˆ ˜— – € – ˜— ym p ‘ y f ˜ ‘ – j – y y d  t † — y‘ – ˜ s „ — € – ˜— ym p — –m  € „ – ˜— „ d “ € – ‘ ‘ „— y … – ˆ y „ € – — • ˆ – ˆ – € ˜ ˜ – j € – d – ˆ ˜— – – j km s „ € d – p € –mm “ d € y „ d ‘ „ ˜ ym „ — – ‘ ” † — – ˜ ‘ – € ™  ˆ — k € ˜ ˜ ‘ „ — — – mm “ ˜ — – m ˜ ‘ „ ˆ — – ep „ € d — € – ˜ — ym p — – ˆ € – € ™ l – ˆ – m s “ d “ p — “ d ˜ — – ‘ ˜ – – y y € ™ e d— € – d  e – j € „ … – ˆ — € – ˜— ym p — – ˆ – y y — ‘ “ ˜ y ˆ „ € d – ‘ – ˆ „ e ˜ ™ j – ˜ ˜ – p f — ym d – ’ © Y ! # ©      © !   ©  © " ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢  ‚ – d “ ˜ ™ – y y “ ep — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d – m k ˆ „ j y ˆ – p ‘ “ m s j – —  “ € • “ m € – € „ m ™ j “ ˆ ˜ ‘ “ ˜ ˜ – j € – d — – — “ e d u y – ˆ € – € ™ ˜ – ˜ —  — ‘ „ p  p – mm – i † ¢ f† m “ ˜ – € – ˜ — d j – ’¡ ¡ – ˆ „ e ˜ ™ j “ m ˜— – ‘ „ ˜ “ — j ˜ d „ ˆ – j km s „ € d – p – € ˆ y „ — ™ € € y „ d – y y— — “ m p – ˆ „ e ˜ ™ j – ‘ ” ©   1 © (2π )M |Σk | $ $ # ¢ → P (−i |θk ) = x → − T −1 − − → →→ 1− e− 2 ( xi −µk ) Σk ( xi −µk ) ‚ ‘ –  — — y “ l — “ p – m — ‘ “ ˆ f p – •# k =1 K → πk × P (−i |θk ) x i=1 N → log P (−i |θ ) x  → P (−i |θ ) = x log L(θ |D) = © L(θ |D) † – j e ˜ € “ l „m ‘ „ — € – — j u“ j ˜ y “• y y ™ € – — j u“ j ˜ – f ˜ y ˆ „ € d ‘ y ˆ y –m y “ — – j € – ˜ – ˆ – j j „ — – ‘ y € – € ™ ˆ — ‘ „ p – ˆ ˜ – j € – d ‘ „ ˜ p ‘ „ … – ˜ ˜ – p – ˆ – j e ˜ € “ l „m y ˆ ‘ „ ˜ “ —m ˜ y n © L(θ |D) = P (D|θ ) = i=1 N → P (−i |θ ) x ‚ “ ‘ „ f — – €˜ y“ — – ˆ — ‘ y — –m ˜ ‘ – j j “ ˆ ‘ – d ™ ˆ ‘  — ™ € ™ ‘ ™ l ™ ˜ ™ ˜ ‘ „ — ˜ – s „ — – m – y y – — k e ˜ „ d  e m ˜  “ … ‘ „ m  z † ¤¢ £  ¨  ¤    ¥  –™˜„ ‘ f —–™ ‘ ‘„ ˆ uy“ ˜€„ d d“€ ¤ !  ¡     ¥   £        ¤ ˆ p ‘ „ ˆ – m € “ d ‘ w † € “ d – m k ˆ „ j y ˆ – p ‘ “ m s j – —  “ € • “ m € – —  j u “ j – ˆ ˜ — – … ˜ p – s „ m —  y ƒ † ‘ „ ‘ — ˜– f ‘ „ ˜ y s € ˜— ˆ “ m € “ d ™ € ™ ‘ ™ l ™ ˜ ™ “ ˜ – s „ m — – — „ d ‘ w † – m k ˆ „ j y ˆ — ™ ep “ p — – € ˜ k j “ € “ d — – ˆ – j j „ p — – y • — € „m “ ˜ ‘ „ — — € – ˜— ym p u y “ — ˜ – s „ — – ˆ — – p ‘ “ ‘ – ˜ € “ d d “ — – n  © © L(θ |D) ©© © © D  zik = 0 zik = 1 θ −i → x θk k =1 πk = 1 K → θk = (−k , Σk , πk ) µ θ = (θ1 , ..., θK ) † ‘ „  ˜ € “ d d “© ˆ ™ ˜m s “ s „ € d € y – m ˜ – f – p ‘ “  € “ •„ p „ –ˆ€˜–—pˆ€˜“ j– ˆ€ y––lm‘f“m™ j–€˜‘‘y–p–˜€‘y––—m™€€“dd–€—–y™y—€™˜—p–“™€‘“‘p„fˆ —– ˆ –mk ˆ „€jy„‘dy p ‘—„–ˆ‘ ‘˜–y—€˜——y‘“„lp—‘‘w˜ y s † – ™ — m ˜ y – € ˜ © ˆ ™ ˜m s “ s „ € d – ‘ “ ˜ € – p – ‘ y ˜ ‘ “ “ — ‘ „  ˜ y s € ˜ —  ˆ — – p – ˆ – ‘ y p “ ep f – p ‘ “  € “ •„ p – ˆ – p  € ˜ “ j € y – m ˜ – – € ˜ ‘ – p € y – m € “ d — – ™ —  € ™ ˜ p “ € “ p f o Y Y  f —   “ — ˜– dz ˆ ‘“ – —– ‘ ‘–—— y“ l — ‘„˜ y s€˜—ˆ – ˆ – l ‘“m™ j ‘ y ‘„m–— —–™€™ ‘™ l ™˜™ ˜‘„ —–mm–© yy Σk πk k ∈ [1, K ] θk − → µk θ K ¢ o ¡ " ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £ ! ! € – —m “ ‘ ™ d – ˆ ‘ „ ¡ “ … € y –m € y— p ‘ „ ˆ ˜‘ – p ‘ – € ™  ˆ – — — – € k ˜ € p — ˜‘ – € ™  ˆ — – n † — – u–m d j „ p — ym d — –m — –m k ˆ „ j — –m € – —m “ ‘ ™ d ˆ ‘ – ˜  y y – j € – ˜ ‘ y — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ “ m s j – — “ € • “ m – ˆ m y pm “ p y “ ˜ ‘ – ˜ y „ “  y y “ m – p € y „ d — ™ — „ d „ € d ™ ˜ ™ ˜ ‘ „ — – € k ˜ € p — € y – — ym ƒ † — – € „ ˜ “ ™ m “ — ‘ „  ˜ “ — m “  ˜ ‘ — € y –  — ym d – m k ˆ „ j – y y “ ep € y „ d ˜ ‘ “ € ™ ˆ  — ‘ „ p ‘ – f — – ™ ‘ ‘ „ ˆ u y “ ™  € d „ € d d “ — y m d – m – m k ˆ „ j – m €  —  „ ep € y „ d – y y  ˜ —  ˜ “ ˜ — – € k ˜  € p ‘ y € – —  m  ˜ y ˜ – f — ™ e p € – ep – € — € – ˜— ym p – ˆ – € s j „ ‘ –m € – € “• ˜‘ “ — “ … ‘ – — –m k ˆ „ j — € y –— ym d € – € ™ ‘ ™ l ‚ — – j j — –m p ‘ „ ˆ ˜ ‘ „ — — ‘ „ ˜ “ ˜ jm — – p € –mm “ d € y „ d — ‘ „ ˜ ym „ — — – n † – ™ — „ d „ € d –m “  ˜ ‘ ‘ „ ˜ ym „ — “ m –m s — ‘ – — ˜‘ – j – m “ l ™ ˜ — – ˜ – f — ™ ep € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ y ˆ ‘ „  ˜ “ p  p ™ d — “ m – ˆ – y y  ˜ “ j ™ m s „ € d “ m – ™ ˜ ‘ „ € … ‘ „ p ˜— – – y y ˜— ˜ “ ˜ — – ˆ „ e ˜ ™ j – ˜ ˜ – p f — ‘ “ – j – ˆ „ e ˜ ™ j “ m € y „ d – y y – j j – ˆ f — ‘ # † ¢ f ˜ € – “ • „ h ˆ ‘ “ u y – m – i¡ — ‘ “ – j – ˆ „ e ˜ ™ j “ m € “ d ™ —  m  ˜ y – m k ˆ „ j – m ˜ ‘ –  ˜ s „ ‘ „ — € „m “ f — –m “ l ™ — – ˜ y „ ˜ ˜‘ „ — – l ‘ “ m ™ j y ˆ — ™ ˜m s “ s „ € d — –m – y y ˜ – f ™ ˜ ˜‘ – ˆ – p € ˜ “ j “ m — – ˜ y „ ˜ ˜ ‘ – ˆ ‘ „ d — – € € „ p — € – ˜ — ym p — – ˆ – p ‘ “  € “ •„ p – ˆ — – p  € ˜ “ j — – m – y y – — k e ˜ „ d  e m ˜ “ … ‘ „ m — f ˜ – – ‘ ‡ † — ‘ “ – j – ˆ „ e ˜ ™ j “ m € “ d ™ —m ˜ y  ym – p – —m “ € ™ ‘ ™ l –m k ˆ „ j – p f ˜ “ … ‘ ‡ † … ˜ — „ d ˜ ‘ – j – ˜ p  € ˜ — – p ‘ “ m s j – — “ € • l „m “ m – ˆ — – •— — – p p y— — € y –m “• ™ u m y – — ‘ y – € y – € ™ … ‘ ˜— – u y – ˆ – € ˜ ‘ – – p ‘ – € ™  ˆ “ m f – € ˜ y “ m ‘ „ ˜ “ € ™ ˜ – ‘ y ˆ – y y — € „m – ˜ € € “ — – € y ˆ ™ p „ € d – ˜ ˜ – i    # # © !   !   log L(θ |D) © i=1 N  πk     Σk δ →µ→µ →x → z ik (−i − −k )(−i − −k )T x N i=1 z ik   © © © 1 Σk = N × πk → × −i x N × πk −= → µk © ©© ‚ ˜ y — – j j „ p € y „ —  j ˜ ‘ „ —  y y – p ‘ “  € “ •„ p – ˆ – p  € ˜ “ j – ‘ y ˆ ˜ – – € ˜ ‘ – p ‘ y ˆ ˆ ‘ – d ™ ˆ – m k ˆ „ j y ˆ – ˜ ‘ “ — „ d j „ p – y y “ ep f — – ‘ ‘ –  — — y “ l – ˆ – l ‘ “ m ™ j ‘ y ˆ — “ p – m — ‘ “ ’ − → µk Σk K N πk = N i=1 z ik † – p ‘ “ — — “ ‘ ‘ „ p – ˜ ˜ – p – ˆ ‘ „ ˜ p ‘ „ … ‘ – –m k ˆ „ j y ˆ u y “ j ˜ d „ — – € ˜ k j “ € “ d — –m € – m y p m “ p ˜ – f — € – ˜ — y m p u y “ — ˜ – s „ — – ˆ — ‘ „  ˜ “ ˜ p – “ — – m — – ™ u € – — „ d d y — ‚ £         ¡ †  !  θ k =1  z ik = 1 K ‚ – ˜ ‘ “• y — ™ ˜™€ d „ € d “m –•€–— ‘ „ p ‘ „ f ™ ‘ ‘ „ ˆ ˜– s „ ‘ y € y „ d —“ j f — ‘“ ˆ — – ‘ ™ ˆ ˜‘ „ — — – ˆ — –mm – p – y y — € „m “ f –mm “• € – ˜‘ m — ‘ “ ˆ — – ‘ ™ ˆ p ‘ „ ˆ ˜ ‘ „ — — – ˆ — € y –m “• — – n ©  {0, 1}  z ik [0, 1] −i → x zik → πk × P (−i |θk ) x − → j πj × P ( xi |θj ) z ik = !©  ‚ – ˆ € y –m “• “ m – ˆ – p ‘ “ € ™ d — – m — € „m “ – — „ d ‘ w † –m k ˆ „ j – p — y „ — — –m “ j ˜ d „ — € – ˜— ym p u y “ — ˜ – s „ — – ˆ — ‘ „ ˜ “ ˜ p – “ — – ˆ – m s j – — ‘ – m € – ep € – ep ˜ – f – m k ˆ „ j y ˆ — – € ˜ k j “ € “ d — – m — ™ u € – — „ d d y — ‚ ¤¢ £ ¥  †  zik © θ z ik  ¢ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ ""  © T© ™ j — € y –  — y m d f ˜ ‘ – ˆ ™ p ™ € d – € ˜  d “ ep y ˆ † ‘ „  ˜ p – — “ m — ‘ “ ˆ – ™ ˜ ‘ – — ™ € d – y y  ˜ “ € ˆ “ y y € y – € € – m –m d j – u– € “ d f – ™ € ™ ˆ— ‘ „ p l ‘ € – ˜— ym p ‘ y ˆ – ‘ € – ˜‘ ™ ˜m “ y y “ m – ˆ – € y — – j – ‘ y – ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ C A ¢ H E  @ £   E  ¦ D H B C E  A I F 8 ¥ † – y y  ˜ —  ˜ “ ˜ — l ‘ € – ˜ — ym p y “ — – ™  p „ — — “ — – y y  ˜ —  € ™ ˜ p “ € “ i o† o ¦ ¥ ¤£ ¥ ‘„ ‘ ‘„ ‘ y„ y„ y„ y„ ‘„ ‘ ‘„ ‘ y„ y„ ‘„ ‘ !©  — – p ‘ “  € “ •„ p – ˆ — – p  € ˜ “ j ˜ – — – ˆ  „ € ˜ ‘ – p — € – ˜— ym p – ˆ – € s j „ ‘   DH G" — – y y € ˜ ‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– –—€ d ˆ€“ e – j ˜ ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜— ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d € „€ d “ — – p ‘“ ——“ ‘ ‘ „ p  O(M 2 × N × K ) K K D  ( © W  F ©  G  G  † – ep „ € d d “ – ˜ ˜ – p – ˆ — – y y  ˜ —  € ™ ˜ p “ € “ p — – ˜ ‘ – € ™  ˆ — – m – j y — ™ € o† o y “ – m s “ ˜ – m f ˜ ‘ – j – m “ ‘ v † — ‘ „  — ‘ – j ˆ – ˆ – € s j „ ‘ y ˆ ‘ „  ˜ p ‘ „ … ‘ – – y y  ˜ “ € ˆ “ y y ˜ — – ™ ˜  u – m d j „ p “ — ˜ – € – l € – • ‘ „ p – ˆ ˜ ‘ “ •“ — ‘ „  ˜ “ € ™ ˜ — – — y – € s j „ ‘ – ˆ ˜‘ – • y „ — – ˜ — — – p ™ ‘ –mm – f ˜‘ “ ˆ ‘ – d – i † — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ € – ˜— u– ˜ y – d  y y ˜ y € s –m € – € ™ l – ˆ –m s “ d “ p ˜‘ – j –mm – € y ˜ “ ‘ ˜ ‘ – j –m “ l ™ ˜— – –mm ‡ † — – ˜ ‘ – € ™  ˆ — k € ˜ ˜ ‘ „ — — –mm “ ˜ — – m ˜ ‘ „ ˆ — – ep „ € d — € – ˜ — y m p — – ˆ € – € ™ l – ˆ – m s “ d “ p ˜ — – ˜ – f – ™ l ‘ „ m m “ – j € „ … – ˆ — € – ˜ — y m p — – ˆ € –  ˜ ‘ – ˆ ˆ –m s “ d “ p ˜ — – – y y ˜— ˜ “ ˜— – ˆ „ e ˜ ™ j – ˜ ˜ – p f — ‘ “ – j ˜‘ – j – € “ € ˜ ‘ „ p f – € ˜‘ „ p € “ ƒ † — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ – ‘ ˜ € – d “ m € – — j ˜ d „ € y „ d – ™ — j ‘ j – € ˜ ˜ „ ˆ – € k ˜ € p – p – ˆ € y –m “• “ n † –m k ˆ „ j y ˆ — ˜‘ “ ˆ ‘ – d ™ ˆ ‘ — – € ˜ k j “ € “ d – ˆ – € s j „ ‘ –m ˜‘ “ ˜‘ – — ™ € d – € f — – y y  p ™ d— d „ € ˜ – € ˆ ˜— – p f — – u–m d j „ p — ym d — –m k ˆ „ j — –m – —m “ ‘ ™ d – j € – ˜ ˆ ‘ „ p – — –m – y y — ˆ ‘ “ ˜ f –m k ˆ „ j y ˆ ™ ˜ p  p ™ d— “ m – € ˆ ˜— – p f — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ “ m s j – — “ € • “ m – ym “• ™ – € k ˜ € p – p – ˆ – j € – ˜ € – j – € d – n  ©  ! M  Mθ !     ©    © BIC (θ |D) = −2 × log L(θ |D) + Mθ × log N  # $ #   ‚ ˜ y— – j j „ p  ‘ ™ ˆ f ¢ f ˜ € “ ‰ e p z¡ ‘ „  € – ˜  € i ‘ „  ˜ “ j € „ … ‘ r ‘ “  — –  “ x i r x – € k ˜  € p – m – ˜ y „ ˆ — ‘ “ — ˜ — – – € ˆ “ p – p — ‘ “ ˆ ™ — m ˜ y — ym d – m ˜ – y ‘ ‘ „ p — ym d – m – € k ˜ € p – m —  “ j f ¢ f† m “ ˜ –  p “ ‘ € –  x¡ ˆ „ „ em –  n ˆ – ˜ – m d j „ i ˆ – ˜ “ € l – ˜ ‘ r n i r – € k ˜ € p – m ‘ –  s y „ f ¢ f –  “ #¡ ‘ „  € – ˜ € i ‘ „  ˜ “ j € „ … ‘ r –  “ # i r # – € k ˜ € p –m € – ˜ p –m d j – u– € “ d ˜ y – d ‘ „ f m u y – p  j € “ ƒ † — – u–m d j „ p — ym d — –m — –m k ˆ „ j — –m     YYY  Y$ #     "  ¤ £ ©¨ ¦ ¡ ¡¤ ¢ ¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £ ! † – ˆ „ e˜™ j “m – ˆ — – y y˜—€™ ˜p “ €“ p — – ˜‘– €™  ˆ — –m – j y— ™ € † o y “ –m s “ ˜ –m – y y — “ p € – ‘ € – ˆ – p ˜‘ “ € ™ ˆ— ‘ „ p ‘ – ˜— – i † — € – ˜ — ym p —m – ˜ – ˆ € – € ™ l – ˆ – m s “ d “ p “ € – — – ˆ „ e ˜ ™ j “ m — € „ m “ f – p ‘ “  € “ •„ p – ˆ – p  € ˜ “ j – ‘ y ˜ – – € ˜ ‘ – p ‘ y € – ˜ — ym p – y y “ ep ˜ ‘ “  p „ — — “ f — – ™ ‘ ‘ „ ˆ — – ˆ – ‘ ‘ –  — — y “ l ‘ „  ˜ y s  € ˜ —  ˆ – ‘ y € y — ™ — “ s – y y  ˜ —  ˜ “ ˜ — ˜ — – ˜ ‘ y ˜ — –  —  „ ep – € k ˜  € p – m – €  “ € ˜ ‘ „ p y “  z † — – ™  € “ • — k € ˜ ˜ ‘ „ — — – m m  “ ˜ — – m ˜ ‘ „ ˆ — – ep „ € d — € – ˜ — y m p — – ˆ – p “ … – €  “ … ˜ y – d – ‘ – ˆ „ e ˜ ™ j “ m — € „ m “ f – y ‘ – ˜ – € ˜ — –  y y – y y  ˜ “ € ˆ “ y y € y – € € – m ˜ — – p  — f  — ‘  # †  —  „ ep –m s p – € k ˜ € p y ˆ ˜ ‘ – j – ˜ € „ … — k € ˜ ˜‘ – ˆ ‘ – d ™ ˆ — – ˆ „ e ˜ ™ j — –mm – ˜ – ˆ — – y y ˜— € ™ ˜ p “ € “ p — – n † ˜ ‘ – j – l ‘ “ ep – p € “ d – ™ € „ m ™ j “ ˜ — – ‘ „  ˜ ym „ — “ m  — – ˜ ‘ “• y — ‘ „  ˜ “ € ™ ˜ m € y „ d € – • € – — ‘ „ p –m ˜ – f — – ˆ  „ ˆ ™ j — – ˆ ‘ y m € –  ˆ „ j — € „m “ – ˜— — ‘ „ p — –m s— — „ d — ‘ „ ˜ ym „ — — – ˆ – p “ d— – m € € y „ p € “ d € y „ d – ˆ „ e ˜ ™ j – ‘ y ˜ ‡ † – ep „ € d — y m d – m ˜ — – – ˆ  „ ˆ ™ j – m ˜ ‘ „ ˆ € – ˜ — y m p y “ — ™  p „ — — “ — € „m “ ˜ ‘ „ — — ˜ – s „ — – € ˜ y “ — – n † € –  ˜ ‘ – ˆ — € – ˜— ym p — – ˆ u y “ € ˜ ‘ – p — ˜ ‘ „ d – ˆ ˜ ‘ “• € – — – — “ s “ m – ˆ — ˜ – s „ – ˆ –m s j – — ‘ – ‘ y – € ˆ ˜— – p f — – ˆ  „ ˆ ™ j – ˆ –m s j – — ‘ – ‘ y ˆ ‘ „ d— – € € „ p ‘ „ ˜ ym „ — – ‘ y y y € – € ™ ˆ  — ‘ „ p – ˆ ˜ — – — “ p u y – € s j „ ‘ – ˆ — ‘ “ ˆ – ™ l “ —  • ‘ – – y y  ‘ ep – ˜ – € ˜ y “ – ‘ ” †  —  „ ep – m s  p – € k ˜  € p – m ˜ ‘ – —  j  ˜ d „  y y — –mm – p — – ™ • € – — ‘ „ p ˜‘ „ — — –mm – y y — –m  j € “ d f — ‘ „ ˜ ym „ — – ˆ m – ˜‘ – ˜ „ d –m s j – — ‘ – – € ˜ y “ ‘ y – ˜ ‘ “• y— – d “ ˜ ™ m € ‘ – ˜ s „ € y „ d — ‘ „ ˜ ym „ — — – ˜‘ – € ™  ˆ — – p – ‘ s j „ p ˜ – f — –mm – ˜‘ – ˜ „ d — ‘ „ ˜ ym „ — – ˆ — ‘ „  ˜ “ m y d „ d — – ˆ ˜ " ˜ y m d – € k ˆ  — ‘ „ p  ¤ ¡    £ §  ¤  £   ¥ §¨  — – ˆ – y y  ‘ e p – ˜ “ m f ‘ ‘ ‡ † – ˜ ‘ “ € y „ p –m “ j ˜ d „ ‘ „  ˜ y m „ — “ m – ˆ — – ‘  —  „ • — ‘ „  ˜ y m „ — u y “ – e p € – ep – € “ — – € y — – j ˜ – € y … y “ ˜ ‘  – € ˜ — – € —  y d f — ‘ „ ˜ ym „ — — – ˆ – p “ d— – m –m s— — „ d ˜‘ – j – l € “ m — ym d –m € € y „ p € “ d – ˆ ‘ “ f — – € ˜ y “ — – ˆ — – ‘ y — –m — – ˜ ‘ – € ™  ˆ — k € ˜ — ‘ „  ˜ y m „ — — – ˆ € – € ™ ˆ  — ‘ „ p € “ d – p ‘ – j j „ p ¨ ¡      ¡ ¢¤ ¥ – ˆ – y y  ‘ e p – ˜ “ n † ˜‘– j j– ˆ ™ p ™ € d – ™ €˜‘ „ p ‘– € ™ ˆ ‘ „  ˜ ym „ — – ‘ y € – € ™ ˆ  — ‘ „ p – ˆ ˜ ‘ “ ep d j – — ‘ – f – ˜ ‘ “ € y „ p – m “ j ˜ d „ ‘ „  ˜ ym „ — “ m – ˆ – ‘ —  „ • ‘ „  ˜ ym „ — – ‘ y – ˜ ‘ “• y — – d “ ˜ ™ m € y „ d – ‘ ‘ „  ˜ p – m ™ — ¡  ¤ £¢¥¤ £¢¤¥ “ m – ˆ – y y  ‘ ep – ˜ “ m f  — ‘ # † — – ™  ˜ ‘ – ˆ — –m “ j ˜ d „ — ‘ „ ˜ ym „ — — – ˆ ‘ „ ˜ “ ˜ „m d u– m ˜ – f — ‘ „ ˜ ym „ — — – ˆ – p “ d— – m – ˆ – ˜‘ “ ˜ € „ d j – ˜ € “ d – ‘ y ˆ – € “ — — – p ™ ‘ ‘ „ ˜ “ € „m d u– m – € ˜ ‘ – — j „ € d j „ p ‘ „ s ‘ y € – • y „ € ˜ – € k ‘ “ j – ˆ — – ˜ y € ˜— ‘ „ p ˜ ‘ – • y „ — ˜‘ „ — – € • y ¤ ‘ – — – — j — – ˆ „ e ˜ ™ j — –m f – y y ˜ “ € d ‘ ‡ † € – ym “• ™ – ˜ ‘ “ € y „ p ‘ „ ˜ ym „ — –mm – • y „ ‘ – ‘ y € – — „ d „ € d € y „ d – ™ — m ˜ y – y y  ‘ ep – ˜ “ m — ‘ “ ˆ – € ˆ ˜ — – p f — – m s — — „ d — ‘ „  ˜ ym „ — — – ˆ – p “ d — – m € € y „ p € “ d – ˆ ‘ „ ¡ “ … € y – m — ‘ “ ˆ — € „ m “ – ˆ  — ™ € — – ˜ ‘ “ ˜ —  u – — – y y  ˜ — “ ep „ ˜ — — – ˆ „ e ˜ ™ j — – ˜ ‘ – € ™  ˆ — – ˆ ™ ˜m “ ‘ l € „ n † –m s p – € k ˜ € p y ˆ – y • – ˆ ˜ ‘ „ d y ˆ — –m “ j ˜ d „ — ‘ „ ˜ ym „ — — – ˆ – ˜ € “ d ˜ “ … –mm – — – ˜ ‘ “• y — – d “ ˜ ™ m € y „ d – ™ • € – — ‘ „ p ˜ – f – ™ € ™ ˆ — ‘ „ p ˜— – ‘ „ ˜ ym „ — –mm – • y „ ‘ – ‘ y f – d “ ˜ ™ – y y “ ep © † € y – ˜ “ —  m  ˜ y m € “ d  ‘ € y „ … ˜ ‘ “ ˜ ™ — € – ˜ — y m p – ˆ – € s j „ ‘ – m f — € – ˜ — y m p u y “ — ˜ – s „ — – ˆ — ‘ „ ˜ “  p „ — — “ — – ˆ –m s j – — ‘ – m ˜ — – – € ˆ “ p – p — ‘ “ ˆ – ™ € ™ ˆ— ‘ „ p ‘ „ ˜ ym „ — – ‘ y f ˜‘ – j – y y d  t † € € y „ p € “ d –m € y „ d — – ™ —m ˜ y ˜‘ – j –m “ € ™ ‘ ™ l ˜ ‘ „ — — – y y ˜— € y – e — – ˆ f ˜‘ – j – € k ˜ ‘ – y € y „ p € “ d – € ˜ € y „ d – ˜— “• d „ € ˜ d y „ p y “ – s ˜ ‘ – • y „ — ˜ ‘ “ ˜ ™ — ‘ „ ˜ ym „ — — – ˆ – p “ d— – m f ˜‘ – j j – ˆ • †  —  „ ep – m s p – € k ˜ € p – m – —  j u “ j  y y – ™ € ˜ ‘ „ p ‘ – € ‘ „  ˜ ym „ — “ m € – ‘ ‘ „  ˜ p – m ™ — ˜ – — – m s — — „ d — ‘ „  ˜ y m „ — — – ˆ – p “ d — – m €  € y „ p € “ d ˜ ‘ – j – m d j  — – ˜ —  — ‘ „ p – y y  ‘ ep – ˜ “ m f — “ p – p — ‘ “ ’ † ¢ f ‘ e € – x¡ — – ™ l “ —  • ‘ – – € ˜ ˜ ‘ – • y – d – € k ˜  € p – p ˜ ‘ “ —  j  ˜ d „ — ˜ – s „ — – ˆ ‘ „  ˜  ˜ € “ d “ m – ˆ – y y  ˜ — “ e p „ ˜ — – ep € – ep – € – ˆ — – ˆ „ e ˜ "    © K ©     ©  ©   © ©  © ©© ©     © !© © ©    © © ©   ©  ©© ©   ! © !  ©  K ©   ©  YY  T  " ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ † — – ™ ‘ ‘ „ ˆ — –m — ‘“ ˆ €– ˜— u– ˜ y– d  y y ˜ y € s y “ – p “ … – € “ … – ˆ –m s “ d “ p ˜ ‘ – j –mm – € y ˜ “ ‘ ˜‘ – j –m “ l ™ ˜— – –mm ‡ † — – € ˜ y “ — –m — ‘ “ ˆ — ‘ y — –m — y‘ – ˜‘ „ p – € ˆ ˜ — – p — – y y € ˜ ‘ – p ‘ „ p — € – ˜— ym p — – ˆ – y y — ‘ “ — – ™ € “• — k € ˜ — – j € „ … – ˆ — € – ˜ — y m p — – ˆ € – € ™ ˆ  — ‘ „ p – ˆ ˜ – j € – d – m m – y y ˜ — – – ˆ „ e ˜ ™ j – ˜ ˜ – p – ˆ ˜ ‘ “ ˜ € „ d j  – l “ ˜ ‘ “ •“ ‘ ” † — ˜ – s „ — – p € y „ d – ™ € ™ ˜ ˜— – – — “ e d – j j – ˜ ˜ – p ˜ – ˜‘ “ € y „ p € – ˜— ym p y “ — ™ € l ™ ˜‘ ˜‘ „ — — ˜ ‘ “ ˆ ‘ „ d— – € € „ p — ˜ – s „ — – m — € „ m “ f   ‘ „ “ € – ˆ ˜ – – € ˜ ‘ – p – ˆ – € k e d — € – d  e m — ‘ “ ˆ — ˜ ‘ – — ™ € d ˜ ‘ „ — — ˜ – s „  £ ¡ — ‘ „ j y “  — – € ˆ ˜ — – p f ™ ˜ — ‘ – ˆ – ˆ – € k ˜ € p – m – ˜ p – d — – € – l “ ‘ —  „ • ‘ „ —  — † € – ˜— ym p ‘ y p y “ ™ ‘ l— — “ – € „ p ‘ – ˜— – ‘  y y – — “ s “ m – ˆ — ˜ – s „ — – ˆ ‘ y m € – ‘ ‘ „ ˜ p –m ™ — † ‚ € – ˜ — ym p ‘ y — ™ ‘ l— — “ ˜‘ – „ — — ˜ – s „ — –m — y „ ˜ – y y – p y y — y — – ™ € ™ ˜ — – — “ e d u y – ˆ – ˆ ‘ „  — — – p p y — – ‘ y ‘ – – ˜ —  — ‘ „ p – ˆ „ e ˜ ™ j “ m f o† o – € y l “ m € “ d – ™ € ˜ — ymm r † –— ‘– ˆ – j j „ p – ‘ „ “m €– €™ ˆ— ‘ „ p € y „ d – l“ ‘  —  „ • – p — ‘ “ ˆ — y ‘ – ˜ ‘ „ p – € ˜ ˜ ‘ – • „ ˆ  y y — ˜ – s „ ˆ j y j ‘ j – € s j „ ‘ – m ‚  £  ¡ ˜ – † ˜ – s „ m – ˆ – l “ ‘— „ • y ˆ — y ˆ “ € –m ‚   † ‚ ˜ – s „ ‘ y ˆ – l “ ‘— „ • y ˆ ™ ˜— ‘ – ˆ – ˆ ‘ „ ˜ „ ‘ “ m ˜ ‘ “ m " € ˜‘ „ p ˜ – – j e ˜ € „ lm “ m – ˆ – ™ € ˜‘ – ‘ – — ‘ € y „ … — – € ˜ k j “ € “ d u y – ˆ – ˆ ‘ „ ˜ “ —m ˜ y m € y — – ™ — “ s — ‘ “ ˜ — – f ¢ f† m “ ˜ – € – ˜ — ‡¡ # i z x ’ – ™ m – d d “ f – € ˆ “ p – p — ‘ “ ˆ – p ‘ – € ™ … ™ € – ˆ – ˆ „ e ˜ ™ j “ n † ™ ˜— ‘ – ˆ – ˆ – € k ˜ € p ‘ “ ˜ € – p ‘ y ˆ € ˜ € “ d ¤ £ £  ¡ ¨ § – € k ‘ “ j – ˆ — € – ˜— ym p — – ˆ ‘ „ ˜ “ j € „ … “ m € y— – ™ ˆ ‘ „ … f – y € “ d d “ ˜— – — – ˆ „ e ˜ ™ j – ˆ –mm j “ … – € ˜ y “ – ‘ y f — € – ˜— ym p — – ˆ ‘ „ — • – ˜ ˜ – p € y— – ™ — “ x † ™ ˜— ‘ – ˆ –m s “ … – ˆ — ‘ „ l ™ € — – ˆ € “ d — – ™ € y „ ˜‘ – f ™ ˜— ‘ – ˆ – ˜ y “ e – ˆ — – ‘ k l „ j „ e — ‘ „ l ™ € — – ˆ – j j „ p € – € ™ ˆ— ‘ „ p — –m – ˆ ˜— – — € – ˜— ym p — – ˆ –m s — — „ d ‘ „ — • – € ˜ y “ – ‘ ” EH  D C G @ I B A   @  ¦ D H B C E  A I F  ¥  ¢    ©    © −i → x   # # ©   © ©   ©    ©   ©  −i → x ©  © © ©  ©  † – y y  ˜ — “ ep „ ˜ — l ‘ € – ˜ — ym p y “ — – ™  p „ — — “ — – y y  ˜ —  € ™ ˜ p “ € “ i ‘„ ‘ ‘„ ‘ y„ y„ y„ y„ ‘„ ‘ y„ y„ y„ ‘„ ‘ $" ¥ ¢   ¢ "† o ¦ ¥ ¤£ — – y y € ˜ ‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– – —€ d ˆ€“ e – j ˜  ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜ ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜ ‘ – — ™ € d !©   O(M × N × K ) K ‘ „ ˜ ˜ € “ d — €– ˜— ympDH–Gˆ"–€ s j„ ‘ €„ D€d(“ ©—W–p ‘F ©“——G“G‘‘„p ¢ ¨ © ¦ ¡¤ ¥ ¡ £ £ ¦ ¢ ¦ ¡ ¢ ¢ ¡ © ¤ ¨ ¦ £ £ 32 1 ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢¡ ¢ ¢¡ © † – m d j – u – m € y — ™ ˜ — ‘ – ˆ “ m € y — ™ — “ s l ‘ € – ˜ — ym i o† o ¦ € y– y l ‘ „m Y # o $ ¥  YYY Y Y Y o YYYY u € d !" ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ ©  † — – j € „ … ‘ y – j j „ p — – ™ € ™ ˆ  — ‘ „ p ˜ ‘ –  „ — — – ˜ ‘ “ — „ d j „ p — – m — – ˜ y „ ˜ – y y – p y y — y — – y ‘ – ˜ s „ — – ˜ ‘ “ — „ d j „ p — y „ — u y – ˆ — – ˆ – ‘ y p “ ep € y — ˜ y ˆ ‘ „ p – € ˜ — – ˜ — – ˜ – m ˜ – f – ™ j € d d y — ˜ — – – — ‘ – ˆ — ‘ „ j “ m – m ymm – p “ m f — – € ˜ y “ — – m € y „ d ˜ – f — € – ˜ — ym p — – m — € „ m “ ˜ ‘ – j € „ … — – j € „ … ‘ y — – u – ‘ ‘ „ p — – ˜ ‘ “ — „ d j „ p — – n † ™ ‘ – j ˜ — – — – u – ‘ ‘ „ p — – ˜ ‘ “ — „ d j „ p — – p – ˆ ™ ˜ j € „ … ‘ y ˆ ˜ — – ˜ ‘ y f ‘ ‘ ‡ † — – ™  ˜‘ – ˆ ˜ ‘ „ — f — – ™ ˜ p – ‘ ‘ „ p — – — ‘ – ˆ — –m ymm – p – ˆ — –m s j – — ‘ – — –m – € ˆ ˜— – p f — – u– ‘ ‘ „ p — – ˜‘ “ — „ d j „ p — –m f – ˜ y— ‘ ‡ † — – ™ ˜ p – ˜ ™ ˆ ˜‘ „ — j y j ‘ j ™ ˜— ‘ – ˆ – ˆ – € k ˜ € p –m ˜ ‘ “ — “ …— ˜ “ — — –m ym m – p — – m ˜ – f – ™ ym “• ™ ˜ — – – m ymm – p – y y “ ep – ˆ ™ ˜ — ‘ – ˆ “ m —  y ƒ † ‘ „  — ‘ – j ˆ – y y “ ep € y — — € – m y l ™ € — –mm “• € – ˜ ‘ — – ˆ € “ d ™ d y „ p ™ ˆ ˜— – — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d— – m f — d j – ˜ € – j – € d ‘ y — ‘ “ ’ † ‘ „ ‘ y „ – — ‘ – ˆ – j j „ p – ™ € ™ ˆ — ‘ „ p ˜— – –mm € l “ m – ˆ –m ymm – p – ‘ y — ˜ ‘ “ ‘ j € – ˜ ™ ˆ j y j ‘  j ™ ˜— ‘ – ˆ “ m ˜ – € – —m ˜ y –mm € l “ m – ˆ –mm “ ˜ “ m € –  p ™ d — ˜ „ ˆ € y – ˜ “ —m ˜ y m f — “ p – p — ‘ “ ’ †¢ f € –  ˆ  ’ ˆ ‘ “ p – mm – ™ € x¡ — € – ˜ — ym p — – m € – j € „ … € y „ d — – ™ ˜ p – ‘ ‘ „ p — – — ‘ – ˆ — – m ymm – p – ˆ — – m s j – — ‘ – — – m € –  ˜‘ – ˆ ˆ — y d f — –m ymm – p — – — y – € s j „ ‘ – ˆ ‘ – – p “ d— – m € – ‘ ‘ „ ˜ ˜ € “ d € y „ d –mm € l – ‘ y € – —m ˜ y ˆ ˜ — – — – ™ ‘ ‘ „ ˆ – ˆ – m s j – — ‘ – ‘ y ˆ l ‘  € – ˜ — y m p – m € – y ˜ p – – € y „ d – m s  — — „ d – ep „ € d d “ – € ˜ y “ – ‘ ” D H B C E  A I F  ¥        ©  ©  !©  CIIH B  CI B A   @  ¦ ©   © © ©  YY © ¦ † ™ ˜— ‘ – ˆ “ m € y — ™ — “ s l ‘ € – ˜— ym p y “ — – ™ p „ — — “ — – y y ˜— € ™ ˜ p “ € “ i y„ y„ ‘„ ‘ y„ ‘„ ‘ y„ ‘„ ‘ y„ ‘„ ‘ ‘„ ‘ y„ ¥ † o ¦ ¥ ¤£ — – y y € ˜ ‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– –—€ d ˆ€“ e – j ˜  ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜— ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d € „€ d “ — – p ‘“ ——“ ‘ ‘ „ p !©   O(M × N 2 ) ‘ „ ˜ ˜ € “ d – l “ ‘ — „ • y ˆ ™ ˜— ‘ – ˆ – ˆ – € k ˜ € p   DH G"  D  ( © W  F ©  G  G  © † – ˆ „ e ˜ ™ j –mm – ˜ – ‘ y ˆ — – y y ˜— € ™ ˜ p “ € “ p — –m ˜‘ – j –m “ ‘ – j y— ™ € † o y “ –m s “ ˜ – n † — – ˜‘ – € ™  ˆ — k € ˜ ˜ ‘ „ — — ™ ˜— ‘ – ˆ — –m ˜ ‘ „ ˆ — € – ˜— ym p — – ˆ € –m s p – ˆ — “ d ˜‘ – ˜ ˜ – j € – d – ‘ —m f –m “ s „m l – € k ‘ “ j – ˆ — ™ u ˜ ‘ “ ˜ ™ — – € ˜ k j “ € “ d — – p f — ym d – ’ † — – € ˜ k j “ € “ d — – p € y „ d — – ™  € d „ € d d “ — ym d — – m — € y – m “• — – m € –  p ™ d — – ˆ € y – ˜ “ — m ˜ y ‘ y – m p £ ˆ — k € ˜ ˜ ‘ – • y „ — ˜ — – m € w †  £  ¡ ˜ –   — – € ˜ k j “ € “ d u y – ˆ — – m € y „ d — – ™ u — € y – m “• u y “ – m s — ‘ – — ˜ — – – mm – f — € y – mm “ € “ ƒ † ¤ £¡  £¢ ¤ !  ¤ ¤ ˆ – j k m s „ € d y ˆ – € y „ — – mm – f — ym d – ’ † – — “ s “ m – ˆ — ˜ – s „ ˆ – € s j „ ‘ y ˆ ‘ „ ˜ p ‘ „ … ‘ – – y y ˜ “ € ˆ “ y y ˜— – ™ ˜ u–m d j „ p “ — f – € ˜ ‘ „ p € “ ƒ  #" ! ¢ ©  © !  ¦ ¤ £ £ ©  ¢ ¦ ¤ £ £ ¦ ¢ ¦ ¡ ¢ ¢ ¡ © ¤ ¨ ¦ £ £     — ˜– s „ u y– ˆ — –m – €˜‘– – p ‘“ ˜— ˆ “m p €“ ‘ y ˜– f ˜– s „ ‘ y ˆ ‘ „ d— – €€ „ p ˆ y ¤ ‘ ‘ y m– y y –m — ‘ “ ˆ f — – ™ ‘ ‘ „ ˆ — – ˆ ˜ –m d j „ p – e d “ € l –m — d j – ˜ € – j – € d ‘ y — ‘ “ ˆ ˜ y € ˜ — ‘ „ p ‘ „ f ˜‘ – j – y y d  t † – e d “ € l ‘ y — ‘ “ ˆ — ™ ˜ p – ‘ ‘ „ p — ˆ y ¤ ‘ – ˆ — –m s j – — ‘ – — – ˆ – j j „ p — € – ˜ — ym p — –m € – € ™ ˆ— ‘ „ p – ˜—— ‘ „ p — – e d “ € l — –m € y— – ™ — “ s l ‘ € – ˜ — ym p – ˆ – ˆ „ e ˜ ™ j “ m f † o – € y l “ m € “ d – ™ € ˜— ymmr @ B ¦  C I B A   @  ¦ D H B C E  A I F  ¥ C£    ¨ † — –mm € l — –m € y— ™ — “ s l ‘ € – ˜— ym p y “ — – ™ p „ — — “ — – y y ˜— € ™ ˜ p “ € “ i y„ y„ ‘„ ‘ y„ ‘„ ‘ y„ ‘„ ‘ y„ ‘„ ‘ ‘„ ‘ y„ ¥ † o ¦ ¥ ¤£ — – y y € ˜ ‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– – —€ d ˆ€“ e – j ˜  ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜— ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜ ‘ – — ™ € d € „€ d “ — – p ‘“ ——“ ‘ ‘ „ p !©   O(M × taille de grille) — – ™ ˜ p – ‘ ‘ „ p — –m ymm – p – ˆ — –m s j – — ‘ – ™ ˜— ‘ – ˆ – ˆ – € k ˜ € p ˜ – –mm € l – ˆ –mm “ ˜   DH G" D  ( © W  F ©  G  G  † – ep „ € d d “ – ˜˜– p – ˆ — – y y˜—€™ ˜p “ €“ p — –m – j y— ™ € † o y“ –m s“ ˜ – n † — – ™ ‘ ‘ „ ˆ — –m — ‘“ ˆ €– ˜— u– ˜ y– d  y y ˜ y € s –m € – € ™ l – ˆ ˜ – — – y y € ˜ ‘ – p ‘ „ p y „ — – ™ € “• — k € ˜ — – j € „ … – ˆ — € – ˜— ym p — – ˆ € –  ˜‘ – ˆ ˆ — –m s “ d “ p ˜‘ „ — — –mm – — “ j f € – ˜— u– ˜ ‘ – — — y d — – ˜‘ – € ™  ˆ — ™ ˜— ‘ – ˆ – — € – ˜— ym p — – ˆ – y y ˜ “ … –m – ˜ d j „ p ‘ – ˜ ‘ – j – m p £ ˆ ˜ ‘ – ‘ ‘ – € d ˜ – ¤ £ ¡  £ ¢ ¤ !  ¤ ¤ m – ˆ ˜ ‘ – € y „ — — – mm – ‚ ™ ˜ — ‘ – ˆ “ m € y — — – ™ — “ s — – ˆ „ e ˜ ™ j — – m – y y — – — — – m s “ … ˜ – — – p € „ … — – j j — – m ˜ ‘ „ — – y y  ‘ ep – ˜ — – p f — € y – mm “ € “ ƒ † £¤ ¤ £ £   ¥    ¡  y ˆ – j km s „ € d –m ˜ — – p ‚ — ™ ˜— ‘ – ˆ — –m s “ … d „ € ˜ – ˆ ˜‘ – ‘ k j “ –mm “ ˜ – ˆ ‘ “ € l d „ € ˜ – ˆ — –m ymm – p — – ˆ f – — € – •‘ m  £¤ ¤ £ £   ¥   ¥ ¡  y ˆ – j km s „ € d –m ˜ — – p ‚ ™ ˜ — ‘ – ˆ “ m – ˆ – ™ ˜ y € s ‘ „ ˜ “ j ˜— – – ‘ y ˜ ‘ – ‘ k j “ –mm “ ˜ – ˜ ˜ – d d „ € ˜ – ˆ — –m ymm – p — – ˆ ‚ – ™ —m ˜ y –mm € l “ m – ˆ –mm “ ˜ “ m – ˆ ˜ – ™ ˜— ‘ – ˆ – ˆ – € k ˜  € p y ˆ u  „ ep y ˆ – m m – p ˜ — – – y y  ˜  € p — y m d “ m – y y  ˜ “ j ™ m s „ € d “ m f — – ˆ „ e ˜ ™ j — – p € y „ ƒ † — – ™ ‘ ‘ „ ˆ — –m — ‘“ ˆ €– ˜— u– ˜ y – d  y y ˜ y € s –m € –m „ — ˆ ˜ – j € – d “ m – p f — “ p — – € ˜ y “ ˆ — ‘ “ ˆ — “ j f – ™ —m ˜ y –mm € l – ˆ –mm “ ˜ “ m – ˆ u „ ep y ˆ – p ‘ – y y ™ — ‘ „ p “ m ˜ — –  p – p f — “ p – p — ‘ “ ’ † – ˆ u € d ‘ y ˜ – – ˆ € y – y l ‘ „m – ‘ y € “ d ™ — € ™ ˜ p “ € “ p ˜ – s „ m – ˆ — “ p –m ˜— – i † — ™m „ — ˜‘ – „ — — ˜ – s „ — ‘ “ ˜ € – p – y y ˜ – j ˆ “ –mm – y y € – • € – — s „ — € „m — k ˆ ˜ y – d ‘ w † –m d j – u– m € y— – ˆ „ e ˜ ™ j – ˜ ˜ – p – ˜ ‘ – — ™ € d † o – € y l “ n ! ©  © © © ! © ©©    © ©© ¥ ¥ © Y Y Y  " $  Y ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ ! ¨ ' ( ' & ¥ 3 2 1 ¤ ¤¨¤£00" ¤)§¤ ¦¥ ¦¥ ¦¥ ¦¥ ¦¥ ¦ ¦¥ ¦ ¦¥ §!'§"¨0!)¨ (§0¤00'( ¤ ¨¨  ¨¨¨¤  ' §§0'0'!¤£¥¤£ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¢¡ ¢ ¢¡ © † –m d j – u– m € y— — –mm € l — –m € y — ™ — “ s l ‘ € – ˜— ym i Y # € y– y l ‘ „m Y # € y– y l ‘ „m o $ ¥ "† o ¦  YYY Y Y Y o YYYY u € d o $  YYY Y Y Y o YYYY u € d  ¦ ¤ ¡ ¡¡  ¢ ¦ ¤ £  £ ¦ ¢ ¦ ¡ ¢ ¢ ¡ ©  ¤ ¨ ¦ £ £ YY † ¢ o f “ m – q ˆ ‘ “ “ j € – ¡ — € – ˜ — ym p — –m € –  ˜‘ – ˆ € y „ d – p “ d— – — y „ — m – ˜ ‘ y — ‘ “ ˆ ˜ ‘ – j – ‘ ‘ „ ˜ ˜ € “ d – ˆ – ˆ „ e ˜ ™ j – ‘ y € – —m ˜ y ˆ — y d f j y j u“ j – p ‘ “  € “• – ˆ – p “ d — – — y „ — ‘ y — ‘ “ ˆ — ˜ – s „ — –m € – ˜ – „ € d – ˜— — ‘ „ p m “ € ˜ p – d— l ‘ € – ˜— ym p y ˆ ˜‘ – j – ˆ ‘ „ … –m f — ˜ – s „ — –m – € ˜ ‘ – ™ ˜ € “ m j— – ˆ – p € ˜ “ j – ‘ y ˆ ˜‘ “ ˜ € “ ƒ     I @ BE C ¨   © ¦ 32 1 DH B C E AI F ¢ $ ©  ¥ † – ˆ „ e˜™ j – ˜˜– p – ˆ — – y y˜—€™ ˜p “ €“ p — –m – j y— ™ € † o y“ –m s“ ˜ –m f ˜‘ – j –m “ ‘ v † — “ p — – ˆ € y –mm – j –m — ‘ “ ˆ – y y ˜ “ € ˆ “ y y — ‘ „ j y “ ‘ „ ¡ “ … – ˜ y „ ˜ – ˆ “ € – —  p – mm – p —  “ j f ¢ f ‘ “ € ˆ ‘ “ ep “ j “ g ˆ ‘ “ –  ˜ ˜ – ƒ¡ — ‘ “ ˆ — – ™ • y „ € ˜ – € ˜ ˜ ‘ – • y – d ™ ˜ u – m d j „ p – ˜ ˜ – p – €  y ˆ ™ € € y „ d — – ™ —  m  ˜ y — – y y  ‘ ep – ˜ — – ˜ ‘ – € ™  ’ † ‘ – ˜ — – ™ ˜ u–m d j „ p “ m f t z q – ˆ ‘ „ ˜ p y € ˜— ‘ „ p – ˆ – y y— — “ m p – ˆ „ e ˜ ™ j – ‘ y ˜ ‘ —m ˜ y ‘ – ‚ m “  ˜ ‘ t z q – e d “ € l –m € – j € „ … € y „ d – ™ —m ˜ y – ˆ „ e ˜ ™ j “ m – ˆ ™ ˜ u–m d j „ p “ m ˜‘ – j –m “ d p ‘ € d ˜— –  p – ˜ jm “ n † –m d j – u– m € y — — – e d “ € l — –m € y — ™ — “ s l ‘ € – ˜— ym i † o ¦  YYY ! O(N 2 log N ) © € y– y l ‘ „m ¥ Y # o $    YYY ¡ ¡ ¡ ¢ ¡ ¢ ¢ ¢ ¢ ¢ ¢ £ £ £ £ £ £ £ £ £ Y Y Y o — € – ˜— ym p — – ˆ ‘ „ ˜ “ j € „ … “ m € y „ d — l ‘ „m — ym d — –m — p €“ — – ˆ ‘ „—— – € d d yz YYYY u € d   † ™  p ™ d — m y – — ‘ y – € y – € ™ … ‘ ˜— – € y –m “• “ m ˜‘ „ ˆ — p € “ — –m € – • € – — ‘ „ p – ˆ ˜ „ — f — l ‘ „m — ym d — –m — p € “ — –m € – j € d d y— – ˆ ˜ „ — € l “ — ˜ y – d m f — € – ˜— ym p — – ˆ ‘ „ ˜ “ j € „ … “ m € y „ d ™ —m ˜ y – € ˜ k j “ € “ d –m ‘ „m – — — y ƒ † –m “ j ‘ j ˜— – f — “ p – € ˜ „ ‘ — ‘ “ ˆ — – p ‘ “ ˜— ˆ u y “ ˜‘ “ ˆ ‘ „ d— – € € „ p f — p € “ — – ˆ — € y –m “• — – ˆ – j j „ — “ m ˜‘ „ ˆ – u – ‘ ‘ „ p – e d “ € l ‚ t z q ¤¤¥ § £ £ £   ¨    £ ¡ – m – ˜ y — ‘ – – • € ™ ˆ ‘ – ‘ w † — ™ € ™ ˆ  — ‘ „ p ©   ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ † – ™ • –m ™ ˜— – ™ ˜ u–m d j „ p € y –m f —m – p € ˜ “ j —m y pm “ p — – ˆ € y — — – ™ — “ s — – ˆ „ e ˜ ™ j — – ˆ ˜‘ “ —m ˜ y — y d f — ˜ – s „ ˆ — – € “ d — –m — – ˜ y „ ˜ – € ˜ ‘ – ™ ˜ € “ m j— – ˆ – p € ˜ “ j – ‘ y ˆ ˜ ‘ “ ˜ € “ d f – € ˜‘ „ p € “ ƒ † – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜ — ym p — – ˆ € –  ˜‘ – ˆ ˆ — –m s “ d “ p ˜‘ – j –mm – ˜ ‘ – ˜ „ d ˜‘ „ — — –mm – f — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d — – m € – j € „ …— ‘ “ € ˜ – ˆ — –m s “ d “ p ˜ ‘ „ — — –mm – y y — y ƒ † — – ™ — „ d „ € d ™ ˜ ™ ˜‘ „ — – ˜ ‘ “  € “ • — – — y – € s j „ ‘ – ˆ f ˜ ‘ – j j – p ™ € ˜ ‘ – j – •  ˜ “ m – € ™ l € – j ™ ˜ ‘ “ “ – ˆ „ e ˜ ™ j – ˆ – d ˜ – i † — € – ˜— ym p —m – ˜ – ˆ € –m s p – ˆ –m s “ d “ p — “ d ˜— – ‘ m y – — — ‘ “ – j – y y — € „m “ f – y y ‘ „ pm – y y ‘ „ ˜ “ ˜ ‘ – € „ ˆ — € – ˜— ym p — – ˆ € –m s p – ˆ ˜ – j € – d — ‘ “ – j – j e ˜ € „ lm “ m p – • “ – y y  ‘ ep – ˜ – m m – ˜ – ‘ y € – —  m  ˜ y f – m d j – u – € “ ƒ † m “  ˜  ‘  – p “ d — – m — ‘ “ ˆ – y y — – ˜ ‘ “ ˜ € „ d j  — y m d ˜ ‘ „ — — € – ˜— ym p – € ˜‘ – — – p ‘ – € ™  ˆ — –m m – y y –m — ‘ “ ˆ – p “ d — – — y „ — ‘ y — ‘ “ ˆ – ˆ „ e ˜ ™ j “ m – ˆ – d “ ˜ ™ – y y “ ep € – p “ m d – — – ˆ ˜ — – — – ™ ‘ ‘ „ ˆ — – ˆ — ‘ „  ˜ “ j € „ … — ‘ “ € ˜ — – m m – ˜ – ˆ € – —  m  ˜ y ˆ ˜ € ™ ˜ ‘  n † – € ˜ k j “ € “ d – p ˜‘ – j – y y ˜ “ j „ ˜ y “ € ‘ ™ ˆ € y „ d — – ™ € ™ l l y— ™ ˜ ™ ˜ ‘ „ — – y y  ‘ ep – ˜ — € y –  — y m d —  “ j f — ™ e p € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m ˜ ‘ – • y „ — ˜ € –  y y – € – ˆ „ e ˜ ™ j – m m – ˜ – ‘ y f ˜  ˆ ˜ ‘ – j – € ˜ y # † ˜ ‘  – ˜ ˜ “ ˜  „ — € y – ˜ “ —  m  ˜ y m € “ d — ™ ep € – e p – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m – y y – p y y — y f ™ m s  p – d y „ € l — y „ — – y y “ ep € y — – ™ € ™ ˜  ˜ — – – m m – —  y d f u y – ˆ ‘ – — ˜ – s „ — – ˆ – — “ s “ m € – —  •  ˆ € y „ d – ™ —  m  ˜ y — € „ m “ ˜ — – ˜ ‘ – j – ‘ ‘ „  ˜  ˜ € “ d – ˆ – y y  ‘ ep – ˜ – ˜ ˜ – i † — ˜ – s „ — –m – € ˜ ‘ – ‘ „ ˜ “ € “ d ™ — “ m ˜‘ “ — j u“ j – € y d y „ p – ˆ – ‘ „ “ m € – • y „ € ˜ € y „ d – e d “ € l – ˆ ‘ „ ˜ ˜ € “ d – ˆ – j e ˜ € „ lm “ ‘ y € – —m ˜ y ‘ – s y „ f – p “ d— – m – • y „ ‘ – m — ‘ “ ˆ — ‘ “ – j – d ˜ – ˆ – j e ˜  € „ l m “ ‘ y € – —  m  ˜ y ˜  „ — ˜ y – d ‘ w † — ™ ˜ – „ € d — ˜ – s „ — – ˆ – m s j – — ‘ – m € – ‘ ‘ „  ˜  ˜ € “ d € y „ d — – ™ —  m  ˜ y – € ˜ ˜ ‘ – • y – d — – y y  ‘ ep – ˜ — – ˜ ‘ – € ™  ˆ f – ˜  y — ‘ ‡ † — ™p „—— “ — – € d „ € d — € y– ˜p –• — – p – ˆ –˜€“ d – ‘ y €“ d ™ j€ „ … – p “ d— – m — ‘“ ˆ — ™ ˜– „ € d — € „m “ ˜‘ „ — — ˜ – s „ — – n † — ™ p „ — — “ — – € d „ € d — € y – ˜ p – • — –m – y y — ‘ “ f ™ ˜ € “ m j— – ˆ – p € ˜ “ j “ m – ˆ — – € d „ € d — € y–m“• — –m €–m ypm“ p p ‘ „ ˆ – ˜—— ‘ „ p – ˆ „ e˜™ j “m – ˆ – d“ ˜™ – €k j– € d “ n  © © !  © © ©   © ©     © ©  © © © © ©  ¥   $† o ¦ ¥ ¤ £ !©  — € – ˜— ym p – € ˜‘ – –mD“H j" ‘ j –p ‘“˜—ˆ G  — – y y € ˜‘ – p ‘ „ p — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜— ‘ – ˆ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– –—€ d ˆ€“ e – j ˜ ‘ “ m “ ˜‘ – j ™ € p ‘ – ˜— ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d  – e d “ € l – ˆ – j € „ … — y „ — ‘ „ ˜ ˜ € “ d o  ! ©  † — – e d “ € l — –m € y— ™ — “ s l ‘ € – ˜— ym p y “ — – ™ p „ — — “ — – y y ˜— € ™ ˜ p “ € “ i y„ y„ y„ y„ ‘„ ‘ ‘„ ‘ ‘„ ‘ y„ ‘„ ‘ ‘„ ‘ y„ O(M × N 2 log N ) €„ D€d(“ ©—W– p ‘F ©“——G“G‘‘„p £ ¡ ¨ ¤ ¤ ¡¦ ¢ ¡ © ¤ ¨ ¦ £ £ ! # # — – — ‘ – ˆ — – m ymm – p – ˆ — – m s j – — ‘ – — – m € – ep € – ep – € – ˆ ˜ — – ¢ f† m “ ˜ – m “ ‰“ € l #¡ — ‘ „  — ‘ – j ˆ — – m € y — ˜ ‘ “ ˆ ‘ – p — “ l ‘ € – ˜— ym p – p “ d— s y — y ˆ m “ € ™ ‘ ™ l – d p ‘ € d –m f — –mm € l – ˆ ‘ „ ˜ “ —m ˜ y m € y— ™ — “ x † u y– j –m ˜‘– ˆ ‘ „ d— – €€ „ p € y–m  y y — – p “ d— – — y „ — — –m – ˜ y— ‘ – –m s p ‘ „ — y d f — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ — ˜‘ – — ™ € d — € – ˜— ym p — –m ˆ € „ s “ ˆ – ep € – e p ‘ „ f  £   £ ¤   !  ¤ ¨ ¥ ¡   ¤  £  ! £ ¤¢   — – ˆ „ e ˜ ™ j — – m € y „ d f – €  “ € ˜ ‘ „ p y “ † — – p “ d— – — y „ — — – p — ‘ “ ˆ — ˜ ‘ – — ™ € d — € – ˜ — ym p — – m – ˜  y — ‘ – – e p € – ep ‘ „ ˜ – f — € – ˜ — y m p – ˆ ‘ „  ˜ p – ˜ ™ ˆ “ m € y „ d — ™  € d „ € d d “ — y m d — – m — – p “ d — – — y „ — — – m ˆ € „ s “ ˆ – e p € – ep ‘ „ f  £   £ ¤   !  ¤ ¨ ¥ ¡   ¤  £  ! £ ¤ ¢   — – ˆ „ e ˜ ™ j — – m € y „ d † ‚ ˜‘ – y l ‘ ˜— ˆ – — — – ˆ „ e ˜ ™ j – ˆ — –mm j “ … — – ˆ ‘ “ € l u y – ˆ f – € ˆ “ p – p — ‘ “ ’ † l ‘ € – ˜ — ym p – p “ d s y— – ˆ — “ p – p — ‘ “ ˆ –m € “ d ‘ w † — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ m “ ‘ l € „ – p “ d— – m – ˆ — – p “ d— – — y „ — — ˜ ‘ – € ™  ˆ — ‘ “ ˆ — ™ ˜ – „ € d – € ˜ ˜ ‘ – • y – d — € – ˜— ym p — –m – y y ˜ “ … –m – ˜ d j „ p ‘ – – € ˆ ‘ – € d – ˆ ˜ y s € y „ d ˜‘ „  y y ™ l € – j ™ ˜‘ – j j – p ™ € ˜‘ – j – • ˜ “ m – € ˜‘ „ l ‘ € – ˜ — ym p – ˆ — – ˆ „ e ˜ ™ j — € y –  — ym d f o† ‘ „  ˜ p – — “ m — ‘ “ ˆ ™ y y „ • ™ — ‘ „ •“ m — y „ ‘ – j j „ i ED @ G D C   @ ¦ DH B C E AI  C  @   A £ ¢ ¥ ©  © ©  ©    !    © ¨ † — – ™ ‘ s j „ p — – ˆ „ e ˜ ™ j u y “ — ™ p „ — — “ u y “ – m s “ ˜ u y “ “ m – p € y „ d — ‘ „ „ • ‘ – € — y „ † — – y y  ˜ —  € ™ ˜ p “ € “ p — € y – m – ˆ … ˜ “ m y ˜ d “ p ™ € y “ – m s “ ˜ ‘ y p y “  p — ‘ „ — „ d „ € d – ‘ — y „ ‘  „ y y € y „ d ˜ — – p f ‘ „ — “ ‘ s j „ p “ m – ˆ — € „m — – ™ —m ˜ y — – ˆ „ e ˜ ™ j — – ˆ p ‘ „ ˆ ˜‘– ˆ ‘– d™ ˆ – € ˆ “ p – p — ‘“ ˆ – € • y ¤ ‘– — –— j — – ˆ „ e˜™ j — – ˆ — – y y˜—€™ ˜p “ €“ p — – n † ˜ ‘ “ ˆ ‘ – p — “ – y y  ep € “ € ™  e – j e ˜  € „ l m “ ‘ y ˜ ‘ “ —  m  ˜ y ‘ – € – ‘ ‘ „  — y … — – m —  y d f — ‘  —  „ • — – ep „ € d — ym d — – ˆ – e d “ € l y ˆ ‘ „  ˜ ˜ € “ d € “ d – ˆ „ e ˜ ™ j – ‘ y ˜ ‘ “ — m ˜ y ‘ – — € – ˜ — ym p   ¤  – ˆ ˜ ‘ “ ˜ € „ d j m “  ˜ ‘ –m s j – — ‘ – ‘ y € – ™ € p – ˜—— ‘ „ p  • y— – d p ‘ € d –m f ¢ f† m “ ˜ – — d  € “ ¡ — ‘“ ’ †–‘ ˜ — – — – m m  € l — – m € “ d – p “ d — – m – ˆ ‘ „  —  •  ˆ “ m – ˆ ‘ „  ˜ y m „ — ™ € “ m — y m d f –  ep € “ € ™  e “ m — ‘ “ ˆ ˆ ‘ – p — – ˆ ‘ „ — y m d —  y d f – p “ d — – m – ˆ – m s j – — ‘ – m – ˜ ‘ – — ™ € d – € – m y m m – p – y y  ‘ y – ‘ y f –  ep € “ € ™  e “ m – ˆ ˜ y “ e ‘ ‡ † — – m m  € l — – m € y — – ™ — “ s – ˆ „ e ˜ ™ j – ‘ y – ™ ‘  s j „ p ˜ — – – y y  ep € “ € ™  e – ˆ „ e ˜ ™ j “ m f — “ p – p — ‘ “ ’ † – ™ — „ d „ € d ™ ˜ ™ “ ‘ „ —  “ ‘ s j „ p – ˆ – ˆ „ e ˜ ™ j – € ˜ y “ – ‘ y f ¢ f† m “ ˜ – l ‘ “ ¡ — ‘ “ ’ † — ‘ “ – j – j e ˜ € „ lm “ m ˜‘ “ —m ˜ y ‘ – — € – ˜— ym p — y „ — u y – ˆ € – • y „ € ˜ — ˜ – s „ ˆ – l “ ˜ ‘ “•“ ˆ ˜ ‘ – ‘ ‘ –  ˜ ‘ „ p  y y u y – p ˜ ‘ – j – m m – y ˜ ‘ – • ™ ˜ ‘ “ —  € „ •“ … ‘ – f – € k ˜  € p ‘  “ ˜ € – p ‘ y ‘ „ m – — € – —  •  ˆ € – ˜ — y m p ‘ y €  —  „ ep ‚ ˜‘ – ˜ ˜ “ ˜ „ — — € – ˜— ym p – ˆ ™ €— ™ ˆ – € s j „ ‘ –m – y y – p y y — y € – € ™ ˜ † — ˜ – s „ — –m — y „ ˜ ˜ ‘ “ d y „ € l – € € – ˜— ym p – y y ‘ y ‘ y € – j € „ … † ‚ –m d j– u– €“ d – ˜ —  — ‘ „ p ¢ f † m “ ˜ – e p “ s ‘  – ˜ z¡ — ‘ “ ˆ – ™ — „ d „ € d ‘ „ —  “ ‘  s j „ p – ˆ – ˆ „ e ˜ ™ j – ‘ y f  — ‘  # † – € y – € ™ … ‘ ˜— –  y y – p “ d— – ‘ – ˜ – — d j – ˜ ‘ – ™ ˜ u–m d j „ p – ‘ y “ € – ‘ € – ˆ – p – y y — € „m “ f ‘ „ ˜ ˜ € “ d € “ d l ‘ € – ˜— ym p –m – y y –m d y „ — — y m d ˜ — – – y y  e p € “ € ™  e l ‘  € – ˜ — y m p – m ‚ — – €  “ ˜ ‘ – j ™ m d j „ p ˜ ‘ – m s j – —  y y — – y y  ‘ ep – ˜ u y – ˆ — – p – ˆ — – l “ ˜ ‘ “•“ — – ˆ ‘ „  ˜ “ ˜ „ m d u – m € “ d ™ • ˜ „ j ™ ˜ ™ “ l ‘ € – ˜ — ym p y ˆ ‘ „  ˜ ˜ € “ d € “ d — ‘ „  — € – • — – m ˜ – — – y y  ep € “ € ™  e — ‘ „  — € – • — – m – € ˜ ‘ – — – ˆ  € s  e — – ˆ „ e ˜ ™ j — – ‘ “ ˜ € – p – ˆ ˜ ‘ – j – d d „ m – • ™ ˆ – n C G H B  ¡ £ ¦ D H B C E  A I F  ¥  © ©©  # # # © ©©    $ # #    YYY ¥ ©  ¥    © " ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ ©  † — – ˜ € – • y „ p ˜ ‘ – „ — — –m ymm – p — –m — – ˜ y „ ˜ – y y – p y y — y – ˜ y— – ˆ  — ‘ “ ˜ – — – ™ l “ — •‘ – ™ ˜ ™ ˜ ‘ – “ — ‘ „— ‘ – j ˆ — –m — – ˜ y „ ˜ – y y – p y y — y – p ˜ – f – — ‘ – ˆ ‘ „ ‘ – m y m m – p – ˆ — “ d – € • y „ p – ‘ ‘ „  l ™ € – m m – • y „ ‘ “ m  — – ep y “ l “ m y „ – ˜ „ € ˆ “ m € y— – ˜‘ “ € y „ p ‘ „ l ™ € “ m € l € “ m ™ ˜ – ˜ ‘ – j – € „ ˜ “ ™m “ ‘ „— ‘ – j ˆ – ‘ y € – ‘ ‘ „ ˜ p –m ™ — ˜ ‘ – j – €  „ ˜ “ ™ m “ – ™ ‘ ‘ „  ˜ p – m ™ — – m y m m – p – ‘ y p – •“ € – € € “ j ™ ˆ ‚ ‘ „ ˜ y „m l – j e ˜ € „ lm “ ‘ y € “ d ˜ “ … – — — – — “ e d u y – ˆ — – p – ˆ – € k j – € d “ n ©  ¥ ¥ ¥ † –m “ j ‘ j – € y ˜ € – • y „ p – ‘ y € ‘ – ˜ s „ € y „ d — ‘ „ l ™ € — – p  j € “ d ‘ „ ˜ p –m ™ — – ‘ y € – y ˜ p – !– †  — – m “ j  u “ j — ‘ „  l ™ € — – ˆ € “ d € – ˜ — y m p – y y “ ep – ˆ – € y ˜ € – • y „ p – ‘ y € – y ˜ p – ! – †  ‚ — – d “ ˜ ™ u y – ˆ ‘ – – — „ d j „ p ™ ˆ – — — € – ˜— ym p — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – — “ e d – € k ‘ € – ˆ “ m f ‘  ‘ ‡ † —– mm€•l —–m € y— ™—“ s l ‘€–˜— ymp ‘ y ˜‘“—m˜ y ‘– —–™˜p– ‘ ‘„p —–— ‘– ˆ —™––my˜˜—m‘m––—ˆp‘„–p™ˆ˜–™—ˆ–„“m se—j˜˜™‘–j—–‘‘“–m˜—€–––ˆmd–€–˜–‘j“˜j‘„y–pˆ— —–™dl“–y˜€™©ˆ—m–fp˜““˜dˆ—–—©ˆ–p‘—f“y—p–„p–—“p—d“–—dˆ–—–—–my—s„yj—„—–——––‘p€–˜—©y‘m““–© ˆˆyy———€y€–m„˜dm—“yf –mp˜©—‘y–—mm‘©€‡y–ym s˜–p — ‘ “ ˆ — ym p ‘  ™ ˆ ˜ — – † — ‘ „ — ‘ – j ˆ  – p “ d— – — y „ — y “ – • y „ ‘ – p — ‘ “ ˆ — ym p ‘ — ‘ „— ‘ – j ˆ  – p “ d— – — y „ — ˜ y „ ˜  — ˜ – f ‘ y j j „ p ‘ – — ‘ „— ‘ – j ˆ — – € k j – € d € y –m ˜ ‘ „  p  u y – p — — ‘ „— ‘ – j ˆ — y„— uy“ — ‘„— ‘––ˆj—–ˆp“ d—–— y„— uy– ˆ – ‘—˜s“jˆ„ˆp‘‘“p„ f——–‘p„“d——‘––jy„ˆ— —–† —ˆ™ j–—€—d—“˜dd“yˆ€—yˆ—„–‘dp““pfd———–y–pƒ“— yd„—–— — € ˜ – — ™ • € – — ‘ „ p — – p “ d — –  — y „ — – € ˜ ‘ – – € y d y „ p – ˆ – ‘ „  “ m – ˆ € – ˆ  p ™ ˆ € y „ d   § £¤ £   ¥ ¢ ¤ €¡˜ˆ £˜‘–jn™’jq„…– ‘dyp ‘˜‘€„d— –—m ˜€–y—s„˜‘—“–—m“s— ––—m s‘–“…f “˜€‘––—j–y—y€––•€‘y —˜––jf–p—“–d——‘––ˆ— y—„–—m y–mmm–—p‘“—ˆ–m—™€y“ ds — € — ˜ –  s „ — – ˆ – p “ d — –© m – ˆ – € y ˜ € – • y „ p “ m ˜ ‘ – —  j u “ j  y y — – p “ d — – — y „ — — – m € – ‘ ‘ „  ˜ p – m ™ — – ˆ ˜ — – –™——‘„„dp„€ d—˜–‘y–y‘‘˜€e–pd–˜—–“pm “fd——––——‘y–„ˆ— ——€––m˜€—–y‘mp‘„—–˜ˆp–€m™ ‘— –€˜y‘„„ƒp †––ˆp“—d–—m s–©m˜ d––ˆp—‘y„——˜‘‘–“j˜™ˆ u–yy–yp“ e€p–€€™yˆ— ——˜y– „s—„—–—m–m€–˜m‘s“˜p–€y„€„dd ‘––™—–€m€˜“yjp™‘ˆ„ ˆ‘ w˜—†–—––—˜‘‘“– ˆˆ ‘—–€p–—˜“— y–mep p—„–€ˆd˜d‘“–•– ‘yy„€f˜– –d—p—‘m–€ydy—––pm €—y‘—“–ˆ™——–“px“ d—– (m − 1) Sm−1 Sm m (m − 2) m (m − 1) Sm−1 Sm−1 (m − 1) £ ¢ £ ¡ m (m − 1)  £   £¤  ! ¡ " ¤ ¢   ¤   ¡  ¤ ¢ ¤ ! £   ¢¤  ¥  ¤  ¡   £  ! ¥ ¤   ¡ ¨ ¢ £ ¡© ! ¤   ¥    £ ¤  ¤ ¨  §    ¡ ¥ ¨ Ck   £   £ ¤   ! ¡ ¤ ¢   ¤  ¡  £ ¡  £  ! ¥ ¤   ¡ ¨ ¢ £ ¡   ¤  ¤ © ! ¤ ¨ ¤  £¤ £ ¡   ‚ – ˜ ‘ “•  y— – ‘ „ ˜ „ ‘ „ j – ˆ ™ ˜ ™ € d „ € d “ m – —m ˜ y – j e ˜ € „ lm “ m f – — “ e d – € k j – € d “ m ˜‘ “ ‘ € – p ‘ „ i † — ™ • y „ € ˜ — € – ˜— ym p — – ˆ –m “ j ‘ j ‘ „ ˜ d € p — – ˆ – ‘ y € – € ™ ‘ ™ l † o — – p “ d— – — y „ — — – p — ‘ “ ˆ — € – ˜ — ym p — –m € –  ˜ ‘ – ˆ † — – — ‘ – ˆ — € – ˜— ym p — – ˆ ˜ ‘ – • y „ € ˜ – — —m – y y — –m — ‘ “ ˆ — – p “ d— – — y „ — — –m € –m s p † ‚ — –— “ e d — „ €˜ ‘– –— „ d j „ p ™ ˆ –— – ™— „ d „ € d – ˆ „ e˜™ j “ n † – — ‘ – ˆ – j j „ p – ™ € ™ ˆ— ‘ „ p ˜ „ — –mm – y y € y „ d –mm € l “ m – ˆ –m ymm – p – ‘ y — ‘ “ ˆ — y‘ – ˜ ‘ „ p – € ˜ ˜‘ – • „ ˆ  y y — ˜ – s „ ˆ j y j ‘ j – l “ ˜ ‘ – p € y „ d –m – ˜‘ – — ™ € d – € ˜ – f € y – y l ‘ „m – j j – ˆ — –mm “• € – ˜‘ ‘ – — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d— – m – ˆ ‘ „— ‘ – j ˆ – y y “ ep ˜ ‘ “ ‘ ‘ „  ˜  ˜ € “ d ‘ – – y ‘ – ˜ s „ f € – —  m  ˜ y – m m  € l “ m – ˆ – m m  “ ˜ “ m € – ‘  j € – ˜ ™ ˆ € y „ d ™ —  m  ˜ y € – ˜ ‘ – ‘ y ˜— – ‚ — € y – ˜ “ —m ˜ y — – € ˜ k j “ € “ d u y – ˆ – ˆ – ™ ‘ ‘ „ ˆ “ m p ‘ „ ˆ – ˜— — – p ™ ‘ m f — –mm € l — –m € y— – ™ — “ s – ˆ „ e ˜ ™ j – ˜ y „ ˜ – j j „ i † m “ ‘ l € „ – p “ d— – m – ˆ — – p “ d— – — y „ — — – ˆ — ‘ “ ˆ — – ™ ˜ p – ‘ ‘ „ p    © Ck τ © © © ©    ©    ξ ξ   ©  ¨ ¡ ¡ ¥ ¡ ¤ ¤ ¦ ¡ ¢ ¡ ©  ¤ ¨ ¦ £ £ ¤ ¤ ¤ ¡ ¡¦ ¢ £ – ˆ … ˜ “ m y ˜ d “ p ™ € y “ – m s “ ˜ – ˆ  p  € – ˜ ‘ – — ™ € d – ˆ – € y — – j ‘ – — “ d — – j j „ — – ‘ — y „ ‘ f — – ˜ ‘ – € ™ ! ˆ — k € ˜ — – y y  ˜ —  € ™ ˜ p “ € “ p — – ˆ ˜ ‘ “ “ – € ˆ “ p – p — ‘ “ ˆ — – ™ — „ d „ € d — – ep „ € d d “ — – ˜ ‘ – € ™ !  ˆ — – n † – m s j – — ‘ –  — y „ — ‘ y € – ‘ ‘ „  ˜ p – m ™ — ‘ –© ˆ y – m y “ f — ‘ „  — ‘ – j ˆ — – p € y — — € – ˜— ym p — – ˆ — – € s j – j — – ˆ ‘ „— € – d— ˆ “ m – ˆ ‘ „ ˜ p ‘ „ … ‘ – — ‘ „ — ‘ – j ˆ — –m € – € ™ ˆ ‘ „ d – ˆ – € k l  l y— m f “ m – p € y „ ƒ † — € – ˜— ym p — – ˆ — – ˜ ‘ – ‘ ˜ € – d — ‘ „— ‘ – j ˆ — –m € – ‘ ‘ „ ˜ p –m ™ — € y „ d — € y – ˜ “ —m ˜ y — – € ˜ k j “ € “ d – ˆ € – — — “ d – — – ˆ — — y “  ym ˜ – j € – d " Y Y  f† m “ ˜ –  ‘ „ p ‘ – j „ ’ i # n f ‘  ‘ ‡ † — – € k ˜  € p — – m – ˜ p – d — – € – ‘ — € – ˜ — y m p – ˆ ‘ „  — y … – ‘ y p y “ — y m d – y y —  „ … – y y “ ep  j j˜—y€m–jyd–‘——‘j—„–ˆ——‘–‘–˜‘„j–˜‘ˆ“p˜—€––pˆd„€–—j‘‘‘„p„–—•‘˜“p––jmf –™˜—ˆ‘€“–yˆˆ„‘d––p€™—s—“jm–„˜yy‘y˜–e—m–p€–m“m"y€™€–˜—e‘–„˜€p—˜–ym“–y™–‘—ym‘˜˜–y”f–—†ˆ€—––„˜ye—y˜y™mj“u“‘yn“ˆ†™——–pm˜„‘y—–—–‘“— p — – ˆ € – — m ˜ y© ˆ – — „ d „ € d m f — – ˆ „ e ˜ ™ j — – ˜ ‘ – € ™ ! ˆ — – p € “ d —  y y – € — – € ˜ k j “ € “ d — – m € –  p ™ d — – ˆ € y – ˜ “ — m ˜ y© m € y „ d – m p £ ˆ — k € ˜ ˜ — – m© y y ˜ ‘ “ ˜ “ ˜ — ‘ „ p f o Y Y  f† m “ ˜ – d ƒ g #  ˜ ‘ “ y ¡ † — – ™  p „ — — “ — ‘ „  — ‘ – j ˆ – ˆ – € s j „ ‘ – m – y y  — ‘ “ — ˜ –  s „© ˆ – € s j „ ‘ – m ˜ ‘ – —  j u “ j  y y u y – p — € „ m “ ˜ ‘ „ — — ™ • € – — ‘ „ p — € – ˜ — ym p — – n † — – ™  p „ — — “ — – ˜ ‘ – ‘ ˜ € – d — ‘ „  — ‘ – j ˆ — – — € y  € y –  € ™ … ‘ — ˜ — – — – € s j – j — – — – ˆ ‘ „  ˜ ‘  ™ ˆ – ˆ – mm “ • € – ˜ ‘© m  — ˜ – f – — “ s “ m – ˆ — ˜ –  s „ — – ˆ — ‘ „ j y “© ˆ ™ — „ d j „ p ˜ — – m© — ˜ ‘ – ‘ ˜ € – d – j j „ p ™ € ™ ˆ  — ‘ „ p ˜ — – € – ˜ — ym p ‘ ” † ˜ – — – € ˜ k j “ € “ d u y – ˆ € y — – ™ — “ s – y y  ˜ — “ e p „ ˜ — – ep € – ep – € – ‘ y ˜ ‘ – j – m “ l ™ – —  m  ˜ y  Y Y  f † m “ ˜ – p y  d „ p „ € ƒ i w ’ —€™˜p“€“p – j j„p –™€™ ˆ— ‘„p ˜—– p–mm–p —€„m“ f ‘„—† –‘ˆ– j„ˆˆ™ jm y€ y“—™p–„ˆ——“— ‘€–„˜j— y–mpˆ y—ˆ˜‘–“y˜—yˆ˜ “ ˜ ‘ „ — – ˆ „ ˆ ™ j y ˆ — ‘ —  „ • — – ep „ € d — ym d — – ˆ  — f — ™ ‘ ‘ „ ˆ ‘ „  — ‘ – j ˆ – ‘ y ˜ – – ˆ „ ˆ ™ j ‘ y € y „ d ‚ € y – ˜ “ — m ˜ y© m € “ d ™  p ™ d — – m “ j ‘ j ™ ˜ — ‘ – ˆ – ˆ – € k ˜ € p ‘ y ˜ ‘ – ˜ p – d — – € — € – ˜ — ym p — –m — –mm – y y — –m € y — — –mm – p ˜ ‘ „ — — € – ˜— ym p — – ˆ — – ˜ ‘ – ‘ ˜ € – d — ‘ „— ‘ – j ˆ — –m f t r ’  r v € y „ ƒ † –m “ j ‘ j ˜— – — ‘ „— ‘ – j ˆ — – p € y — – € ˜‘ – p € y –m  — € – ˜— ym p — – ˆ — – € s j – j — – ˆ – ‘ ‘ – „ j – p ‘ “ ˜— ˆ “ m — –mm – y y — –m € y „ d — –mm – p – ‘ ‘ „  ˜ p – m ™ — ˜ – — € – ˜ — ym p — – ˆ — – ˜ ‘ – ‘ ˜ € – d — ‘ „  — ‘ – j ˆ – ˆ ‘ – „ j – € s j „ ‘ – m € –  p ™ d — – ˆ € y – ˜ “ — m˜ y© m  – ˆ ‘“ j– ˆ z ” n i w g ƒ f uy– j –m ˜‘– ˆ ‘„ d—–€€„p  ym  yy — ‘„— ‘– jˆ —– ˆ –m s j–— ‘–© m € – ˜ — y m p – y y “ ep  € –  p „ — — “ € y „ ƒ † — – ˆ „ ˆ ™ j — – m € y — – ™ — “ s – y y  ˜ — “ ep „ ˜ — – e p „ € d d “ – ‘ y ˜ ‘ – — m ˜ y  Y Y  f – – n ˆ ‘ “ „ „ t r ’  r v ˜ – # # # f† m “ ˜ – m “ ‰ € “ l l # z ” n i w g ƒ f  — ‘ # † — ™  ˜ ‘ – ˆ  — ˜ –  s „© ˆ — – d y „ € l u y “ — – ™  p „ — — “ — – ˜ ‘ – ‘ ˜ € – d — ‘ „  — ‘ – j ˆ — – ˆ ‘ „  ˜ p – m ™ — – ˆ — – m “ ‘ l € „ — – y y  ‘ – ˆ ‘ep––p˜—–—ˆ– ˆ– e˜p‘„–€—d„dd“„€–d‘ y–€–ˆ—“mp˜ y–pl—‘‘€“–ˆ˜——y–m™pd–dp„“md–—•s™yˆ— —––ˆˆ „—–eˆ˜™„j˜™—jn–†ˆ— ‘–m„m—j‘“–…j–jk—–umy€–yˆ— “–n˜‘“ ˆ e  ¡ ¢ ¢  ¡  ¡ w ¢ ¡  k ¢ α w α  α K ¢ ¡ ED @ G D C   C G ¦ DH B C E AI  C  @ ¨   A £   ! ¥ † — – ˜ ‘ – € ™  ˆ — k € ˜ — ™ ˜— ‘ – ˆ — – ˆ ˜ ‘ “ “ — € – ˜ — ym p — – ˆ € –  ˜ ‘ – ˆ  ˆ — “ d ˜ ‘ – ˜ ˜ – j € – d – ‘ — m f — € – ˜ — ym p — – m — y „ ˜ — ‘ y j j „ p ˜ ‘ “ ˜ ™ ™ ˜— ‘ – ˆ – ˆ – € k ˜ € p – ˆ ˜ – –m ymm – p – ˆ –mm “ ˜ – ˆ — – € ˜ k j “ € “ d u y – ˆ — –m f € –m y p ˜ € “ d ‘ ‡ †  p – ™ —m “ ™ € ˜— –  y y – ˜ u– ˜ ‘ „ p y ˆ – ˜ d j „ p ‘ – – — € d “ m – ˆ ‘ „ ˜ d – p u– m f — – ™ ‘ ‘ „ ˆ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ ˜ –m d j „ p – p “ d— – m — ‘ “ ˆ — – ™ d d „m – • ™ ˆ — –mm € l — –m € y— — – ™ — “ s — – ˆ „ e ˜ ™ j — –m – y y — – j j — –m ˜‘ „ — — – y y ˜— € ™ ˜ p “ € “ p — – — f — –mm € l – ˆ ‘ „ ˜ “ —m ˜ y m € y— – ™ — “ s ˜ ‘ “ ˜ ™ – ˆ „ e ˜ ™ j – ˜ ˜ – i † — –m ymm – p – ˆ — ‘ „ j –m ˜‘ – € • y „ p  y y — –mm – p € “ d ˜ ‘ “ ¡ ‘ – j j „ p ‘ – — – ˜‘ “ ˆ ‘ „ ˆ – € — ‘ „ l ™ € — –m – j € d d y— ‘ „ f –m “ j ‘ j – € y ˜ € – • y „ p – ‘ y € ‘ – ˜ s „ € y „ d f ˜ ‘ – j –m “ ‘ — y ƒ © © © ©    ¢ ¡ © ¤ ¨ ¦ £ £¤ ¤ ¥ ¦ ¤ ¥ ¢ ¡¨ ¢ © † — – j e ˜ € „ lm “ —m – ˜ – ˆ ‘ „ — “ € “ d j „ p “ m – ˆ y „ f l ‘ € – ˜— ym p – ˆ — – j e ˜ € „ lm “ ˆ — ˜ “ ˜ m y — ™ € — – ˆ ‘ „  ˜ “ y m “ • ™ m – ˆ – ‘  “ j „ ˆ – m — ‘ “ ˆ u y “ •“ € ˜ — „ ‘ – ˜  y — ‘ – – ˜ ‘ – — ™ € d r r r –  ˜ € “ d “ n † – m s “ ‘ ‘ „ —  “ € ™ ˜  u – m d j „ p – ‘ y ˜ ‘ “ “ ˜ – f – m s  — ‘ – e ™ € d j „ p ˜ “ ˜ m y — ™ € ‘ y –  ˜ € „ — ‘ – ˜ ‘ “ — —  ‘ € y „ … f € y – ˜ “ —m ˜ y m – ˆ ˜ € “ d “ m – ˆ — y y – € — – € ˜ k j “ € “ d – ˆ – € s j „ ‘ –m ˜‘ “ — j ‘ j f l ‘ € – ˜— ym p – p “ d— s y — – ˆ — – ˆ „ e ˜ ™ j — – mm – • y „ ‘ – ˆ – € • y ¤ ‘ – – —  j “ m — ‘ “ ˆ u y “•“ € ˜ — „ ‘  — ‘ “ – ˜ ‘ – — ™ € d r r –  ˜ € “ d “ n † — ˜‘ „ d — ˜ ‘ – € ™  ˆ — – p € y— — ‘ „ ˜ y s € ˜‘ „ p — „ ‘ p ‘ „ ˆ — ‘ „ ˜‘ – — ™ € d — y „ ‘ f – — k e ˜ – ˜ ˜ – p – ˆ – ˜ y— “ m — ‘ “ ’ † — – ˆ „ e ˜ ™ j — – ˜ ‘ – € ™  ˆ – ˆ ‘ „ — “ € “ d j „ p “ m ˜ – — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ym “• ™ m – ‘ € – p ‘ „ p l ‘ € – ˜ — ym p y ˆ – € ˆ “ p –m — ‘ “ ˆ – ˜‘ “ ˜ € „ d j – ˜ € – • y „ – y y ˜ “ j ™m s „ € d – ‘ y f ‘ ‘ ‡ †  ˆ ‘ „ … „ € d d “ – € ˜ ˆ p ‘ „ ˆ – ˜ € ™ j  y y f ˜‘ – p ™ € – ˆ y ˜ ™ ˆ ˜ – y— ‘ y – y ˜ ˜ — ‘ „ p — ™ ™ € p ˜ ‘ „ — — € – ˜— ym p — –m m – y y –m — ‘ “ ˆ – ˜ u– ˜ ‘ „ p y ˆ – ˜ d j „ p ‘ – – — € d “ n † — “ p — ‘ “ ˜ € – p — ‘ “ ˆ – y y ˜ “ j ™m s „ € d  — — y “ ˜— – – € • y ¤ ‘ – — – — j — – ˆ „ e ˜ ™ j — – ˆ ™ ˜ u–m d j „ p “ n † ˜ € ™ ˜ ‘ ‘ „ — ™ € lm “ j – ™ ˆ € „ s “ ˜‘ – j – € “ € ˜— – € y – ˜ “ — m ˜ y m € “ d –m s “ ˜ ™ € d € – ˜ ‘ ˜ ‘ – j –m p “ … – j € „ … – ‘ y — y „ — — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d “ m f – j j – ’ † € y – ˜ “ — m ˜ y m € “ d — – ˆ „ e ˜ ™ j — – ˆ – ™ € ˜ ‘ – ‘ – — – € ˜ k j “ € “ d — – m € – u – ˆ ™ ˜m y p £ ˆ “ m – ‘ € – p ‘ „ p ™ € ˜‘ „ p ‘ – € ˜ ‘ – • y „ — — k € ˜ – j km s „ € d ‘ ” † l ‘ € – ˜— ym p y ˆ – € ˆ “ p –m — ‘ “ ˆ — – ˜ € – • y „ – € „ p ‘ – ˜‘ – ˜ — – € — – y y ˜ “ j ™m s „ € d — € y –— ym d f — – ˜ ‘ “ ˜— u– — – ˆ „ e ˜ ™ j – ˆ ˜ ‘ “ ˜ € „ d j – € s j „ ‘ –m ™ € lm “ q † – ™ —m ˜ y – € ˜ ˜ „ ˆ  y y l ‘ € – ˜ — ym p – ˆ – ˆ „ e ˜ ™ j “ m € – ‘ j € – ˜ ™ ˆ “•  y y – ™ — • ‘ „ ˜ pm d d “ m p ‘ „ ˆ ˜ — – i † ™ —m ˜ y – € ˜ ˜ „ ˆ  y y l ‘ € – ˜— ym p – p “ d— s y — –m ˜— – p — € „m “ f — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ m “ ‘ l € „ – p “ d— – m – ˆ — – p “ d — – — y „ — — – ˆ — ‘ “ ˆ — ‘ ™ ˆ – € ˜ ˜ ‘ – • y – d — € – ˜— ym p — –m — f ‘ ‘ ‡ † — ™ —m ˜ y ˜‘ „ —  y y m “ € ˜ p – d— l ‘ € – ˜— ym p –m y „ — –mm € l — –m € y— f ™ ˜— ‘ – ˆ “ m € y — — ™ — “ s — l ‘ € – ˜ — ym p — –m ˜‘ „ — – p — € „m “ f – y y ‘ „ pm – y y – j € „ … – ˆ – € ˜ ˜ ‘ – • y – d  y y — € – ˜— ym p — – ˆ € –m s p – ˆ ˜ — – …  ˜ p – s „ m  z † ™ —  m  ˜ y – € ˜ ˜ y – d – y y  ˜ — “ ep „ ˜ — l ‘  € – ˜ — y m p – m — € „ m “ f — ˜ – s „ ˆ – l “ ˜ ‘ “ •“ ˆ – € „ p ‘ – ˜ ‘ – ‘ ‘ – ˜‘ „ p — – j km s „ € d — –m – y y — € „ n † — ‘ “ – j l ‘ € – ˜— ym p –m – y y ™ € d „ € d d “ — ym d ˜— – – y y ˜ — ˜ “ ˜— l ‘ € – ˜ — ym p –m — € „m “ f – ™ l ‘ „mm “ – j € „ … – ˆ – € ˜ ˜ ‘ – • y – d — € – ˜— ym p — –m – y y — € „ n † ™ —m ˜ y ˜— –  y y — ‘ “ – j l ‘ € – ˜— ym p –m ˜‘ – • y „ — ˜— – p — € „m “ f ˜‘ – — „ d – — ‘ „ ˜ y p ™ u– ˆ — d j – ˜ – ˆ — – j km s „ € d — – ˆ – € “ € ˜ ‘ „ p y “  z † — “ p —m – ˜ – ˆ — ‘ “ ˆ ™ —m ˜ y – € ˜ ˜‘ – j –m “ l ™ ˜ y – d — – e d “ € l — –m € y— ™ — “ s l ‘ € – ˜— ym p – n † ‘ „ ˜ “ ˜ jm – ‘ y — “ d ˜— – ‘ – ˆ „ e ˜ ™ j “ m – ˆ – ™ • –m ™ ™ ˜ u–m d j „ p “ m — “ p —m – ˜ – ˆ — ‘ “ ˆ – y y —  y d f – mm “ ˜ – ˜ ˜ – d – ˆ ˜ ‘ „ — — – ™ ‘ ‘ „ ˆ – ˆ u y – — – m – y y — € „ m ™ — m ˜ y ˜ ‘ – • y „ — ˜ — – ¡ £ ¨ ¤ ¨ § £   ˜ – – ˜ ‘ “ ˆ ‘ – p — “ – j € „ … “ — — y „ — f – y y  ep € “ € ™  e l ‘  € – ˜ — y m p – n † — – — — – m s  “ … — – — ˜ – — – p € „ … — – — – ˆ k — — „ d – ‘ y p “ ep f — – ™ ˜ ‘ – — ™ € d — ‘ „ • “ — y „ ‘ – y y — – y y  ‘ e p – ˜ – ˆ ™ ˜ ™  € “ • – ˆ ‘ “ € l “ m  j € “ ƒ D @IH  ¥  © ©   © © ! ! ©  © © ©    © ©  © ©      ©    ©  ©  © ¦ † — ™ ™ €p ˜‘ „— — € – ˜ — ym p — –m m – y y –m — ‘ “ ˆ – ˜ u– ˜‘ „ p y ˆ – y y ˜ “ j ™m s „ € d “ m – ˜ d j „ p ‘ – ˜‘ – ‘ ‘ – € d — –mm – y y … y “ — f – y y  e p € “ € ™  e l ‘  € – ˜ — y m p y “ ƒ g #  f – y y  ˜ — “ ep „ ˜ — l ‘  € – ˜ — y m p y “ i w ’ f — ‘ “ – j l ‘  € – ˜ — y m p y “ i # n ˜ – tr ’ r v f z ” n i w g ƒ ˜ ‘ – j j – ˆ ™ p ™ € d – ™ ˜‘ – — ™ € d l ‘ € – ˜— ym p – ˆ – ˆ „ e ˜ ™ j – ‘ y – ™ ep “ ˜ ˜ “ € – € ˜ ˜ y – d — – e p „ € d d “ — – p – ˆ – ‘ y p “ ep f ˜ ‘ – j – m “ s „ m h † — – m “ € ™ ‘ ™ l — ™ ˜ ™  € d „ € d — € y – m   $ ©   ¡ ¡ £© ¥ ¦¥ £ ¡ §¨ £ ¤ # ¡ ¤ ¨ ¡ ¤  ¡ § ¢ ¢ ¨ ¡££ ¡ ¤   £ §¤ ¥ ¦ ¢ £ § ¡ ¢¤ ¢ ¡ § © © !  † — –mm – ™ € — ‘ „ ˜ “ pm d d “ ˆ — “ p – m — ‘ “ ˆ — – ep „ € d d “ — – p – ˆ ˜ € ™ ˜ ‘  m ˜ ‘ “ € ˜ — y m m  f — – ™ € y ˜ p y € ˜ —  j – — — – ™ ‘ ‘ „ ˆ — – ˆ ˜ – — –mm – ™ € — – ™ ‘ ‘ „ ˆ — – ˆ – p “ … — y‘ – ˜ s „ — ˜ “ ˜m y — ™ € — –m — ‘ „ € ˜‘ „ j — y „ ‘ — y d f — – ™m " € ˜ ‘ „ p — ‘ „ ˜ “ y ˜— — – ˜‘ – € ™  ˆ — ‘ “ ˆ – — — – ˜— y s „ € € y –m € – • € – — s „ € y „ d f — –mm – p  ˜ € “ — – ™ ‘ ‘ „ ˆ — – ˆ – p “ … — – ˜ y ˆ ‘ „ p ˆ € „ s “ ˆ ˜ ‘ „ — — ‘ „  ˜ “ ˜ ‘ – j  € ™ d u – — – i † — – € ˜ y “ ˆ — – ™ € “ d j „ p f — – e p „ € d d “ — – ˜ ‘ – € ™  ˆ — – p p – •“ — – ™ ‘ – j — ‘ „ •“ — y „ ‘ – y y — ‘ „  ˜ “ ˜ ‘ – j € ™ d u – — – ˜ ‘ – € ™  ˆ — – m – ˜ ‘ – — ™ € d – € ˜ d “ ep – m f ‘ ‘ ‡ † — – ™ € y˜p y€˜—  j–— — – ™ ‘ ‘ „ ˆ — – ˆ – p “ … ˜– f ™— •€– d y— ‘ „ ‘ – y y – €˜ y“ – l“ — —˜‘– € d d“ ‘ y ˆ – € ˆ “ p –m — ‘“ ˆ f — – ˆ „ e˜ ™ j — – p – ˆ — – m s — — „ d — ‘ „  — ‘ – ˜ u – — € y –  — ym d — ‘ „ ˜ ‘ – — ™ € d — y „ ‘ f – € ˜ d “ ep – m — ‘ “ ˆ —  y ƒ † l ‘ € – ˜— ym p – p “ d— s y — y ˆ – € ˆ “ p –m — ‘ “ ˆ — – ˜—m s “ s „ € d — –m k ˆ „ j — – ˆ € – —m ˜ y ˆ ˜ € ™ ˜‘ m € –m y p ˜ € “ d ‘ – — € „m“ — ‘ „ €˜‘ „ j — y „ † – ˆ — – ™ • € – — s „ — ‘ „ ˜ “ ˜ jm — – ‘ “ ˜ € – p ˜ ‘ “ ˆ ‘ „ d ™ € ˜ – & % % – ™ m – d d “ l ‘ € – ˜ — ym p – p “ d — s y — – ˆ – ˆ „ e ˜ ™ j – j k  u y – ˆ – ‘ y – ˜ y — ‘ – – ˜ ‘ – — ™ € d – € ˜ d “ ep – n † ™ —  • € – d y — – l “ — —  ˜ ‘ – € d d “ ‘ – ¢ o f ‘ “ m ‘  y ¡¡ † i – ˆ – € k  ‘ “ j “ m f – l “ d y „ p ™ ˆ € “ d ™ € „m ™ j “ ˜ „ — – ‘ – d y „ € l ‘ y p y “ — ym d – y y – p y y — y — y— — – p „ € d – p € – € ™ ˜ ˜ – f ™ € ™ ˆ— ‘ „ p — ˜ – s „ ˆ – d y „ € l –m € y „ d ˜‘ – ‘ ˜ € – d — ym d –m ˜ ‘ – j – ‘ ‘ „ ˜ ˜ € “ d –m ˜ ‘ “ ˜ ˜ – j € – d ‘ „ — ‘ – j ˆ “ m – d “ ˜ ™ – y y “ ep € – € ™ d – € – ˜ —  — ‘ „ p m “ € ™ ‘ ™ l – d  p ‘  € d ‘ „ z † – ™m– d d“ f – € ˆ “ p – p — ‘“ ˆ – ™ d d „m – • ™ ˆ — ‘ „ •“ — y „ ‘ – y y l ‘  € – ˜ — y m p – ˆ – ˆ „ e ˜ ™ j – € k  j – € d “ m  — ‘  “ – ˜ ‘ – — ™ € d – € ˜  d “ ep – n † ˜ y € s y ˆ – y y — ‘ “ — – ˜ ‘ “ y y ‘ “ j — – ™ ‘ ‘ „ ˆ — – ˆ € ‘ – ˜‘ „ p ˜ ‘ “• y „ d — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — – l € “ m – ˆ ™ ˜ ‘ „ € … ‘ „ p – € ˜ ˆ ™ ˜m s — — „ d “ m ™ l “ —  • ‘ – ˜ ‘ – j – m “ l ™ — ‘ „ •“ — y „ ‘ f ‘ ‘ ‡ † — – ™ € y ˜ p y € ˜—  j – — — – ™ ‘ ‘ „ ˆ — – ˆ — “ p u y “ – ™ ˜ d “ ˆ “ l ‘ € – ˜— ym p – p “ d— s y— – ˆ – ˆ „ e ˜ ™ j – ‘ y ˆ – € • y ¤ ‘ – – —  j “ m € y — — € y – mm “ € “ d – ˜ € „ d u y “•“ € ˜ — „ ‘ – ˆ –  ˜ € “ d – ‘ ” † — m –  € „ l ™ ˜ “ p – y y ‘ –  s — — y “ — – y y € ™ j y‘ — ˜ y s € ˜ ˜ “ — – ˆ € “ d — – ˜ € p ™ ˆ ˜‘ „ — — – ™ ‘ ‘ „ ˆ — –m f — –mm – ™ € — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — – ˆ ˜ € “ d y m d “ m — ‘ “ ˆ – y y ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ ™ f — ˜ y s  € ˜ ˜ “ ˆ — – d ˜ — ˜ ‘ – € ™  ˆ ˜ ‘ “ d y „ € l – € — – ‘ k l „ € ™ ˜ ™ e — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — – ˆ ™ ˜ ‘ „ € … ‘ „ p – € ˜ ˆ ™ ˜m s — — „ d “ m ™ € ™ ˆ  — ‘ „ p ˜ ‘ – j – m “ l ™ — ‘ „ •“ — y „ † € y – ˜ “ —m ˜ y m € y „ d –mm – y— • ˜ – –m s— ‘ – e ™ € d j „ p – j € „ … — y „ — — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜ ‘ – — ™ € d “ m ™ — — – € ™ ˜‘ — – j j „ — — y „ ‘ — y „ ‘ ˜ – f € y – ˜ “ —m ˜ y m – ˆ ˜ € “ d “ m – ˆ — – —  y y – €  € „  € d “ — – p ‘ “ — —  “ ‘ ‘ „ p – ˆ – € s j „ ‘ – m € – —  j ‘ j ˆ … ˜ p – s „ – j j „ p ™ ‘ ‘ „ ˆ p ‘ „ ˆ — – j j „ — — y „ ‘ — y „ ‘ f — – ˆ „ e˜™ j — „ ‘ – ˆ – € • y ¤ ‘– –— j “m — ‘“ ’ † — – ™ ‘ ‘ „ ˆ — –m — ‘“ ˆ — ˜‘–— ™ € d — ˜ d– p ‘ „ p — y „— — –m u y– j y“ €– ˆ ‘– e™ € d d“ €– ˆ“ m € y „ d –— y–p ™ € d – ˆ“ – ‘ y ˆ – € k •“ — – m s — ‘ – e ™ € d j „ p – j € „ … — y „ — — ˜ “ ˜m y — ™ € — – ˆ ‘ „  ˜ d € p — – ˆ “ m ˜ – f — – ™ ‘ ‘ „ ˆ — – ˆ – € y ˜ p y € ˜ — “ m € y —  € „ € d “ — – p ‘ “ — — “ ‘ ‘ „ p – ˆ ˜‘ – j – € “ € “ € y – ˜ “ —m ˜ y m f – € ˆ “ p m – ˜ ‘ y — ‘ “ ˆ € w † — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ — ˜‘ – — ™ € d — ˜ d – p ‘ „ p — y „ — — ‘ “ ˜ € – p € – l € – j ™ — ‘ “ – € “ … – ˆ ˜ – — – ™ ‘ ‘ „ ˆ — – ˆ – € y ˜ p y € ˜— “ m – ˆ ‘ „  — ‘ – e ™ € d j „ p “ m € – ˆ  “ ˆ — € „ m “ ˜ ‘ “ ˜ ™ … ˜ p – s „ m f — – p ‘ “ — —  “ ‘ ‘ „ p – ˆ ‘ „  ˜ p “ € ˜ u – ˆ – € ˆ “ p ‘ y — ‘ “ ˆ l ‘ € – ˜ — ym p –m € – —m ˜ y ˆ – y y ˜ d „ m — ‘ “ ˆ — ‘ „ ¡ “ m d — y „ ‘ — y „ ‘ f — € y –mm “ € “ ƒ † l ‘ € – ˜ — ym p – p “ d— s y— y ˆ – € ˆ “ p –m — ‘ “ ˆ ˜ ‘ – y ˜— – — — ‘ „ ˜ y s € ˜‘ „ p — „ ‘ f ˜ ˆ ˜ ‘ – j – € ˜ y # † — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ m “ ‘ l € „ – p “ d— – m – ˆ — – p “ d— – — y „ — — ˜ ‘ – € ™  ˆ — ‘ “ ˆ — ‘ ™ ˆ – € ˜ ˜‘ – • y – d — € – ˜ — ym p — –m ¤ „ — “ p u y “ – € ˆ ˜— – p f — ™ ™ € p ˜ ‘ 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§  C0 C2 C0 C1 Y † – ep y “ l — “ s ‘ – – y ˜ — – — – d y „ € l – m ˜ – ˜ y “ e ‘ – – y ˜ — – — – d y „ € l –m – ˜ „ € ˆ – y ˜ — – — – d y „ € l –m ‚ € y – ˜ “ —m ˜ y ‘ y € “ d ˜ y— – j j „ p — ‘ ™ ˆ ˜ ‘ – j –mm – € y ˜ “ ‘ ˜ ‘ „ € – — — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ — ˜‘ – — ™ € d — – d y „ € l — „ € ˜ — –m f –m d j– u– €“ d † – € y l “m €“ d ™ ˜‘–— ™ € d — “ p –m — ‘ „ €™ ˆ— ‘ „ p — y „ ‘ z † ‘ „—p ™ ˆ – ˆ — – € s€“ — –m – y y ˜ ‘ “ ˜ € „ d j — ym d … — — – € d u – m –  ˜ ‘ – ˜ „ d ‘ y ˜ ‘ „ — – m l k € — – m € w † € y – ˜ “ — m ˜ y m – m s — ‘ – e ™ € d j „ p ˜ “ ˜m y— ™ € ‘ y € ‘ € y „ … € y „ d — – ™ —m ˜ y ˜‘ – • y „ — — – € y ˜ p y € ˜ — — – ˆ ˜‘ „ — ‘ „— p ™ ˆ – ˆ — – € s € “ — – n † † † x — – u– ‘ ‘ “ ‘ – – ™mm “ ˜ ™ ˆ f ¢ f – mm „ x¡ ¤ ¨   ¢ £  ¥ ¤  £     £ £ ¤ ¤  ¨  £ m ˜ – ¢ f l ‘ “ e i ˆ ‘ “ y z¡ £¤ £ £ ¡ ¤ ! ¤¢  ¥     £  §¥  ¡  ¤  ¥ ¢ — – m ˜ ‘ „ — — ‘ „  ˜ p – „ € d — – m m – ˜ – ˆ € – y ˜ p – – € y „ d — – ™ —  m  ˜ y — y m d — – m — – y y  ‘ ep – ˜ u y – ˆ — – n † ‘ „  ˜ “ j € „ … ‘  – ‘ y p y “ “ € – ˜ € „ d d “ ‘ — ˜ – s „ u y – ˆ – € ˜ ‘ – – ™ • € – — s „ ™ ˜  j  u„ € d “ m — € „ m “ f — ˜ – s „ u y “ ™  p „ — — “ ˜ — – ‘ – m s  — ‘ – e ™ € d j „ p ˜ ‘ “  ˜ ‘ – ˆ  ‘ y p y “  — f – € ˜ ‘ „ p € “ ƒ † ‘ „  ˜ “ j € „ … ‘ m – ˆ – ˜ € „ d d “ ¤ ¨ ¢ © ¨ ¡ ¤ §   £¤¥   m € y — ˜ ‘ – j y p „ ˆ ‘ y ˆ – y y ¤ ¡     ¥ £  m € y — ˜ ‘ – j y p „ ˆ ‘ y ˆ – ep „ € d — ym d ˜ „ — ¤ ¡      ¡  ¤ §   £¤¥   m € y — ˜ ‘ – j y p „ ˆ ‘ y y y ˜ “ … – m — € „ m “ f — ™ € ˜  ˜ — ˜ ‘ – j y p „ ˆ — – ˆ ˜ ‘ „ — — ˜ – s „ — – m  — f – j j – ’ † ‘ „  ˜ “ j € „ … ‘  m – ˆ – ˜ € „ d d “ ‘ –  ep ‘ y ˆ – y y ˜ “ ep ‘ y ˆ – ep „ € d — y m d ˜  „ — ‘ „  m ‘ y y y ˜  “ … – m — € „ m “ f u y “ j  ‘ “ — – ˆ ˜ ‘ – ˜ ‘ – — ™ € d – € — ˜ – s „ — – m  — f – m d j – u – € “ ƒ † — ˜ – s „ — – p – ˆ ‘ y p “ e p – ˆ – m s  — ‘ – e ™ € d j „ p ˜ ‘ “  ˜ ‘ – ˆ  ‘ y €  „ •“ ˆ – ˜ — — – p ™ ‘ “ m – p —  “ j f — ˜ – s „ – € ˜ ‘ – ˜ ‘ – j – ‘ l „ m ™ m y „ ™ ˜ j u„ € d “ m ˜ ‘ – j ™ —  “ € – • € – — s „ ˆ ˜ – j € – d — ˜ – s „ — – ˆ — ‘ „— ‘ – j ˆ — „ € ˜ y „ u y – ˆ ‘ – ‘ „ ˜ “ —m “ y— • – ‘ y f — “ p € – j – € d –m — ‘ “ ’ † — –m l k € – ˆ – ˆ “ m — ˜ “ ˜m y— ™ € — –m € – ˜‘ – — ™ € d y „ † o ‘ „— p ™ ˆ – ˆ – € s € “ ‘ y ˆ – ˆ “ m — ˜ “ ˜m y— ™ € — –m € – ˜‘ – — ™ € d † — ‘ „— ‘ – j ˆ — „ € ˜ y „ u y – ˆ m – y— • – p “ d— – ‘ y — ‘ “ ˆ — – ™ ‘ ‘ „ ˆ — –m € – ˜ – „ € d †   ! ©  © ©  "   ¥ ©   © ©YYY  ! ©  © ©©  © ©  ©  ©   C2 C0 C1 © © ©  ¥ ¥ ! # ©© ¡ ©©   ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡ " © ©¤ ©¤ ©£ ©¤ ¤ ©¤ ¤ ©¢ ¤ ©¡ ¤ ¨£ ¨¡ ©¢ ¨¡ ¤ ¨¢ ¨£ ¨¨ ¢ ¨¨ ¢ ¨¨ ¡ ©£ ¤ ¨¨ ¨ ¨ ¨¨ ¡ ¨ £ § © ¤ © 321 ¤ ¥ § © ¢ ¤ ¦ © ¡¦ ¤ ¦ § © £¥ ¤ ¦ § © ¥ ¤ ¥ ¦ § © ¢¥ ¤ ¥ ¦ § © ¡¥ ¤ ¦ § 32 1 © © © – ’ † — ˜ “ ˜m y— ™ € — –m ˜‘ – j –m p “ … — ym d € – ˜ ™ € d € – ˜‘ ˆ € y – ˜ “ —m ˜ y m — € y –mm “ € “ d ˜ – j € – d — € – ˜— ym p — –m € – ˜‘ – — ™ € d – € € y „ d — – ˆ  „ — dmm – € – d e — – ˆ – y y ˜ " ˜ ym d — –m l ‘ “ ˜ p – € € – d e — – ˆ € – —m ˜ ” † — € – ˜ — ym p — – m € – ˜ ‘ – — ™ € d – € € y „ d — – m l k € – ˆ ‘ „  ˜ “ — m ˜ y m – ˆ ˜ € ™ ˜ ‘r o† ¦   ¥ "    © ©  ! ! † — € – ˜— ym p u y – ˆ — – ˆ ‘ y m € – — • ˆ – ˆ ˜ – – € y „ d ˜ ‘ – j –m “ ˜ “ … “ € y“  y y – p f — – u“ u y– ˆ — – ˆ ‘ y m € y— y „ ‘– – l“ d y „ p ™ ˆ ‘ y € y— €– ˜€ „ d “ € •– ˆ ‘ „—p ™ ˆ – ˆ – € s € “ m – ˆ ˜— – ˜ € – j – € d –m f — “ p u y – ˆ — –m — ‘ “ ˆ f ˜ – – ‘ ‡ † u y – ˆ ‘ – — € – ˜— ym p — – ˆ ‘ y m € – d y „ p ™ ˆ – ˆ – y y ˜‘ – j – € ˜ y “ – € “ … ˜ ‘ „ € € y „ d – ‘ ‘ „— p ™ ˆ – ˆ — – € s € “ — –m – y y — € „m “ f — € – ˜— ym p – € ˜‘ – — ˜ ‘ – j – ep y “• – ep — – ˆ – ˜ d j „ p ‘ – – —  € d “ m ˜ ‘ – j – mm – € y ˜ “ ‘ ˜ ‘ – ˜ ˜ – j € – d — – m l k € — – m € “ p ‘ „  —  p ™ ˆ – ˆ — – € s € “ — –m ˜‘ “ —m ˜ y –mm – p – y y – ™ € d „ € d d “ — ym d ˜— – — –m l k € — – ˆ € “ d — € – ˜— ym p — – ˆ ‘ „ ˜ d € p — – ˆ “ m f o† – € y l “ m — ‘ “ ˆ — ™ ˜ ‘ – — ™ € d ˜ ‘ „ —  y y u y – p – j j „ p — “ p — – ˆ — ‘ “ ˆ f — € y – mm “ € “ ƒ † — – ™ ‘ ‘ „ ˆ – ˆ y– –m € y— y‘– ˜ s „ ‘ „—p ™ ˆ – ˆ – € s€“ ˆ –m d j– u ‡ † ¦ ©  "  © ¥ © " C1 C0 C2 Y > 60 ©  © ©  " © © X > 60 † — € – ˜— ym p — –m € – — € ™ ˜ p “ € “ p € y „ d — – € “ — — – p ™ ‘ — ˜ y s € ˜ ˜ “ ˆ – € s j „ ‘ –m € – — j ‘ j – ˆ p ‘ „ ˆ ˜ – j € – d — –m l k € – ˆ ‘ „ ˜ “ —m ˜ y m f — “ p m – ˜ ‘ y — ‘ “ ’ † € – ˜— ym p y “ – p ‘ “ ‘ – ˜ € “ d d “ € y –m € – — € ™ ˜ p “ € “ p € y „ d € y— – ˆ — – € s j – j — – ˆ — € y –m “• — –m € – € ™ ˆ— ‘ „ p – ˆ –m ˜ y — “ d ˜— – ‘ m y y — € „m “ f € y— ˜ – € y— — – € y – € ™ d y— — € y –m “• — – ˆ € “ d  ‘ ™ ˆ ˜— – m y y € – y y € “ j – € ˜ y – d ‘ „ f — “ p – p — ‘ “ ˆ € – ˜— ym p –m ™ — € ™ ˜ p “ € “ p ˜— – ˜ ‘ – j j „ p – • € – — s „ ‘ „ m — € w † † – € y l “ m € “ d ™ ˜ ‘ – — ™ € d  ym – p – € ˜ ˜ “ € € y „ d — “ p m – ˜ ‘ y — ‘ “ ˆ y‘ – ˜ s „ ‘ „— p ™ ˆ – ˆ – € s € “ ‘ y f – € ˜‘ „ p € “ ƒ ˜– X  Y C1 © Y ∈ [20, 40] C1 X Y Y  — – € y – € ™ … ‘ — € y –m “• — – ˆ € “ d  ‘ ™ ˆ ˜— – ‚ € y— ˜– € y— X Y  C2 : X ∈ [20, 40] C2 € – ˜ — ym p –m ˜ – ¦ ¤ £ ¢   ¦ ¤ £  ¡ ¡ ¦¨ ¡ ¨ £ £ ¦ ¢  ¦ ¤ ¥ ¡¢ ¥ ©¨ ¡¨ ¡¤ ¦ ¢    © © “ m € – m s  p € – y „ ep ™ ˜ ‘ – j – m “ l ™ ˜ y – d – ˆ „ e ˜ ™ j “ m — € „ m “ f ‘ „  — ‘ – j  ˆ – ‘ y € y — ‘ „  ˜ p – „ € d € y – m – ˆ — € „m — – ™ € “ d ™ — ˜ ‘ – j –m s “ … – y y ˜‘ „ — – ‘ — – ™ ‘ ‘ „ ˆ — –m — f – € k ‘ “ j – j j “ m – ’ † — – d y „ € l — – ˆ ‘ „ ˜ “ € “ d ™ — – ˆ — – ‘ „ — –m — ‘ “ ˆ — ™ ˜ y € s — ˜ – s „ ˆ ‘ „ ˜ € “ d d “ m – ˆ – — y “ p ˜ y “ …™ ˆ ‘ – – ˆ „ e ˜ ™ j – ˜ ˜ – p – € ˜ ˜ – j – ˆ – y y — € — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ ˜ y € s – ˆ – p – — ™ € d “ m f – € ˜ ‘ „ p € “ ƒ † — – ™ € “ d ™ — ‘ – s ™ ˜— ‘ – ˆ – ˆ — ™ ˜m s “ s „ € d – ˆ — ‘ „ ˜ y s € ˜ — ˆ u y – ˆ ˜‘ – • y— ‘ „ — ‘ – j ˆ – ‘ y € y— — – ™ ˜– „ € d — – ™ ‘ ‘ „ ˆ – ˆ –m s j–— ‘– ‘ y ¤ „ f † – € y l “m – ˆ –m d j– u– m — ‘“ ˆ — “ p –m ˜— – i † — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ ˜ y € s – ˆ — “ d “  ‘ m — ˜ – f — ‘ „ — ‘ – j ˆ — – ‘ “ ˜ € – p € y— ˜ ‘ – ˜— u– — – ™ ‘ ‘ „ ˆ — –m – € ˜‘ – — – ˜ ‘ “ ˜ € „ d j  — ‘ „  ˜ “ € “ d ™ — — – ˆ – y y — € „ m – p “ p £ – ˜  “ … ˜ y „ ˜ — € „ m “ ˜ — – – y y  ‘ ep – ˜ – ˜ ˜ – i   ©©  ""  ©  © n− Distmax (E ) = M axi=11 (vi+1 − vi ) ‚ € – ep € – ep p ‘ „ ˆ ˜ ‘ –  • – € “ m – p f ˜ ‘ “ — —  „ € p – € ˆ € „ € “ d — – ™ l ‘ “ € ˜ ‘ „ — — € y – m “ • — –m  z † – p ‘ “ ˜— ˆ – ˆ ‘ “ € l — ym d “ m € “ d — – ™ € “ d ™ — — – • ˜ y p ™ — ‘ „ p — € y –m “• u y – ˆ — –m – € ˜‘ – ˜‘ “ d y „ p ‘ – u y – ˆ ‘ – – m s j – — ‘ – ˜ – p € – —  •  ˆ – ˜ —  — ‘ „ p – m s  — — „ d – ep „ € d d “ – € k  j – € d – ‘ y f – y y  € ™ j y ‘ ‘ „— ‘ – j ˆ – ‘ y ˆ — “ p –m — ‘ “ ’ † ‘ „ — ‘ – j ˆ –m y – — – ‘ y € y— — € y –m “• – ˆ –m s j – — ‘ – ‘ y € – ‘ ‘ „ ˜ ˜ € “ d € y „ d — – ˆ „ e ˜ ™ j — – — y – € s j „ ‘ – ˆ € – l “ — • ‘ – y “ – • ‘ – p ˜ y – d ‘ w † – • ˜ “ € ™ ˜ ‘ „ ¡ “ … – ˆ ‘ „— ‘ – j ˆ –mm – y y – ˜ € „ d j ‘ ˜ – — ˜ – s „ ˆ –m s j – — ‘ – — y „ — m – y y – ˜ € „ d j ‘ € y— ™ —m ˜ y “ € – —  y y f ‘ „— ‘ – j ˆ – ‘ y € y — ˜‘ – j – ‘ ‘ „ ˜ ˜ € “ d – ˆ – j e ˜ € „ lm “ ‘ y ˜ — – – ˆ „ e ˜ ™ j – € ˜ „ ‘ – ˆ – — “ s – ˆ – y y € s “ n D  H  D C H G C D A B A  E D C C D D  H EH E B @ ¡  8   {v1 , ..., vn } ©  n © © vi  © £ E ¦ £ © † € y – ˜ “ —m ˜ y m – ˆ ˜ € “ d “ m – ˆ — – — y y – €  € „ € d “ — – p ‘ “ — — “ ‘ ‘ „ p – ˆ – € s j „ ‘ –m € – — j ‘ j – ˆ ˜ – f — ™  ˜‘ – ˆ — € – ˜— ym p — –m € – ˜‘ – — ™ € d – € € y „ d — –m l k € – ˆ ‘ „ ˜ “ —m ˜ y m € y— € – — “ s — y „ ‘ – ˆ  —  „ ep p ‘ „ ˆ — ‘ „ •“ — y „ ‘ f – ˆ „ e ˜ ™ j – € ˜ „ ‘ – ˆ – € • y ¤ ‘ – – —  j “ m € y „ d f ˜ ‘ – j – m “ ‘  v † — – ˜‘– €™  ˆ — k € ˜ — ™ ˜  — ‘ – ˆ — – ˆ ˜ ‘ “ “ — € – ˜ — y m p — – ˆ € – m s  p – ˆ — – m s “ d “ p — “ d ˜ ‘ „ — – ‘ – m “ s „ m l – € k  ‘ “ j – ˆ  ‘ ™ ˆ ™ ˜  — ‘ – ˆ – ˆ m  y – — ‘ y ˆ ‘ „  ˜ “ —  m  ˜ y m € y — — – ™ — “ s — – ˆ „ e ˜ ™ j — – m m – ˜ – ˆ f — y m d – ’ † — –  —  „ ep — € y – m “ • — – ˆ — ˜ ‘ “ ˆ ‘ – d ™ ˆ ˜ ‘ „ — — ˜ “ ˜m y — ™ € — – m ˜ – f € ‘ € y „ … — – m p £ ˆ ˜ ‘ – • y „ — ˜ ‘ „ — — – € ˜ k j “ € “ d —m – ˜ – ˆ € w † € y – ˜ “ —m ˜ y m € “ d  ‘ € y „ … – € ˜ ˜ „ ˆ  y y — – ˜ y ˆ „ € d — –m l k € — – ˆ ™ ˜ — ‘ – ˆ – ˆ m y – — ‘ y ˆ ‘ „ ˜ “ —m ˜ y m € y— — – ™ — “ s ˜ ‘ „ — — – ˆ „ e ˜ ™ j u y – ˆ — – p f ˜‘ “ ˆ ‘ – d – i † ‘ „ ˜ “ j € „ … € y –m € y „ d – y y ˜ ™ ‘ ™ l – j e ˜ € „ lm “ ‘ y – —m ˜ y – ˆ ‘ „ p – — “ m – y y — € „m “ — –m l k € — – p – ˆ – ˜‘ “ ˆ ‘ – p — “ ‘ „ ˜ “ ™ € p – ‘ y € y — – ™ — “ s ˜ — – – € k  j – € d “ n † — – ™ ‘ ‘ „ ˆ — – m € – ˜ ‘ – — ™ € d – € € y „ d — – m l k € – ˆ — – m s j – — ‘ – — – ˆ € – ep € – ep – € € y „ d — – ™ — „ d „ € d ˜ ‘ „ — — – ˆ „ e ˜ ™ j — – ˆ f ¢ o f† m “ ˜ – — “ € “ z¡ ˜ – ¢ f – € € „t¡ — ‘ “ ’ † ˜‘–— „ d€– d y— –— — – €˜‘– p u y– ˆ — –m m– y y –m € y „ d — “ p – j k u y – ˆ –m — ‘ “ ˆ ˜ y „ ˜ y ˆ — “ d ˜ ‘ „ € – — –m – ‘ — –mm – f — € – ˜— ym p – ˆ — – € ˜‘ – p — – ˆ ‘ „ ˜ “ ˜ ‘ – — ™ € d “ m € “ d ˜ ‘ “•“ ‘ – — – —  j – € ˜ ˜ ‘ – • y – d — € – ˜ — ym p — – € ˜ ‘ – — – p ‘ – € ™  ˆ — – m f o† – € y l “ m – ˆ — “ p € – j – € d –m — ‘ “ ˆ — f — ym d – ’ † ˜‘ “ ˜ € „ d j ˜— – — – ˆ  „ € ˜‘ – p — – p ˜‘ “• € p ™ ˆ — ˜ y s € ˜ ˜ “ ˆ – € s j „ ‘ – m – y y — € „ m ˜ y „ ˜ € y — f ˜ ‘ “•“ ‘ – — – —  j — “ d  — ‘ “ ˜ ‘ „ — – ‘ — € – ˜ — ym p – € ˜ ‘ – — – p ‘ – € ™  ˆ — – m f — “ p u y – € s j „ ‘ – ˆ — ‘ “ ˆ — “ j f — – ˆ  „ € ˜‘ – p — € y –m € “ d — € – ˜— ym p — –m ˜ ‘ – • y „ — — — y “ – ˜ ‘ – — ™ € d – € ‘ w † — ™ € ™ ˆ— ‘ „ p — ˜ y s € ˜ ˜ “ ˆ – € s j „ ‘ –m ˜‘ “ — y ˆ ™ € ‘ – ‘ „ ˜ d € p — – ˆ € y –m ˜‘ – j –mm – € y ˜ “ ‘ — ym d € – m d j— – ˆ f – ˜ y— “ m — ‘ “ ˆ — ‘ „ € € – • –m — y „ ‘ f € –m y p ˜ € “ d ‘ – ˜ – j € – d  y y – p f — ™ —m ˜ y ˜ ‘ „ — — – ˆ  „ — dmm – € – d e — – ˆ – y y — € „m – y y u y – ˜ „ p — ‘ „ j ˜— – — –m l ‘ “ ˜ p – € € – d e — – ˆ € “ d — ™ ˜‘ – — ™ € d – € ˜ ‘ „ —  p u y – p – y y — € „m — € – ˜ — ym p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ m € –m y pm “ p f — ym d   © © © ©© © !    YY  ©    # # # " !© ! © ¥ ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡ † ‘ „— ‘ – j ˆ – ‘ y € y — — – l “ d y „ p ™ ˆ — – ˆ — € „m — –m s— — „ d — – d y „ € l – ˆ — – € s j „ ‘ — ˜ ‘ – € ™  ˆ € – € ™ ˆ— ‘ „ p — € „m “ y „ f – ™ € ™ ˜ “ € – — – d “ ˜ ™ – ˜ ˜ – p – y y — y d ‘ „ — ‘ – j ˆ – j j “ m € y — — – ™ ‘ ‘ „ ˆ — –m – ˜ y— ‘ – € – —  •  ˆ – € – ˜ ˜  y y f — – d y „ € l u y – ˆ ‘ – — – ™ ‘ ‘ „ ˆ — – m € – —  •  ˆ – ˆ €  —  „ ep ˜ y – d ‘ „ f ‘ ‘ ‡ † — – ™ l “ — •‘ – – € ˜ ˜‘ – • y – d f – y y ˜— ˜ “ ˜— l ‘ € – ˜— ym p – ˆ y „ — ‘ “ – j l ‘ € – ˜ — ym p – ˆ – ˆ „ e ˜ ™ j “ m –mm – ˜ f l ‘ € – ˜— ym p – ˆ –m “ € ™ ‘ ™ l – ˆ „ e ˜ ™ j – ˜ y „ ˜ y „ f † i — – u– ‘ ‘ “ ‘ – – ™ mm “ ˜ ™ ˆ f ¢ “ f† m “ ˜ – “ ˆ  ’¡ € – e —  v – ˆ – ˆ „ e ˜ ™ j “ n † — – ™ € ™ ˆ  — ‘ „ p – € ˜ ˜ ‘ – • y – d ‘ „  — ‘ – j ˆ – ‘ y € y— ˜‘ – j – ‘ ‘ „ ˜ ˜ € “ d – ˆ — – u–m d j „ p — ym d — – ˆ „ e ˜ ™ j — – € ˜ y “ ˆ f — € y –mm “ € “ ƒ €y„ d    ! #   n− j = ArgM axi=1r (vi+r − vi )    ©  vj +r + vj 2 !  ©  ‚ – ˜‘ “• y— € y –m “• “ m – € ˆ ˜— – p f – p ‘ “ ˜— ˆ – ˆ ‘ “ € l — ym d “ m € “ d — ™ € “ d ™ — — ˜ – s „ u y – ˆ — – ˆ y –m j y “ – ™ y ˜— — € „m “ ˜ “ € – — – ˜‘ “ ˆ ‘ „ d— – € € „ p ‘ „ ˜ “ € “ d ™ — – ˆ – ‘ „ “ n   n− Distmax (E ) = M axi=1r (vi+r − vi ) 32 1  p ‘„ ˆ ˜‘– •–€ “m–p f  –m d j– u– €“ d – ˆ   m“ l™ €–€™ ˆ— ‘„p  — ‘—„• – ˆ –€ s‚ j€–„e‘p€––mep – — „ d ‘ „© m  — f — “ p – p — ‘ “ ’ † — – d y „ € l — – ˆ ‘ „  ˜ “ € “ d ™ — – ˆ — – ‘ „  — – ˆ — – ep „ € d — ˜ –  s „ — – ˆ € – m s p – ˆ ™ ˜m s “ s „ € d “ m — ‘ “ – € „m ™ j “ ‘ w † — ‘— „ • ˜ ‘ – j – ˜ p – € ˆ — ˜ –  s „ – € ˜ ‘ – – p ‘ “ ˜— ˆ “ m € – — j j  uu““ j –p€‘–“e˜p—€–ˆe“pm –€ˆ– eyp–€–m eyp“–€f –l“–‘˜———„‘•„p‘‘“„˜€˜–“p˜‘jym— ‘–˜“˜ˆ–p— y‘€––˜‘m„m“pd—˜€–y„sd„ ‘u„y–˜ˆym–„€—˜‘– ‘– ”–m“ j † † " – € y l  “ m € “ d ™ € ˜ — ymm ˜ — – — “ p m – ˜ ‘ ” † — – € ˜ ‘ – p — € y – m ‘ –© y y ˜ " ˜ ym d — – d y „ € l — – ˆ ‘ „ ˜ “ € “ d ™ — – ˆ — – ‘ „  — –m — ‘ “ ˆ ˜ ‘ – y ˜— – — – p ‘ “ ˜— ˆ – ˆ ‘ “ € l — ym d “ m € “ d — – ™ € “ d ™ — — – • ˜ y p ™ — ‘ „ p — € y – m “ • — – m – y y – m s “ s „ € d — ym d ˜ — – m© — – j  j f ‘ „  ˜ “ € “ d ™ — – ˆ – ‘ „  – € y – mm – j  ""¦ n † ‘ „ — ‘ – j ˆ – ‘ y € y— — – ™ ˜ – „ € d — – ™ ‘ ‘ „ ˆ — – ˆ ‘ „ ˜ “ € “ d ™ — – ˆ –m d j – u ‡ † —–™ ‘‘„ˆ€„ „ i ¥ Y Y Y! Y Y" Y ‘ „˜“ €“ d™— – ˆ – ‘ „ ¡ $ ¡¢ YY † Y "† Y †Y !† Y  r C0 C1 ¡ ¢ £¤ ¥ ¦ © ¦ ¡¤ § © ¥ ¤ ¡£ £ ¦ ¨ ¡¤ §¤ ¡ ¡ ¢ ©¨ ©¨  ¡ € — ˜€‘j–„˜——‘——–—“p–j‘p ‘“f˜“—˜—ˆ–ˆ€—k–—˜m–€ˆfp–€‘k„—˜€˜–€“˜p——y€jm–pj“u“€–˜j‘d “—ym–ˆp–‘€‘˜“‘„˜–—˜“ˆ–yy—my“–•ˆ—™—©“‘mm„p—˜‘—““—jˆj„–€y‘dyjj„yp“dm –˜—m–“pp ––€–•˜€„ky‘˜„€€—˜p–‘€“—ˆ€‘–w™˜—ymmdp ˜ € “ d ™ ˆ – ˆ € – ˜— ym p –m – y y — ˜‘ “ — — – € ™ ˜ ‘ — ym d —m ˜‘ „ — — ™ j € „ … — € – ˜— ym p u y “ – • y „ ‘ — –m †  – • ˜ “ p  ‘ l — ˜ — – — € – ˜ — ym p u y “ – • y „ ‘ u y – ˆ — – m – € ˜ ‘ – ‘ „  ˜ “ € “ d ™ — “ m – y y – p  ˜ — – †  ‚ – ˜ d j „ p ‘– – € ˆ ‘– € d  ˜‘ „— — ‘ „˜“ €™ ˆ— ‘ „ p u y – ˆ f “ m – p € y „ ƒ † — – ˜ “ ˆ  ˆ ‘ “ p — – m — – ˜ y „ ˜  j € “ d ‘ „  —  •  ˆ – € y – m m  – j “ m €  —  „ ep  — € „ m “ – ˜ —  — ‘ „ p – ˜ ‘ “• y — – d “ ˜ ™© n † ‘ „  — ‘ – j ˆ – mm – y y – ˜ € „ d j© ‘ € y — ™  p „ — — “ ˜ — –  ym  y y — ˜ –  s „ — – ˆ – m s j – — ‘ –© m ˜ ‘ “ ˜ –  „ € d ‘ – ™ j € „ …  ™ ˆ € – ˜ — ym p m – y y – ˜ € „ d j© ‘ € y — ™ — m ˜ y – € ˜  ˜ y – d – j e ˜ € „ lm “ ˜ – i † –m s j – — ‘ – ˜ – p – ˆ u y – ˆ ‘ – ‘ „ ˜ ˜ € “ d – ‘ y – ˜ € „ — ‘ – ˜ ‘ € y „ …  y y ˜ – — € y –m “• – ˆ –m s j – — ‘ – § ¨ ‘‘y„–˜™˜€€˜“‘d– ‘– ˆ– ˜––jjeˆ˜“€„ ylym“ f ‘ y© ˆ£—¥‘¤„—¡„ d£——ˆ‘„— yj„j‘ „–‘yy— y—„‘‘„€–™yˆy—‘‘„„p— ‘—–yj„ ‘ˆf y– ‘“–y•€‘y—–p˜‘– j– ‘ ¤ ¤ ¢¡ 32 1 D  H  H § H G C D A G E B E D H I B C A I @ § £ © ¥ 8 † – p ‘ “ ˜— ˆ – ˆ ‘ “ € l — ym d “ m € “ d — – ™ € “ d ™ — ˜ – – l “ ‘— „ • ‘ “ ˜ € – p ‘ y — ‘ “ ˆ — – y‘ – ˜‘ „ p — € y –m “• u y – ˆ — –m – € ˜‘ – u y – ˆ ‘ – € – — • ˆ – ˜—— ‘ „ p  y y –mm – p ‘ „— ‘ – j ˆ – ‘ y € y— ˜ ‘ – j – ‘ ‘ „ ˜ ˜ € “ d – ˆ – € “ ˜‘ – j ™m ™ ‘ „ ˜ “ € ™ d „ – j j „ p — ‘ „ € ™ ˆ — ‘ „ p — y „ ‘ f – ˜ y — “ m — ‘ “ ˆ  „ y y € y „ d ˜— – i † – ˆ „ e ˜ ™ j “ m – ˆ –m “ € ™ ‘ ™ l ™ ˜ u–m d j „ p “ m € – ‘ j € – ˜ ™ ˆ p ‘ „ ˆ “• – € “ ˜ ‘ – j ™m ™ ‘ „ ˜ “ € ™ d „ – ˜ ˜ – p – ˆ ™ ˜ u–m d j „ p “ n † m “ s „m l – j e ˜ € „ lm “ m ˜ ‘ “ ˆ ‘ – d — „ … — – — y – € s j „ ‘ – ˆ – ™ ˜ y p ™ u– “ € – —  y y – € “ ˜‘ – j ™m ™ – ˆ „ e ˜ ™ j – ‘ y ˆ ˜ l “ — m € w † – ˆ „ e ˜ ™ j “ m – ˆ ‘ „  ˜ y p ™ u – ˆ — d j – ˜ – m ˜ ‘ – j j – ˆ  • ™ – p ‘ – y ‘  u  „ ep — – p – ˆ – m s j – — ‘ – n © ©  "¦ † — – ™ € “ d ™ — ˜ ‘ – j –m s “ … ˜ – — – ™ ˜ y € s — – ™ ‘ ‘ „ ˆ – ˆ –m d j – u ‡ † —–™ ‘‘„ˆ€„ „ i ¥ Y Y Y! Y Y" Y ‘ „˜“ €“ d™— – ˆ – ‘ „ ¡ ˜ y € x C0 C1  ©©© © YY † Y "† Y †Y !† Y  ¡ ¢ £¤ ¥ ¦ ! ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡ ©  y y – p f ™ € ™ ˆ — ‘ „ p m “  ˜ ‘ € – ˜— ym p y ˆ – ˜ € – ‘ m – € y – € ™ … ‘ ˜ — – ‘ „— • ˆ – ˆ ‘ „ — ‘ – j ˆ “ m € y— – m “ j  u “ j –  ˜ € – ‘  m — – m m – y y — – m € y „ d — ‘ „  —  •  ˆ — – m – y y € – € ™ ˆ  — ‘ „ p – ‘ – ˆ ˜  —  „ ep ‘ „ f  — ‘  # † – j e ˜ € „ lm “ m — ‘ “ ˆ — – ™mm “ ˜ ™ ˆ ˜ ‘ “ ˜ ™ — – d “ ˜ ™ — – ˜ ‘ – € ™  ˆ — – — f – ™ ‘ ‘ „ ˆ ‘ „ — • ˆ – ‘ y ‘ „ ‘ y „ € – ˜ d – p p “ ˜ y “ … m — ˜ ‘ “ ‘ € y „ ˜ – € ‘ „  ˜ p ‘ „ … “ m £     ¢ ¤¢¢ — ‘ „ j j „ ‘ — y „ † — ˜ y ˆ „ € d — € – ˜— ym p u y “ – • y „ ‘ — – ˆ — – ‘ € – ˜‘ — – p ‘ “ ˜ —  ˆ — – m € – —  j  ‘  j  — ‘  “ ‘ –  s p ‘ „ ˆ – e p € – ep ‘ w † ˜ ‘ “ — — – € ™ ˜ ‘  ˜ — – € – ˜ — y m p – m — ‘  „ j p ‘ „ ˆ ˜ – f — ™ — € – d— ˆ ˜‘ „ — ™ € ™ ˆ— ‘ „ p € – ˜ — ym p y ˆ — – € s j – j — –m — ym d f – ˆ ‘ “ € l ˜ — – – ˜ € – ‘ m — ym ƒ  ©©  © ! Pd (Ck ) ©  µkd = Inertied (Ck ) = Inertie(Ck ) = Nk {i∈Dk } xid |Dkd | {i∈Dk } (xid − µkd )2 |Sk | {d∈Sk } Inertied (Ck ) © ©© ! ‚ — – y y  ˜ —  € ™ ˜ p “ € “ p — ‘ „  — ‘ – j ˆ — – — € y — € – ˜ — ym p y ˆ – ‘ ‘ – „ j –  ˜ € – ‘ m € “ d – ™ ym “• ™ ˜— – ™ ˜ ™ ‘ ™ l „ j „ e – ˜ ˜ – i † — – y y ˜— € ™ ˜ p “ € “ p — ‘ „ — ‘ – j ˆ — – — € y— € – ˜— ym p y ˆ ™ ˜ ™ ‘ ™ l „ j „ e m – ˆ – — — “ s – ‘ y – y y € “ j  p –mm – p — ‘ „— • ˆ – ‘ y € – — y …– € – ˆ ˜— – – € k ˜ € p ˆ ‘ „ p – — y ˆ – ™ ˆ n † ™ —m ˜ y – € ˜ ˜ y – d i r x – € k ˜  € p – m — € „ m “ f — ™ € ™ ˆ  — ‘ „ p ˜ ‘ „ — — ™ e p € – ep – € — – d y „ € l – ˆ — – € s j „ ‘ — ˜ ‘ – € ™  ˆ p – •“ — ˜‘ – j – ‘ ‘ „ ˜ ˜ € “ d — ˜‘ – € ™  ˆ  z † ™ € d „ € d d “ ˜ “ … ˜ y „ ˜ ˜— – — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ “ m s j – — “ € • “ m – ˆ – € k ˜ € p –m — € „m “ f u y – ˆ ‘ – — – ™ ‘ ‘ „ ˆ — –m € – ‘ ‘ „ ˜ ˜ € “ d € y „ d ™ —m ˜ y ˜— –  y y – y y  ˜ —  ˜ “ ˜ — l ‘  € – ˜ — y m p ‘ y ˜ — – p  z † – ™ —  m  ˜ y ˜ ‘ – j – ‘ ‘ „  ˜  ˜ € “ d – ˆ – ˆ „ e ˜ ™ j “ m p – •“ ˆ € „ p p “ ‘ – ˜‘ – „ — — –mm – y y ˜ y “ … m — “ j f — – ™ l “ — • ‘ – – € ˜ ˜‘ – • y – d ‘ „— ‘ – j ˆ – ‘ y € y— ˜‘ – j – ‘ ‘ „ ˜  ˜ € “ d ‘ y ˆ – ‘ € – ˜‘ ™ ˜m “ y y “ m – ˆ — – € y— – j — – ˜‘ – € ™  ˆ f y “ – • ‘ – p f — „ … – ‘ y – € „ p ‘ ‡ † — ˜  y ˆ „ € d — € – ˜ — y m p u y “ – • y „ ‘ — – m ˜ ‘ “ € “ d ™ — – p ‘ “ ˜ —  ˆ “ m € – —  j  u “ j  — ‘  “ ‘ –  s – e p € – ep ‘ w † – ˜‘ “ — — – € ™ ˜ ‘ – ™ l y ˜— – – ˜‘ “ ˆ ‘ „ d— – € € „ p ‘ „— • ˆ “ m — ym d f – ˆ ‘ “ € l ˜ — – ™ ˜ ˜‘ “ y y – ˜ ˜ – p — ym ƒ Sk ©©  !  ! © Distmax (Dkd ) |Dkd |    Interet(Pd (Ck )) = Nk ×  © © ‚ — ˜ – s „ ˆ – l “ ˜ ‘ “ •“ ˆ ˜ ‘ “ ‘ – ˜ ‘ „ p — € – ˜ — y m p — – ˆ ‘ „  —  •  ˆ “ m € – —  € „ •“ … € y „ d m “  ˜  ‘  € – ˜ — y m p y ˆ — ˜ – s „ ˆ – € s j „ ‘ – m € “ d – ™ € ™ ˆ ‘ „ d ˜ – f — – ˜‘ – € ™  ˆ — –mm “ ˜ – ˆ — ‘ „ — ‘ – j ˆ — – ˆ € y — — ‘ „— • ˆ — – ˆ € – € “ d j „ p – ˆ – € ˜ ˜ – j € – d € y „ d f ‘ „ ˜ d € p — – ˆ – ˆ ˜ –m d j „ p – p “ d— – m — “ d ‘ „ ‘ ˜ – f – • y „ € ˜ – — m – y y –m — ‘ “ ˆ – p “ d — – m – € k ˆ— ‘ „ p ‘ „ ‘ „— ‘ – j ˆ “ m € y— – ˆ — ˜ – s „ — – ˆ — – ™ ‘ ‘ „ ˆ € „ „ p — – ˆ ‘ „ ˜ ‘ ™ ˆ – ˆ –mm “• € – ˜ ‘ m – ˆ –mm “ ˜ “ m f € “ d – ™ —m “ j € „ ‘ f — ™ j € „ … — –mm “• € – ˜ ‘ u y – ˆ — –m – € ˜‘ – ‘ „ ˜ “ € “ d ™ — – ˆ – p ‘ “ ˜— ˆ “ m ˜‘ – j –m d j— ˜ y „ ˜ ˜— – – ™ —m ˜ y – € y— – j “ m f – € k ˜ € p € – j – € d –m € – ym “• ™ € y „ ƒ † ‘ „— ‘ – j ˆ “ m € y— ! Ck © Ck d   © © S Ck |Dkd | Distmax (Dkd ) d — ™ ™ € p ˜ ‘ “ ˆ ‘ „ d — – € € „ p — –mm “• € – ˜ ‘ u y – ˆ — –m ˜‘ „ — ˜ – † m “  ˜ ‘ € – ˜— ym p y “ ˜‘ “ ‘ – ˜ € “ d d “ — ˜ –  s „ — – ˆ ‘ „— ‘ – j ˆ “ m € y— — – ™ ‘ ‘ „ ˆ € „ „ p — – ˆ – m s j – — ‘ –© m – ™ € ˜ ‘ – ‘ –  ‘ € y „ … “ ‘ „ ˜ „ z – mm – y y “ m  § £  ¥ ¤   ¡¨£ – ˆ „ e ˜ ™ j “ m € “ d – y ‘ – ˜ s „ ‘ „  ˜ ˜ € “ d “ m  ‘„˜p–m™— — ‘„— ‘– jˆ †—––€ˆk˜–m€sp jˆ–‘—„‘p––—— yy„ˆ— ‘‘„y˜m“yy–m—“–•y™©ym ˜€–y„f ‘d„™—€‘™–ˆj— ‘ˆ„p–m˜y—––— —–€‘„ym“©y—y–—™y‘–—j–™“‘€ Dkd Ck π 1 π 2 d ¢¡ # ¡¢ Pd (Ck ) = {π1 , π2 } © ¦ ©£ © ¥ ¤ ¡£¡ ¥ ¨ ¢ ¨ ¡ ©¡ £ ¤ £ £ ¡£ !  † ‘ „— ‘ – j ˆ – ˜ ˜ – p € y— € – y ˜ p – – ‘ „— • ˆ “ m € – y ˜ p – – –mm – y y “ m € y— ‘ „— ‘ – j ˆ “ m f € – — • ˆ € – ˜— ym p y ˆ „ € ™ j y ‘ –m d ! ˜ – f ‘ „— • ˆ “ m ‚  ! D ¨©D%   ¨D¨ D    Best k π = Pd (Ck ) ¨DV© ¢ ¨G P = (C1 , ..., CK ) K Best = (0, 0, ∅) BestN ote = 0 k=1 K d=1 M π = Clustering1D(Dkd ) N ote = Interet(π ) (N ote > BestN ote) (AcceptDivision(π )) BestN ote = N ote Best = (k, d, π )   „˜ „˜ ‘ „— • ˆ – € y –mm – j “ m – ™ p „ — — “ ˜ € ™ ˜‘ ˆ – € y— – j “ m – ˜‘ “ € y „ p ‘ „— • ˆ – € y –mm – j “ m † — € – ˜— ym p – ˆ – ™ — „ d j „ p ‘ „ ˜ ˜ € “ d – ‘ y ‘ „— • ’ ˜ p –m – z ¢ ©  ¡ ¡ B D R V! ©D F ©E¨H &  ©  † – j e ˜ € „ lm “ m € “ d — – ˜  € p ™ ˆ ˜ ‘ „ — — – m “ d  p ‘  € d — – d “ ˜ ™ — – — ˜ – f £      ¢  ¢¤ ¨ ¤ – ™ j j „ ‘ ˜ — – f – ˜ ‘ “ € y „ p ‘ „  ˜  ˜ € “ d “ m – ™ ‘ ‘ „ ˆ ‘ “ ˜ ™ f € – € ™ ˆ — ‘ „ p – ˜‘ “• y— ‘ „— • ˆ “ m – ˆ ‘ „ ˜ p –m ™ — – ˆ – ˆ „ e ˜ ™ j “ m f ˜‘ – j –m “ ‘ v † — ™ € ™ ‘ ™ l u y “ ‘ — € – ˜ — y m p – ˆ – € s j „ ‘ y “ ˜ ‘ “ y y – ‘ ! “ € ˜ ‘ – u  „ ep – p – y y — – p ‘ – € ™  ˆ — – m — ‘ „  ˜ “ ˜ ‘ – j € ™ d u – — „ ‘ ˜ ‘ “ ˜ ‘ – — ™ € d – € ˜ d “ ep – m — ‘ “ ˆ — m “ ˜ ™ ˆ ‘ – — ym d — ‘ „ € € – • — y „ † ˜ y ˆ ™ € ˜— – — ‘ „— • ˆ – ˆ – € s j „ ‘ –m f –m “ j u“ j – ˜ € – ‘ m ˜‘ “ —m ˜ y ‘ – f – € “ € ˜ ‘ „ p y “ ˜ – f — – ™ ˜ d – p p “ ˜‘ – j –m p “ … — ym d ˜‘ „ — — ‘ „— • ˆ — –m — € „m “ f –m “ j ‘ j – ˜ € – ‘ m – —m ˜ y ‘ „ — f — ‘ # † – ˜ € – ‘ ‘ „ — – y‘ j ˆ — ™ € ™ ‘ ™ l — € – ˜ — ym p u y “ – • y „ ‘ u y – ˆ — – ˆ ‘ y m — ‘ „ j y “ — ‘ „— • ˆ – ‘ y € – ˜ d – p p “ ˜ – f – j e ˜ € „ lm “ m — ‘ “ ˆ ‘ q € “ d u“ q € – p “ m d j – € ˜‘ – • – €  y y – p f ™ € ™ ˆ— ‘ „ p m “  ˜ ‘ € – ˜— ym p y ˆ – ˜ € – ‘ m – € y – € ™ … ‘ ˜ — – ‘ „— • ˆ – ˆ ‘ „— ‘ – j ˆ “ m € y— – ˜ € – ‘ – ˜ ˜ – d — ym d “ m — – m m – y y — – m € y „ d — ‘ „  —  •  ˆ — – m – y y € – € ™ ˆ  — ‘ „ p – ‘ – ˆ €  —  „ ep  — — y “ ˜ y – d ‘ w † –  ˜ € – ‘  € y – m u y – ˆ — –m — y „ ˜ ˜ ‘ – — y ˆ ™ € — € – ˜— ym p u y “ – • y „ ‘ — –m — – y y ‘ „— • ˆ – ‘ y € – ˜ d – p p “ ‘ ˜‘ – • – €   $ © © © © © π2 !  ˜– π1 © ¡ ¢ πi ¢ – € ˜ ‘ – –m “ j u“ j – ™ —m “ j € „ ‘ – ˜ € – ‘ € y— – ˆ – ˜ € – ‘ € y — – ˆ ™ ˜  •“ € l – ˆ – € ˜ ‘ – p Dkd ¡   „˜   € y — – ˆ –  ˜ € – ‘© m f €–m ypm“ p ! D F  ©¨ & € – ˜ — ym p y ˆ – ™ — „ d „ € d ‘ „— • ˆ “ m ‘ „  —  • ’ ˜ d – p p # ¦ D R V ©   EH Y$ Dkd πi ¢ − µi )2 Ck Sk Ck † ‘ „— ‘ – j ˆ “ m € y — † ‘ „ ‘ — Y f – ™ ˜ d – p p “ ˜ — – ‘ „  —  • ˆ “ m  —  ! D ©   %  ‘€ y˜–€   ¨D I1 maxInertie = M ax(kd |,I2 ) |D maxInertie < Inertie(Ck ) {j ∈Dk |xjd ∈πi } (xjd Ii = Pd (Ck ) = {π1 , π2 } Inertie(Ck ) i=P 2 1 {j ∈Dk |xjd ∈πi } xjd µi = |{j ∈Dk |xjd ∈πi }| d ¡  ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡  © € “ d – d “ ˜ ™ – y y “ ep – m s  — — „ d – l “ d y „ p ™ ˆ € y – m m  – j y ˆ ‘ „  ˜ p – m ™ — “ m € – p “ m d j – € – ˆ ˜ — – – ™ ˆ  n † £      ¢  ¢¤ ¨ ¤ – j e ˜  € „ l m “ m — ‘ “ ˆ – y y  ˜ — “ e p „ ˜ — ‘ „  — ‘ – j  ˆ – ‘ y – €  y ˆ „ € ˜ ‘  ˆ — ‘ „ — „ d „ € d — y „ ‘ f – p “ p £– – j e ˜ € „ lm “ ‘ y ˜‘ “• € – — ‘ „ p ‘ – ˜ y „ ˜ — – p ‘ – l u– u y – ˆ — – p – € “ …— ˜ “ — € y „ ƒ † — – ˆ • ‘ „ ‘ — ‘ „ ˜ p – — € – ˜ ‘ — – ˆ € „ •“ ˜ ‘ – j – mm – y ˜ ‘ – • ™ ˜ ‘ – • y – d  y y — – ‘ k l „ j „ e — – m s — — „ d — € – ˜ — ym p – ˆ – m s j – — ‘ – ‘ y ˜ " ˜ ym d – y ˜ ˜ — ‘ „ p m y y — “ j f ˜ € “ d ™ ˆ – ˆ — – ™ ‘ ‘ „ ˆ — – ˆ –m s j – — ‘ – m – ˆ ˜ p € ˜— ˜ ‘ – j – ‘ ‘ „ ˜ ˜ € “ d ‘ y – y ˜ p – – m y y — “ d ˜‘ “ € ™ y y – € – ‘ ‘ – f – j e ˜ € „ lm “ m — ‘ “ ˆ – — — –m d y „ — – ‘ “ ˜ € – p – ‘ y – € y ˆ „ € ˜‘ — ‘ „ m y „ • — y „ ‘ f — y m d – ’ † £ ¢ ¥  ¤   ¤ y „ f ¡ ¤ ¢    £ ¤ ¤ £ ¢ ¥ ¤ £ ¢¤ ¥ “ m – ˆ – d  p ‘  € d – m ˜ — – i † – ˜ ‘ “ ˆ ‘ – p — – ˆ – ep € – ep – € – ‘ y ˆ — € „ m — – m s  — — „ d u  „ ep — € y – m m  – j — – m € – • € – — ‘ „ p – ˆ ˜ ‘ – ‘  ˜ € – d — ym d ˜‘ – • y „ — ˜— – m € w † — ˜ ‘ “ € y „ p — € – ˜ — ym p — – ˆ ‘ y m – ˆ ‘ „— ‘ – j ˆ – ‘ y € y— –m s— — „ d – l “ d y „ p ™ ˆ € y – m m  – j – m € – e p € – e p – € – ˆ – ˜ ‘ – ˜ ‘ „ p – — £      ¢  ¢¤ ¨ ¤ – j e ˜  € „ l m “ m – ˆ – — “ s – ˆ ‘ „  — € – • “ n C B @ A ¡ I @ D ¢ C £ EH B  ¦ I ¤  8 © ! ©© © © © © © © © ¡ k © ¦ ¢ £ † † " – € y l  “ m € “ d ™ ‘ ‘ „ ˆ ˜— – – € “ ˜‘ – j ™m ™ –m d j – u– ‘ y € y— – j e ˜ € „ lm “ ˜ – p – ˆ ˜ “ ˜m y— ™ € – n † $ – € ˜  d “ ep – m — ‘ “ ˆ — – ™ ˜ ‘ – — ™ € d — ‘ „  ˜ “ ˜ ‘ – j  € ™ d u – — – m — ‘ “ ˆ ™ ˜ y p —  ˆ ˜ — – — ˜ “ ˜ m y — ™ € — – m € y — – € ˜ k j “ € “ d – p – ˆ – p ‘ – y ‘© n † – ˜ ‘ – ˜ ˜ “ ˜ ‘ – j – € “ — — – p ™ ‘ — “ d ˜ — –© ‘  y y – € y –  € ™ d y — – ‘ € „ s – ‘ y© ˆ —  “ j — € – ˜ — ym p – ˆ y ˆ ‘ – ˜ ˜ “ – € s j „ ‘ y ˆ — “ d ˜ l “© — – ‘ mr † — ™ € ™ ‘ ™ l — € – ˜ — ym p – ˆ j y j u “ j –m€“s© mj€„“‘d y–“™˜ˆ‘‘–„—™d€—d–€˜€—„–p f˜– –™ j˜—j– „–‘jfe–˜—“€„slm–“ˆ ˜––•p†˜–“†ˆ"€™––˜€€˜yk–lˆj„“e“€˜“m ™d€j“mdy“––m—™f€–˜˜‘n—–y†jomm––m˜“—j‘–e v˜€„ l –™m“˜pj“€‘“pj y‘y„˜—d–mm€“p•—€–¨–ˆ˜‘–ˆ —––—ˆ“ e—d˜€–„˜d˜d– iy—† €—–‘˜„—y—m‘p––jpˆ ˜—––ˆs„m“‘jy© ‘ˆ –jp ‘–“m ‘s–j˜–€“— ‘d–d©“m © m˜‘˜“‘m–s—p€ ‘p–“€f “p ‘€–„˜——‘y–mpjyˆˆ “¤m –£€ d y—£ ‘„ —˜–€¥„¢sm“¤„!f ‘–y‘pyy“˜—–‘‘˜–—“˜u€˜“–‘©d‘–d˜“ms©„— f‘—wf‘†“ ‚ ––ˆ˜‘– ‘‘–˜„y€–y—d‘˜–——jy€m™ˆd˜ jd –˜ y„˜ € y„ d – yy m–˜ ˜“„p— ‘–„‘— ‘– jˆ –m€m––•˜—yy„m‘p ––p‘ y– ˆ – y€–y˜˜——ym€p™˜‘py“€“p ˜‘‘“„˜—y‘„–“j‘ˆ–©–y€y˜ y€“–•–‘€€y“ f˜–yy–yd˜—m€™—˜“p“q€ M axK ¢ ¡ e ∈ Sl \ d xje d Sl Cl Dle −j ∈ D \ D → x l d ∈ Sl Cl Sl = Sk ∪ d ‚ ‘ „ — ‘ – j ˆ “ m — ym d – ˆ — – y y ˜— € ™ ˜ – ˆ — ‘ „— ‘ – j ˆ — –m € y— – y • – j k u y – ˆ “ m ˜ ‘ „ ˆ ˜ – d S p “ € “ p — ‘ „— ‘ – j ˆ — –m ˆ ‘ – € d j „ p Ck ¥ Dl = {j ∈ Dk |xjd ∈ πi } © ‚ –mm “• € – ˜‘ m ˜‘ – ‘ ‘ – ˜ € “ d d “ ‘ „— ‘ – j ˆ “ m € y— — – ™ ‘ ‘ „ ˆ € „ „ p — –m ˜‘ „ ˆ – ˆ — ˜‘ – j ™m ™ — –m ˜ ym p ‘ – ˆ — ˜ – s „ — –m € y— – y • “ m ˜ ‘ „ ˆ ‚ € – ˜ — ym p –m – ‘ € y „ ˜ – €  y y ‘ „ ˜ p ‘ „ … “ m ˜ „ z † ‘ „— ‘ – j ˆ – ™ ‘ ‘ „ ˜ p –m ™ — ‘ „— • ˆ – ‘ y ˆ € ˜ € “ d — € – ˜— ym p “ m € y — € – ˜— ym p y ˆ u y “ – • y „ ‘ – ˆ ‘ „ ˜ “ € ™ ‘ ™ l – ˆ – ˆ „ e ˜ ™ j “ m ˜‘ – j –mm – € y ˜ “ ‘ — ‘ „ — y ˆ ™ ˆ ‘ – — y „ † – ˜ ‘ – ‘ ˜ € – d ˜— – ‘ ‘ „ — • ˆ – ‘ y p y “ — ym d – y y – y y ˆ ‘ y „ € – y ˜ p – – – ˜ ‘ “• y — ‘ „— • ˆ “ m – ‘ l — ™ ˆ f – ˜‘ “ € y „ p ‘ „ ˜ ˜ € “ d – ‘ y – ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ ™ f  y y – j e ˜ € „ lm “ ‘ y ˆ — ‘ „ — „ d— ˆ — y „ ‘ f y “ – • ‘ – p © H G ¡ C  @  C G C £ E H B  ¦ I ¤ 8 8 πi Dk Cl Pd (Ck ) = {π1 , π2 } Ck ©  C H   C   A   D H H § $  SubCluster (Ck , d, πi ) Dl D d ©  ! © § ¥ d £ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ © £ © ¥ ¦ ¤ ¦ ¡ ¢ ¤ ¥ ¤ § ¡¨ ©  ¢ ¢ £   © © © ! EDG © $ — – p ™ ‘  y y f £      ¢  ¢¤ ¨ ¤ – j e ˜  € „ l m “ m – ˆ f – m m – f ˜ ‘ –  • „ € d — ‘ „  — ‘ – j  ˆ – ˆ – € s j „ ‘ y ˆ ‘ „ ˜ p ‘ „ … ‘ – – € “ ™ ‘m ™ ˜ u–m d j „ p “ n † – l “ ‘— „ • ‘ “ ˜ € – p ‘ y — ‘ “ ˆ — – y‘ – ˜‘ „ p — € y –m “• – € ˜‘ – –m “ j u“ j – p ‘ “ ˜ — ˆ “ m € –m s p € y „ d ‘ „ — ‘ – j ˆ – ‘ y € y— — € y –m “• — – ˆ ˜‘ – j – p ‘ “ ‘ ‘ „ ˆ € „ m – ˜— — – p ™ ‘  y y ¢¡ § £  ¥ ¤   ¡ ¨ £ – j e ˜  € „ l m “ m – ˆ ˜ ‘ –  • „ € d — ˜ – s „ ˆ – € s j „ ‘ – m ‘ „ m – — ‘– ™ ˜ u–m d j „ p “ n † ‘– ˜— – – ˆ — “ p — – ˆ – € d –m — ‘ “ ˆ ™ ˜ u–m d j „ p “ n † — ˜ “ ˜m y — ™ € — – m € y — — – € ˜ k j “ € “ d — ˜ ‘ – € ™  ˆ — – p – ˆ – p ‘ – y ‘ m – € ˜ d “ ep – m — ‘ “ ˆ — ‘ „ € € – • — y „ † – ˆ „ e ˜ ™ j “ m – ˆ ‘ „ ˜ “ € ™ ˜ – ‘ y € y „ d — ™ € ™ ‘ ™ l — € – ˜— ym p – ˆ m “ ˜ „ ˜ – € s j „ ‘ –m € y— – € y – € ™ d y— – ‘ € „ s f¡ ˜ – f – j e ˜  € „ l m “ – € ˜ „ ‘ – ˆ – y y  ˜ — “ ep „ ˜ — – ˜ ‘ “ — „ d j „ p “ m € – € ™ l € y „ d — ™ —  m  ˜ y ¡ f ¤¦ ¢ ¥ ¤ ˜ – £ ¢ ¢ £ ‚ – ™ € ˜ ‘ – ‘ – — – € ˜ k j “ € “ d —  „ € ˜ – ˆ – ™ ‘ ‘ „ ˆ “ m – ˜ — — – p ™ ‘ m r † – j e ˜ € „ lm “ – — “ s “ m – ˆ — ˜ – s „ — – ˆ –m s j – — ‘ – m ˜ ‘ – € • y „ p ˜ – — – y y ˜— € ™ ˜ p “ € “ p — ‘ „— ‘ – j ˆ — € y –m € y— — – — ‘ – ˆ — ym d — –m ˜‘ „ —  y y u y – p – ˆ –m s j – — ‘ – — y „ — –m — ˜ ‘ “ ˜m y— ™ € — € – ˜— ym p — – ˆ –m s j – — ‘ – m — ‘ “ ˆ € – ‘ ‘ „ ˜ p –m ™ — † ¡ — – ™ — „ d „ € d — ‘ „— • ˆ — – € y –mm – j — –m  j € “ d – € „ ˜ “ ™m “ – l “ € ˜ ‘ y ˜ ‘ “ — “ … ‘ – f ¤¦ ¢ ¥ ¤ —  „ … – ˆ – € s j „ ‘ ‘ “ ˜ € – p ‘ y     ¢ – — “ s – ˆ – j e ˜ € „ lm “ m € – ˜ y p ™ u – † ‚ – ˜—— ‘ „ p ˜ – – j e ˜ € „ lm “ m € “ d ˜  € p ™ ˆ ˜ — – f   ¨  ¡  ¤  £ ¢ § £  ¥ ¤   ¡ ¨ ¢ – ˆ – j e ˜  € „ l m “ – € ˜ „ ‘ f f ˜‘ – j –m “ ‘ v † – j e ˜ € „ lm “ m € “ d – ™ —m “ j € „ … ˜ –  ¢¤ ¨ ¤ ¨  £  – ™ j j „ ‘ ˜ — – – ˜ ‘ “ ˆ ‘ „ d — – € € „ p ‘ „  ˜ p – m ™ — – ˆ ‘ „  ˜ p ‘ „ … “ n † — ˜ € – • y „ p – € „ p ‘ – ‘ „ ‘ — ˜ – s „ ˆ — ym d –m ˜‘ – € • y „ p ˜ – — – y y ˜— € ™ ˜ p “ € “ p — ‘ „— ‘ – j ˆ — € y –m € y — — – — ‘ – ˆ — ym d — –m ˜‘ „ —  y y u y – p — ™ € ™ ‘ ™ l — € – ˜— ym p — –m — y „ ˜  j € “ d € – ‘ ‘ „ ˜ p –m ™ — – ˆ – ˜ y— ‘ – ˜ l “ — m f – — “ s – ˆ – j e ˜ € „ lm “ m – ˆ — ‘ „  ˜ y p ™ u – — € y –  — ym d — k € d # † ™ y ˜ p – – ˜ — – ˜ ‘ – j – ‘ ‘ „  ˜ ˜ € “ d – ˆ u „ ep ‘ y y y —  „ … – y y “ ep — – •  ˜ “ ‘ € – ˜ m “ — ‘ „  ˜ y m „ — – ˆ – m k m m “ € “ d – ep € – e p – € “ m f – m s “ ‘ ‘ „ —  “ € – y y  j e ˜  € „ l m “ ˜ „ p ‘ y € y „ d f € – m y j  — – ˆ  — ‘  “ ˜ – j € – d – m m ‡ † — ™ y ˜ p – – — ˜ € ™ ˜ ‘  ˆ — – € y — – j — – ˆ u  „ ep u y “ — ˆ ˜ – ‘ „  — ‘ – j  ˆ – ‘ y € y— ˜ ‘ – j – ‘ ‘ „ ˜ ˜ € “ d – ˆ – j e ˜ € „ lm “ – € ˜ „ ‘ – ˆ ™ ˜ pm d j— “ m — ˆ — –m s— — „ d — “  s — –m € – ˜  j  m – ˆ ˜ y s € y „ d “ – ˆ „ e ˜ ™ j “ m – ˆ – y y  ˜ — “ ep „ ˜ — ‘ „  — ‘ – j  ˆ “ m f – m “ € ™ ‘ ™ l – € k  ‘ “ j – ’ † – j e ˜ € „ lm “ m € “ d – j e ˜¡„ lm “ m € – p “ m d j – € ˜ y ˆ ‘ „ p – ™ ˆ  – ˜ ˜ – i € † — –m s— — „ d — – l “ d y „ p ™ ˆ — € y –mm – j – ˜ y— “ m € “ d £ ¢ ¢ £ ™ j j „ ‘ — –m  j € “ d ˜ € ™ ˜‘ ˆ – € y — – j € y – m — – m m – ‘ ‘ „  ˜ € „ d „ € d — ™ ˜  m  s “ s „ € d — – ˆ p – •“ ˜ € „ — y “ – l “ €  ˜ ¤   ¤ ¨ ¡ ¥ ¤ § ¥   ‘ y M © O(N log N ) N O(M × N log N )  M axK ¢ ©     ¡ ¡ "  ©   E D G ©  © ! !© ©  K  ¢ ©  ©      © ©  © ¨G DH V – y y  ˜ —  € ™ ˜ p “ € “ p ‘ „  — ‘ – j  ˆ – ‘ y p y “ ˜ – – ˆ — ˜ – s „ — – m — y „ ˜ ˜ ‘ “ ‘ – ˜ ‘ „ p € – ˜ — y m p ‘ y p – •“ – ™ —m “  ˜ ‘ ˜ – — € – ˜ — ym p – ˆ – ™ — „ d j „ p – ˜‘ “ € y „ p ‘ „ ˜ ˜ € “ d “ m † — ™ € ™ ‘ ™ l — € – ˜— ym p – ˆ – € s j „ ‘ –m € y— – € y–€™ d y— – ‘€ „ s f ˜– f — ˜– s „ ˆ –m s j–— ‘– ‘ y ¢ D K © © © ¥ ¥ ¥   ! DH V D ¨©D% – ˆ – ™ — „ d j „ p f – ˆ – ™ ™ € p ‘ „ ˜ ˜ € “ d “ m D † — € – ˜— ym p k P (k, d, π ) = SelectDivision(P ) (K < M axK ) (π = ∅) K =K +1 P [K ] = SubCluster (P [k], d, π1 ) P [k] = SubCluster (P [k], d, π2 ) (k, d, π ) = SelectDivision(P ) K = 1 P [1] = (D, ∅) P ¡ ! D F  ©¨ & … —  • ’ S D R V ©   EH $ → → D = (− , ..., − ) x1 xN M axK ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡  † ˜‘ – j –mm – ˜ € “ d € – ep y “ • – ep – — ˜ ‘ – • y – d u y “ ‘ — € – ˜ — y m p — – m – y y —  y d — – ™ ‘ ‘ „ ˆ — – ˆ – m s j – — ‘ – m – ˆ ‘ „  ˜  ˜ € “ d – ‘ y — “ d – —  m “ ™ € – ‘ m r †  ¤ ¡      ¥  ¢ ¥ ¢  £   £ ¤   ! ¤ !  £  ¤ ¥   ¤ ¥ ¤ ¥  £ ‘ y € y — — – m m “ • € – ˜ ‘  — – ˆ € “ d — ‘ ™ ˆ — – € “ m y l ‘ “ ˜ p – € € – d e — € – ˜— ym p – ˆ –m s j – — ‘ – ‘ y ˜ y ˆ „ € d m y y ˜ — – –m “ d p ‘ € d “ n † — – y y ˜— € ™ ˜ p “ € “ p – ˆ – € s j „ ‘ ‘ “ ˜ € – p ‘ y € “ d — ˜ ‘ – € € y p ‘ „ p — – — – ˆ – y l ‘ ˜— ˆ – —    © © © EDG 0 0( 01 0 01 01 01 01 01 01 01 321 01 01 01 01 01 01 01 01 01 1 001 0¨'1'1(110'1¨1¤1§1"1"3¨1'10¨!1§¨11¨¤'00§0§§0'¤0'!0'!2¥¨£ 01 01 01 01 01 01 01 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) % )) ¨ %) ¨ %) ¨ %¨ ¨ © © &¨ © © & ¦ © &¦ © &¦ © %& ¤ ¢ © ¨ © © ¨ %& ¦ 0¢ &¦ & ¥¤ &¦ © © ¨ % ¦¢ © © © © &¤ % ¦¢ ¨ % ¦¢ % ¨¦  ¨ ¨ &¦¢ & ¦ ¥¢ © © © © %& ¤¡ © % ¤0¡ % ¤¢ ¨ ¨ © © &¦¢ % ¤¢ &¤¢ © %& ¦¡ &¦ © ¨ © % £¤¡ ¨ &¤¢ © % ¤§¡ ¨ &¦ © © © % ¦¢ ¨ &¦ ¢ &¦ © © ¨ % ¦¤ ¢ © © © ¨ % ¤¤¡ & ¦ ¤¢ &¦ © &¤¢ ¨ % ¤¡ %&¤¨¤"¨'¨¨#©!¡ % ¤¡ ¨ % ¤¨¨'¨"¨0¨¢ &¦¢ © %&¤¨¢ %&¤©¡ & ¤ §¢ © % ¤¡ &¦¢ &¦ ¢ % ¦¡ % 0¤¡ $¤0&0&'(&&#(&©"&!¥§¥¨¤'!¦(¥§§'0¥¨¤§0¨¦§§§¨¡ §(¡ $¥§¤¡ §¢ $!&0¤$§$&¥¦!¦¤!¤£¤§¨§ " ¨¢©¨¢¨¢ ¨  ¤£¥¤£ $¥§¢ $'¥¤¡¢ § $¥¦¡¢ § $¥§ §¢ $¥¦¡ $¥§ § § $(&0$¥¦¡¢ $¥§ $¥¦§ §¢ $¥§ $¥¦¡¢ $¥¦¡£ $¥§¢ $¥§ §£¡ $0¥(¦§¡¢ $¥§ § $¥¦¡¢ $¥§ $¥¦¡¢ § $¥¦¡¢ $¥¤¡ $¥§ $¥¤¡¢ $¥§ D @IH ¡ ¡ ©  8 † ˜ – ¤¦ ¢ ¥ ¤ f£ ¢ ¢ £ – ™ € ˜ ‘ – ˆ — – € ˜ k j “ € “ d — – ˆ ‘ „  ˜ p ‘ „ … ‘ – – € “ ™ ‘m ˜— – – ˆ „ e ˜ ™ j “ m f ˜‘ – j ™ — p ™ € d — ym ƒ † ˜ – ˜‘ “• – ˆ — –m s “ – lm l ™ ‘ — – ™ — „ d d y— ˜ ‘ „ — — € y –m “• — –m ˜ ‘ „ ˆ f – ˆ „ e ˜ ™ j “ m – ˆ – ™ € ˜‘ – ˆ — – € ˜ k j “ € “ d — – ˆ — – ˜‘ “ ˆ ‘ – d ™ ˆ — – ˜‘ “ ˜— ‘ „ p — – ˆ € “ d – ™ ‘ € „ s € “ p f – € y – € ™ … ‘ ˜ — – ™ ˜ u–m d j „ p € y –m f – j e ˜ € „ lm “ m – ˆ — – ˜‘ “ — „ d j „ p — – € ˜ y “ u y “ ˜ ‘ “ y ¡ † – p “ d — – m – ˆ ‘ „  — ‘ – j  ˆ – y y “ e p € y — ¢¡ § £  ¥ ¤   ¡ ¨ £ – j e ˜  € „ l m “ m € – ˜ y p ™ u – ˆ – ˜  — M axK © © ¢ N © M © © © †     ¢ – j e ˜ € „ lm “ m – ˆ ‘ „  ˜ y p ™ u – ˆ – m d j – u ‡ † ¡ ¥ © "¦ § § ¦ § Yo ¦ ¦ Y" o $ ¡ ¡ £© ˜ ‘ – j –m p “ … — ‘ „ j — ˜ “ ˜m y — ™ € — –m — € „m “ ˆ ‘ – € –mm – ˜ – f — ‘ „— ‘ – j ˆ – ˆ – € s j „ ‘ y ˆ ‘ „ ˜ p ‘ „ … ‘ – – y y ˜ “ € ˆ “ y y ™ ˜ u–m d j „ p – ‘ y “ – — m “ ‘ “ –mm – ˜ – ‘ y y y — y d — ˜‘ – y y ™ — ‘ „ p — ym d —m y pm “ p – ˆ — d j – ˜ — – ˆ € – ‘!“ € ˜‘ – ˆ — ‘ „ j ‘ “ ™ ‘ – y y — € – € “ ‘ jm ™ € d – — “ e d – ˜ ˜ – i † ˜‘ – j –m “  ˜ ‘ — “ p –m — “ d ˜ “ ˜ ™ ‘ – p – y y — € „m “ f — – u“ u y “ –m kmm “ € “ d ˜— – –m “ € ™ ‘ ™ l ‘ „ ˜ “ ˜ ‘ – € „ m ˜‘ „ ˆ — € – ˜— ym p — – ˆ € – • y „ € ˜ – € – ˆ f – € k d – € – ˆ ˜ ‘ – j – l ‘ “ ep ‘ y “  • f – € ˜ ˜ – j € – d ˜ “ € • – ˆ – — “ e d – ˜ ˜ – i † † † x — – u – ‘ ‘ “ ‘ – – ™ ˜ ‘ – — ™ € d – ˆ „ e ˜ ™ j ¤ ¨   ¢ £ ¥ ¤  £     £ £¤ ¤  ¨  £ ˆ – — “ e d – € k  j – € d – ‘ y – ˆ „ e ˜ ™ j “ m € – ˜ y „ “ ˆ –m s— — „ d ˜— – m f — – u“ u y “ — –m kmm “ € “ d – € ˜ ˜ ‘ – • „ ˆ — –mm – y y –  ‘ l —  y y – p ‘ „ ˜ d € p — – ˆ – ˆ ‘ „ — ‘ – j ˆ – y y ‘ y – ‘ y € y — — ˜ – s „ — – ˆ ‘ „ ˜ p – „ € d “ m – ˆ — € „m — – m s  —  • – € ˜ ˜ ‘ – •  „ ˆ — € – ˜ — y m p – € ˜ ‘ – — ‘ „  ˜ “ € “ d ™ — — – m – y y – ˜ ‘  “ € ˜ ‘ „ p “ m € – ep ¦ m – € € y „ ƒ † – € “ ˜‘ – j ™m ™ ˜ ‘ – j – ‘ ‘ „  ˜ ˜ € “ d – ˆ – j e ˜ € „ lm “ m € – y „ ep ™ ˜ “ …  y y – p f ‘ „  — ‘ – j ˆ – m y – — – ‘ y € y — — ˜ – s „ — – ˆ ‘ „ ˜ p – „ € d – ‘ y p y “ € y— –m s— • ˜— – ‘ ‘ „ ˜ “ € “ d ™ — – ˜ ˜ – p f ‘ „ ˜ d € p — – ˆ – ˆ ˜ –m d j „ p – p “ d— – m — ‘ “ ˆ — ™ € “ d ™ — ‘ – s ˜‘ „ — — € – ˜— ym p — –m ˆ ‘ “ y y – j j f — –m d j – u– — – p — y „ ˜ — ‘ “ ’ † † – € y l “ m – ˆ u y – p – j j „ p — “ p — – ˆ ˜ ‘ – j – ˜ p – € € „ p € – ˜ “ € ˜ ˜ y – d – ‘ f — – y y˜—€™ ˜p “ €“ p — – p €“ d – ’ † ™ — • € – d y — – l “ — — ˜ ‘ – € d d “ ‘ – † i – ˆ ™ ˜ p “ p £– m – ˆ ‘ „ …  y y –mm – p –m s “ € “ d j „ p ˜ “ … ˜ y „ ˜ ˜— – – — k e ˜ „ d  e – ˜ ˜ – i † — ™ € “ d ™ — ‘ – s ˜‘ „ — — € – ˜— ym p u y – ˆ — – p –mm – y y “ m € y — ‘ „ ˜ p – „ € d – ˆ ‘ „— ‘ – j ˆ – ‘ y — ‘ „ j y “ — € y „ y „ ˜ – ˜— u– m f — ˜ p ‘ ˜— ˆ — € – ˜— ym p – ˆ –m d y „ p ˜ y „ ˜ € y „ d – y y ˜— – ™ — “ s ˜ — – –mm – y y “ m € y— –m “ ˜ ‘ – j “ ˆ ‘ „ … – — k e ˜ „ d  e n     © ©     © © ©   © © ©   ©  $"  © EDG  " ©   EDG  !  © ˜– f ‘ „— † ‘ „— ‘ – j ˆ – ˜ ˜ – p € y— € – y ˜ p – – ‘ „— • ˆ “ m %  • ˆ “ m € – y ˜ p – – –mm – y y “ m € y— ‘ „ — ‘ – j ˆ “ m f € – — • ˆ € – ˜— ym p y ˆ „ € ™ j y‘ –m ! – ™ p „ — — “ ˜ € ™ ˜ ‘ ˆ – € y— – j “ — –mm – ‘ ‘ „ ˜ € „ d „ € d ‘ „— • ˆ – ‘ y ˆ – l “ € ˜ – ˆ ™ ˜  m  s “ s „ € d “ m p – •“ f — ‘ “ ˆ — – ™ p „ ˜— — ‘ „ — • ˆ — – ˆ – ‘ y m ˜ ‘ – j – € „ ˜ “ ™m “ € – ‘ € y „ ˜ – g  !  d © © © D ©   k π = Pd (Ck )   ¨D¨ D ¨yD“V–©m s“ ˜ y ˆ ˜‘–j™m™ –m €– j„˜€ d d yz    ¨D¨D¨ D € y„   –€˜˜– q — ‘ “ ˆ f ˜ ‘ – j – p “ m d j –© m – ˆ – p  ˆ ‘© m € – m s i ¨DV©    Bests i=1 ALEA D IV N ot AcceptDivision(Bests[i]) ieme Bests M inBest = 0 k=1 K d=1 M π = Clustering1D(Dkd ) N ote = Interet(π ) (N ote > M inBest) i Bests Bests[i] = (k, d, π, N ote) M inBest –m “ j ‘ j – ˜ „ ‘ – ˆ ‘ „— • ˆ “ m – ˆ f   „˜ „˜ — ‘ “ ˆ — – ™ p „ ˜— — ‘ „ — • ˆ u y “ – ™ p „ — — “ –m “ j ‘ j – ˜ „ ‘ – m “ j u “ j ˜ — – – ™  p „ — — “ ˜ € ™ ˜ ‘ ˆ – € y — – j “ m ˜ ‘ „ ˆ — ‘ „  —  • ˆ ¡ £ ¢ ¢ £ — –m ˜‘“ p „ ˜— y“ –m s“ ˜ € – —m “  ˜ ‘ ˜ – f — € – ˜ — ym p ¢ Bests ¢   ¢ (0, 0, ∅, 0) © ¡ † – ˆ „ e ˜ ™ j “ m – ˆ – y y  ˜ — “ ep „ ˜ — – ˜ ‘ “ — „ d j „ p “ m ˜ ‘ “ m " € ˜ ‘ „ p – ˆ – ™ — „ d j „ p ‘ „ ˜ ˜ € “ d – ‘ y  – yy˜—“ ep „˜— f  ‘„—€–• ‘„— • ’˜p–m–z  D R V! ©D F©E¨H & "$ ¡ K ALEA D IV f Bests[1..ALEA D IV ] ← P = (C1 , ..., CK )  ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡ K † — € – ˜— ym p ! D ©   % – ˆ ™ — „ d j „ p f ˜ “ ˜m y — ™ € l ‘ € – ˜— ym p –m  ¨D¨D¨ D – € ˜ y “ ‘ y — ‘ “ ˆ — ym p ‘ € – ˜ — ym p – ˆ ¨DV ©€– j€ d d yz    DHHHVGH HG¨D   ¨D¨ D P = (C1 , ..., CK ) P ¡ Ck ¢ Ck ∈ P Cl ∈ P Dk ⊂ Dl P = P ∪ Best SET = SET \ Best N bU ncover = N bU ncover − N bCover U ncover = U ncover \ Best ¨DV©   Densite = N cover/V olume Densite > BestDensite Best = Ck BestDensite = Densite N bCover = N cover — –m s “ € “ d j „ p ˜ ‘ – „ — — – y y ˜— € ™ ˜ p “ € “ p — ‘ „— ‘ – j ˆ – ˆ — – € s j „ ‘ — ˜‘ – € ™  ˆ € y— — ‘ ™ ˆ — € – ˜— ym p – ˆ — – j ym „ • — –m – y y – € ˜ ˜ – j € – d € y „ d – ™ — m ˜ y ˜ ‘ “ ˜ ™ – ‘ ‘ – „ j “ m f — – y y  ˜ —  € ™ ˜ p “ € “ p — ‘ „  — ‘ – j ˆ — – — € y — – ˆ ‘ „  ˜ ‘ ™ ˆ – ˆ — –mm “• € – ˜ ‘ ˆ — –mm “ ˜ — – ˆ – y y € ˜ ™ j „ ™ l – ‘ ‘ – „ j —˜€–• y„p ‘„ ‘ – ˆ —˜– s„ ˆ –€ s j„ ‘ —˜€–• y„p ‘„ ‘ – ˆ —˜– s„ ˆ –€ s j„ ‘ – ˜ ‘ “ € y „ p ™ ˜— ‘ – ˆ – € y –mm – j – — ‘ – ˆ — ym d –m ˜‘ “ € y „ p € – ˜— ym p – ˜ ‘ “ ˜m y— ™ € ‘ „ ˜ ˜ € “ d “ m — ˜ € – • y „ p ‘ „ ‘ — ˜ – s „ — – ˆ – ˜ —m — ™ ‘ ‘ „ ˜ p –m ™ — — € – ˜ — ym p — –m € “ d — ˜ € – • y „ p ‘ „ ‘ — ˜ – s „ ˆ – € s j „ ‘ † € € • y „ p – € – — ˜‘ “• y „ d — ˜ “ ˆ ˆ ‘ “ p — € – ˜ — ym p – ˆ –m s j – — ‘ – ‘ y f ˜– f — ˜– s „ ˆ –m s j–— ‘– ‘ y © !  ¢ ¢ © Ck ¢ ¡ ¡ Best ¢ ¢  ©   © ¢ ¢ ¢ ©  N bU ncover = N → → U ncover = {− , ..., − } x1 xN P =∅ N bU ncover > 0 Best = ∅ BestDensite = 0 N bCover = 0 Ck ∈ SET N cover = |Dk ∩ U ncover | V olume = |Sk | d∈Sk |Dkd | Ck   HH G   ¡ ¡  DH V ¡ ¡ ¡ ¡  D R V! ©D F ©E¨H & → → D = (− , ..., − ) x1 xN SET ˜ p –m – z m “ ‘ v $ ¡  ¡ £© ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ © © ¨ ¤ ¨ ¤    © ©¦ ¡321 ©§ ¡  ¥¡  ¤  £ © ©¦ ¡ ©§ ¡  ¥¡  ¦  ¥ ©¦£¡ ©¦ ¡ § ¡ § ¡ ¥ ¡ ¤¡ £¡ ©¥¢¡ ©¥¢¡ §¢¡ ¦¢¡ §£¡ ¤ ¡ £¡ ¢¡ ¡ ¤ ¡ £¡ ¢¡ † EDG $"¦ € y „ d — – y y˜“ j™m s „ € d — “ p – ˆ — –m d j– u ‡ † K † — € – ˜— ym p ¥ ! D ©   %  ‘€y˜¨–D€  – ˆ ™ — „ d j „ p f ˜ “ ˜m y— ™ € l ‘ € – ˜— ym p –m F inalSelect(D, SET ) P = (C1 , ..., CK ) SET = ∅ n=1 N B RU N S SET = SET ∪ Divisif (D, ALEA D IV, M axK )  „˜ † – ˆ „ e ˜ ™ j “ m – ˆ ‘ „ ˜ “ € ™ ˜ – ‘ y € y „ d — ™ € ™ ‘ ™ l — € – ˜ — ym p – ˆ – € s j „ ‘ – m € y — – € y – € ™ d y — – ‘ € „ s “ m  ˜ – f – y y  ˜ — “ ep „ ˜ — e –„ˆd„j˜„™pj““m m˜–‘ˆ“m—"‘€˜„‘˜„“p€™˜© ˆ – € s j„ ‘ –mf —˜– s„© ˆ –m s jf––—ˆ‘„–e‘˜™yj “m – ˆ – yy˜—“ ep „˜— –˜‘“— E D  G  $ D R V! ©D F ©E¨H & M axK N B RU N S → → D = (− , ..., − ) x1 xN ALEA D IV  $ ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡ † EDG  ‘„ ‘ ‘„ ‘ y„ y„ y„ y„ y„ y„ y„ ‘„ ‘ ‘„ ‘ l ‘ € – ˜— ym p y “ — – ™ p „ — — “ — – y y ˜ — € ™ ˜ p “ € “ i ¥ † " ¦ ¥ ¤£ — – y y € ˜‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜ — ‘ – ˆ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– –—€ d ˆ€“ e – j ˜ ‘ “ m “ ˜ ‘ – j ™ € p ‘ – ˜ — ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d € „€ d “ — – p ‘“ ——“ ‘ ‘ „ p !©   O(M × N log N ) — –m lk € – ˆ –m s j–— ‘– — € – ˜— ym p – ˆ – € s j „ ‘ –m € y— – ‘ € „ s   DH G" D  ( © W  F ©  G  G  "  ©!  EDG  ©  "  © EDG  EDG ©   ED G  ©© ©   $  © ED G  ©  © ©  † – ep „ € d d “ – ˜ ˜ – p – ˆ — – y y  ˜ —  € ™ ˜ p “ € “ p — – ˜ ‘ – € ™  ˆ — – m – j y — ™ € † y “ – m s “ ˜ – m f ˜ ‘ – j – m “ ‘ v † –m kmm “ € “ d ‘ „— € – • – ‘ y ˆ – € • y ¤ ‘ – – — j “ m € “ d — – ™ € „m ™ j “ – € „ p ‘ – – € ˜ p ‘ „ ˆ ˜‘„€€ y„ d – ˆ — – p ‘ “ j € „ …€ – d — – n † — – € ˜ y “ — – ˆ — – ‘ y — –m ˜‘ – j j “ ˆ ‘ – d ™ ˆ ‘ — – ™ —m “ ™ € – € ˜ ˜‘ – • y – d —m y pm “ p – ˆ — – d “ ˜ ™ — € y –— ym d – y y — — y “ — ‘ „ ˜ „ † – p “ d — – m – ˆ — ‘ „— ‘ – j ˆ — –m — – ˜ y „ ˜ — ym d ‘ „ ‘ ˜ – f — € – ˜— ym p — – ˆ — – y y ˜ — € ™ ˜ p “ € “ p — ‘ „— ‘ – j ˆ — –m € – € ™ ˆ — ‘ „ p – ˆ — € „m “ ˜ £ y— m € “ p ¤ !  ¥ ¦¥  ¤ ¤ ¨  ¤ ¤  ¡ ¤  ¡ £ ¤ ! £  ¢ ¨    ¨ ¢ “ m f — ™ ™ € p — € – ˜ — ym p — – m —  „ … – ‘ y f — ym d – ’ † † ‘ „ ˜ p – — “ m – ˆ ‘ † …p –m s “ ‘ ‘ „ — “ € — k € ˜ ™ ˜ u–m d j „ p – ‘ y ˆ ˜— – † — ™ € ™ ‘ ™ l — € – ˜— ym p — – ˆ ‘ y p “ ep € y „ d — – y y  p ™ d — — ‘ „  — ‘ – j ˆ – ˆ j y j ‘ j ‘ y € “ d — –  ‘ ™ ˆ — – m l k € – ˆ – j € „ … — y„— – y y — ym d ˜ ‘ “ ˜ y “ ˆ  p – p ˜ – f – ‘ k l „ j „ e ¤ ¨   £¤ £ ¥  ¢   ¨ ¡  ¥ ‘ y ˜ ‘ € y „ … – — — “ — “ d – j j „ p ™ € ™ ˆ— ‘ „ p ˜ ‘ “ ˜m y— ™ € € – ˜ — ym p ‘ y € y— d y„p —k€ d“ €–p ‘“m–€ – ˆ –m p “ … — k € ˜ ˜ — – m f — “ s d „ € ˜ y “ – • ‘ ‘ y ˜ ‘ – j –m “  ˜ ‘ – ™ u ˜— – – ‘ € „ s – ˜ ˜ – p — f € w † – ˜‘ – ˜ ˜ “ ˜ ‘ – j – €  “ — — – p ™ ‘ — “ d ˜ — – ‘  y y f — ™ e p € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m € y — – € y –  € ™ d y — – ‘ € „ s – ‘ y p ‘ „ ˆ ˜ — – € y – ˜ “ — m ˜ y m € “ d – j e ˜ € „ lm “ m – ˆ – ™ € ˜ ‘ – ‘ – € u – € ˜ k j “ € d m y – — – n † – ˆ „ e ˜ ™ j – “ “ m – ˆ – y y  ˜ — “ ep „ ˜ — – ˜ ‘ “ — „ d j „ p “ m ˜ ‘ “ m " € ˜ ‘ „ p ˜– —–€˜k j“€“ d u y – ˆ u y “ — – ™  p „ — — “ – € ˜ ˜ ‘ – • y – d ˜ y “ … ™ ˆ € “ d — € y – m “ • — – ˆ – y y f – € ˜  d “ ep – m — ‘ “ ˆ – ™ ˜ ‘ – — ™ € d f u y “•“ € ˜ — – p – ˆ – m “ ˜ ‘ – j € ™ d u – –  ˜ € “ d “ m — ‘ “ ˆ — ‘ „ € € – • — y „ † – m d j – u – € “ d – m “ j u “ j –  ˜ € – ‘ € y – m y „ f — ™ e p € – ep – € — € – ˜ — y m p – ˆ ˜ p “ u – – € s j „ ‘ – m € –  p ™ d — – ˆ ˜ ‘ – € k  y y – €  y y — – ˆ „ e ˜ ™ j — – — y – € s j „ ‘ – ˆ ˜ ‘ – j – € “ € ˜ ‘ „ p f € y – ˜ “ — m ˜ y m – ˆ ˜ € “ d “ m – ˆ  ¥  ¥   ¤¢ £     £ £ ¢ – ˆ y – d — k € ˜ – ˆ ‘ “ j – ˆ m y y ˜ — – – ˆ — ˜ € ™ ˜‘ u y “ d p ‘ € d — – ˆ ‘ y m f — € y –mm “ € “ ƒ † y ˆ € – d  — ‘ “ ˜ “ € – — – ˆ „ e ˜ ™ j “ m – ˆ ˜ € d— – m f ‘ ‘ ‡ † – j km s „ € d y ˆ ™ ˜m “ ‘ „— ‘ – j ˆ “ m –m “ l ™ ˜ „ — –mm – y y € – ˜ “ e y „ — — € „m “ ˜ “ € € y „ d ‘ „ ˜ – f € – € ™ l – € “ ˜‘ – j ™m d d y — – y y ˜ “ j ™m s „ € d – ‘ y — € „m “ ˜ “ € – — € – € ™ ˆ — ‘ „ p ‘ „ ˜ p – „ € d – ˆ –m “ j u“ j ™ ˜m “ ‘ „— ‘ – j ˆ “ m – ˆ ‘ „ ˜ “ p  p ™ d— “ m f — ym d – ’ † – ™ ˜‘ – j l y “ ˜‘ – j – ˜ € „ … — € „m “ ˜ “ € – — – ˆ „ e ˜ ™ j “ m – ˆ ™ ˜ u–m d j „ p “ m — “ j f ‘ „— ‘ – j ˆ – ‘ y ˆ — ym d € y— — ‘ „ ˜ p – „ € d — – ˆ € – € ™ ˆ — ‘ „ p – ˆ ˜— – ™ ˜m s— — „ d – € ˜ y “ – ‘ ” † — –m s “ ˜ ™ € d € – ˜‘ ©   N B RU N S $$ © ©   ALEA D IV   ¡ ¡ £© † –m s— — „ d — ˜ ‘ – ‘ ˜ € – d — ˜ y s € ˜ ˜ “ ˆ j y j ‘ j ‘ y € “ d — – ‘ ™ ˆ — –m l k € – ˆ – j € „ … — y „ — –m s “ ˜ ™ € d € – ˜ ‘ ˜ “ ˜m y— ™ € ‘ y – ˜ € „ — ‘ – ˜‘ “ — — ‘ € y „ … ˜ – f — – ˜—m s “ s „ € d — –m k ˆ „ j – ˆ ‘ „ ˜ “ —m ˜ y m € y— – ™ — “ s f — € – ˜— ym p — –m ˜‘ “ — € ™ ˜ p “ € “ p — ‘ „ — ‘ – j ˆ – ˆ — –m s j – — ‘ – — y „ — — € y –— ym d ˜‘ – j ™ ‘ “ ˜m y j — € – € ™ ˆ — ‘ „ p – ˆ – m s “ d “ p – ˆ „ e ˜ ™ j – ‘ y ˆ u  „ ep – m ˜  “ … p ‘ „ ˆ — ‘ „ •“ — y „ ‘ f – € ˆ “ p – p — ‘ “ ’ † — ˜ € „ … — ˜ ‘  „ d — – — ˜ ‘ “• € – — ‘ „ p ‘ – ˜ y „ ˜ – ˆ — – ˜ jm u y “ € – mm “ d – ˆ … ˜ p – s „ € y „ d p – •“ – € • y ¤ ‘ – – —  j — ‘ „ •“ — y „ ‘ – y y l ‘ € – ˜ — ym p – p “ d — s y — – ˆ – ˆ „ e ˜ ™ j – € ˜ y “ – ‘ y ˆ € „ s “ ˆ — ‘ „ ˜ ‘ – — ™ € d — y „ ‘ f — ‘ „  ˜ “ ˜ ‘ – j  € ™ d u – — „ ‘ – ˆ — m  “ ˜ ™ ˆ — – m — ‘ “ ˆ € – € ˜ ‘ – € – ˆ ˜ ‘ “ •“  „ y y € y „ d ˜ — – i † – ™  € ™ • — “ d ˜ — – ‘ –mm – ‘ ‘ „ — ‘ – j ˆ ‘ y ™ ˜m s “ € “ d ™ — – ˆ – — k e ˜ „ d  e m — –mm – y y — –m € y „ d — –mm – ™ € — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — € y –— ym d € y— — ™ l ˜ j — ˜ “ ˜m y — ™ € — – ˆ ˜‘ – ˜ s „ f — – ˜ jm — – — € “ d – ˆ f ˜‘ “ ˆ ‘ – d – i † — – ˜ ‘ – ‘ ˜ € – d ‘ „ ‘ — ‘ „— ‘ – j ˆ — – ‘ “ ˜ € – p € “ d — – ˜ € p ™ ˆ ˜ ‘ „ —  p — –mm – p – y y — € „m — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ — ˜‘ – — ™ € d — € – ˜— ym p — –m € –  ˜‘ – ˆ ˜‘ – • y – d – ‘ l ‘ € – ˜— ym p – ˆ — – y y— — “ m p — – ˆ „ e ˜ ™ j — – ˆ – y y ˜ – f — ˜ “ ˜m y — ™ € — – j j — –m € ‘ € y „ … € y „ d ˜ ‘ “ ˜ € „ d j — ym d d y „ p y “ – s ‘ „ ˜ y p ™ u– ˆ — d j – ˜ ‘ y ˜ ‘ – ˜— — – p ™ ‘ l ‘ € – ˜— ym p – p “ d — s y— – ˆ — – ˆ „ e ˜ ™ j — – € ˜ y “ ˆ – y y — € „m “ f –m s “ ‘ ‘ „ — “ € ˜ “ … ˜ y „ ˜ — d j – ˜ ‘ y ‘ – — ‘ € y „ … — € „m “ ˜‘ „ — — ˜ “ ˜m y— ™ € — –m f € –m y p ˜ € “ d ‘ ‡ † – ™  € ™ • ˜— – ‘ „ ˜ d € p — – ˆ – ˆ ‘ „ — ‘ – j ˆ – ‘ y — ‘ „ j y “ € y — — € – ˜ — ym p — – ˆ ™ ˜m s “ € “ d ™ — – ˆ – — k e ˜ „ d e m — –mm – y y — –m € y „ d — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s — – ˆ ™ ˜‘ „ € … ‘ „ p ˜ — – m y y — € „m ˜‘ “ — — – € ™ ˜‘ ˜ “ … ˜ y „ ˜ ˜— – f — – y y  ˜ —  € ™ ˜ p “ € “ p — – — € “ d – ’ † – € ˜  d “ ep – m — ‘ “ ˆ — – ™ ˜ ‘ – — ™ € d ˜ ‘ „ — l ‘  € – ˜ — y m p – p “ d — s y — – ˆ – ˆ „ e ˜ ™ j – m m – • y „ ‘ – ˜ ˜ – p p – •“ — – ˜  y ˆ ‘ „ p — ‘ „  ˜ “ ˜ ‘ – j  € ™ d u – — – n    © ED©  G    E©D G  ©    © EDG ©  ©©   $ © © © !$ ¤ ¤ ¡ ¡¦ ¤¡ £ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ § © ¥ ¦ ¤ £ £ ¦ ¦ ¤ £ © ¦ ¦ ¤ ¤ ¤ £ ¦ ¦ ¡ ¢ © ¦ ©£ © ¥ ¦ ¥ ¤ £ ¢ ¡ #$  — – € „ ˜ “ ™m “ — € y –m “• — – ˆ ˆ ‘ – € d ˜ – ‘ „— ‘ – j ˆ – ‘ y € y — — – ‘ ‘ – „ j — € y –m “• — – ˆ € “ d  ‘ ™ ˆ ˜— – € – ˜— ym p ‘ y ¤ „ f † – € y l “ m — ‘ “ ˆ ™ ˜‘ – — ™ € d — “ p –m –m d j – u– € “ d — ‘ „ ‘ – € ƒ † l ‘ € – ˜— ym p – p “ d— s y— y ˆ – € ˆ “ p – m — ‘ “ ˆ ˜ ‘ “ ˜ € „ d j  – l “ ˜ ‘ “ •“ ‘ y ˜ ‘ – j – m “ l ™ – y ˜  ˜ — ‘ „ p — € – ˜ — y m p — – ˆ ˜ ‘ – j – e p y “ • – ep – m ˜ ‘ – j – mm – € y ˜ “ ‘ ˜ ‘ – ˜ ˜ – j € – d — – ˜ — m s “ s „ € d — – ˆ „ e ˜ ™ j — – m – y y ˜ “ … – m f — € y – mm “ € “ ƒ † € y – ˜ “ —m ˜ y m – ˆ ˜ € “ d “ m – ˆ  € „ € d “ – p ‘ “ — — “ ‘ ‘ „ p – ˆ € – — — “ d – — ˜ y – d  y y – ˆ „ e ˜ ™ j – ‘ y € – — „ d „ € d – ˆ — „m “ ˜ – j € – d — y „ ‘  p – i † – ˜—m s “ s „ € d – € ˆ “ p ‘ y — ‘ “ ˆ –m k ˆ „ j – ˆ ‘ „ ˜ p –m ™ — – ˆ – j km s „ € d ‘ y – j j „ p y • – € ˜ ˜ y – d l ‘ € – ˜— ym p – p “ d— s y — – ˆ — – ˆ „ e ˜ ™ j u y “ — ˜ ‘ – € ™ e ‘ — – € ˜ k j “ € “ d — – ˆ ‘ „  ˜ “ p  p ™ d — “ m – ˆ – m p £ ˆ – j k m s „ € d – m – y y — ‘ „ € ˜ ‘ „ j — y „ ‘ f € – m y p  ˜ € “ d ‘ ‡ † — ˜ ‘ “ ˜ € „ d j — – l “ ˜ ‘ “•“ — € y –  — ym d – ˜ ‘ – — ™ € d  y y f — – ˜ — m  s “ s „ € d — –m k ˆ „ j – ˆ ‘ „ ˜ “ —m ˜ y m € y— – ™ — “ s ˜ — –  p — ‘ „ — „ d „ € d — y „ ‘ – y y – ˆ „ e ˜ ™ j “ n      © ©    C EHIH  @   B  CI ¥ G  ¨  ¨ © ¢ † € y – ˜ “ —m ˜ y m –mm – y— • ˜ – –m s— ‘ – e ™ € d j „ p – € k ‘ “ j – ˆ — ˜ “ ˜m y— ™ € — –m € – ˜ ‘ – — ™ € d – ˆ ˜ ‘ “ ˜ ˜ – j € – d –m “ ‘ l € „ – ˆ „ e ˜ ™ j – ‘ y – — „ d „ € d ‘ „ ˜ p – — – € k ‘ € – ˆ “ m f ‘ ‘ ‡ † ™ ‘ ‘ „ ˜ p –m ™ — ˜— – — – ™ ‘ ‘ „ ˆ u y “ u y – j –m ˆ ‘ „ d— – € € „ p  y y  ym – p f — – ™ ‘ ‘ „ ˆ — –m € – ˜‘ – — ™ € d – € € y „ d — ™ l “ — • ‘ – – € ˜ ˜‘ – • y – d  y y — –m k ˆ „ j — ˜‘ – € ™  ˆ — –m  j € “ d f ˜ ‘ – j j „ p – ˜ y— ‘ – — ‘ „ € ˜‘ „ j — y „ † l ‘ € – ˜— ym p – p “ d— s y— y ˆ — “ p y “ – ˆ „ e ˜ ™ j “ m € – ˜ d “ ˆ “ ˆ ‘ “ f – l “ — — ˜‘ – € d d “ m ˜ ‘ “ ˆ ‘ – d — ˜ y s € ˜ ˜ “ ˆ ‘ „ ˜ p –m ™ — “ m – ˆ € – y ˜ p – – ˆ — –m s— — „ d — – € k ‘ “ j — – ˜‘ – € ™  ˆ — – ˆ – ˜ y— ‘ – – ˜ “ € ˜ † ‘ „ ˜ p – — “ n † ™ — j ˜ d „ — y d ™ —m “  ˜ ‘ ˜— – — ‘ „ —m ˜ y — y „ ‘ – y y –m k ˆ „ j –m ˜‘ – j j „ p — ‘ „ ˜ ‘ – — ™ € d — y „ ‘ f – ˜ y — “ m € y „ d — – € “ — — – p ™ ‘ — ‘ „  ˜ “ ˜ „ 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˜ p – ˜ ™ ˆ “ m – ˆ – j km s „ € d –m f ‘ ‘ ‡ † — € – ˜— ym p — – ˆ ‘ y p “ ep — – y y  p ™ d — ‘ „  ˜ d € p — – ˆ – ˆ — – p “ d — – — y „ — — – m € –  ˜ ‘ – ˆ  ˆ ˜ ‘ “ ˜ ˜ – j € – d – m k ˆ „ j ‘ y ˆ – € • y ¤ ‘ – – — j “ m € “ d f l ‘ € – ˜ — ym p – p “ d— s y— y ˆ — “ p y “ — – ˆ „ e ˜ ™ j — – p € – ˜ d “ ˆ “ ˆ ˜ € ™ ˜‘ m — ‘ „ € € – • — y „ ‘ f — € y – mm “ € “ ƒ † € – l € – • ‘ „ p – ˆ ˜ ‘ “•“ — – d “ ˜ ™ — – — y – € s j „ ‘ – ˆ € – ˜ — — – p ™ ‘ ˜ ‘ – • y – d — – ep „ € d d “ — – mm – ˜ – ˆ – y y y ‘ ‘ „ p – € ˜ — – m f ˜ ‘ “ ˆ ‘ – d – i † — € – ˜ — ym p u y – ˆ — – m € –  ˜ ‘ – ˆ  ˆ — – m s “ d “ p ˜‘ „ — — – ˜—m s “ s „ € d — –m k ˆ „ j – ˆ ‘ „ ˜ “ —m ˜ y m € y— — – ™ — “ s — – ˆ „ e ˜ ™ j — –m f – € ˜ ‘ „ p € “ ƒ † — ™  ˜ ‘ – ˆ – € ˜ — “ d – ‘ ˜‘ – • y – d — € – ˜— ym p u y – ˆ — –m f — € – ˜ — y m p — – ˆ ‘ „  ˜ p – — € – ˜ ‘  m ˜ ‘ – j – m m – € y ˜ “ ‘ — “ d ˜ – j € – d – ‘  y y — ‘ “ – j – d ˜ – ˆ – ep „ € d d “ – ‘ y € y — — – ™ — “ s ˜ € “ d ym d 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y „ d f — “ p m – ˜ ‘ y — ‘ “ ’   © ©       © ©  !©  ©   ©  ©  ©  ©  ©© © 32 1 ""¨¨''§0¤0§¤¨!¤¨¨''¨¨''¤¤''(( ¥¥££ "¨'§¤§¨"¨¨§§¨¨'(¨¨¥¤'£¤'( ¥£    © † — – ˜—m s “ s „ € d — – ˆ „ e ˜ ™ j u y “ ˜ ‘ – j – € “ € ˜‘ „ p m “ j ˜ ‘ – ˜ € „ d j „ p – — l ‘ € – ˜— ym p – p “ d— s y— – ˆ — – ˜‘ “ ˜— u– — – ˆ „ e ˜ ™ j — –m ¤ „ — “ p ‘ ” † ¦ ¥ ¦ ¦ £  £ + + ++ ++ ++ ++ + ++ + + + + +++ +++ + + + + + + + + + + ++ + + ++ ++ ++ ++ + ++ + + + + +++ +++ + + + + + + + + + + ++ ¦ £ ¢  ©    + + ++ ++ ++ ++ + ++ + + + + +++ +++ + + + + + + + + + + ++ + + ++ ++ ++ ++ + ++ + + + + +++ +++ + + + + + + + + + + ++ † l ‘ € – ˜— ym p – p “ d — s y— y ˆ – y y ˜ “ j ™m s „ € d “ m ˜ “ … ˜ y „ ˜ ˆ ‘ „ d— – € € „ p  y y – p f – € d „ € d ˜ — –  y m  y y ‘ „  — ‘ – j  ˆ – m y – — – ‘ y – p “ d — – — y „ — ‘ y — ‘ “ ˆ ‘ y p “ ep —  ‘ ™ ˆ — € – ˜ — y m p u y – ˆ — “ p – p — ‘ “ ˆ ‘ – s p ‘ „ ˆ “ ‘ w † € – ˜— ym p – € ˜ y “ m € y „ d ˜ ‘ – j – — € – •‘ ˜ – f ‘ „ — ‘ – j ˆ – € ˜ y “ – ‘ y € y—  ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ ¡¢  Y! ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡  —™p „€—–—˜“— y—m˜p––sm „— ‘—“–ˆˆ —–ym smpj‘– ——‘˜––©ms–„‚˜© ‘ˆ—–––—˜€™‘s€“dj•–„€y‘— –—m–m–˜—„€„‘m™“ˆ ˜‘„— —–™p „——“ — ‘ „  ˜ “ ˜ „ ‘ — – n † †  – € y l  “ m — ‘ “ ˆ – ˜ „ € ˆ  ˜ ‘ “ y ˜ — – — € – ˜ — ym p – m — ‘ „ ˜ „ ‘ f – m d j – u – € “ ƒ † – m m –  € „ l ™ ˜ “ p ‘ „  — ‘ – j  ˆ – y y “ ep – ˆ ™ ˜m “ ˆ „ j – y y “ ep € y — – p ‘ – y y ™ € … – ‘ y© ˆ ˜ – f – y y  € ™ j y ‘ ‘ „  — ‘ – j ˆ – ˜ y „ ˜ € y — ˆ € “ ˆ ‘ “ ˜ — ‘ „  ˜ “  • ™ ˆ – ‘ y© ˆ ˜ – – € ˜ ‘ – p ‘ y© ˆ f – ™ ˜ „ ‘ € – ˜ — ym p y ˆ ™ ˜m s “ s „ € d – ‘ y© ˆ ™ — „ d j „ p f –m k ˆ „ j ‘ y € “ d ™ € ™ ‘ ™ l ™ — „ d d y — p ‘ „ ˆ ˜— – € – ˜— ym p ‘ y f ˜‘ – j –m “ ‘ v „ j —– ˆ –m s j–— ‘–© m € y— —–p ‘– yy™€… – ˆ € y–†˜™p€–™•ˆ‘—y‘„€p“ d˜ y—s–™€˜˜‘˜–“—© m™€–dˆ–€—–fm—s–m—“—„jd„—‘™˜˜mmy“jˆ —™‘˜„“p˜ y—s˜y€˜s—€ˆ˜˜“—– ˆ—–m‘„€my–z— †—–™–€™p‘‘™“l€“˜•‘„€—y—––m ™˜‘–‘„ ˆ–—€–˜‘m ––pyy€ y––—mk€e“˜d„ d—–e™©˜m‘—–‘—™„€—d“–…€ —f y—–„‘‘‘f–—m—–—y€“„ ll — ‘ „  ˜ y s € ˜ —  ˆ — – ˆ ‘ „ m – — — – ™ € ™ ‘ ™ l ˜ ‘ „ — — – ™ ‘ ‘ „ ˆ — – m – y y – — k e ˜ „ d  e© m — ‘ „ —  “ … — y „ ‘ f — – y y  € ˜™“j© ˆy‘—–—d˜y˜su€y˜–˜“ˆ —€“–md €—y–z™—†€—™m–˜p€“„€l“™p˜“–p€˜—˜˜y‘s–•€˜y˜–“d ——––m™ ˜‘–‘„—ˆ– y—y–m€–™ yjyy‘p——˜‘y„s€€™˜ˆ˜“— ‘—„–pm —‚ y—„˜y s€˜ † ‘ „ ˜ ˜ € “ d “ m  ™ p „ — — “ – y y € ˜ ™ j “ € “ d –m k ˆ „ j y ˆ – m s j – — ‘ –© m — € „ m “ – ˜ ‘ – — ™ € d – € f — ™ e p € – ep – € — € – ˜ — y m p – ˆ m “ ˜ „ ˜ – € s j „ ‘ – m ™ ‘ ‘ „ ˆ ˜‘ “ ˜ T † – ™ p „ — — “ ˜— –  ym  y y –m l k € “ m ˜ – f € – ˜— ym p – p € y „ d — – ˜ ‘ – ‘ ˜ € – d — ‘ „— ‘ – j ˆ — – ˆ –m s j – — ‘ – — y „ — –m f € – ˜— ym p y “ ™ p „ — — “ – y y € ˜ ™ j “ € “ d –m k ˆ „ j –m – ˜ „ ‘ ‘ w † — € – ˜— ym p u y “ — ™ p „ — — “ — – ˜—m s “ s p„–€—d “—–m mk— ˆ‘“„ˆj ——––™m˜‘€––˜—‘™–€ —d™€—d––mm€–p€ y˜„‘d– ‘—‘––€€“d—–—€–p–™€‘˜ d—–“mem–pp –—p‘„—˜‘y“„ˆ“—–—™y—„m‘˜ y—–m—m–‘y„y˜u“y˜„“ ‘f —† –n‘„˜ C1 D1 = {5, 6, 7, 8, 9} N1 = 5 ¥ ¥ mod ∈ M odalitesd C1 d πk µkd F reqskd(mod) d σkd Ck θk −−→ −− F reqs µ σ Ck S k Rk θ = {θk |k ∈ [1, K ]} P = {Ck |k ∈ [1, K ]} K θk  D H E @ E  ¤  ¦¢  © † ™ € ™ ˆ— ‘ „ p – j km s „ € d –m € y — – ˜‘ “ — — – € ™ ˜‘ ‘ „ ˜ “ j € „ … ‘ m – ˆ € – ˜ € „ d d “ – j j ˆ ‘ “ y y ˜ ‘ – • y – d — ‘ „ ‘ – ˜ s „ — y „ ‘ – y y — ˜ “ ˜ m y — ™ € — – m – y y – € ˜  d “ ep – p – ˆ ‘ „  ˜ p – — – € k  ‘ € – ˆ “ m — ‘ “ ˆ — ‘ „ € € – • — y „ ‘ — € „m “ f – ™  € ™ • — “ d ˜— – ‘ — ‘ „ — ‘ – j ˆ — – ˆ – p ‘ “ ˆ ‘ – d ™ ˆ ‘ ˆ – — k e ˜ „ d e – ˜ ˜ – p –mm – y y “ m € y „ d — – ™ ‘ ‘ „ ˆ – ˆ – — “ s – ‘ y – ™ ˜ ‘ „ €… ‘ „ p ˜— – – ˆ „ e ˜ ™ j “ m – € “ € ˜‘ „ p y “  z † — – ™ ˜ • ™ ˜‘ „ — — –mm – p € ˜ “ j — ‘ „ ˜ “ € ™ d „ — –m ˜ – — ˜ – s „ — – ˆ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d— – m – ˆ — ‘ „— ‘ – j ˆ – ˆ – € s j „ ‘ –m € y „ d y ˆ y –m y “ — – € ˜ k j “ € “ d – ˆ — ‘ „ j – ˜— — – p ™ ‘ –m k ˆ „ j y “ – • y „ ‘ –m € “ p – y y— — “ m p – j e ˜ € „ lm “ m – y y – ˆ d “ € — ym d — € „m “ ˜— – – j e ˜ € „ lm “ m f ‘ ‘ ‡ † — – € ˜ y “ — – ˆ ˜ ‘ – j j “ ˆ ‘ – d ™ ˆ ‘ – ™ —  € ™ ˜ p “ € “ p ˜ — – ‘ „  — ‘ – j  ˆ – y y “ ep – y y —  y d f — ˜ – s „ — – ˆ ‘ „  ˜ d  € p — – ˆ – ˆ m “ ‘  l  € „ – p “ d — – m – ˆ – p “ d — – — y „ — ‘ y — ‘ “ ˆ – m l ‘ “ ˜ p – € € – d  e – m l k € – ˆ – j € „ … — y „ — € – ˜ — y m p – y y “ ep ˜ ‘ “ ˜ ‘ – — ™ € d – € ‘ – –m s— ‘ – e ™ € d j „ p – € „ ™ e ˜ – ‘ y –m k ˆ „ j y ˆ € – • € ™ ˆ – ˆ — — y “ ˜ – j € – d — y „ ‘ –mm ‡ † — – ™ ‘ ‘ „ ˆ — –m ˜ ‘ “ —  € ™ ˜ p “ € “ p — ˜ y s € ˜ ˜ “ ˆ — – d ˜ — ˜ ‘ – € ™  ˆ € – € ™ ˆ  — ‘ „ p – ˆ ™ ˜m s — — „ d “ m ˜ ‘ – j – m “ l ™ – ‘ ‘ „ ˆ — y „ ‘ – — k e ˜ „ d  e – ˜ ˜ – i † ™ € d „ € d d “ — ym d –m ˜— –  ym  y y — ‘ „— ‘ – j ˆ — – ˆ –m s j – — ‘ – — y „ — –m € – ˜ — y m p – y y “ e p € y „ d € – m s  p – ˆ ˜ – j € – d — y „ ‘ “ m – p ˜ ‘ – j j „ p — ‘ „ € € – • — y „ ‘ f ˆ € „ s “ ˆ ˜ y „t † — – l “ ˜ ‘ “•“ u y – € s j „ ‘ – ˆ – ˜ € „ d d “ – p ‘ “ ˆ ‘ – d ™ ˆ ‘ ˆ – — k e ˜ „ d  e – ˜ ˜ – p —  “ j f o† o ‘ „  ˜ p – — “ m — ‘ “ ˆ ™ ˜ ‘ – — ™ € d f — ‘ „  — ‘ – j ˆ – € ˜‘ – — –m s— — „ d — ‘ „ ˜ “ m ™ € € „ p — –m – ˜ d j „ p ‘ – ˜ ‘ – j –m “ l ™ ˜ ‘ “ ‘ – € d – y y — — “ m p –m k ˆ „ j –m – y y – ep  € — ‘  „ j — € „ m “ ˜ — – – m k ˆ „ j – n † ‘ „  — ‘ – j  ˆ – y y “ ep € y — — – ˜ ‘ “ ˆ ‘ – d ™ ˆ ‘  — ‘ „  ˜ y s  € ˜ —  ˆ O(M 2 )   ©  © © © ©   ! © M O(M ) © © ©  ! ¦ ¡ ¢ ©¨ ¡¨ ¢  © ‚ ˜ y— – j j „ p — € – ˜— ym p u y “ — ™ p „ — — “ — ˜ – s „ — – ˆ –m s j – — ‘ – m – ˆ ‘ „ ˜ p ‘ „ … ‘ – — ™ —m “  ˜ ‘ – ˜ y — ‘ – ˜ ‘ „ — – l ‘ “ m ™ j y ˆ — – ˜ ‘ “ — „ d j „ p — – ˆ — – € ˜ k j “ € “ d — – n – y y€™ j y‘ — ˜ – –mm – € „ l ™ ˜ “ p — ˜ – –mm – € „ l ™ ˜ “ p — Dk dist(xid , µkd ) = 0 1 (xid − µkd )2 θk d d d xid = µkd xid = µkd d=1 →µ Dist(−i , −k ) = x→ dist(xid , µkd ) M ‚ – ˜‘ “• y— – p ‘ “ ˜ —  ˆ “ m – ˆ ‘ „  ˜ “ —  j  ‘  j “ m – ˆ — ‘ – — y “ f – ep „ € d — y m d – m ˜ — – – ˆ  „ € ˜ ‘ – p – m ˜ ‘ „ ˆ € – ˜ — y m p y ˆ — – € s j – j u y “ ™ p „ — — “ – ˜ y — ‘ – ˜— – ˜ – s „ – y y “ e i † — € – ˜— ym p — – ˆ u y “  ˜ ‘ — – ˆ  „ € ˜‘ – p – ˆ ˜ ‘ “• € – — – — “ s “ m – ˆ — ˜ – s „ ˜ ‘ – j – € „ ˜ “ ™m “ ˆ € „ s “ ˆ – ‘ ‘ „ ˜ p –m ™ — ‘ w † — ‘ “ – j – d ˜ – ˆ – ˆ „ e ˜ ™ j – ‘ y ˜ ‘ “ —m ˜ y ‘ – — – ™ ‘ ‘ „ ˆ — – ˆ € ˜ € “ d ˜‘ – j – € „ ˜ “ ™m “ ˜ “ … – — –m k ˆ „ j y ˆ ‘ „ ˜ “ —m “  ˜ ‘ n † y‘ ‘ „ p ˜ — – — ™ e p € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m – y y — ‘ „ — „ d d y — — y „ ‘ f — d j – ˜ € –  j – € d ‘ y — ‘ “ ’  Dk  K −i → x  CI ¥ G  £ Ck 32 1 − → µk Ck K © © A G D  H E @  H I @ H EH D † — – € ˜ y “ — – ˆ € – ˜— ym p – p € – y l ‘ ˜— ˆ  ˜ £ y— “ m – p – y y — y d ‘ „ — ‘ – j ˆ –m y – — – ‘ y € y— – ‘ ™ ˆ f – ™ p „ — — “ ˜— –  ym  y y –m l k € “ m ˜ — –  ™p „——“ – yy€˜™ j‘“€“‘d– –mk ˆ „ j –m – ˜‘– — ™ € d– € — ‘ „  — ‘ – j  ˆ u y – ˆ — – p € y — — – ep „ € d ˜ ‘ „ — — ˜ –  s „ — – m – y y — y d f € – ˜— ym p y “ ™ p „ — — “ ‘ „ ˜ d € p — – ˆ – ˆ – p “ d — – — y „ — –m – ˜‘ – — ™ € d – €  †  ¦ ¥  ¢ ¥ S1 = {d1 , d2 } ¥ θ1 = (π1 = 0.33, µ11 = 0.9, σ11 = 0.05, µ12 = 0.4, σ12 = 0.1) C1 R1 = {d1 ∈ [0.8, 1]} © † –m d j– u– m € y— — ‘ „˜“ ˜ „ d1 C1  !† Y ¥ ¥ Y "      Y  $ #! o "o Y  R1 d2 ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ ¡¢ ! ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡ –mm – € „ l ™ ˜ “ p — d – y y€™ j y‘ — F reqskd(xid ) xid −µkd σkd d 2 © √1 e 2πσkd −1 2 ! ©© © P (xid |θkd ) = Sk = S ∀ k ∈ [1, K ] © ‚ “ ‘ „ f – ™ y ˜ p – – ˜— – ‘ — ˜ y s € ˜ ˜ “ ˆ ‘ „ ˜ p –m ™ — – ‘ y p y “ y y — y d – p “ d— – m – ˆ ‘ „ ˜ d € p — – ˆ – ˆ — ‘ „ — ‘ – j ˆ — – ˆ –m s j – — ‘ – m € y — — ‘ ™ ˆ ˜‘ „ — — € – ˜ — ym p — –m – y y ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ ™ f y “ – • ‘ – p © d∈Sk P (xid |θkd )  → P (−i |θk ) = x !   ‚ € – ˜— ym p ˜ y „ ˜ ˜ – ˜ – s „ ˜ y „ ˜ € y „ d — ˜‘ “• y— —m y pm “ p — –m € – y ˜ p – – ˜ – –m k ˆ „ j y ˆ — – € ˜ k j “ € “ d — –m — ™ u € – — „ d d y — – ˜—— ‘ „ p – p ‘ “ € ™ d — ‡ ˆ – d “ ˜ ™ m f ˜‘ – j –mm – j € „ … — ym ƒ † — € – ˜— ym p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ — ™ ˜m s “ s „ € d — –mm – • y „ ‘ — – ˆ ‘ „ ˜ p ‘ „ … ‘ – –m k ˆ „ j y ˆ — – € ˜ k j “ € “ d — –m € y „ – € ˜ ˜ – j ‚ ‘ „ ˜ “ — j u“ q q – d “ ˜ ™ † –m k ˆ „ j y ˆ — ˜ ‘ “ € y „ p — – € ˜ k j “ € “ d — – ˆ ‘ „ ˜ p ‘ „ … ‘ – — € – ˜ — ym p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ — ™ ˜m s “ s „ € d — –m € –m y pm “ p ‚ – p ‘ “ € ™ d— ‡ ‡ – d “ ˜ ™ † ‚ – d “ ˜ ™ – y y “ ep – m k ˆ „ j – m € – —  j  ˜ d „ ˆ ˜ ‘ “ ˜ ˜ – j € – d — – — “ e d u y – ˆ – ˆ ‘ „  — — – p p y — – ‘ y ‘ – – ˜— — ‘ „ p q ‡ – ˆ „ e ˜ ™ j “ m f m – d d “ € € y „ ƒ † † ’ — – u– ‘ ‘ “ ‘ – — –m s ‘ „ d— ˆ ˜ ‘ „ — — k € d “  p — – ™ ˜‘ – — ™ € d — ‘ „  ˜ “ y y ™ u y “ ™ ‘ – j ˜ ‘ “ “ — – y y  € „ ™ e ˜ — ‘ „  ˜ “ € ˜ — ‘ „ j ™ ˆ — – n † — ‘ „  — ‘ – j  ˆ — – ˆ – p ‘ “ ˆ ‘ – d ™ ˆ ‘ ˆ – — k e ˜ „ d  e m – ˆ – ˜ d j „ p ˜ ‘ “ ‘ – ˜ ‘ – —  “ j f o† o ‘ „  ˜ p – — “ m — ‘ “ ˆ – ™ ˜ ‘ – — ™ € d ¢ f† m “ ˜ – € – ˜ — d j – ’¡ q ‡ – ˆ „ e ˜ ™ j “ m ˜ ‘ “ — m ˜ y ‘ – – ˜ y — ‘ – ˜ “ … – — – m k ˆ „ j y ˆ ‘ „  ˜ “ —  j ˜ d „ n C I ¥ G  A G D  H E @  H H E ¡ 8 Ck   −i → x ©©  ©   ©  ©  ©  © θ θ θ    $ $ # © £ £ ¨ ¢ † $ – j e ˜ € „‘l–m“‘©ym €“ d™–p™„˜—‘—–“—™–€ydy˜€—˜–™ fj“€—“€d–˜–—mykmˆp„–jˆ ™y‘ˆ‘„‘ˆ„˜–“€—sj“„˜‘ ‘‘©yˆ €–y€„ydˆ™˜p– „f€ d —“–m™f‘˜‘‘„–ˆj––mˆ“ ‘–mvs j–— m D K –mm – € „ l ™ ˜ “ p − µkd )2 ∀d ∀d ∀ mod ∈ M odalitesd – y y€™ j y‘ xid – y y€™ j y‘ |{i ∈ Dk |xid = mod}| ∀d Nk i∈Dk (xid i∈Dk Nk N o F reqskd(mod) = Nk σkd = Nk µkd = πk = ! ¤ £ ¥ ¢ § £ ¥ ¡¢ ©¨ ¡ ¦ © § ©¨ ¡ † m “  ˜ ‘ – y y € ˜ ™ j “ € “ d –m k ˆ „ j  ! D ¨©D%  ¨D   ¨D¨ D HH G  D WH D  ¨DV©  –y y€™„j˜ y‘      „˜   m “  ˜ ‘ –m k ˆ „ j y ˆ — – € ˜ k j “ € “ d — – ˆ ‘ „ ˜ “ —m “  ˜ ‘r  ¨D — ˜ –  s „ — – ˆ – m s j – — ‘ –© m  € – ˜ y „  “   ¨D¨ D xid Dc c ¢ Nk 2 i∈Dk (xid −µkd ) Nk P i∈Dk ¢ ¡ −i → x „ € ™ j y‘ € – ˜ — ym p y ˆ θ = (θ1 , ..., θK ) →→ θk = (πk , −k , − , F reqsk ) µ σk mod ∈ M odalitesd F reqskd(mod) = |{i∈Dk |xid =mod}| Nk σkd = µkd = k=1 K π k = Nk N d=1 M d P ¡ → push(Dc , −i ) x ¨DV©  i=1 N minDist = +∞ c=0 k=1 K →µ Dist(−i , −k ) < minDist x→ →µ minDist = Dist(−i , −k ) x→ c=k          © ¨ D    X D R V ©D F ©E¨H & „˜ ˜ – s „ m – ˆ – ep „ € d — y m d – m ˜ ‘ “ € y „ p € – ˜ — y m p – m „˜ — – ep „ € d — y m d — – m — – ˆ  „ € ˜ ‘ – p u y “ — ˜ – s „ — – ˆ ‘ „  ˜ “  p „ — — # – ˆ — ˜– s „ — – ˆ –m s j–— ‘– m  j € “ d ˜ ‘ – j – € „ ˜ “ ™m “ ™ ‘ ‘ „ ˜ p –m ™ — € – ˜ — ym p y ˆ m “  ˜ ‘ – ˆ  „ € ˜‘ – p „˜ u y “  ˜ ‘ — – ˆ  „ € ˜‘ – p — – ˆ – € „ ˜ “ ™m “ ‘ „ ˜ p –m ™ z ! † — ™ ep € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m ˜ – f — ˜ – s „ ˆ – m s j – — ‘ – ‘ y € – —m “  ˜ ‘r ¢ © −i → x ¡ ¢ ¢ ¡ ¢ © ¡¢ ¡ D Ck K ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ k=1 K − = Random(D) → µk → → D = (− , ..., − ) x1 xN ¡  "! ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡ † ˜ y “ … ™ ˆ € “ d   – ™ u  ˜ — – – m k ˆ „ j y ˆ ‘ „  ˜ “ —  j ˜ d „© m – ˆ – ˜ y — € y „ d “ m € y „ d – p ‘ “ m s j – —  “ € •  l „ m “ m – ˆ — – • — — – p p y — — € y – m “• u y – ˆ – € ˜ ‘ – m “ j ‘ j ˜ € “ p ™© m ˜ ‘ “ —  p ™ € d f – ˆ € y – m “ • “ n † – ˜ y — “ m — ‘ “ ˆ  s  Y Y  f† m “ ˜ – € – mm ˆ ‘ “ i l ‘ € – ˜ — ym ¢ – p “ d — s y % m “ p  ˜ —  ˜ “ ˜ % % !¢ %– j e—˜‘„€„jlmj“©„m‘€—“yd„–‘™–jyyy—™–€j˜e—˜–€–„€lm€“p©™mˆ— ‘–“ˆˆ—–‘€„•‘y–¤• —‘y–† —„–y‘—‘––jy˜ sy™„˜–™—ˆ“˜„“e˜ym˜yy™—j–™m€m“–—mp–fm˜€‘ˆy–‘—j„‘–dm„—“–˜‘€“€v—„pm˜˜y– ‘ ‘—„˜—“–˜ˆ— l—–‘p€‘––˜y—yy™mp— ‘y„ˆp —––€mˆ$“p–€–˜m d— “‘e“pˆ y–ˆyy–m—“—˜“‘mp– j—“€d™ d˜u‘–“˜–™©˜‘€“–dep“„m —d‘d““ˆ –—j„k€€u–y•– ˆ— y–„˜‘˜–fi– yy˜ ¢ δ ¡ → → Dk = {−i |ArgmaxK P (−i |θj ) = k} x x j =1 ©  © ©  † – € ˜ y “ m ‘ „  ˜ “ € ™ ˜ – ‘ y ˆ — “ d ˜ ‘ – l ‘ “ ep – ‘ — € – ˜ — ym p u y “ — ™  p „ — — “ — ˜ – s „ ˆ — –m s j – — ‘ – — –m – y y — € „m – ˜ € € “ — – ˆ „ e ˜ ™ j “ m f ˜ ˆ ˜‘ – j – € ˜ y # † – € ˜ y “ m ‘ „ ˜ “ € ™ ˜ – ‘ y ˆ ˜ ‘ – € ™  ˆ — “ d ˜ — – ‘ ˜ – s „ – y y “ ep – ˆ – m s “ s „ € d — y m d – m € – ˜ — y m p – m – y y — € „ m – m k ˆ „ j y ˆ ‘ „  ˜ “ —  j ˜ d „ m € – d d „ ˜ — – ˜ —  — ‘ „ p f — ‘ “ – j – d ˜ – ˆ f – m s — — „ d ˜ € € “ ˆ – € k ˜ € p – € ˜ y “ ‘ ” ©  i∈D © © !© © LL(θ |D) = Dk → log P (−i |θ ) x © © † … ˜ — „ d – y y — € „m – ˜ € € “ — – ˆ „ e ˜ ™ j “ m f ˜ ˆ ˜‘ – j – € ˜ ˜‘– j– ˜p€˜— € y „ d y # † – € ˜ y “ m ‘ „ ˜ “ € ™ ˜ – ‘ y ˆ y – d – € „m ™ j “ — — – ™ ‘ ‘ „ ˆ u y“ ˜€ „ d d“ € €“ d –mk ˆ „ j y ˆ – p ‘ “ m s j – — “ € • l „m “ m – y y – p y y — y — – ™ € ™ ˜ ˜ ‘ „ — — – d “ ˜ ™ u y – ˆ — – p f ˜‘ – j – y y— — “ m i δ  © ©  © ∀d –mm – € „ l ™ ˜ “ p − → → P (θk |−i ) × (xid − µkd )2 x ∀d − → i∈D P (θk | xi ) ∀ mod ∈ M odalitesd – y y€™ j y‘ i∈D – y y€™ j y‘ → P (θk |−i ) x {i∈D |xid =mod} P (θk | xi ) i∈D i∈D 1 N ©  LL(θ |D) LL(θ |D)t+1 − LL(θ |D)t < δ i∈D F reqskd(mod) = σkd = → P (θk |−i ) × xid x ∀d − → i∈D P (θk | xi ) µkd = πk = → P (θk |−i ) x ‚ ˜ y— – j j „ p € – ˜— ym p ––y˜y€““ edpd“€©yˆ„ d—™˜–mmsk ˆ“ s„ „j€ dyˆ—–—m–€—˜–k™j“€€“–d—„—d–md€yy—„ –˜–—€˜—˜‘–„jp ‘„˜–˜f“——€–j˜— uy“mpquy–“ˆ —–˜–d“s˜„™© —m –—ˆ y–ƒp ‘“ ‘ u   †  Ck θ → πk × P (−i |θk ) x → P (θk |−i ) = x −i |θ ) → P (x → P (−i |θ ) = x k =1 K → πk × P (−i |θk ) x ! ¤ £ ¥ ¢ § £ ¥ ¡¢ ©¨ ¡ ¦ © § ©¨ ¡ † ™ — j ˜ d „ – y y € ˜ ™ j “ € “ d –m k ˆ D„HjV ! ¨DD ¨©D%  ¨D¨D¨ D  Dk Dc ¢ θ = (θ1 , ..., θK ) –ˆ ! dif f = LL(θ |D) − LL change = 0 k=1 K i ∈ Dk → maxP = P (−i |θk ) x c=1 K → c=k P (−i , θc ) > maxP x −i , θc ) → maxP = P ( x → −i → move(−i , Dk , Dc ) x x change = 1 ¨DV©  €–˜p– “™€ ¡    HH G      ˆ ‘ “„ ˜   ¢ „˜ ˜  € € “© ˆ — – € k ˜ € i    ¨D¨ D    ¨D¨ D HH G  D WH D ¡ i∈D P P i∈D i∈D → P (θk |− ) xi  P F reqskd (mod) = {i∈D |xid =mod} mod ∈ M odalitesd P σkd = i∈D P d ∈ Sk i∈D K → P (θk |− ) xi → P (θk |− )×(xid −µkd )2 xi P − → i∈D P (θk | xi ) → P (θk |− ) xi ¨DV© – yy€™ j y‘ HHG     „˜  ‘ „ ˜ “ — j u“ q    ¨D¨ D ©   µkd = → P (θk |− )×xid xi → P (θk |−i ) x ¢ d k=1 1 πk = N ¡  i=1 N k=1 K → P (θk |−i ) x † ‘ „˜“ y y ™ m ‘ „m–— €–m ypm“ p „˜ „˜ –p ‘“€™ d— ‡ — – ™ ‘ ‘ „ ˆ u y“ ˜€ „ d d“ € €“ d ˜‘“ € y „ p –mk ˆ „ j y ˆ – p ‘“m s j– —“ € • l „m ˆ ‘“ € – ˜ — ym p – ˆ ‘ „  ˜ “ ˜ p – “ ˆ ™ l ‘ “ ep ˜ ‘ „ — ˜ – s „ — ‘ “ ˜ € – p  — ˜ ‘ “ j € d u – ‘ – ™ m „ „ s — – • ˜ y p ™ — ‘ „ p — – p ‘ “ m s j – — “ € • l „m u y – ˆ – € ˜‘ – – p ‘ – € ™  ˆ † — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ “ m s j – — “ € • l „m “ m – ˆ — – • — – p p y— — € y –m “• — –m € y— m y – — –m ˜ – f — € – ˜— ym p – ˆ ™ — „ d j „ p — – ™ ‘ ‘ „ ˆ — – p – ˆ m “  ˜ ‘ – y y € ˜ ™ j“ €“ d –mk ˆ „ j ‘ y f — ˜– s „ ˆ –m s j–— ‘– ‘ y ¢ ¢ ¢        ! ¡ !© change  © θ = (θ1 , ..., θK ) δ ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ ¡¢ ¢ ¡ dif f = +∞ change = 1 dif f > δ LL = LL(θ |D) K  DH V ¡ ¡ D R V! ©D F ©E¨H & → → D = (− , ..., − ) x1 xN € – — j ˜ d w ¡  ! ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡ ‚ — LL(θkd |Ck , d) > α × LL(θUc |Ck , d) € – ˜— ym p –m € y „ d – ˜‘ – ‘ ˜ € – d ˜— – –mm – € „ l ™ ˜ “ p ‘ „— ‘ – j ˆ “ m – y y € – — „ d –“ˆ• ˜u–y– ˆ —–m – yy ˜––˜‘“˜— ‘„p – ‘ y –€ y€ˆy„–€€˜™‘d©yˆ— –—m€sy—„—„yd„˜˜˜——–– m f —–•˜“ l™ ‘ ˜‘„–—j—j€ y„–im d LL(θkd |Ck , d) LL(θUc |Ck , d) LL(θkd |Ck , d) = 0<α<1 Ck i∈Dk log F rekskd (xid ) ‚ “ ‘ „ f — –mm – € „ l ™ ˜ “ p — ‘ „— ‘ – j ˆ — –m € y— m “  j „ ‘ ˜m y j –m k ˆ „ j – € ˜ „ ‘ € y „ ƒ LL(θUc |Ck , d) = −Nk × log |Categoriesd | 1 |Categoriesd |  P (xid |θUc ) = – j j „ p – ‘ ™ ˆ ˜— – –mm – € „ l ™ ˜ “ p ‘ „— ‘ – j ˆ – ‘ y € y— d ‚ p ‘„ˆ “ ‘ w ‚ ˜ y — – j € „ … ‘ y ‘ „  ˜ y s € ˜ —  ˆ – ‘ ” †  ‚ — € – ˜— ym p ‘ y € y „ d – ˜ ‘ – ‘ ˜ € – d – j j „ p – ™ € ™ ˆ — ‘ „ p ˜— – – y y € ™ j y‘ ‘ „— ‘ – j ˆ – ‘ y p ‘ „ ˆ ˜ – f f ˜‘– yy™— ‘„p €“ ƒ – y y€™ j y‘ d  θ Uc LL(θkd |Ck , d) > 0 Ck LL(θUn |Ck , d) = 0 P (xid |θUn ) = 1 ∀ i ∈ D ∀ d Y ‚ “ ‘ „ f ˜ – – € ˜‘ – — – ™ —m “ j € „ ‘ ˜‘ „ — — – y y  € ™ j y ‘ — ‘ „  — ‘ – j ˆ — – m € y — — ˜ – s „ — – ˆ — € y – m “• — – m – y y – — k e ˜ „ d  e m ˜ “ … — ‘ „ •“ — y „ ‘ – j j „ p f – y y  € ™ j y ‘ ‘ „  — ‘ – j ˆ – ‘ y € y — – j € „ … ‘ y – m k ˆ „ j ‘ y ˆ — “ p – m — ‘ “ ’ †  i∈Dk θ Un θ Ck ©© LL(θ |Ck , d) = log P (xid |θ )  ‚ ˜ y— – j j „ p – ™m y pm “ p ˜— – ‘ „— ‘ – j ˆ – ‘ y ˜ – € – ˜ — ym p ‘ y € y— –m k ˆ „ j ‘ y ˆ – p ‘ “ m s j – — “ € • “ n † € – ˜— ym p –m € y „ d – ˜ ‘ – ‘ ˜ € – d ‘ „ ‘ – j j „ p – ™ € ™ ˆ— ‘ „ p ˜— – ‘ „— ‘ – j ˆ “ m — € „m “ f –m k ˆ „ j – € ˜ „ ‘ – ˆ – mm – p – y y – ˜ ‘ “ ˜ € „ d j — ym d ˜ — – – j € „ … ‘ y – m k ˆ „ j ‘ y ˆ – p ‘ “ m s j – —  “ € • “ m  — f  — ‘ # † – j € „ … ‘ y – m k ˆ „ j ‘ y ˆ – mm – p ˜ € „ d d “ € € “ d m “  j „ ‘ ˜m y j y „ ‘ –  — — y “ l – m k ˆ „ j – € ˜ „ ‘ – ˆ – p ‘ “ m s j – —  “ € • “ m ‘ „  — ‘ – j  ˆ – y y “ ep € y — € – € “ d j „ p – ˜ —  — ‘ „ p “ m – p € y „ d ‘ „  ˜ y m „ — – ‘ ” † € – ˜ — y m p – y y “ ep € y „ d — – ˜ ‘ – ‘  ˜ € – d — y m d — – m — ‘ „  — ‘ – j ˆ — – ˆ –m s j – — ‘ – — y „ — ‘ y € – ‘ ‘ „ ˜ p –m ™ — ˜ – ‘ „m — ym d € –mm “ ˜‘ “ ˆ ‘ – d – p — ‘ „ • y „ d — y „ † € – ˜— ym p y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ m – ˆ ‘ „ ˜ “ — € ™ ˜ p “ € “ p “ m € y „ d ‘ „— ‘ – j ˆ “ m – ˆ ˜ ‘ “ ˜ € „ d j — ˆ „ d ‘ y – € y ˆ ‘ “• ˆ € “ ˆ ‘ “ ˜— ‘ „ ˜ “  • ™ ˆ –m s “ … – ‘ y y y — € „m “ ‘ „ — ‘ – j ˆ “ m – ˆ –m s “ … — ˆ „ d ‘ y – € y ˆ ‘ “• – ™ • –m ™ ˆ € “ ˆ ‘ “ ˜— ‘ „ ˜ “  • ™ ˆ – ‘ y f – y y € ™ j y‘ ‘ „— ‘ – j ˆ – ‘ y € y „ d –m d j – u– € “ ƒ † € – ˜ — y m p – y y “ ep € y „ d ‘ „  — ‘ – j  ˆ – y y “ e p – ˆ – p ‘ “ ˜ € „ d j  m – y y k — ‘  € ˜ ‘  – € k  ‘ “ j – ˆ – € k ˆ ‘ „ d –m k ˆ „ j –m f ˜ – – ‘ ‡ † — € – ˜— ym p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ m – ˆ ‘ „ ˜ “ ‘ j € – ˜ ™ ˆ “ m — ‘ “ ˆ – p ‘ “ ˜ € „ d j – j j “ m — “ d ˜ ‘ „ ‘ — ‘ „— ‘ – j ˆ — –m — – ˜ y „ ˜ f –m k ˆ „ j m – ˜ ‘ y ˜‘ “ —m ˜ y ‘ ‡ £ d  ©    ©   © ! ©  © ©   ©©  E A H B E E @ G D  H E  C I $! ¦¨ £ ¢ © ¨ ¨ ¡¡ ¥ ¡¢ ¢¢ ©¨ ¤ ¤ £¢   — – m ˜ – f — € – ˜ — y m p — – ˆ ‘ y p “ ep — – y y  p ™ d — ˜ ‘ „ — — € – ˜ — y m p u y “ — – ™  p „ — — “ — ‘ „  — ‘ – j  ˆ – ˆ — –m s j – — ‘ – — –m f — „ … – ˜ ˜ – p — “ j f — – j j — –m ˜ ‘ „ — — – d “ ˜ ™ — –m f –m k ˆ „ j y ˆ ‘ „ ˜ “ — j ˜ d „ m € y „ d f – ˜ y— ‘ ‡ † – j e ˜ € „ lm “ m € “ d – ‘ € y „ … ˜— – ‘ „ ˜ “ —m “  ˜ ‘ ˆ – € y ˆ ™ p „ € d –mm – • y „ ‘ “ n † — € – ˜— ym p — – ˆ — – ˜ ‘ – ‘ ˜ € – d — ym d — –m — ‘ „— ‘ – j ˆ — – ˆ ‘ „ ˜ p –m ™ — “ m € – y ˜ p – – ˆ ‘ “ f –m k ˆ „ j y ˆ ‘ „ ˜ “ —m “  ˜  ‘ m – ˆ — € „m – ™ ˜ y „ “ ˜— – – d “ ˜ ™ – ‘ y –m y – — f – y y j e ˜ € „ lm “ y “ – • ‘ y # † — ˜ y s € ˜ ˜ “ ˆ ‘ „ ˜ p –m ™ — – ˜ ˜ – p – € • y ¤ ‘ – ˜ ‘ “ ˜ ˜ – j l ‘  € – ˜ — y m p – ˆ – ˆ „ e ˜ ™ j “ m ¢ s f † m “ ˜ – € –  m m  ˆ ‘ “ i¡ j e ˜  € „ l m “ q & ‘ “ ‘ ˆ – ˆ ˆ – s j – ‘ „ ˜ p –m – % – p “ d— s % & % % — ‘ „ j j „ ‘ — y „ ‘ f – ˜ y— “ m — ‘ “ ’ † ™ € „m ™ j “ ˜ ‘ – j – ˜ ˜ – ‘ ˜— – – ˆ „ e ˜ ™ j “ m – ˆ ‘ „  ˜ y p ™ u – ˆ — d j – ˜ – m – y y — € „ m “ — – ™ l ‘ “ ep y – d ˜ ‘ “ • y „ € ˜ ‘ – — – ˆ „ e ˜ ™ j “ m – ˆ — – p ‘ “ j € „ … € – d — – m f y ‘ – ˜ – € — ‘ „ •“ — y „ ‘ – y y u  „ ep € –  j – € d – m ˜ — – i † q ‡ – ˆ „ e ˜ ™ j “ m – ˆ ‘ „  ˜ “ € ™ ˜  – y y “ ep € y „ —  j – € ˜ ˜ ‘ – • y – d — € – ˜ — y m p u y “ — ™  p „ — — “ — ˜ ‘ – ‘  ˜ € – d — ˜ y s  € ˜ ˜ “ ˆ — –m s j – — ‘ – — y „ — — – p f – € “ € ˜‘ „ p y “ ‘ – s y „ f –m k ˆ „ j y ˆ ‘ „ ˜ “ — j ˜ d „ m – ˆ l ‘ „m y “ ˜ y „ ˜ — ™ • € – — ‘ „ p ˜ ‘ „ — — ™ ‘ ‘ „ ˜ p –m ™ — — ˜ y s € ˜ ˜ “ — – p — y d f –m k ˆ „ j y ˆ ‘ „ ˜ “ —m “  ˜ ‘ ˆ – — “ e d “ m — ‘ “ ˆ ˜ ‘ – j – y y ‘ y – ™ y ˜ p – – – € ˜ ˜ y – d — € – ˜— ym p — – ˆ — ˜‘ – ‘ ˜ € – d — ˜ y s € ˜ ˜ “ — – ˆ ‘ „ ˜ p –m ™ — – ˜ ˜ – i † — ‘ „ — ‘ – j ˆ — –m € y— — € – ˜— ym p — – ˆ — – € s j – j — – ˆ ‘ „ ˜ y s € ˜ — ˆ “ m – ˆ – j € „ … “ m € “ d – ™ ‘ j € – ˜ ™ ˆ ˜ ‘ – j – y y k — ‘ € ˜ ‘ ˜— – € – ˜ — y m p – y y “ e p € y „ d — ‘ „  — ‘ – j  ˆ — – ˆ – p ‘ “ ˜ € „ d j  m f ˜ ‘ – j j – ˆ ™ p ™ € d ™ y y  m d u – — ‘ „ •“ m — y „ ‘ – j j „ p f – y y — y d — – ˜‘ – ‘ ˜ € – d — ‘ „ — ‘ – j ˆ — – ˆ –mm – p – y y – ˜‘ “ ˜ € „ d j — ‘ „ j “ € – — — – ˜‘ – ‘ ˜ € – d ‘ „ ‘ — ‘ „— ‘ – j ˆ — –mm – ˜ – ˆ – p ‘ – y ‘ m f ˜ ‘ “ ˆ ‘ – d – i † — € – ˜— ym p — ‘ “ ˜ € – p € y „ d — – ™ ‘ ‘ „ ˜ p –m ™ — – € ˜ ˜ ‘ – • y – d — – ˜‘ – ‘ ˜ € – d ‘ „ ‘ — ‘ „— ‘ – j ˆ — – y ym – y y — € „m “ f — “ p –m — “ d ˜— – ‘ – p  z † – € ˜ y “ m € – ˜— ym p ‘ y ˆ — – ˜ ‘ – € ™  ˆ – € ˜ ˜‘ – • y – d — – ™ ‘ ‘ „ ˜ p –m ™ — — ‘ „— ‘ – j ˆ — –m — – j j f — – ˜‘ – ‘ ˜ € – d — ‘ „ — ‘ – j ˆ – ˆ – € s j „ ‘ – j j –m ˜ ‘ „ — € – ˜— ym p — –m — y „ ˜ – y y — ‘ „ — „ d — y „ ‘ f –m k ˆ „ j – p € “ ƒ † — ˆ „ d € y –m ˜‘ – — j u“ j  y y — ‘ „— ‘ – j ˆ — –m p ‘ „ ˆ ˜ ‘ – ˜‘ „ p € – ˜— ym p y “ — – ™  p „ — — “ — ‘ „  — ‘ – j ˆ — – ˆ – m s j – — ‘ – n † € – ˜ — ym p – y y “ ep € y „ d € – € ™ ˆ  — ‘ „ p — – ˜ ‘ – ‘ ˜ € – d — ‘ „ — ‘ – j ˆ – ˆ ‘ – s j „ p – y y ˆ ‘  y y – € ˜ k j “ € “ d –m –m k ˆ „ j y “ — ‘ „ ˜ y „ “ — y „ ‘ f ‘ ‘ ‡ ©  #  © ©    !  © © !©      YY ©  © !  © © i∈D (xid 1 N Wkd Sk –mm – € „ l ™ ˜ “ p — ©© © © © © © ©  Σ2 = kd d − µkd)2  †  R R mod = Argmax{m∈M odalitesd } F reqskd (m) F reqskd (mod)−F requencesd (mod) 1−F requencesd(mod) 2 σkd Σ2 kd p – •“ f  p – •“ 1− – y y€™ j y‘ — Sk © Ck  Wkd = d ‚ — –mm – € „ l ™ ˜ “ p — ‘ „ — ‘ – j ˆ — –m € y— € – ˜— ym p y ˆ –m s “ s „ € d — ym d “ m – € „ l ™ ˜ “ p “ m – ˆ – • ˜ “ m – € – p ‘ – y y ™ € … “ m ˜ – ˜€ „ d d“ € €“ d –m“ s „m l ˜– –m“ p „m ˆ €“ ˆ ‘“ ˜— ‘ „˜“ •™ ˆ f — – y y € ™ j y‘ — ‘ „— ‘ – j ˆ — –m € y — –€˜‘– ˜€„ d d“€ y“ ˜‘– ˆ ‘„ d—–€€„p — ˆ „ d — – i † — € – ˜ — ym p — – ˆ –m s j – — ‘ – m € y — — ‘ „— ‘ – j  ˆ — – ˆ –m s j–— ‘– m – ˆ — ˆ „ d – ˆ – p€˜“ j – ‘ y –mk ˆ „ j y“ — ‘ „ ˜ y „ “ — y „ ‘ f “m– p € y „ ƒ † –m k ˆ „ j y ˆ — ˜‘ “ € y „ p — – € ˜ k j “ € “ d — – ˆ ‘ „ ˜ p ‘ „ … ‘ – ˜ ‘ – j – ˜ p – € ˆ — € – ˜— ym p — – ˆ — – ˜‘ – ‘ ˜ € – d — ‘ „ — ‘ – j ˆ — – ˆ ‘ „ ˜ p –m ™ — “ m – y ˜ p – –  y y – • ˜ “ ‘ € – ˜m “ – € ˜ y “ – ‘ y p ‘ „ ˆ — ‘ „ — „ d „ € d — y „ † ˜‘ “ ˜ € „ d j – € “ ˜ ‘ – j ™m d d y— m y pm “ p – ˆ — d j – ˜ ‘ y ˜ y ˆ ‘  y y – p f — ˜ – s „ — – ˆ — € y –m “• — – ˆ –m s j – — ‘ – m € € y „ p € “ d – ˆ – ˜— — – p ™ ‘ –mm – y y ˜ — – — € – ˜— ym p — – ˆ — – ˜ ‘ – ‘ ˜ € – d — ‘ „— ‘ – j ˆ — –m € – ‘ ‘ „ ˜ p –m ™ — € y „ d ‘ „ ˜ ym „ — – ˜ ˜ – p – ˆ ˜‘ – ‘ ™ • ‘ „ p ‘ ‘ y f — –mm – € „ l ™ ˜ “ p — ‘ „  — ‘ – j ˆ — – m € y „ d – € ˜ k j “ € “ d – p € – u – ˆ ™ ˜m y p £ ˆ “ m – € ˜ y „ f ˜ ‘ “ ˆ ‘ – d – i  © W  !    © Wkd µkd © ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ ¡¢  ©  α !! ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡ YYY   Y$ #    ! !   ™ — m ˜ y — ym d – m ˜ – y ‘ ‘ „ p — ym d – m – € k ˜ € p – m —  “ j f ¢ f† m “ ˜ –  p “ ‘ € –  x¡ ˆ „ „ em –  n ˆ – ˜ – m d j „ i ˆ – ˜ “ € l – ˜ ‘ r n i r – € k ˜ € p – m ‘ –  s y „ f ¢ f –  “ #¡ ‘ „  € – ˜ € i ‘ „  ˜ “ j € „ … ‘ r –  “ # i r # – € k ˜ € p –m € – ˜ p –m d j – u– € “ d ˜ y – d ‘ „ f m u y – p  j € “ ƒ † — – u–m d j „ p — ym d — –m — –m k ˆ „ j — –m € – —m “ ‘ ™ d – ˆ ‘ „ ¡ “ … € y –m € y— p ‘ „ ˆ ˜ ‘ – p ‘ – € ™  ˆ – — — – k ˜ € p — ˜‘ – € ™  ˆ — – n † — – u–m d j „ p — ym d — –m — –m k ˆ „ j — –m € – —m “ ‘ ™ d ˆ ‘ – ˜  y y – j € – ˜ ‘ y — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ “ m s j – — “ € • “ m – ˆ m y pm “ p y “ ˜ ‘ – ˜ y „ “  y y “ m – p € y „ d — ™ — „ d „ € d ™ ˜ ™ ˜‘ „ — – € k ˜ € p — € y –— ym ƒ † ™ — „ d „ € d –m k ˆ „ j y ˆ ‘ „ ˜ “ —m “ € ™ ‘ ™ l ˜ – ™ ˜ p  p ™ d— – € ˜ ‘ – – y y— — “ m p — j „ € d j „ p –m y “ – • ‘ – p p ‘ „ ˆ – • y „ € ˜ – € ‘ w † ™ ˜ u–m d j „ p “ — ˜ – — – ™ ‘ ‘ „ ˆ u y“ ˜€ „ d d“ € €“ d –mk ˆ „ j y ˆ – p ‘“m s j–—“ € • “m – €˜‘– — j „ € d j „ p ‘ y €–• y „ €˜ — “ p – p — ‘ “ ˆ ˜ y “ … mr † – ˜‘ “ — £ y— — “ d ˜— – ‘ — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ “ m s j – — “ € • “ m – ˆ ‘ „ ˜ “ —m ˜ y –m d j— “ m — € „m “ f — – ˜‘ – € ™  ˆ — ™ ˜ u–m d j „ p – ˆ — –m k ˆ „ j – ˆ –m s j – — ‘ – ‘ y  j € “ d — – ™ ‘ ‘ „ ˆ u y “ ™ € d „ € d d “ — ym d –m –m k ˆ „ j –m € – ym “• ™ – ˜ “ e y „ — ‘ „ – € “ € ˜ ‘ „ p y “  z † — – ™ ‘ ‘ „ ˆ u y “ ™ € d „ € d d “ — ym d –m –m k ˆ „ j –m € – ‘ j € – ˜ ™ ˆ € y „ d ˜ ‘ “ — £ y— ˜— – l ‘ € – ˜— ym p y ˆ – ‘ € – ˜‘ ™ ˜m “ y y “ m – ˆ – € k ˜ € p ‘ y – y ˜ ˜— ‘ „ p  y y – p ‘ “ m s j – — “ € • – ˆ – € k ˜ € p –m € – —m ˜ y f — – € ˜ k j “ € “ d – ˆ – € s j „ ‘ – j j –m – €  ˆ ˜ — – p ™ ˜  u – m d j „ p – j j “ m ˜ ‘ “ “ — – m k ˆ „ j — ˜ ‘ – € ™  ˆ € – € “ d j „ p – ˆ ˜  l “ — m  – j j „ p f — “ p m – ˜ ‘ y — ‘ “ ’ † – m s “ m s j – —  “ € • — ym d – m – m k ˆ „ j – m € —  „ ep —  y d f — – ™ ‘ ‘ „ ˆ — – m € – ˜ ‘ – — ™ € d – € € y „ d — –m s— — „ d — –m k ˆ „ j — € y –— ym d € – € ™ ‘ ™ l – ˜—— ‘ „ p – ˆ „ e ˜ ™ j “ m f ˜ ˆ ˜‘ – j – € ˜ y # – p ‘“m s j–—“ € • l „m “ — † ™•€–— ‘„p ˜—– —–™ ‘ ‘„ ˆ uy“ ˜€„ d d“€ €“ d – — j ˜ d „  y y –m k ˆ „ j –m — y d f — – € „ ˜ “ ™m “ — ‘ „ ˜ “ —m “  ˜ ‘ — – ˜ ‘ – € ™  ˆ – ˆ € ˜ € “ d — „… — € y – — ym d ™ ˜ y p ™ u– p ‘ „ ˆ ˜— – –m k ˆ „ j y ˆ ‘ „ ˜ “ — j ˜ d „ ˆ — y— — – p „ € d – n † ™ € d „ € d d “ — ym d –m ˜ “ ˜m y— ™ € –m € – ‘ ‘ „ ˜ p –m ™ — – ˆ ˜ – — „ … – ˆ – € s j „ ‘ ‘ “ ˜ € – p ‘ y – ˆ „ e ˜ ™ j “ m € – € ™ ˜ ˆ – € “ — — – p ™ ‘ p ‘ „ ˆ ˜— – m f m “ s „m l j y j ˜ d „ m – € ˆ ‘ – ˜ ˜ “ ˆ € – ˜ ‘ – ˜ € y „ ƒ † m “ p „m j y j ˜ d „ ‘ y – € ˜ ˜ y – d ˜ “ ˜m y— ™ € –m – y y – € ˆ ˜— – p f – ™ — „ d „ € d –m “  ˜ ‘ ‘ „ ˜ ym „ — “ m – ˆ ˜ ‘ – j – ˜ € „ … ˜‘ – ˆ ‘ – d ™ ˆ –m k ˆ „ j y ˆ ‘ „ ˜ “ — j ˜ d „ — y d ‘ „ ˜ “ —m “  ˜ ‘ – ˆ „ e ˜ ™ j “ m – ˆ ‘ „ ˜ y p ™ u– – ‘ y ˆ — ˜ “ ˜m y— ™ € — – n C I ¥ G  C G D  H E  C I £       ©   ©   !     D  !  LL(θ |D) ©! θ     © © ©  ©    ©  N B I T ER © ¢ £ † θ = (θ1 , ..., θK ) m “  ˜ ‘ – y y € ˜ ™ j “ € “ d –m k ˆ „ j  ! D¨©D%  R → − , − , F reqs , W , S ) →σ θk = (πk , µk k k k k — ‘ “ ˆ € – p „ ˜— — –m ˜ – j y j u“ j — ˆ „ d – ˆ — ‘ „— ‘ – j ˆ — –m € – ‘ ‘ „ ˜ p –m ™ — † ‘ „˜“ y y ™ m ‘ „m– — €–m ypm“ p „˜ „˜ f € – —m “  ˜ ‘r † — € – ˜— ym p u y “ — – ™ p „ — — “ — – ˜ ‘ – ‘ ˜ € – d — ‘ „— ‘ – j ˆ – ˆ – € s j „ ‘ –m ˜ – ! & f — ™ ep € – e p – € — € – ˜ — y m p – ˆ – € s j „ ‘ – m f — ˜ – s „ ˆ – m s j – — ‘ – ‘ y € – —m “  ˜ ‘r )  † —™ ˆ „ j p ‘ „ ˆ ˜‘ – • y „ € ˜ ‘ – — ‘ „ ˜ “ — j u“ q – ˆ ˜ – – p ‘ “ € ™ d— ‡ ˆ — – d “ ˜ ™ — – ˆ — € „m — ™ y ˜ p – – —m y pm “ p Sk  ¨D ©         © K © #! DK k=1 K d=1 M Wkd D R V ©D F ©E¨H  © R → → D = (− , ..., − ) x1 xN ! ¤ £ ¥ ¢ § ¤ ¥ ¡¢ ©¨ ¤ ¤ £¢ © – ˜ d j „ p ‘ – ˜ ‘ “ ‘ – € d ‘ – — € – ˜— ym p u y “ — – ™ p „ — — “ — –m l k € — – p € – j € „ … – ˆ € – ˜ • ™ ˆ ‘ “ – € “ — — – p ™ ‘ p ‘ „ ˆ ˜— – — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ ˜ ‘ “ ˜— u– ˜ y € s y ˆ ‘ „ ˜ p – ˜ ™ ˆ – ˆ – € “ ‘ jm ™ € d – — “ e d – ‘ ” † — € – ˜— ym p u y “ — ™ p „ — — “ — ˜ – s „ — – ˆ –m s j – — ‘ – m – ˆ € ˜ € “ d ˜ “ … – — — –m l k € — – p – ˆ ‘ „ ˜ p y € ˜— ‘ „ p “ n † — – ˜‘ – ‘ ˜ € – d — ym d — –m  j € “ d — ‘ „— ‘ – j ˆ – ˆ j y j ‘ j ‘ y € “ d — – ‘ ™ ˆ — –m l k € – ˆ – j € „ … — y „ — € y – ˜ “ —m ˜ y m — € – ˜— ym p — –m € – ˜‘ – — ™ € d – € – ˆ ˜‘ “ ˜ ˜ – j € – d – ˆ „ e ˜ ™ j – ‘ y – ˜ y— ‘ – — ‘ „ — „ d „ € d — y „ ©  Rk Ck   Dk Rk  E @ EI A  B  C G D H E @ E D C  B¡  ©  ¢ † ˜ “ … ™ ˆ y €“ d ™ u ˜ — – – €  „ ˜ “ ™ m “ ‘ „  ˜ “ —  m “  ˜  ‘  p – •“ – m k ˆ „ j y ˆ ‘ „  ˜ “ —  j  ˜ d „ m – ˆ — ‘ „ ˜ “ € ™ ˜ ˆ – € s j „ ‘ – n † – j e ˜ € „ lm “ m € “ d ™ ˜ ‘ – — ™ € d  ym ˜ — – & % % – j e ˜ € „ lm “ m f — € – ˜ — ym p u y “ — – ™ p „ — — “ — – ˜‘ – ‘ ˜ € – d — ‘ „ — ‘ – j ˆ – ˆ j y j u“ j – € s j „ ‘ –m ˜‘– j–m“ l™ ™ ‘ ‘ „ ˆ ˜ ‘ “ ˜ ™ ˜ ‡ † – j e ˜  € „ l m “ m € “ d ™ ˜ ‘ – — ™ € d ˜ ‘ – j – m “ ‘ ˜ — – ¢ % % – j e ˜  € „ l m “ m f — ™ e p € – ep – € — € – ˜— ym p – ˆ j y j u“ j – € s j „ ‘ –m ˜– f –mk ˆ „ j – ˆ ‘ „˜ “ —  j  ˜ d „ ˜ – ‘ „  ˜ “ —  m “  ˜  ‘ – ˆ „ e ˜ ™ j “ m – ˆ – d “ ˜ ™ – ‘ y ˆ — ‘ „ ˜ “ € ™ ˜ ˆ – € s j „ ‘ –m —™ ‘ ‘„ ˆ ˜‘“˜ † – ˆ „ e ˜ ™ j “ m – ˆ — – p ‘ “ j € „ …€ – d — –m € – € „m ™ j “ ˆ ˜‘ “ ˜ ˜ – j € – d — –m kmm “ € “ d — ‘ „ ˜ y p ™ u– ˆ – € • y ¤ ‘ – – — j “ m –  ˜— y  y y – p f — – € ˜ y “ — – ˆ — – ‘ y — –m — – ˜‘ “ ˆ ‘ – d ™ ˆ ‘ ˜ ‘ „ — — –m k ˆ „ j – ˆ — ‘ „ ˜ “ € ™ ‘ ™ l — – ˜‘ – € ™  ˆ — – i † ™ • € – — ‘ „ p ˜ ‘ – j –m “ ‘ ˜— – i r x € y –m “• “ m ˜ ‘ “ — j ‘ j – m k ˆ „ j – m —  y d f ˜ – — – € ˜ k j “ € “ d — – m € y „ d — € y – m “ • — – ˜ ‘ – € ™  ˆ p – •“ — – m k ˆ „ j — € y –  — y m d p ‘ „ ˆ – €k ˆ— ‘ „ p ‘ „ f — – ™ ‘ ‘ „ ˆ u y“ u y– j –m ˆ ‘ „ d— – €€ „ p  y y –mk ˆ „ j –m €–• y „ €˜ € y „ ƒ † € y – ˜ “ —m ˜ y m – ˆ ˜ € “ d “ m – ˆ  € „ € d “ – p ‘ “ — — “ ‘ ‘ „ p – ‘ y p y “ ˜ ‘ “ ˜— — – p ™ ‘ – ‘ – ˆ „ e ˜ ™ j – ‘ y € – — „ d „ € d – ˆ ˜ – j € – d — y „ ‘  y y – p f –m k ˆ „ j – ˆ ‘ „ ˜ p –m ™ — – ˆ – j km s „ € d ‘ y – j j „ p y • – € ˜ ˜ y – d — – € ˜ k j “ € “ d — – p € y „ d — – ™ € d „ € d d “ — ym d — –m — € y –m “• — – ˆ ‘ „ ˜ “ ‘ j € – ˜ ™ ˆ “ m – y y — € „m “ ˜— – l ‘ € – ˜— ym p – p “ d — s y— – ˆ — – ˜ ‘ “ ˜— u– — – ˆ „ e ˜ ™ j — – € ˜ y “ u y “ ˜ € „ d d “ € € “ d — – ˜ — m s “ s „ € d — – m k ˆ „ j — – ˆ ‘ „  ˜ “ — m ˜ y m – ˆ ˜ ‘ “ ˜ € „ d j – l “ ˜ ‘ “•“ ‘ ” † € – ˜ — y m p – y y “ ep € y „ d € – ‘ ‘ „  ˜ p – m ™ — — ‘ „  — ‘ – j  ˆ – ˆ – € s j „ ‘ – m – y y  ˆ ‘   y y – € ˜ k j “ € “ d y ˆ ‘ „  ˜ “ p  p ™ d — “ m ˜ ‘ – j – m “ l ™ – ˜  — — – p ™ ‘ & % % – ˆ „ e ˜ ™ j – j k  u y – ˆ “ m ˜ – f — ™ e p € – ep – € — € – ˜ — y m p – ˆ – € s j „ ‘ –m € –  p ™ d— – ˆ ˜ ‘ – ˜— — – p ™ ‘ — – ˆ „ e ˜ ™ j u y – ˆ — – n † — – ™ ‘ ‘ „ ˆ u y “ — ™ € d „ € d d “ — ym d — –m –m k ˆ „ j y ˆ — – € ˜ k j “ € “ d — –m ˜ ‘ – j – y y ˜ “ j „ ˜ y “ € – ‘ j € – ˜ ™ ˆ € y „ d ™ —m ˜ y – € ˜ p ‘ „ ˆ ˜ y – d i r x – € k ˜ € p – p f — ‘ „ — „ d „ € d — y „ ‘ – y y & % % ˜ – ¢ % % — – ˆ „ e ˜ ™ j u y – ˆ — – ˆ — “ p –m — ‘ “ ’ –mm – € „ l ™ ˜ “ p — – y y€™ j y‘ — Y    © Y   © ©    © © !  M axK © K R ©   ! © © © 2 |M odalitesd |  © © T  K    N B I T ER M axR N B I T ER R k =1 d∈Sk Mθ = K d d ‚ — “ p – € ˜ „ ‘ — ‘ “ ˆ ˜ y— – j j „ p ™m y pm “ p ˜— – ˜ – f –m k ˆ „ j y ˆ — ˜ ‘ “ ˆ ‘ – d ™ ˆ ‘ — – € ˜ k j “ € “ d – ˆ – € s j „ ‘ –m – ˜‘ – — ™ € d – € † — – ™ ‘ ‘ „ ˆ u y “ ˜ € „ d d “ € € “ d –m k ˆ „ j y ˆ – p ‘ – ‘ ˜ € – d “ m € – — j ˜ d „ € y „ d – ™ — j ‘ j – € ˜ „ ˆ – € k ˜ € p – p – ˆ € y –m “• “ n  Mθ  o†  ‚ ˜ y — – j j „ p  ‘  ™ ˆ f # $ # f  ˜ € “ ‰ ep z  ‘ „  € – ˜ € i ‘ „  ˜ “ j € „ … ‘ r ‘ “  — – “ x i r x – € k ˜ € p – m – ˜ y „ ˆ — ‘ “ — ˜ — – – € ˆ “ p – p — ‘ “ ˆ Y# BIC (θ |D) = −2 × LL(θ |D) + Mθ × log N ¢ ¡ ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ ¡¢ ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡ † θo — – ™ ‘ ‘ „ ˆ u y “ ™ p „ — — “ m “ j ˜ d „ – y y € ˜ ™ j “ € “ d –m k ˆ „ j ! D ©   %    ¨D¨ D   ¨D¨ D  ¨ D V © © M axR minBIC = +∞ θo = ∅ K=1 M axK R=1 M axR i=1 N B I T ER θ DKR θ Dθδ BIC (θ |D) BIC (θ |D) < minBIC minBIC = BIC (θ |D) θo = θ  o† ‘ „  ˜ “ y y ™ m ‘ „ m – — € – m y p m “ p f f € – — j ˜ d w ff € – —m “  ˜ ‘r „˜ „˜ „˜ m “ j ˜ d „ ˜ ‘ “ € y „ p –m k ˆ „ j i r x – € k ˜ € p y ˆ – ˜ ‘ “ € y „ p –m “ j ‘ j € y –m “• † — € – ˜ — ym p u y “ — – ™ p „ — — “ — – ˜‘ – ‘ ˜ € – d — ‘ „ — ‘ – j ˆ – ˆ j y j u“ j – € s j „ ‘ –m ˜ – f — ™ ep € – ep – € — € – ˜ — ym p – ˆ j y j u “ j – € s j „ ‘ – m f – ˆ „ e ˜ ™ j “ m – ˆ – d “ ˜ ™ – ‘ y ˆ — ‘ „ ˜ “ € ™ ˜ ˆ – € s j „ ‘ –m f — – ™ ‘ ‘ „ ˆ u y“ ˜€ „ d d“ € €“ d –mk ˆ „ j y ˆ – p ‘“m s j–—“ € • l „m “m – ˆ              ¢ ¢ ¡ M axk ¡ © © ! D F  ©¨ & — – • — — – p p y — — € y – m “ • — – m € y — m y – — – m f — ˜ –  s „© ˆ – m s j – — ‘ – ‘ y & %  % ¦ ¦ D R V ©   E H † — – ™ ‘ ‘ „ ˆ u y “ ™ p „ — — “ m “ j ˜ d „ – y y € ˜ ™ j “ € “ d –m k ˆ „ j ! D ©   %  ¨D¨D¨ D N B I T ER → → D = (− , ..., − ) x1 xN δ  θo  ¨ D V © © minBIC = +∞ θo = ∅ K=1 M axK i=1 N B I T ER θ DK θ Dθδ BIC (θ |D) BIC (θ |D) < minBIC minBIC = BIC (θ |D) θo = θ  o† ‘ „  ˜ “ y y ™ m ‘ „ m – — €–m ypm“ p f f € – — j ˜ d w f € – —m “  ˜ ‘r „˜ „˜ m “ j ˜ d „ ˜ ‘ “ € y „ p –m k ˆ „ j i r x – € k ˜ € p y ˆ – ˜ ‘ “ € y „ p –m “ j ‘ j € y –m “• † — ™ e p € – ep – € — € – ˜ — ym p – ˆ j y j “ j – € s j „ ‘ – m u ˜ – f – ˆ „ e ˜ ™ j “ m – ˆ – d “ ˜ ™ – ‘ y ˆ — ‘ „ ˜ “ € ™ ˜ ˆ – € s j „ ‘ –m f — – ™ ‘ ‘ „ ˆ u y“ ˜€ „ d d“ € €“ d –mk ˆ „ j y ˆ – p ‘“m s j–—“ € • l „m “m – ˆ ! & — – •— — – p p y— — € y –m “• — –m € y — m y – — –m f — ˜ – s „ ˆ –m s j – — ‘ – ‘ y ¢ ¢        ¡ ¡ © © © ¦ D R V ©D F ©E¨H → → D = (− , ..., − ) x1 xN M axk N B I T ER δ ¢%% 2 # ¦¨ ¡ ¨ £ £ ¦ ¢  ¦ ¤ ¥ ¡¢  ©¨ ¡¨ ¡¤ ¦ ¢   ©  !! ©  © ©   EDG ©  ©  ©     © EDG ©  © EDG © ©  © — y „ ‘ — y d f — –m l k € — – ˆ ‘ „ ˜ “ p m d j — – ˆ – ˜‘ – ˆ ™ p ™ € d – — “ e d “ m – ˆ — € „m ˜ y ˆ ™ € ˜‘ – j – ˜ € „ … ˜ ‘ – • y „ — ˜— –  y y –m s j – — ‘ – — € – ˜— ym p u y “ — – ™ p „ — — “ — –m l k € — – ˆ ‘ „ ˜ ‘ ™ ˆ “ m — ‘ “ ˆ — – ™ —m ˜ y — ‘ „ — ‘ – j ˆ – ˆ — –m d y „ p — – ˆ –m s j – — ‘ – m € y— — –m l k € — –m p ‘ „ ˆ — ‘ „ ˜ – „ € d — y „ ‘ f – € “ … – p € y „ ƒ † ˜ ‘ – j –m “ l ™ ˜‘ – — — “ € “ d d “ — € – ˜— ym p — ˜‘ – € ™  ˆ — – ˆ — ™ ˜ p  p ™ d— — –m — –mm – y y — –m € y— ˜ – f ˜‘ – j –mm – y— • ˜ ‘ – — — “ € “ d d “ — € – ˜— ym p — –m – € ˜‘ – — – p ‘ – € ™  ˆ — –m — –mm – y y — –m € y— — ‘ „ ˜ p – „ € d — –m ˆ € „ s “ ˆ ˜ ‘ “ ˜‘ – — ™ € d ‘ – f — ™m s p — € – ˜— ym p — –m ˜ ‘ “ ˜‘ – — ™ € d – € — –m l k € — – ˆ — ‘ „— ‘ – j ˆ — „ € ˜ y „ u y – ˆ ‘ – ‘ „ ˜ “ —m “ y — • – ‘ y € y – ˜ “ —m ˜ y m € – — „ d „ € d – ˆ ˜— – – ™ ˆ n † – — y – ˜ „ p d „ € ˜ — “ d ˜ „ — – ‘  y y — ˜ “ ˜m y— ™ € — – ˆ – y y e d “ € l ‘ „ ˜ “ —m “ y— • – ˆ – ˆ „ e ˜ ™ j – ‘ y € – — „ d „ € d – ˆ ™ ˜m s— — „ d “ m ˜‘ – j –m “ l ™ – ‘ ‘ „ ˆ — y „ ‘ — € – ˜— ym p — – ˆ –m “ j ‘ j ‘ „ ˜ d € p — – ˆ – ˜ ˜ – p f ‘ ‘ ‡ † €“ d y‘– ˜ s „  y m – p – y y  ˜ ‘ – ˆ  — € „ m “ – € k •“ —  y y f – m d j – u – ‘ y € y — ‘ „  ˜ “ p  m d j  — – m m – ˜ – ‘ y ˆ ˜ “ ˜ m y — ™ € – m – ˜ ‘ – — ™ € d o† – € y l “ n † ™ • € – — ‘ „ p ˜ — – € – ˜ — ym p – y y “ ep – ˆ — – ˜ ‘ – ‘ ˜ € – d — ym d — – m  j € “ d — ‘ „  — ‘ – j ˆ — – ˆ j y j ‘ j ‘ y f — ‘ # † ˜ € „ d d y— ‘ „ — — “ d –  ˆ „ j – ‘ ‘ „— — – € d d y— € y –m — –m l k € “ m – ˆ — ‘ „— ‘ – j ˆ — –m f — ˆ „ d € y –m – ˆ ˜ ‘ “ — — „ € p – € ˆ € „ m — ‘ “ ˆ f – j € d d y — ‘ „ f € – ˜— ym p – y y “ ep € y „ d —  y ƒ † – y y – l € “ m — y m d – € ˜ ˜ y – d – m l k € “ m — ‘ “ ˆ — y m p ‘  — ˜ – s „ — – ˆ – m s j – — ‘ – m –m l k € “ m – ˆ ˜ ‘ “ € y „ p ˜ € „ d d y— –m ˆ € „ s “ ˆ –m y pm “ p ‘ „ f – € “ … – p € y „ ƒ † – l “ — — ˜‘ – € d d “ m ˜ ‘ “ ˆ ‘ – d – ™ ‘ – j ˜ “ ˜ ™ – — “ e d – ˜ ˜ – p f — ‘ “ ˆ – y y — € „m “ f ˜‘ – j – ˜ “ € ˜ ˜ — „ d ‘ y ˆ ˜ l “ — m f  p y y … y “ — f – ˆ –m “ j ‘ j ‘ „ ˜ d € p — – ˆ – ˆ – — “ e d “ m — ‘ “ ˆ – y y – j j –m — € „m “ ˜— – ˜ € d— – n † — ˜‘ – ‘ ˜ € – d — ym d — –m  j € “ d — ˜ y s € ˜ ˜ “ ˆ j y j ‘ j ‘ y € “ d — – ˜ € p ™ ˆ ˜‘ – „ — — –mm – y y ‘ “ f — –m l k € — – p – ˆ ‘ „ ˜ “ p m d j— – ˆ – — “ e d – ‘ y – ˜ y— ‘ – € – ˜ y „ “ ˆ — ‘ „ — „ d „ € d — y „ † – ™ € ™ ˆ— ‘ „ p –mm – € „ l ™ ˜ “ p ‘ „— ‘ – j ˆ “ m € y — € – ˜ — y m p – m — ‘ “ ˆ – €  “ ˜  € „ “ j –  € „ l ™ ˜ “ p “ m ˜ ‘ “ •“ ‘ – ˜ – j m  ‚ € y – ˜ “ —  m  ˜ y m € y „ d — ‘ – — – ˆ € y – ˜ € „ d ˜ “ … ˜ y „ ˜ ˜— – –mm – € „ l ™ ˜ “ p ‘ „— ‘ – j ˆ – ‘ y € y— ™ ˜ ‘ – — ™ € d ˜ d – p ‘ „ p –m f — ‘ “ — “ j f € – ˜ — y m p y ˆ € y –  € ™ ˜ ‘  m – m “ j  ‘  j – p ‘ – y y ™ € … – ‘ y ˜ ‘ “ “ — ™ ˜  m “ ˆ „ j — – ˆ ‘ „  ‘ y m ‘ –  s y „ f € – ˜ — ym p y ˆ — – € s j – j — –m  j € “ d — – ˜ ‘ – — ™ € d — ™ ˜m “ ˆ „ j — – ˆ ‘ „ ‘ y m –m l k € “ m € – p „ — — “ — ‘ „ € € y „ d — y „ ‘ ‚ – m s “ ˜ y p —  ˆ ˜ — – u  „ ep – p ‘ „  — ‘ – j  ˆ “ m € y — – ˆ — ˜ – s „ — – ˆ — € y – m “ • — – ˆ – m s j – — ‘ – m  j € “ d – ˜ ‘ – y y ™ € … — y m d “ m ™ ˜  m “ ˆ „ j “ m – m l k € “ m € –  p „ — — “ ˆ  —  „ ep — ‘ „ •“ — y „ ‘ f – m m –  € „ l ™ ˜ “ p ‘ „ — ‘ – j ˆ – ‘ y € y— ˜ – ‘ „— ‘ – j ˆ “ m € y— – ˆ — ˜ – s „ — – ˆ — € y –m “• — – ˆ –m s j – — ‘ – m ˜ ‘ “ ‘ – ˜‘ „ p m “ j ‘ j –mm “• € – ˜‘ m –m l k € “ m – p „ — — “ ‘ „ f – y y € ™ j y ‘ ‘ „— ‘ – j ˆ – ‘ y € y— ‚ — ˜ – s „ ˆ — –m s j – — ‘ – u y “ – • y „ ‘ — – p – ˆ € ˜ € “ d — – ™ j € „ … ˜‘ „ — — –m “  ˜ ‘ — –m l k € — –m f – ˜ y — ‘ ‡ † –m s— — „ d – y y ˜‘ – ‘ ˜ € – d — — y “ ˜ “ ˜m y— ™ € ‘ y ˆ ‘ „ ˜ “ ˜ ‘ – — ™ € d “ m € y „ d ˜  y € s y ˆ ‘ „  ˜ p – ˜ ™ ˆ – m m – ˜ – ‘ y ˆ ˜ € ™ ˜ ‘  m – € ˜  d “ ep – m — ‘ “ ˆ – ™ ˜ ‘ – — ™ € d u y “ •“ € ˜ — „ ‘ – ˆ –m “ ˜‘ – j € ™ d u– – ˜ € “ d “ m — ‘ “ ˆ — ‘ „ € € – • — y „ † — € – ˜ — ym p u y “ — ™ p „ — — “ — ˜ – s „ ˆ — –m s j – — ‘ – — – ˆ — ™ j € d d y— — 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ˆ € – ˜ ‘ – — ™ € d – ˆ ˜ ‘ “ •# † ‘ – p ‘ „ ˆ ˜— – – ˆ „ e ˜ ™ j “ m – ˆ –m “ s „m l ™ ˜ u–m d j „ p “m f ˜ –  – € ˜‘ – ˜ ‘ “  € “• – ˆ — € y –m “• — –m ˜ – ˜ –  – € ˜‘ – ˜‘ “  € “• – ˆ — € y –m “• —– n † ‘ – – m k ˆ „ j y ˆ ‘ „  ˜ “ —  j ˜ d „© ˆ – — “ e d “ m – ˆ – mm – p ˜ – ‘ – ˜ — – – ˆ „ e ˜ ™ j “ m – ˆ ‘ „  ˜ “ — m “  ˜ ‘© ˆ – — “ e d “ m – ˆ ™ ˜ u – m d j „ p “ m f – ˆ ˜ – – ˆ – ™ ‘ ‘ „ ˆ € y –m “• – ‘ y € y „ ƒ † € – ˜‘ – — ™ € d – ˆ — ‘ „ ‘ – • — y „ ‘ – y y & %  % l ‘ € – ˜— ym p – p “ d— s y— – ˆ – ˆ „ e ˜  ™ j – mm – • y „ ‘ “ m – ˆ — – y y  ˜ —  € ™ ˜ p “ € “ p — – ˜ ‘ – € ™ ! ˆ — – m – j y — ™ € †  y “ – m s “ ˜ – m f ˜ ‘ – j – m “ ‘ v % – ˆ —˜“˜m y—™€ – ˆ —–m d j– u ‡ †  ¦ ¡ O(N × R × K ) R M axR 2 ) × M ax2 ) O(N × (M + M axR K M axK R † &% K O(N × M × K ) K ¥ © † — – p ‘“ — —“ ‘ ‘ „ p – ˆ ‘ „˜p “ €˜ u– ˆ ™ u — – j j „— — y „ ‘ — y „ ‘ – y y – € ˆ “ p –m — ‘ “ ˆ –m s “ ‘ ‘ „ — “ € ˜ “ … ˜ y „ ˜ –m s j – — — y „ ‘  y y ˜ “ ˜m y— ™ € f u y – ˆ ‘ – ™ — • ˆ ˜ ‘ “ ˜ ™ n ‘ –  ym – p f — € – ˜— ym p — „ € ˜ “ € – ‘ € y „ ˜ – € & % % f — “ p – j k— „ € ˜ –m — ‘ “ ˆ ˜ ‡ † – € ˜ y “ m – ˆ ‘ y m — – ep „ € d d „ € ˜ — “ d ˜ ‘ „ — – ‘ — m — — ™  ˜ ‘ – ˆ  – € ˜ ˜ ‘ „ € € y „ d — € – ˜ — ym p — – m ‚ ™ ˜ y p —  ˆ – € ˜ ˆ ˜ ‘ –  • — “ p – j k u y – ˆ – n † — ™  ˜ ‘ – ˆ ˜‘ „ — — € – ˜— ym p — –m — “ p € – j – € d –m — ‘ “ ’ † † – € y l “ m – ˆ — –m d j – u– — –m — ‘ “ ˆ — ˜ ‘ – — ™ € d — € – ˜— ym p — –m € – ˜ p – ˜ ™ ˆ – ˆ –m s “ d “ p ˜— – & % % f ˜ ‘ – j – € “ € ˜‘ „ i ©    © ©  E DG   © † – ˜— u– — ‘ „— ‘ – j ˆ – € ˜ ‘ – ‘ „ ˜ “ m ™ € € „ p – ‘ y y y — € „m & % © % – ˆ —˜“ ˜m y—™ g "†  ¦ ¥ ++ ++ + ++ +++ ++ + ++ + # + + ++ + ++ + ++ + ++ + + ++ ++ + + + +++ ++ + + ++ + + + + ++++ ++ + + + + ++ + + ++ + + + +++ + + + +++ + + + ++ + + ++ + + + + + + + ++ ++ + + ++ + ++ ++ + + +++ + + ¡ ¡ £© ¤ –m s— ‘ – e ™ € d j „ p – € k ‘ “ j – ˆ € – y ˜ ˜— – € “ m – ˆ ˜ – — –mm – ™ € — – ™ ‘ ‘ „ ˆ – ˆ — – — “ s – ˆ € ˜ € “ d – p ‘ “ — — “ ‘ ‘ „ p “ m – ˆ – € “ € ˜ u– ˆ — –m s “ d “ p — –mm – ˜‘ „ — † ¤ l ‘ € – ˜— ym p – p “ d— s y— – ˆ ˜ – l ‘ € – ˜— ym p – ˆ — – ˜‘ “ ˜ — u– — – ˆ „ e ˜ ™ j u y “ — – ™ € “ d j „ p — – ˜‘ “ — — – € ™ ˜‘ — –mm – ˜ ‘ „ —  „ y y ‘ – † ¤ — – ˜‘ “ y y ‘ “ j — € y –m “• – ˆ y „ f — – ™ ‘ ‘ „ ˆ — –m — ‘ “ ˆ ˜ y € s – ˆ – p ‘ – — ™ € d “ m f — – ˜‘ – ‘ ˜ € – d ‘ „ ‘ — ‘ „— ‘ – j ˆ – ˆ ˜ y „ “ m — – ˜— y s „ € — –mm – ˜‘ „ — † o ¤ —–™ ‘‘„ˆ – ˆ u y – — –m — ‘ “ ˆ — ˜‘ – — ™ € d — € – ˜— ym p — – ˆ — – y y ˜— € ™ ˜ p “ € “ p — –m ˜ – – € s j „ ‘ –m f — ‘ „ — ‘ – j ˆ – ˆ – € s j „ ‘ –m f — ˜– s „ ˆ – € s j „ ‘ –m ‘ „m–— — – ˆ „ e˜™ j — – p ˜‘– ˜€ „ d j „ p –— ˜‘– j j „ p † ¤ &% % ˜– – ˆ — – € ˜ k j “ € “ d — – ˆ – p ‘ – y ‘ m ˜— – –mm – y y † ‚ — – ˜‘ “• y— — –m ˜ ‘ „ — u y “•  † &%  ©  © ©    "    ©  EDG   % l ‘  € – ˜ — y m p y “ — – ™  p „ — — “ — – y y  ˜ —  € ™ ˜ p “ € “ i  †  ¦ ¥ ¤£ ¥ ‘„ ‘ ‘„ ‘ y„ y„ y„ y„ y„ ‘„ ‘ y„ y„ ‘„ ‘ — – y y € ˜‘ – p ‘ „ p — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – y y ‘ „ pm – y y – j € „ … – ˆ — € – ˜ — ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — ™ ˜ — ‘ – ˆ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ — – ™ € “• — –mm “ ˜ – ˆ — € – ˜— ym p u y “ – p ‘ “ € ™m „ ˜ – ‘ ! “ ep – ˆ ˜ – – m – p ‘ “ € ™ m „ ˜ ˜ y € s y “ – p ‘ “ € ™m „ ˜ – ˜ u– ˜‘ „ p y ˆ – ˜ d j „ p ‘– –—€ d ˆ€“ e – j ˜ ‘ “ m “ ˜ ‘ – j ™ € p ‘ – ˜ — ‘ j € – ˜ ™ ˆ ™ ˜ u–m d j „ p — ˜ “ ˜m y— ™ € — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d !©   O(N × (M + M ax2 ) × M ax2 ) R K — –m lk € – ˆ –m s j–— ‘– — € – ˜ — ym p — – ˆ — – ˜ ‘ – ‘ ˜ € – d — ‘ „— ‘ – j ˆ – ˆ ˜ – — € – ˜— ym p M axR € „€ d “ — – p ‘“ ——“ ‘ ‘ „ p M axK –ˆ –€s j„ ‘ D–Hm G€"y— —– ‘€„ s D  ( © W  F ©  G  G  ¤ £ ©¨ ¦ ©¨ ¡¨ ¦ ¢ ¡ © ¤ ¨ ¦ £ £¤ ¡¤ ¦ ¤ ¤ ¡ ¡¦ ¤ ¦ £ ¢ ¦ ¤ ¥ ¡¢ # ©¨ ¤ ¤ £¢ ¦ ¦ ¡ ¡ $#   !© ©  © ©   © ©   ©  © © © ©  © © ©  ©   © ©  ©© © ˜ ‘ „ — — – ˆ „ e ˜ ™ j u y – ’ † — – ™ ‘ ‘ „ ˆ — – m — ‘ “ ˆ — ˜ ‘ – — ™ € d – € ˜ ˜ ‘ – • y – d  y y — … ˜ p ‘ „ —  ˆ — ˜ d – p ‘ „ p — –m € –m s p – ˆ — € „m “ ˜ – j € – d  p – p € “ p f — – — — “ m p — –m € – ˜‘ – — ™ € d – € € y „ d — ‘ „ ˜ y s € ˜— ˆ – ˆ – l ‘ “ m ™ j ‘ y € – —m ˜ y ˆ ™ € d „ € d d “ — ym d ˜‘ – • y „ — ˜ — – mr † ˜‘ “ — £ y— — “ d ˜— – ‘ ‘ „ ˜ y s € ˜— ˆ – y y ‘ y – ‘ y € “ d – — — “ m p – ‘ y € – j  u„ € d d “ f — “ p u y – € s j „ ‘ – ˆ — ‘ “ ˆ f ˜ – – ‘ ‡ † — – — — “ m p — – ˆ ™ ˜  p  ‘ y ˆ – — k e ˜ „ d  e m — –m s— ‘ – — — ym d d y „ p y “ – s – € ˜ ˜ ‘ – • y – d — ˜ “ ˜m y— ™ € — –m f ™ — • € – d y— — “ p ‘ y — ‘ “ ’ † q ‡ – j e ˜ € „ lm “ m – ˆ ‘ „ ˜ “ j ˜— ‡ ˆ – d “ ˜ ™ m – ˆ — € „m € –  ˆ „ j — –m — “ j “ – ‘ – ˆ – € ˆ ˜— – p f — – y ‘ ‘ „ p ˜ ‘ – j –mm – ˜ € “ d y „ — – y‘ ‘ „ p ˜ ‘ „ —  y y — € – ˜— ym p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ — ™ ˜m s “ s „ € d — –m € – l – ˆ ˜ l “ — m f — “ p – p — ‘ “ ’ † — –mm – ˜ € “ d – y y ˜‘ „ — – ‘ — ‘ „ ˜ “ j € „ … ‘ — – p — – j j ˜ – f — – ™ ‘ j € – ˜ ™ ˆ ™ € d — – — — “ m p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ — ™ ˜m s “ s „ € d — – ˆ – ˜ d j „ p ‘ – ˜‘ “ ‘ – € d ‘ – – y ˆ ‘ – ˜ ™ – € ˜ ˜ y – d –mm – — € „m “ f & % % € y „ d – j j „ p f – y y ˜— ˜ “ ˜— ˜— – – ™ —m ˜ y l ‘ € – ˜— ym p – ˆ – ˆ „ e ˜ ™ j “ m  z † – y‘ ‘ „ p — “ d ˜— – ‘ — – ™ ‘ j € – ˜ ™ ˆ ™ € d — – — — “ m p u y “ – p ‘ “ ‘ – ˜ € “ d d “ ˆ ‘ „ ˜ “ j € „ … ‘ m —m – y y — –m € y „ d — ˜ – s „ — – ˆ –m s j – — ‘ – m € y— ™ € ˜ ‘ – p ‘ „ p — € „m “ ˜— – – l “ — — ˜ ‘ – € d d “ m ˜ – f € – ˜— ym p – j j y “ — ™ p „ — — “ ˜‘ „ — – — — “ m p – j j “ m ˜‘ “ – l “ ˜ € “ d — ˜ – s „ — –m ‚ – ˜ p – € ˆ ˜— – — – ˆ „ e ˜ ™ j — – ˆ ‘ „— ‘ – ˜ u– m — € „m “ f € – ˜— ym p – y y ‘ y ‘ y ˆ ‘ „ d — – € € „ p – — — “ m p – ‘ y  — – €  ˆ ˜ — – p f ‘ „  ˜ y s  € ˜ —  ˆ – y y  ‘ y – ‘ y € “ d – ™ j  u„ € d d “ – € ˜ ˜ y – d – ™ ‘ j € – ˜ ™ ˆ ™ € d – — — “ m p – ‘ y y y – — k e ˜ „ d e m — ‘ „ — “ … — y „ ‘ — f ™ — • € – d y—  j – — — “ p ‘ y — ‘ “ ’ † — ‘ „ ˜ “ j € „ … ‘ — –mm – ˜ – ˆ – ˜ d j „ p ‘ – – € ˆ ‘ – € d € y „ d l ‘ € – ˜ — ym p – ˆ — – ˆ „ e ˜ ™ j — –m € – ˜ d “ ˆ “ ˆ –m s— — „ d ˜— – m f – y ‘ ‘ „ p ˜ — – — – ™ ‘ j € – ˜ ™ ˆ ™ € d — – — — “ m p — – ˆ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ ‘ „ ˜ “ j € „ … ‘ m – ˆ – ˜ € “ d y „ – ˜ y „ ˜ ¤ „ — “ p –m — ‘ “ ’  © ©  ©      H § B C A  A @  D H  D C E  ¤ ¨   † – € y ˜ p y € ˜— € y –m – ˆ € ˜ € “ d n q 1 — ˜‘ – j y p „ ˆ – ˆ ‘ „ ˜ “ p — — “ m p “ m – ˆ – € ˆ “ p –m — ‘ “ ˆ — – ™ € y ˜ p y € ˜ —  j – — — – ™ ‘ ‘ „ ˆ — – ˆ — “ p y “ ™ — — – € ™ ˜‘ — – j j „ — — y „ ‘ — y „ ‘ f ˜‘ – j – € km y p ˜ € “ d — ym ƒ † — € y –m “• — ˜ y s € ˜ ˜ “ ˆ – j € „ … — y „ — – y y ˜‘ – j – € ˜ y “ — – ™ ˜‘ – — ™ € d – € — – ™ ‘ ‘ „ ˆ – ˆ – € ˆ “ p –m — ‘ “ ˆ –m s— — „ d ‘ „ ˜ “ ˜ d “ ˆ “ € y –m € y— – ˜ € „ d – ˆ ‘ „ p – — “ n † — ™ — • € – d y— ˜‘ – j –mm – ˜ € “ d y „ — ™ — • € – d y—  j – — f — ™ — • € – d y— — – l “ — — ˜‘ – € d d “ ˆ – € ˆ “ p –m — ‘ “ ˆ — – ˆ „ e ˜ ™ j — –mm – ˜ – ˆ ‘ „ ˜ “ pm d d “ m € y— – ˜ € „ d – € k j – € d “ n † l ‘ € – ˜— ym p – ˆ — – ˆ „ e ˜ ™ j u y “ — – ™ ˜ € „ d d “ – € ˜ ˜ ‘ – • y – d  y y — ‘ „  — ‘ – ˜ u – ˆ — – d  ˜ u y – ˆ — ‘ „ ˆ € „ s “ — y „ ‘ f – € ˜  d “ ep – p — ‘ “ ’      © © © © ¨ ¡ ¤  ¡ § ¢ ¢ ¨ ¡ ¤ ¨ ¦ ¥ ¨ ¦ ¡ § ¡ ¢ § ¦ ¥¤ ¢ ¡ © – € s j „ ‘ f ‘ „  ˜  ˆ ™ ˆ – p ‘ “ ˜ —  ˆ “ m ‚ — ˜ ‘ – j y p „ ˆ — – m ˜ ‘ “ ˜ ‘ – — ™ € d – € — – € s € “ — – m – € ˜ ‘ – ™ ˜  j  u„ € d “ m € – € “ d j „ p € y „ d — – p ‘ “ ˜— ˆ – ˆ ‘ „ ˜ “ —m ˜ y m € y— ˜‘ – — “ s – — — – ˆ „ e ˜ ™ j — – ‘ “ ˜ € – i † — ˜‘ – j y p „ ˆ — – p ˜ ‘ “ ˜ ‘ – — ™ € d – € — – € s € “ — – m € y — ˜ ‘ – j – ˜ p – €  ˆ ˜ ‘ – m m  “ •“ € ˜ n q 1 — ˜ ‘ – j y p „ ˆ – ˆ ‘ „  ˜ “ p  — — “ m p ©  — “„m s–€ˆ“ ––yjy€˜„“… j— y™m„s—„™€ d˜‘“–—m ™€ d˜–‘€– ynyq“˜1˜“©˜—‘– yjyyp—„–˜ˆ‘“–˜ˆ—–um–d j––ˆu„–e‘˜™yj–˜—‘––ˆ—™˜€€d“ dy†m d –“€nyl†–˜“‘n–p—–€ 7 ' 4 0  6 5  ) 7 0 ) 4  8 ' ( ' 4 0 7  3 0 ¤ @ ¢ £¢ §¦¥ ¥ ¦ † – ˜ ‘ – p — – € „ s € “ – € y ˜ p y € ˜ — € y – m – ˆ ™ ˜ j u„ € d “ m ‘ „ m – — — – d y „ € l — ˜ ‘ – € ™  ˆ ‘ – € – € “ d ™ — – ˆ ˜ l “ — m y y n q 1 — ˜ ‘ – j y p „ ˆ – ˆ ‘ „ ˜ p –mm „ p – ‘ y p ‘ „ ˆ ˆ ‘ „ d— – € € „ p – j e ˜  € „ l m “ m – ˆ – ™ € ˜ ‘ – n † ¢ “ f † m “ ˜ – € –  m m  ˆ ‘ “ i¡ — – ˜ —  m  s “ s „ € d — – m k ˆ „ j — – ˆ ˜ ‘ “ —  m  ˜ y ‘ – ˜ – f ˜‘ – j – y y ‘ y – € y ˜ p y € ˜— € y –m – ˆ € ˜ € “ d n q 1 — ˜ ‘ – j y p „ ˆ – ˆ – ™ — • € – d y— ‘ „ ‘ ˜ – – ™ — • € – d y — ‘ „  ˜ “ p  — — “ m p “ m – ˆ – € ˆ “ p – m — ‘ “ ˆ u y “ •“ € ˜ — „ ‘ ‘ „  ˜ p – — – ˜ ˜ – p — ‘ “ ˆ — ‘ „ ˜ ‘ – — ™ € d — y „ † ¢ f† m “ ˜ – € – „ ‘ – ’¡ – € y ˜ p y € ˜ — € y – m y „ y ‘ – ˜ ‘ „ p € y – m € y — ˜ ‘ “ — “ s – — ‘ – — ™ € y ˜ p y € ˜ —  j – — — ˜‘ – j y p „ ˆ – ˆ ‘ „ ˜ “ p — — “ m p “ m ˜‘ “ ‘ € – p ‘ „ p ™ l € – j ™ “ –mm – • y „ ‘ – y y ˜ “ j ™m s „ € d – ‘ y f ¨  ¤  ¥ ¤¥ £ ¡ ¡ £   ¡¨   ¤ £ ¥ ¤      ¤ ¢ 1 ‡ r ˜ – „ € d y ˆ – € ˆ “ p – m — ‘ “ ’ B A E  A B E H C   C D D  G  A @  D H  D C E  ¤    !© © © YY© YY ¡  ¢ C     ¡ ¤ ¦ £ † – p ‘ “ ˜ —  ˆ – ˆ ‘ „  ˜ p ‘ „ … – ˜ ˜ – p ‘ „ m – — – ep „ € d — ym d – m € – ˜ — ym p y “ ™  p „ — — “ — € „ m “ ˜ ‘ “ ˜ ™ ˜ –  s „ ‘dy“©fˆ—–—p™‘˜“m˜—s“ˆs—„–€ˆd €—“–dm f —––y™yp“˜m—d˜j“˜–—€ —˜“‘d– j—––m©“‘€™–‘™—™lm˜˜y‘„l—‘—€€––˜˜——yymmpp –uˆy“– ˆ—„˜–e˜s™„j—“–mˆ—–pf ‘‘“‘‘–‡˜€“ d ˜ → P (c|−i ) = x k =1 K → P (−i |θk ) × P (c|θk ) x  © © © ‚ ˜ y — – j j „ p — – ™m y pm “ p ˜‘ „ — — – — — “ m p u y “ — ˜ – s „ — – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ — ™ ˜m s “ s „ € d — – m f — – — — “ m p — – ˆ – ‘ y p “ ep – p ‘ “ ‘ – ˜ € “ d d “ ˆ ™ ˜m s “ s „ € d – ‘ y – ™  p „ — — “ ˜ — – € – ˜ — ym p – y y “ ep ¤ „ f — “ p ˆ ‘ „ p – — – m — ‘ “ ’ † – ˜ € „ … — ym d “ m – p ‘ “ ‘ – ˜ € “ d d “ ˆ ™ ˜m s “ s „ € d “ m ™ ‘ – j ˜ ‘ “  “ – — — “ m p “ m ™  p „ — — “ – ˜  y — ‘ – ˜ — – ˜ – s „ m ˜ – f – — — “ m p – y y “ ep ™  p „ — — “ — ‘ „  ˜ y s  € ˜ —  ˆ – ˆ – l ‘ “ m ™ j – y y “ ep ˜ – s „ m – • y „ ‘ – y y “ e p – ˆ – p ‘ “ ‘ – ˜ € “ d d “ ˆ ™ ˜  m  s “ s „ € d “ m € – m y p m “ p – ˆ ˜ l “ — m f — “ p € – j – € d –m — ‘ “ ˆ — € „m “ f & % % € y „ d – j j „ p f – y y ˜— ˜ “ ˜ — ˜ — – – ™ —m ˜ y l ‘ € – ˜ — ym p – ˆ – ˆ „ e ˜ ™ j “ m — f — –m d j – u– u y “ – • y „ ‘ – ˆ ‘ „ ˜ “ p — — “ m p “ m – ˆ y “ – • ‘ y “ f – ˜ y — ‘ ‡ † —–——“mp —– ˆ – ‘ y p “ e p – p ‘ “ ‘ – ˜ € “ d d “ ˆ ™ ˜  m  s “ s „ € d – ‘ y – ™  p „ — — “ ˜ — – € – ˜ — y m p – y y “ ep —  “ j f — – — — “ m p u y “ ˜ ‘ – j – ˜ p – € ˆ — ™ p „ — — “ — “ d — € „m “ ˜ ‘ „ — – ‘ — € – ˜— ym p — –m f — “ p – j k u y – ˆ – p — ‘ “ ’ † ˜‘ “ y ym d ™ € –m ‘ – — ˆ „ d – ˆ — ym d ˜‘ “ ‘ ‘ „ ˆ  ym ‘ – ˜‘ – j –mm – y ˜‘ – • ™ f ˜ y s € ˜ ˜ “ – € ˜ y “ ‘ y – j j „ p – — — “ m p – ˆ ˜ y s € ˜ ˜ “ m ˜ ‘ “ ˜ “ € ˜ ‘ – – — “ s “ m – ˆ –m s j – — ‘ – m € y— ˜ ‘ – j – ˜ p – € ˆ l ‘ € – ˜ — ym p –m € – p ‘ “ m — € „m “ – ˜—— ‘ „ p – • ˜ “ ‘ € – ˜m “ – € ˜ y “ – ‘ ” † € € • y „ p – € – — – ˆ ˜‘ – y y — € — ™m s p — ˜ d – p ‘ „ p — –m ˜ – f — – — — “ m p — – € ˜ y “ — – ˆ u y – p ˜ ‘ “ — „ d d „ — – m ‘ – — ˜ y € ˜ — ‘ „ p — “ d ˜ ‘ „ — – ‘ – — — “ m p – y y “ ep ˜ ‘ “ ˆ ‘ „ d — – € € „ p — – m k ˆ „ j — – m f — “ p – p — ‘ “ ˆ —  “ q † ¢ f  ‘ “ € e — s t ˆ ‘ “ –  ˜ — “ ¡ ˜ ‘ – j j “ ˆ ‘ – d ™ ˆ ‘ – — — “ m p – y y “ ep € y — l ‘  € – ˜ — y m p – m € – ˜ y p ™ u – ˜ ‘ – j – m d j  — – ˜ —  — ‘ „ p – l “ — —  ˜ ‘ – € d d “ ˆ – ep „ € d d “ – € k  j – € d “ n † l ‘ € – ˜— ym p – ˆ – ™ ‘ ‘ „ ˆ – ˆ „ e ˜ ™ j – ‘ y p – •“ ™ —  • € – d y — – l “ — —  ˜ ‘ – € d d “ ‘ y € – y ˜ p – – € y „ d — – m s “ – l “ —  • ‘ – — € „ m “    ©      ©  © ©  ©  # # ©    ©  ! !# ¦ ¤ ¥ ¢ ¡¨ ¢ § ¦ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ ¨ ¡ ! © – ˆ – € s j „ ‘ y “ — € „m “ ˜‘ – ˆ ‘ „ d— – € € „ p — ˜‘ – j y p „ ˆ — ˜‘ – € ™  ˆ — –m — ‘ “ ˆ — ˜ y s € ˜ ˜ “ u y “ – • y „ ‘ — – p – ˆ — € y –m “• — – n † — – ™ € ™ ˆ— ‘ „ p n q 1 — – ™ ‘ ‘ „ ˆ — – ˆ –m s j – — ‘ – m — ‘ “ ˆ — – ™ € ˜‘ „ p ‘ – € ˜ ‘ „ — —m – s “ m – € ˜ ‘ – — – ˜ p ‘ ˜— ˆ — ‘ „ ˜ “ m – € – ˆ – y y — ˜ y s € ˜ ˜ “ u y “ – • y „ ‘ – ˆ ˜‘ “ ˜ y “ p ‘ „ ˆ — ‘ „ ™ € p — y „ † — ˆ y ¤ ‘ u y“ — ™p „ —— “ —m – s “ m – ˆ — – ‘ — – p ‘ – y y ™ — — – ˆ –m s j – — ‘ – m ˆ ‘ „ d — – € € „ p ‘ „ ˜ ‘ ™ ˆ – ˆ – ‘ “ j „ ˆ –m ˜ ‘ „ ˆ f – ‘  p “ € “ m – ˆ ˜ ‘ “ € € “ j ™ ˆ — ‘  j – e p — y „ — ˜ – — ‘  j – ep — – ˆ – m s j – — ‘ – m – y y  — ‘  “ f — ˆ y ¤ ‘ u y“ — ™p „ — — “ —m – s “ m – ˆ — – € “ d — – ˆ –m s j – — ‘ – m ˜‘ „ — ‘ „ ˜ ‘ ™ ˆ – ˆ — – ‘ “ j „ ˆ — –m ˜ ‘ „ ˆ f §£¨    ¤ £ ˜ ‘ “ •  y — — – € k € … — ‘ „  ˜ “ m – € — – ˆ ˜ – — m – € k d — ‘ „  ˜ “ m – € — – ˆ – m s j – — ‘ – m  — — y “ —  “ j f – € s€“ m — ‘“ ˆ — ˜‘–— ™ € d —m– s“m — – ˆ –m s j–— ‘– m ‚ — ™ € ˜ ‘ „ p ‘ – € n q 1 — – € s € “ — – ˆ ‘ y p “ ep — ‘ „  p „ — — “ — y „ ‘ ‚ – ˜ ‘ “• y— “ m — € „m “ ˜— – € – —m ˜ y ˆ — ‘ „ — „ d „ € d — y „ ‘ – y y — – € s € “ — – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d – € “ n † — ‘ „ ˜ “ j € „ … ‘ ˆ ‘ „ ˜ p “ € ˜ u– ˆ – € ˆ “ p ‘ y — ‘ “ ˆ — ˜ “ ˜m y — ™ € — ‘ „ s — k € ˜ – ˆ € ‘ – ˜ s „ ˆ ˜ – j € – d ˜ – f – ep  € — ym d ˜ — – — – € s € “ — – m € – j € „ …— ‘ “ € € y „ d – ™ —  m  ˜ y — € y – m “ • — ˜ y s  € ˜ ˜ “ ‘ – ‘ „  ˜ “ ˜ ‘ – — ™ € d – € “ m f – €  “ € ˜ ‘ „ p y “ f ¢ f † m “ ˜ – ‘ „ € – m m  h¡ — ‘ “ ’ †  ¨ ¤  ¨ ¤ ! ¢   – m d j  — ‘ y ˜ — – — – € s € “ — – m € y „ d  —  „ ep ˜ ‘ „ — m  y y ‘ „  ˜ “ ˜ ‘ – — ™ € d – € “ m —“ j f ‘ „˜“ p —— “mp € y–m – ˆ — € „m n q 1 — ˜‘– j yp „ ˆ — – ˆ – € y˜p y€˜— “m – ˜ d j „ p ‘– – € ˆ ‘– € d € y „ d – ™ ˆ  – mm – ˜ – ‘ y ™ — m ˜ y ˜ ‘ „ — € y – ˜ y “ — – m f ¢ f “  q ‘ – ‘ „ e # ˆ ‘ “ ˜ – p y „ ’¡ — ‘ “ ’ † — € y –m “• — ˜ y s € ˜ ˜ “ ˆ – j € „ … — y „ — — – ™ ˜‘ – — ™ € d — – ™ ‘ ‘ „ ˆ — – ˆ € – ˜ “ € ˜ – ˆ — –m s “ d “ p — – ˜‘ “ ˜— u– ‘ „ ˜ “ p — — “ m p – ˆ — – ˆ „ e ˜ ™ j — – ˆ — ˜ y „ ˜ “ — – ˆ € – p ™ ‘ ™ s — € „m “ — ‘ „ • y „ d — y „ ‘ f — ym d – ’ † ™ m d j— – j km s „ € d –m ˜ – f — – ™ • € – — ‘ „ p – € ˜ ˜ ‘ – • y – d — – € s € “ — – m — ‘ “ ˆ — – y ‘ – ˜ ‘ „ p — – m “ d p ‘ € d — ‘ „  ˜ “ j € „ … ‘ — – m f ˜ ‘ – ‘ 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“ l “ j “ m “ ’ f f e —  ˆ “ l “ x ˆ ‘ “ ‘ “ j € –  ¡ – € ˜ y “ ‘ y ‘ – – € s € “ ‘ y € – l ‘ “ ep € y „ d — – €  “ — — – p ™ ‘ — ‘ „  ˜ “ ˜ y j – ˆ j y j  ‘  j       ©   ©  ©  © ©    © © © © © © YY   ¥ © ©   YY    © !  ¥ ¥  © 32 1 !   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j€– d — “ d ˜— – ‘ ‘ „˜“m– € – ˜ ˜ – p m – y y –m € y „ d –m s j – — ‘ – – € ˜ y “ ‘ y ˆ – — j € – d ˜— – —m – s “ m — ‘ “ ˜ € – p – € ˜‘ – ˜ ‘ “• y — – €k €… y „ —m – €k d – ™ ‘ ‘ „ ˆ ‘ „˜“m– € – ‘ y m– y y –m € y „ d — ˜‘– j yp „ ˆ – ˆ –m s j–— ‘– ‘ y — ˜‘ – € ™  ˆ ˜‘ „ — —m – s “ m — ‘ “ ˜ € – p – ˆ – € s j „ ‘ –m —m – y y — –m € y „ d y „ f — ˜‘ – € ™  ˆ —m – s “ m — – ˆ ˜‘ “ —m ˜ y — ˜ ‘ – j y p „ ˆ – ˆ — –m s j – — ‘ – u y – ˆ ‚ –m d j – u– € “ d € – p ‘ – € ™  ˆ – ˆ – € ˜ ˜ – j € – d ˜ “ € • – ˆ ‘ „ ˜ “ ˜‘ – — ™ € d – € –mm – ˜ – ‘ y f ˜‘ – j – • ˜ y ˜‘r † —m– € y˜“ ‘ — – € s j „ ‘ — – ˆ –m s j– — ‘– m — ‘“ ˆ — € y–m“• — € y–m — y „ ˜ ˜‘– j–m“ ‘ p ‘ „ ˆ ˜‘– ‘ ‘– € d — ™ ™ €p — ˜ y s € ˜ ˜ “ u y “ – • y „ ‘ — – n † ˜‘ – j y p „ ˆ –m — ‘ “ ˆ —m — – — – ˆ – € s j „ ‘ y “ – € ˆ ˜— – p f ˆ y ¤ ‘ y ˆ ™ ˜ € “ m ˆ ‘ „ d— – € € „ p ˜‘ – j y p „ ˆ ‘ y € y „ d ˜ y s € ˜ ˜ “ m – ˆ € y –m “• “ m f ˆ y ¤ ‘ – ˆ ‘ „ ˜— „ d – ‘ y ˆ ˜ ‘ “  ˜ ‘ – ˆ  – y y “ ep € y „ ƒ † p ˜ – f — m ˆ ‘ „ p – — ‘ „ — f — m € –  j – € d ‘ „ — f – ™ ˆ „ p ˜ — – – ‘ p “ € “m ‚ — €–˜‘– ˆ – p ‘– y y ™— – ‘ y €“ d ™ ˆ „ p – €˜ –m d j– u– €“ d ˜ y– d ˆ y ¤ ‘ m– ˜ ‘ y ˆ ˜‘“ ˜‘– ˆ n † — – € s € “ — –m — ‘ “ ˆ — ˆ y ¤ ‘ — – ˆ — – ˜‘ – € ™  ˆ — – ym „ — s “ — ‘ „ ˜ — „ d – ˆ – ˜— u– m y y — ˜ y s € ˜ ˜ “ u y “ – • y „ ‘ – ˆ ˜‘“ ˜ y“ ˜‘– j–m“ l™ — ‘ „ —— ‘ ™ ˆ — y „ ‘ f ‘ ‘ ‡ † — ˜‘– j yp „ ˆ — – p — ‘“ ˆ – p ‘– € yp p „ € y–m ¥ ¥ ¥ 0 0.0 0.1  Y Y ¦ ¤ ¥ ¢ ¡¨ ¢ § ¦ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ ¨ ¡ ! !   © © ˜ " ˜ ym d –m s— — „ d – y y — k ˆ ‘ „— • ˆ – ‘ y — ‘ „ y ˜ p – – — y „ ‘  „ y y € y „ d ˜— – i † — – y y € – —m ˜ y — ‘ „ € ™ …™ € d — y „ ‘ f ˜ ‘ – j –mm – j € „ … — ym ƒ † y “ – • ‘ ˜ y “ e — ym d – ˆ — ˜ y s € ˜ ˜ “ — – ˆ – y y ¤ ¨    — ym d ˜‘ „ —  y y y “ – • ‘ — “ s — ym d – ˆ — ˜ y s € ˜ ˜ “ — – ˆ € y— – ‘ ™ ˆ ˜ „ — – l “ d y „ p ™ ˆ – ˆ –m l k € – ‘ y y y — ‘ „ € ™ …™ € d — y „ ‘ f – d “ ˜ ™ – ˜ ˜ – p © † — € – ˜— ym p — ˜ ‘ – € ™  ˆ — ‘ “ ˆ – ™ € “ d ™ — ˜ „ — – — — “ m p – ‘ y y y — “ d ˜ – j € – d – ‘ — “ j € – ˜— ym p – j j ‘ y — ‘ “ ˆ — – — — “ m p — € y –— ym d – ˆ – p ‘ – — ™ € d “ m – — € „ ˜ y “  y y l ‘ € – ˜— ym p – ˆ – — “ e d – ‘ y ‘ – – ˜—— ‘ „ p ˜ – o – j e ˜ € „ lm “ m € “ d – ˜ € p ™ ˆ ˜ — – – d “ ˜ ™ – € k j – € d “ n † –m s— — „ d – y y –m s— ‘ – e ™ € d j „ p — — y “ – € ˜ – € k ‘ “ j – ˆ ˜ y € ˜— ‘ „ p ‘ „— p ™ ˆ – ˆ – € s € “ ‘ y — € „m“ ˜— – – ˆ „ e˜™ j “m – ˆ –˜€ „— “ n † — –—— “mp — –m €“ d – ™ ˆ y l ˜‘– j– ˜km d j „ p – d“ ˜™ – jk u y– ˆ – ‘ y ˜ – f — ˜ – s „ — – ˆ — – y‘ ‘ „ p — – — — “ m p — –m – —m ˜ y — “ j l ‘ € – ˜ — ym p ‘ y – j j „ p ˜ l “  y y – d “ ˜ ™ – € k j – € d – ‘ y ‚ — –m “ d p ‘ € d — – d “ ˜ ™ u y – ˆ – ˆ – ™ — „ d j „ p — € „m “ ˜— – – ˆ „ e ˜ ™ j –mm – • y „ ‘ “ n † n q 1  i<j  Aj  ©© Ai — ˜ ‘ – j y p „ ˆ – ˆ – ™ —  • € – d y — ‘ „  ˜ “ p  — — “ m p “ m € y „ d € – ˜ d “ ˆ “© m – ˆ ˜ ‘ “ ‘ – ˜ ‘ “ j — ‘ „ — „ d „ € d — y „ ‘ f — –m s— ‘ – e ™ € d j „ p — ˜ “ ˜m y— ™ € — – ˆ € – ˜ ‘ – — ™ € d  & %  % – ˆ ™ ˜ p “ d “ p “ m – ˆ € – p ™ ‘ ™ s – ˆ ‘  # 7 ' 4 0  6 5  ) 0 ) 0  7 8  3 0 6 7 4  8 ' ( 5 8 7 7 ( 1 ¢ £¢ §¦¥ © ¥ ¤ §  $  ¦ † –m s— — „ d — ˜ y s € ˜ ˜ “ ˆ j y j ‘ j ‘ y € “ d – ‘ ™ ˆ ˜ – f † ‘ „ ˜ p – — “m — ‘“ ˆ – ™ ˜‘–— ™ € d – j j „ p – ™ ™ €p f –m lk € – ‘ y – p ‘“ ‘– ˜€“ d d“ ˆ ˜— – ˜ ‘ y ˆ ‘ „ d— – €€ „ p ˆ y ¤ ‘ – y y “ ep m – y y – m — ‘ “ ˆ ‘ „  —  p ™ ˆ – ˆ – € s € “ ‘ y — € „ m “ ˜ — – – € y ˆ ™ p „ € d “ m – ˆ –  ˜ € „ — “ n † — ˜ ‘ – j y p „ ˆ – ˆ – l “ ˜ ‘ “ •“ ˆ ˜ ‘ “ ‘ – ˜ ‘ „ p — € – ˜ — y m p — – ˆ — ‘ „  —  •  ˆ — – m € – € ™ … ™ € d € y „ d € – ˜ — y m p y ˆ — – € s j – j – ˆ – € s j „ ‘ – m € “ d ™ € ™ ˆ ‘ „ d f € – ˜ — ym p m y – — ‘ y p – •“ ‘ „  ˜ ˜ € “ d – ‘ y ˆ – p ‘ “ m s j – —  “ € • l „ m “ m ˜ – — € – ˜ — y m p u y – ˆ p – •“ ‘ „  ˜  ˜ € “ d – ‘ y ˆ – p ‘ “ m s j – —  “ € • l „ m “ m – € ˜ ‘ – ˜ € „ d d “ € – m € “ d ™ y m “ • ™ ˜ — – – ˆ — ˜ y s € ˜ ˜ “ — –m € y— € – ˜— ym p ‘ y ˆ — ˜‘ – j ™m ™ — –m € – ‘ ‘ „ ˜ ˜ € “ d – ˆ ˜ € ™ ˜‘ n † ˜ ‘ – ˜ ˜ “ ˜ „ — — € – ˜ — ym p – ˆ y ˆ ‘ – ˜ ˜ “ – € s j „ ‘ –m – y y – p y y — y f — –m s— — „ d — ˜ y s € ˜ ˜ “ ˆ — –m s j – — ‘ – — –m € y— € y „ d — ˜‘ “ € y „ p — € – ˜— ym p — – ˆ — –m s— — „ d — ‘ „— • ˆ — – ˆ –m s j – — ‘ – m  j € “ d j y j  u “ j ˜ € ™ ˜ ‘  ˆ ‘ „  —  •  ˆ “ m – d “ ˜ ™ – y y “ ep €  —  „ ep – ˜ —  — ‘ „ p – m m ‡ † – j e ˜  € „ l m “ m € “ d — – ™ ˜ ‘ – — ™ € d ˜ ‘ „ — l ‘ € – ˜— ym p – ˆ – ˆ „ e ˜ ™ j –mm – • y „ ‘ – € ˜ „ ‘ – ˆ — –m “ d p ‘ € d — – d “ ˜ ™ — – n † — € – ˜ — y m p u y – ˆ € – ep € – e p – € € y „ d ™  p ‘ “ ˜ — ‘  & % % – j e ˜  € „ l m “ m ˜ ‘ “ —  m  ˜ y ‘ – f — ˜ y s  € ˜ ˜ “ — – ˆ – m s j – — ‘ – m p – •“ — ™ j € „ … — ‘ “ € ˜ ˜ – € – ˜ — y m p –m — ‘ “ ˆ — ym p ‘ — ˜‘ – j y p „ ˆ — – ˆ ™ — „ d j „ p — – ™ ‘ ‘ „ ˆ – ˆ y – y ˆ u y – ˆ ‘ – ˜ ‘ – j – ‘ ‘ „ ˜ ˜ € “ d –m — ‘ „ ˜ „ ‘ — y „ ‘ f ‘ ‘ ‡ † — ˜ y s € ˜ ˜ “ ˆ — –m s j – — ‘ – – ˆ ™— „ d j „ p p ‘ „ ˆ ˜— – † – ‘ p “ € “ m – ˆ ˜ ‘ “ € € “ j ™ ˆ — ‘ j – ep — – ˆ – m s j – — ‘ – m ˜ — – ˜ – f — ˆ y ¤ ‘ – ˆ — ‘ „ ˜— „ d — – ˆ –m s j – — ‘ – m ˜— – f ˜ ‘ “• y — — – € k €… — ‘ „ ˜ “ m – € — – ˆ –m s j – — ‘ – m ˜— – f — – € s€“ — –m — ‘“ ˆ — ˜‘–— ™ € d —m – €k d — ‘ „˜“m– € — – ˆ –m s j–— ‘– m ˜— – f n q 1 — ˜‘– j yp „ ˆ — –m — ‘“ ˆ — ˜‘–— ™ € d —m– s“m — – ˆ –m s j–— ‘– m — ‘ „m– d d“ — y „ ‘ ‚ – ˜ ‘ “ — — „ € p ™ ˜ u–m d j „ p – ˆ — – d y „ € l — ˜ ‘ – € ™  ˆ ‘ – ™ ‘ ‘ „  ˜  ˜ € “ d – € ˜ ˜ y – d f ˜  y ˆ „ € ˜ ‘  — ‘ „ •“ m — y „ ‘ – j j „ i † n q 1 — ˜‘ – j y p „ ˆ u y “ — ™ p „ — — “ — –m s— — „ d — ˜ y s € ˜ ˜ “ — – ˆ –m s j – — ‘ – m – ˜ „ ‘ ™ ˆ † — ‘ „ ˜ “ ˜ „ ‘ — – y ym – y y — ‘ „ —  y ˆ „ € ˜ ‘  f – ™ — „ d „ € d l ‘  € – ˜ — y m p – ˆ – € y ˆ ™ p „ € d “ m € y — — m  “ ˜ ™ ˆ – ˆ — y m d € – ‘ ‘ „ ˆ – ˆ ˜ ‘ “ •# † — ˜ y s  € ˜ ˜ “ ˆ ˜ ‘  – € ˜ — – € – m s j – — ‘ – ‘ y € y — ˜ — – ˜ ‘ y – ˜ ‘ – — ™ € d – € ˆ y ¤ ‘ – y y “ ep m– y y –m — ‘“ ˆ ‘ „—p ™ ˆ – ˆ – € s€“ ‘ y — € „m“ ˜— – – ˆ „ e˜™ j “m – ˆ –˜€ „— “m ˜– f — ™ €™ ˆ— ‘ „ p ˜‘ „— — ˜ y s € ˜ ˜ “ — – p € y— — ˜ – s „ — – p – ˆ u y – ˆ ‘ – ‘ „ ˜ ˜ € “ d – ‘ y ˜ – f — ˜ – s „ — – ˆ – ˜ y ˆ ™ € — ym d ‘ – — ym d – ˆ –  ˜ € “ d – ‘ y f — ˜ y s  € ˜ ˜ “ — – ˆ –  ˜ € “ d – ‘ y – m y – — f – d “ ˜ ™ – y y “ ep f  — ‘  # † — – € s € “ u y “ ™  p „ — — “ — ˜ y s € ˜ ˜ “ ˆ ™ ‘ ‘ „ ˆ –m s j – — ‘ – ‘ y ‘ „m – — ˜ – – •— € y p ™ € ‘ „ ¡ “ … – ˆ u y – ˆ ‘ – ™ — • ˆ ˜ „ — — – ™ ‘ ‘ „ ˆ – ˆ ©©  ©  Ck Ai k ∈ [1, K ] © ©  Ck Ai  Cut(Ck , Ai ) ©  ! Ai ∈ A © ©   © © © ©  ©  ©  © ©  SA = 5 © © © © ©©  A5 A  Y Ck Ck A ¥ A2 A3 A4 A1 ¥ ¥ ¥ ¥ A  ¦ ¤ ¢ £ ¨ ¤ £ ¨ ¦ © §¤ ¦ ¦ ¤ ¢ ¡ ¡ ¢ ¥ ¡£ ¡ ¦ ¡ ¢ © ¦ ¡¤ ¨ ¡  ©  † — „ … “ m — ˜ y s € ˜ ˜ “ ˆ –m s j – — ‘ – ‘ y ˜ ‘ – j –m y – — € y— ™ ˜ y p ™ u– ˜ — – — ˜ — – ˜ — – p – ˆ ‘ y p “ e i † — – ˜—m s “ s „ € d — –m k ˆ „ j — – ˆ € y — – ˜—m s “ s „ € d ˜— – ˜ ‘ y ‘ – s y „ f –m l k € – ‘ y – p ‘ “ ‘ – ˜ € “ d d “ ˆ ˜— – ˜ ‘ y – € ˆ ‘ „ d — – € € „ p ˜ ‘ – • y – d —m ‚ — – ˜ ‘ – € ™  ˆ — – € y ˜ “ ‘ u y – ˆ – ˆ – € ˜ p ‘ „ ˆ ˜ ‘ – • y – d ˆ y ¤ ‘ – y y “ ep € y — — ˜ — – ˜ — – m f m “ ‘ ‘ „  —  p ™ ˆ – ˆ – € s € “ m — ‘ “ ’  ©   ©! © † ‘ „—p ™ ˆ – ˆ – € s€“ m — ‘ “ ˆ ˜‘ “• y— ˜— – ˜ – j j „ p ™ ‘ ‘ „ ˜ p –m ™ — ˜— – –m s “ … — ym d –m € y – € € – ˆ u y “ ˜ y “ ™ ‘ – j “  y y  y m – p ˜ – f – ˆ — ˜ y s  € ˜ ˜ “ ˆ – m s  — — „ d – m s j – — ‘ – – y y “ ep € y — — ™ € ™ ‘ ™ l — – ˜ —  m  s “ s „ € d — –m k ˆ „ j – ˆ – ™ — „ € p ‘ „ ˜ “ ˆm “• ‘ – € y – € € – m — ‘ „ ˜— – ˜ — y „ ‘ — € „m “ f – ™ • y „ € ˜ ™ ˜ ™ “ ‘ ™ ‘ ‘ „ ˆ € – ˜— ym p ‘ y — ‘ “ ˆ — – € ˜ y “ — – ˆ – — — “ m p – ‘ y € – y l ‘ ˜ — ˆ – ˆ –m s “ d “ p –m l k € – ‘ y p y “ — — y ƒ † † – m s — — „ d – y y — k ˆ – ™ ˜ y p ™ u – ˜ — – ‘ „  —  • ˆ – ‘ y f ˜ ‘ – j j – ˆ ™ p ™ € d ™ y y m d u – — ‘ „ •“ m — y „ ‘ – j j „ i † ‘ „ — p ™ ˆ – ˆ – € s € “ m — ‘ “ ˆ ˜‘ “• y— ˜ — – ˜ – j j „ p – ™ —m ˜ y ˜— – –m l k € – ˜ ˜ – p — € „m “ f — – € ˜ y “ — – ˆ – — — “ m p – ‘ y € – y l ‘ ˜— ˆ – ˆ –m s “ d “ p ˜— –  y y – ™ • y „ € ˜ ˜— – –m l k € – ‘ y  z † A © © © © © ©   † — – u–m d j „ p — ym d ‘ „ ˜ “ € “ d ™ — – ˆ — – ‘ „ — – ˆ – €  € p ™ ˆ – ˆ — – m s “ d “ p f — – ep  € — y m d — – m k ˆ „ j — – ˆ ˜ ‘ „ —  y y f — – ˜ —  m  s “ s „ € d — – m k ˆ „ j – ˆ – —m ˜ y – d “ ˜ ™ – ˆ ‘ „ p – — “ m – y y — ˆ ‘ “ ˜ f –m s— — „ d –m s— ‘ – e ™ € d j „ p — ym d –m – € ˜ ˆ ˜ y s –m — ‘ “ ˆ f — – — — “ m p — – m € – y l ‘  ˜ —  ˆ € y „ d — – m l k € — – ˆ € – —  m  ˜ y ˆ – “ — — – – € k  j – € d “ m ‚ — – d “ ˜ ™ u y – ˆ – ˆ – ™ — „ d j „ p – j j –mm – ˜ — – –mm ‡ † — € – ˜— ym p — – j j — –m — ‘ “ ˆ — ym p ‘ – € „ p ‘ – ˜‘ „ —  y y — – ˜‘ – € ™  ˆ — –—— “mp – ˆ — –m d j– u– — –m €– €“ d™— – ˜—— ‘ „ p ˜– – ˜‘– ˆ ™ p ™ € d – d“ ˜™ m – ˆ –˜€ „— “m – ™ €˜‘– ‘ – ˆ ‘ – € d –mm ‡ † – j e ˜ € „ lm “ m € “ d – ˜ € p ™ ˆ ˜— – – ˆ „ e ˜ ™ j – € ˜ „ ‘ – ˆ – d “ ˜ ™ – ˆ ‘ „ p – — “ n † – ˜‘ “• y— – d “ ˜ ™ m ™ —m ˜ y “ € – — ˜‘ – j – ‘ ‘ „ — “ € – j j – n † l ‘ € – ˜— ym p ‘ – ˜ “ … ˜ — – m – j j „ p — –m s— — „ d — ‘ „ — • ˆ — € y –— ym d – ˆ ˜ € ™ ˜ ‘ m € – € “ d j „ p – ˆ – y y   ! ©  © " © © © © nb clus − 1 † — ˜— – ˜  DH©V D ¨©D% p – •“ f ‘ „  —  p ™ ˆ – ˆ – € s € “ m ! O ©  © K =K +1 j y j u “ j ˜  € ™ ˜ ‘© ˆ ‘ „  —  • ˆ “ m € – ˜ y p ™ u – ˜ – € – ‘ ‘ „  ˜ p – m ™ —    ¨D¨ D – ˆ ˜€™˜‘© m €–m yHHpGm“ p   HH G DH V O ‘ „—p ™ ˆ – ˆ – € s€“ m €– ˜ y „ “ m ˜– – ™p „—— “ –m lk € “m €–m ypm“ p D K=1 C1 = D O=∅ K = nb clus k ∈ [1, K ] i ∈ [1, SA] Cut(Ck , Ai ) – ˆ • ‘ „—p ™ ˆ – ˆ – € s€“ m–• y „ ‘ ‘ y €– ™ €p – ˆ — ˜ ‘ – j y p „ ˆ — – m — y „ ˜ p – •“ € – ˜ — ym p – y y  ‘ y m € – — m “  ˜ ‘ ¢ ¢ D † © O C1 nb clus ¡ ¡ ‘ „ ˜ ˜ € “ d “ m – ˆ — € – ˜— ym p – ˆ ˜‘ “ € y „ p – € s j „ ‘ — € – ˜ — ym p – ˆ y ˆ ‘ – ˜ ˜ “ – € s j „ ‘ – m ˜ – n q 1 — ˜ ‘ – j y p „ ˆ – ˆ – m s j – — ‘ –© m ! D F  © ¨ & † n q 1 € y „ d & %  % B ¦ D R V ©   EH  Y ¢ ¡  ¦ ¤ ¥ ¢ ¡¨ ¢ § ¦ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ ¨ ¡ ¨DV©  † P = {Ck |k ∈ [1, K ]} – ˜ ‘ “ € y „ p ‘ „ ˜ ˜ € “ d “ m ˜ – f ‘ „  —  p ™ ˆ – ˆ – € s € “© m ! D ©   % DH V  ¨ D DH V ¨D¨D¨ D  D WH D O ˜– ‘ „—p ™ ˆ – ˆ – € s€“ m €– ˜ y „ “ m ˜– – ™p „—— “ –m lk € “m €–m ypm“ p Cut(Ck , Ai ) ©  © ‘ „— • ˆ “ m € – ˜ y p ™ u– € “ d — € – ˜ — ym p — ˜ ‘ – € ™ ! ˆ — ‘ “ ˆ – ™ € “ d ™ — ™ ˜ ™ “© ‘ – — — “ m p – ‘ y p y “   DH  V ˜–   ˜– HH G   DH V  – ˆ • ‘ „—p ™ ˆ – ˆ – € s€“ m–• y „ ‘ ‘ y €– ™ €p – ˆ — ˜ ‘ – j y p „ ˆ — – ˆ – m s j – — ‘ – m p – •“ € – ˜ — ym p – y y  ‘ y m € – — m “  ˜ ‘ ‘ „ ˜ ˜ € “ d “ m – ˆ — € – ˜— ym p – ˆ ˜‘ “ € y „ p – € s j „ ‘ † n q 1 — ˜‘– j yp „ ˆ – ˆ –m s j–— ‘– m ! † – d“ ˜™ ‚ ™ — •€– d y— n q 1 € y „ d &% % ¢ ¢ CU T = 1 O D ¢ ©  o O i =i+1 CU Tk = 1 K =K +1 K=1 C1 = D C1 O=∅ CU T = 1 CU T = 1 CU T = 0 k ∈ [1, K ] CU Tk = 0 i = 1 CU Tk = 0 i <= SA © ¡ ¡ Y  S¦ D©R V ©D F ©E¨H & ¡ D  ¦ ¤ ¢ £ ¨ ¤ £ ¨ ¦ © §¤ ¦ ¦ ¤ ¢ ¡ ¡ ¢ ¥ ¡£ ¡ ¦ ¡ ¢ © ¦ ¡¤ ¨ ¡ † ‘ „  —  p ™ ˆ – ˆ – € s € “© m ! D  ©   % DH    ¨ D V  ¨D¨ D ‘ „  —  p ™ ˆ – ˆ – € s € “© m € y „   – € ˜ ˜ – j – m s “ … — ym d – m € y – € € –© ˆ u y “ ˜ y “ ™ ‘ – j “  y y – m k ˆ „ j – m € —  „ ep  ¨D — ˜ y s € ˜ ˜ “ — –m € y— ™ € ™ ‘ ™ l – ˜ — m s “ s „ € d – m k ˆ „ j y ˆ – ™ —  „ € p ‘ „  ˜ “ ˆ m “• ‘ – ‘ „  ˜ “ p  — — “ m p – ˆ € y – € € –© m € – m y p m “ p ¨DV©  –™ y˜p– !– ™˜™“© ‘ ‘„—•ˆ – ‘HHyGp y“ ¨ D   ¨D¨ D ¨ D V © —‘–„€˜—y“p™—ˆ– ˆ– ˆ–—–—€“sm€p““© mm €€y–„yl ‘˜–—€˜ˆ˜––jˆ ˜––m s‘“„d— • ˆ “m €–˜ yp™ u– “p ˜—– –m lk€ – ‘ yHHG  HHG    —–˜‘–€™ !ˆ —–——“mp —€ y–—ym d ˜‘–˜‘„p D†HHHVG    ¦ D ©R V! ©D F ©E¨H & O O Ai i ∈ [1, SA] O k ∈ [1, K ] Ck i ∈ [1, SA] class ∈ Ck © – ˜‘ – ˆ ™ p ™ € d – d “ ˜ ™ m – ˆ ‘ „ ˜ ˜ € “ d “ m ˜ – ‘ „— p ™ ˆ – ˆ – € s € “ m † – d“ ˜™ ‚ ™ — •€– d y— n q 1 € y „ d &% %  [1, K ]} O P = {Ck |k ∈  " Y ¦ ¤ ¥ ¢ ¡¨ ¢ § ¦ ¤ ¥ ¦ ¡ ¢ © ¦ ¡¤ ¨ ¡ ©   © ©   © © © ©© © † € y – ˜ “ —  m  ˜ y m – m s  — ‘ – e ™ € d j „ p ˜ “ ˜ m y — ™ € ‘ y €  ‘ € y „ … – ˆ …  ˜ p – s „ m — ‘ “ ˆ — – ™ — „ d „ € d — ‘ „ •“ — y „ ‘ – y y & % % – ˆ — ‘ „ ˜ “ ˜ d “ ˆ “ — – ˆ ˜ € ™ ˜ ‘ m € – • € – — s „ ˜ ‘ – j –m “ l ™ — ‘ „ € € y „ d — y „ † – y y ˜ “ € d ‘ – — ˜ “ ˜m y — ™ € — ‘ „ s — k € ˜ – ˆ € ‘ – ˜ s „ ˆ ˜ – j € – d –mm – y y — ‘ „ € € – • — y „ ‘ f ‘ „ ˜ “ j € „ … ‘ ˆ – ˜ € – d – ‘ y € “ d – — y ˆ “ € ˜ – — — € y –m “• — ˜ y s € ˜ ˜ “ ˆ — –m s j – — ‘ – ‘ – n q 1 — – € s € “ — – ˆ ‘ „ ˜ “ j € „ …— ‘ “ € ˜ – ˆ – ˆ „ e˜™ j “m – y y ‘– x † n q 1 — ˜‘– j yp „ ˆ — – ˆ – € y˜p y€˜— “m € y— — – y • — – ˜‘– €™  ˆ ˜‘“ ˜‘–— ™ € d – € — ˜ y s  € ˜ ˜ “ ˆ — – y y  p ™ d — — – m s j – — ‘ – — € y –  — y m d – ˆ – p ‘ – ˜ —  u – m € y — — ‘ „ •“ — y „ ‘ – y y – p ‘ “ — —  “ ‘ ‘ „ p “ m € – —  m  ˜ y ˆ ˜ € ™ ˜ ‘  m — ‘ „  ˜ “ ˜ ‘ – j  € ™ d u – — „ ‘ ˜ ‘ “ ‘ € – p ‘ „ p ˜ ‘ “ •  y — – € ˜  d “ ep – m — ‘ “ ˆ — ‘ „ €€–• — y „ ‘ f — – ™ € y˜p y€˜—  j–— — – ™ ‘ ‘ „ ˆ u y“ — – ˆ „ e˜™ j — – ˆ ‘ „— ‘– ˜ u– m – ˆ — “ p –m — ‘“ ’ † — – ™ ‘ ‘ „ ˆ u y “ ™  € d „ € d d “ — y m d – m ˜ — –  y y  y m – p €  —  „ ep € y „ d – y y  ˜ —  ˜ “ ˜ — ˜ — – ˜ ‘ y € – — m ˜ y ˜ – — – m s — — „ d — – m k ˆ „ j – ˆ – m s j – — ‘ – ‘ y € – ˜ — – ˜ – ˜ —  — ‘ „ p ¢ f ˜ € – “ •„ h ˆ ‘ “  p “ ‘ € –  x¡ — ‘ “ ˆ – ™  ˆ y ˜ ™ – ˆ „ e ˜ ™ j – € ˜ y “ – ‘ ” † — ˜ ‘ “ —  “ …—  ˜ “ — — ™ l y — “ d ˜ ‘ „ — – ‘ — ˜ “ ˜m y — ™ € — – m – y y ˜ ‘ “ ˜ — – u–m d j „ p — ym d ‘ – — ym d – ˆ — –m k ˆ „ j — – ˆ — y d f ™ € ™ ˆ— ‘ „ p ˆ € „ s “ ˆ ˜— –  y y –m d j— — ym d –m –m k ˆ „ j –m ˜ ‘ – • y „ — ˜— – p f — ‘ „ ˜ “ j € „ … ‘ — –mm – ˜ – ˆ € – ˆ ™ — — „ d – ˆ – € “ € ˜— – m y y — y d f – y y ˜ “ € d ‘ ‡ † ™ —  m  ˜ y “ € – —  y y – m k ˆ „ j € –  j – € d – m ˜ — – p f – € ˜  d “ ep – p – ˆ ‘ „  ˜ p – — – € k  j – € d “ m — ‘ “ ˆ — ™ ˜ ‘ – — ™ € d — ‘ „ •“ — y „ ‘ – y y — – m s  — — „ d — – m k ˆ „ j — – m  j € “ d — € „ m “ f ‘ „  ˜ y s  € ˜ —  ˆ – y y  ‘ y – ‘ y € “ d – ™ ˜ ‘ – — ™ € d – € – € ˜ ˜ y – d – — — “ m p – y y “ ep – y y ˜  “ — € y – ˜ “ —  m  ˜ y m  — f – m d j u – € “ d ™ —  • € – d y — – l “ — — ˜‘ – € d d “ ‘ y ˆ — “ p –m — ‘ “ ’ † € –m y p ˜ € “ d – ‘ “ j „ ˆ ‘ y € y — — – € “ ˜ ‘ – j ™m d d y— — – p ‘ “ — — “ ‘ ‘ „ p — – ‘ “ ˜ € – p – ˆ k — — „ d ‘ „ m – y y — € „m – ™ ˜ d “ ˆ “ – € ˜ ˜ y – d – ˆ „ e ˜ ™ j – ˜ y „ ˜ f –m “ € ™ ‘ ™ l – € k ‘ “ j – ’ D @ I H ¥     © ©  © ©  ! 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"# % 2"  % % $ 9 & 4 $# & ! % I % #$ ¡ § ¡ £ ¨ © ¡ ! 22 ( % 0  "% ! $ 0 C ¡  ¢ ¤  ¨  ¥  £ ¢  ¥  ¤ ¥  ¤ ¤   £ ¨ ¨   ¨ ¤  ¥ '  ¥ ¤    ¢ £ ¨ ¤ ¦¢ ¤ ¢ ¢ ¢ ¡ ¢ ¥ ¨ ¤ ¡ ¨ ¨ ¤  ¨  ¥    ¥ § £   ¡ ¤  ¥ (  ¢ % % ¢ §£  ¡ © ¨ § ¦ ¥£ ¡ ¡ £ % §¦ ¦  "      # 4 ' "2 !  ) 0 # ( ! @ $  B # & #    !  # %  @0 # $  % $ 1 #2 4 0  # % 0 ( #2 0 " & ( # %2 0 " ! 5 ! $# " % % % 2 @ ! # 8 % % 0 # ! R2 % ! ( 8 $ ( #  % 7 % I B " % 4 4 5 & ! # &    @ & ( ) %2 ! " ¤# $ 5 5 0 # H $ # ( # 4 ' "2 !  ) 0 8 $ 0 2 5 %2 !  #  #2 $  9 # & % %  ( # ( "% 4 "2 0 ! " $ 0 &2 !   !   0 # $    %  2 "  2 ! 5 # ( ( 5 0  # S  " % 0 '5 ! ' 5 # '5  !   0 # ! $ 2 # $   ( !  H 0S ( # %  " % # & !  # $  ¢ B # & % %  ( #  "% @2 ! 5 & ( 2 $ 1 # "% %2 " !  %  % # " $ H2 ! " " 0 8 $ 0 5 0 G !2 0 G ! $   # & #    !  & " & " %  2 $ 1 # $ 1 2 % '5 " # " % ! & 3 2 ( #  ! $ # #  2 0 " & ( ( ) 0 " % 0 @ 0 ( " 2 % ! $  G " % 0 @ 2 $ # 8 % % 0S C B !(05 5 # %0 ( # 5 0 5 % # 4 ' "2 !  ) 0 # ! $ 2# $  $ 1 2# %2 0 9 # & % %  ( ( # # 0 H # ) ! 0  8 $ 0 5 0 G !2 0 G ! $   # & # ¤    !  # (  ' " & 4 # " % ! & 32 ( "2 $ # % # %& " % # & !  # $  ¢ B )&%2 ! " # $ 5 ( & (  ' " & 4 " $  " & ( # 0 H 0  " % $ " 2 " # %  5 2 $ 1 9 %  2 " $   # % $S ( " 2  0 4 2 "  S  " # % %  ( ! " % " 2 ! 0  2 4 2 # 0  ! 2 % 7 ( ! $   " % 4 $ 1 2 # # 0  5 # & # 2  2 " $ # ! $ # 4 # ! $ 2 # $   ( !  H 0S ( " % # & !  8 % % 0 ! R 2 4 !  0 C ¤ % #  B "% 4 4 ( & 5 & !  # %2 0 " ! 5 ( &2 0 " & ( # $  % 2 " 0 "% # & !  0  F & $ 1 @& & "& "%  2 $ 1 # (  '"& 4 " # % 2" & !5 0 # %  5 "# 2"!0  !R2 %! ( ""  ¥ ¥¥ ¨ →→ →→ distp (−i , −j ) = −i − −j xx x x p = d=1 p |xid − xjd |p M  # "% ( & 5 & !  8 $ ( #  #2 0 ! & % & ) 2 $ 1 9 2 E# ¨ E %2  ( 5 % 0 " #2 ( 0  →→ →→ dist2 (−i , −j ) = −i − −j xx x x d=1 2 = (xid − xjd )2 M  % % 2 (2 5 $ 5 %0 "#2 ( 0 →→ →→ dist1 (−i , −j ) = |−i − −j | = xx x x d=1    ¤ |xid − xjd| M  %0 ""0 ' %0  ( 5 %0 "#2 ( 0 # " % [email protected] $ # #  "%  # " # " A H  8 $ ( ! " % # " % 0 ! $  5 # $   #  5 % 0 " # 2 ( ( # ! $ # 4 # C B # " A H S ( ! 2 0  ! " % 2 % 7 & ( 5 % 0 "#2 ( ( % 2 " 5 %  G % $ ! 0  & $ 0@ & "# # " A H  ! " % & "2 ! 0 2 42# 0  9 "% 4 $ 12  U −i → x −j → x ©     § ¨   ¨ © ¨   §¦ ©  ¥ ¤ £ ¤ £ ¢ B # ! " # $ 5 # (  ¤ £  ¨  ¤ ¦ ¥ ¨ ¡ ¥ ¦  ¥ % %  H % $ ! $   ! # 2 4 2 % 2 4 F  # ! " # $  5 ! " % & " 2 ! 0  2 4 2 # 0  " B  ! " # $  5 $ (  ¤ £  ¨ ¤ ¥ ¤ ¦ ¥  © £ ¦¢ % %  H % $ ! $   ! " # $  5 $ 0 & 2 5  # # 0 " # " A H S  2 # ! # 2 4 2 8 0 4 F  ! " # $  5 % $ " " A H  % $ ! " % & " 2 ! 0  2 4 2 # 0  B # " % ! & 32 ( # ! " # $ 5 # ( F " % 0 % " ! 0   0 # " A H  8 $ ( ! $   ! # 2 42 %2 4 F " 9 ! " # $ 5 4  4 $ 0 " % 0 % " ! 0   0 # " A H  8 $ ( ! $   ! #2 42 80 4 F  # " A H  ! " % & "2 ! 0 2 42# 0  B ¥  ) %2 ! "# $ 5 % & " 2 ! 0  2 4 2 # ( # "  5 %  5 # 2  ! " " # 2 8  £ B %  2 " $   # % $S ( & " 2  0 $ 1 0  ( " # & % %  ( #  ! " % & " 2 ! 0  2 4 2 # ( %  2 "  % 0  ! 2 % 7 & ( ( " # ) % 2 ! " # $  5 $ (  0 ! " % 5 $ 1 2 " 0 4 &  H  !  # ( % $S C  © © © ¨©        ¥ §¦ ¡  © & % $ '   (   )  ! (   $ '  &  ¢ § ¡§ " 3 S  # !  0 ! " # $2 ¥B ¢ ! $ ) 7 0 C B # " A H  ! " % & "2 ! 0 2 42 # 0  !2 % 7& ( ! $   # %2 #2  @ # " A H  # ( ¦ " 3 S  " 45 % ( % ! 4 4  # %0 Q ¥  ¤ ¡ ¨ ¦ ¥   ¦ £ §¥ ¥ © ¤ ¡ ¢ ¨ ¢ § 0  9 " 2 !  # B & "2 ! 0 2 42# ! $  "2 % 7 & ( 2 $ 1 9 ¦ ¤¥ ¥¤ # &    0 9 # " A H  8 $ ( ! " % % $ 4 4  5 % # %2 # 2  @ ( ! H 4  %  " # S 5 # 2 $  9 # " A H  ! " % ) 0 % 2 # 2  @  ! 2 % 7 & ( ! $   & # 2  2 " $ " #  2 $ # % $ 9 § § § § 9 B  0 " 0 ' $ D¦ # % 0 Q B "# " $ H 2 ! " " 0S  ! $ # 2 !  ) & " 0 5 0  #  $ 1 #  ! $   # 0 H 0  ( # " A H S (  0 "  " ! H 4  %  " % # & !  ! P d MND xid M Nxid + Nxjd × δ(xid , xjd ) Nxid × Nxjd  # & % %  ( ( # 0 H 0  (  H 4 # % S  ! $ # 2 !  ) & " 0 5 $ 1 0 '5 ( %  2 " 5 % $ 1 & ! G #  "  4  5 % "% 4  0 ) & ! ( % !  F "#2# %  5 % 2 " $  # ! " $ 0 %  2# 2# 0 1 ¤2 ! 0   0S ( # Nxid = |{j ∈ D|xjd = xid }| d=1 →→ distχ2 (−i , −j ) = xx δ(xid , xjd ) = xid = xjd xid = xjd 5 @0 d=1 δ(xid , xjd ) M  →→ DJ (−i , −j ) = xx 2# %2 0  $ 5 0 5 # ( ! 0 5 5 0 ¥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¥ " § ! "% # & #2 0 4 !  % "%  # # $ 12 ! & 4 $% # " $ H2 ! " " 0 #  ! $# # " A H  # ( # ! $  0@ #  9 " % 4 $ 12  U B "2 $ ( %  5 "% @ $  # "# # & % %  ( # ( %  2 " 0 # 2  0 4 !  % (  H 0  0 & !  # 0 '  % $ 9 # ! " $ 0 #  $ 1 5 % 0 " !   4 2S ( # $   " % 2 0S % ) ! 0  # $   " # # " A H  # ( %  2 " 2 % 7 & ( (   0 @ ! " % 2S  " %  ( # " $ H 2 ! " " 0 #  $ 1 ! " 2 @ &S ( % 7 ¢  M  # " $ H2 ! " " 0 # ( M d=1 xid  H 4 # % S  ! $# M d=1 (xjd − xj )2 − xi )(xjd − xj )  ¡ " A H S  ( # ! $  0 @ # ( % % U  4 0  5 @0 % % U  4 0  F " ! 0 5 &S  ( # $ % 2 #  5 9 %  # ! 0 T ( %  2 " 0  & ! !  5 0  M 2 d=1 xjd × xjd ) ¦£ ¢ ¤ ¡ ¢ −i → x − xi )2 M d=1 (xid M d=1 (xid M 2 d=1 xid ¢ £ ¡ £ ¤ ¦ ¢ ¡£ £ ¤ xi = xi M d=1 (xid →→ P earson(−i , −j ) = xx −i .−j →→ xx →→ Cos(−i , −j ) = − xx →||.||−j || = || xi → x   # $%2 #  5   ¢¡ ¨ ¥ B ! " # $  5 $ 1 0 '5 ( # '5  !  # $  #  # " A H  #  ! 0  & # 2 $ 1  5 % 2 H $  9 # ( ¨  ! " % 5 ! " % 5 % 0 " # 2 ( 0  ! # 2 42 8 0 4 ( # !   0 0 ! 2 ) 0S #  2 9 (  ' " & 4 0  %   3 B " % " % # & !  ! #  2 $ 1 # " A H  #  ! " % 5 % 0 " # 2 ( 0  !  $ 5 0 5 ( " # # ! " # $ 5 8 $ ( ! "% & "2 ! 0 2 42# 0  ! ! $# 4 ! $   & "22 H2# #   ! R2 4 !  %  B # ! " # $  5 # ( ¥ ¤  ¥ ¨ 2¥ ¤ ¦ ¥ ¨ ¡ ¤ ¦ ¦ ¥ % %  H % $ ! $   & # 2 4 2 % 2 4 ! "  " 2  ( 2 $ 1 9 # ! " # $  5 # " % ! & 3 2 ( ! " % 2 % 7& ( "# 2 $ 1  5 " # ) %2 ! "# $ 5 % "% 0 " !   42 & "2 ! 0 2 42# ( % 2 "  % ! " $ 0 %  #  © ¦         §¦ © £ ¤£ ¢ &' Nk × (Nk − 1) i∈Dk j ∈Dk →→ dist(−i , −j ) xx "% 4  0 ) & 9 ! "# $ 5 $ ( # ! H 4 4 ( # !2 0  ! " % →µ dist(−i , −k ) x→ Nk  &% $ # 1 Diam(Ck ) = " 0 )  ! #2 42 %2 4 F % % U 4 5 % 0 " # 2 ( 9 ! " R 4 0 2 (  " B Radius(Ck ) = i∈Dk ! " # $  5 $ ( ! $ 2 ! & " % 2S  F # %  2 " 0 @ ! # H  # ( & " 2 ! 0  2 4 2 # 0  ! # 2 4 2 8 0 4 ! $   ! #2 42 %2 4 F 9 ( ¨  ! "% 5 %  # " ! "# $ 5 $ ( # ! H 4 4 ! " % % % U 4 5 % 0 " #2 ( 9 # $2 ( 0 !  B ¥ # "% [email protected] $# #  " % 0 " & ! ( 0 5 5 # % 0 ( # & #22 " $ # $  #  # ! $# 4 8 $ ( #  9 2 % 7& ( ! "  " $  # ! " # $  5 # ( ¥ ¤  ¥ ¨ ¤ ¥ ¤ ¦ ¥ ¦ ( £ ¦ 0  9 # ! " $ 0S ( $  " % 4 4 ( & 5 & !  # & #    !  #   5 & 4 4  % # " A H  !"% 5 %0 "#2 ( ( ! $# 4 % $ & % %  ( "%0 " 2 4 ! 0  2 # 2  '5 "   dist #  © ¦      © ¨ &' &% $ # " " !  £ ¤£ ¢ B & ! & ( 2 # %  5 4 R  H  !   ! $   " $ H 2 ! " " 0 " 5 ( 5 % 0 " !   4 2S  ( %  2 " 5 %  G % " $ H 2 ! " " 0 $ 1 0 '5 F # ( 2   % $ # !   0 " 5 3 0 2 5 ¤2 $  B # " A H  ! " % 5 % 0 " # 2 ( (  $ 5  0 5  # % 0 ( %2 0 4  ( $ ( " !  8 % $S ( # 5 % 0 # # 2 0 % %  5 #  "  4  5 % ! ( % !  (  H 2 # #   " % 4  0 ) & " #  2 9 % 7 % I B ¥ ¤ ¡ ¨ ¦ ¥   ¦¦ £ §¥ ¥ © ¤ ¡ ¢ ¨ ¢ § 0  ( "  ! & " % 2S  ( %  2 " 0 ! " # $   £  ¥ © ¤ ¨ § ¦  £¢  ¥ © !¤ ¨ § ¦ ¥ ¤  ¥ © ¤ ¨ § ¦    £¤  ¥B ¢ B A@ 9  ¥ © !¤ ¨ § ¦ ¥ ©¤ ¡ ¡ ¢ ¢ £ £ F "!  0! !0  →→ →→ →→ M N D (−i , −j ) = N N (−i , −j ) + N N (−j , −i ) xx xx xx −j → −i → x x B →→ N N (−i , −j ) xx ( # %2 #2  @ ( ! H 4  %  5 @0 B ¢ ( # '5  !  # " A H  # ! " $ 0S ( " $  A 0 # R !  0 " % 4 # ! @ % 2 " 9  " ¨ $ 1 # ! 2 0  2 4 2 # # $   " % 4  0 2 " 2 % 2 " % 0 " & ¨ " ¢ # " A H  #  9 ! $ # 4   " % $S ( %  2 " 0 # 2  2 " $S  (  ¨¥ ¤ ¡ ¥ ¡ ¡ ¢ ¡ ¦ £ ¤ ¦ £ ¢ ¡¤ ¢ K Nk × Radius(Ck )2 N # ! " # $ 5 ( # $2 ( 0 ! # ( & ! ! 0 5 $ ( & ! & ( %   % % U 4 0  B # 0 5 5 # % 0 ( 5 @0 →µ Pik × dist(−i , −k ) x→  → × −i x N  k =1 Q1(P ) = N i=1 Pik = K # ! "# $ 5 8 $ 0 #" AH § ¤ ¥  ¥ ¨ ¦ ¢ ¤  ¨ ¦ ¦ $ ( # 0 5  # % 0 ( # & "22 H 0 H  !  #  ! 0  & ! & ( %   ! "  " % 4  $ "% @ & " $  2 $ 1 Ck Pik # ( 5 % 0 % " ! 0   0S ( −k = → µ k =1 i=1 2 Ep (P ) −i → x k =1 i∈Dk →µ dist(−i , −k ) x→ E 2 (P ) = K  $ 12"0 ! ( 0 $ 1 ! $ !! S  B ¥  # " % [email protected] $ # #  "%  # ) %2 ! ¥ ¡ ¢¥ 0  ! $ 0@ & ! $   "2 $# % # & #22 " $ # ! $# 4 #  0 2 5 %2 !  # C    © ¦  ¨  ¦ § §¦  ©      ©  ©  £ ¤ £ ¢ P ¤ " # $ 5 % $S ( ¥ ¤  ¥ ¨ ¤¥ ( ¨ ¤ &' &% $ i∈Dk1  "# & " $ # → µ− → − dist(−l , −k1,k2 ) − x → µ→ dist(−i , −k1 ) − x  # ! " # $ 5 !"% (Nk1 + Nk2 )(Nk1 + Nk2 − 1) j ∈Dk1 ∪Dk2 " #& © ¨  0 H  ) 5 % 0 2 ! 0@ 0  $  i∈Dk1 ∪Dk2 →→ dist(−i , −j ) xx j ∈Dk2 → µ→ dist(−j , −k2 ) x = Diam(Ck1 ∪ Ck2 )  # ! "# $ 5 ¤0 ! "%2 Nk1 × Nk2 i∈Dk1 j ∈Dk2 →→ dist(−i , −j ) xx % % U 4 5 % 0 " # 2 ( 0      # ! " # $ 5 ¤ ! " %2 % % U 4 5 % 0 " # 2 ( 0  ! ( 0 5 5 # % 0 ( # & # 2  2 " $ ! "  " % 4  0 ) & " % @ $  # ! $ # 4 # ! " $ 0 # 2  !§ N− → i=1 xi  l∈Dk1 ∪Dk2 V ar (k1, k2) = Intra(k1, k2) = Inter (k1, k2) = N −= → x0  # 0 H 0  ( # " A H  # (  H 4 # % S  ( ( ¨  ! "% 5  " ! "# $ 5 $ 1 0 '5 ! " % # 5 % 0 " # 2 ( # ( 4 4  # 0  ! # 2 4 2 8 0 4 F " # 2 # %  5 4 R  H  !  5 F ! 2   0  ! $   ¦ § § 9 # 2  U ! 0 ¦ ( % 0  0 ' ¥¤ # % 0 ( & #    !  %  2 " $   # ! " $ 0 % $ 9 & # 2 4 2 8 0 4 " # # ( ¨  ! " % 5 ! " % # 5 % 0 "#2 ( # ( & ! & ( %   4 4  # 0  9 # 0 H 0  ( # " A H  # ( "# !  ! "# $ 5 ! 2 % ! ( $ 0 #  £ # & '5 ! ' 5 ! # ! " # $  5 ( # 2 $  9 # 0 H 0  ( " # !  5 @0 # ! 2 0  2 4 2 # # % 2  4 #  # " A H  ! H 4  %  ! $  ¢ # ! " # $  5 # ! 2 4 !  8 $ 0 "% 0 % )2# # 0 % S $ 1 "% 0@ ! # H  9 "% 0 ( %   K ¢ £ ¡ £ ¤ ¦ ¢ ¡£ K −1 K −1 ¡ £ ¡ ¦£ ¢ ¤ ¡ ¢ ¢¡ ¨ ¥ 2   !"% S ( ! $# 4 0 " 9 $ ( % " " 0 " 0 " $# & ! ) %2 ! "# $ 5  & % %  ( "% 0 "  B "% 4 # ! @ %2 " 9 ! " $ 0S  F %  2 " $   # % $S ( ) 0 # # 0  $ ( # !   $ ( !  ¡ & % ) 0 ) " # 2 $ 1 %  2 " 0 4 !  G % 2S ( & " 2 " % 0 $ 1 0  ! $ 0@ & F "#2# %  5 "% 4 ( %  G %  3 B # % 2 " $  # 8 $ ( ! ! 0  4  5 ! $   & #    !  "# ¤ ¦ ¥¨ ¡   ¦  ¢ ¤ ¥ ¤ ¦ ¥ ¨ ¡ ¥  ¡¥ ( ! $ # 4 % $ 9 ¦ § § 9 0  2  ( % 0 % % 2 0 E 2 ! " 0 T¤ " ¦  § § 9 0  2 ¤ # % 0 Q B # & #22 " $ ! "  " % @ $  ¤B %  2 " 5 # 0  # % 0 ( # & " % # & !  ! $ # 4 ¤ § 0  $  2   ! " % S  $ 1 $ @ F A& ( # %  @ 0 # $  % 9 2! 2!  0 $% %  5  H2 5 % $ F ) % 2 ! " # $  5 % $S ( # " 0 "  $ # & ! #  ! ! 0  4  5 " 2 0 ' $  # % S  P  # 0 5  # % 0 ( ! 2 ( ¤ F ¤ " # S 5 9 ) % 2 ! " # $  5 % $S ( ¥ ¤  ¥ ¨ 2¥ ( ¨ ¥ ¤ ¡ ¢ ¥ 0  ( # ! $ # 4 #  % ! 5 %  5 2 $ 1 5 % I P0 ¤ P0 P    © ¦  ¨  ¦ § §¦  ©      ©  ©   "# &' &% $ " & $ # 1 #& © £ £ ¤£ ¢ B % 2# % 42 ( 0  ! $# # 0 H 0  ( # " A H  # (  H 4 # % S  ( # ! $ 0@ # ( ( !0 ( %0 "# % 2"02 @& ( 0 " 9 % 2# % 42 ( 0  ! $ # ! "# $ 5 $ 0 "% 0 % " ! 0   0 # " A H  # ( # ! $  0@ # ( ( ! 0 ( % 0 " # % 2 " 0 2 @ & ( 0  " # P d Ck d σd σkd 1 CU (Ck ) = P (Ck ) √ 2π d=1 M 1 1 − σkd σd  ¦ ¢ ¨ ¢ ¥ 9 B  0 " 2 ! 0 % % D¤ # % 0 ( # $ 1 2 ! & 4 $ % # %  2 # % 42 ( # ( # 0 5 $ 0 "2 $ # 4 4  5 & "  0 ( 0 & " & " % 0 U0 ! $ # 4 " " 5 P (xid = m|i ∈ Dk ) > P (xid = m|i ∈ D)  0 %  # $ 1 #  ! $   8 $ 5 !2 ( F " # S 5 9 & " 2  2 H 0 " 5 2 ( & !  0  ! " 6  ! 5 5 0 " %  @ 2 $ 1 # ! " # $  5 #  ! # 2 !  @0 G ( " % 0 " & 2 5 2 & ( 2S  d=1 {m∈M odalitesd } CU (Ck ) = P (Ck ) P (xid = m|i ∈ Dk )2 − P (xid = m|i ∈ D)2 M Nk × CU (Ck ) N ! '#2 § ( ¤ ¨ ¥ ¤ ¥ ¨¡ ¤  ¦ §¥ ¨ ¡ ¤ 0 ! $# &# 0 H k =1 P U (P ) = ¤ ¨ ¥ ¤ ¥ ¨¡ K −− →→ k =1 i∈Dk dist( xi , µk ) K −− →→ k =1 Nk × dist(µk , x0 )   K ¦ ¨ ¢ ¥ 9! ' # 2 §¤ ¤ ¦ ¥¨ ¥¨  ¡ ¦ & 4 4  % ! $# 4 0  $  B H (P ) = # 0 H 0 ( # " A H  # (  H 4 # % S  ( (¨  !"% 5  "  %& ( P  9 ¦ § § 9 #2  U!0 ¦ ( %0  0 ' ¥¤ # % 0 ( & #    !  % ! " 8 %  2 " 0   # 2 " % ! " % 2 %  2 # & '  5 " % 0 8 2 4 ( 2 ! H U ' ! $ # 4 0  B − → x0 K k =1 Nk × (Nk − 1) × (Nk − 1) × Diam(Ck )2  # ! "# $ 5 ¨ Q2(P ) = K k =1 Nk ( # ! " R 4 0 2 ( # ( & ! ! 0 5 $ ( & ! & ( %   % % U 4 0  B  ¥ ¤ ¡ ¥ ¡ ¡ ¢ ¡ ¦ £ ¡ ¦ £ ¢ ¡¤ ¢ B " % 0 @ 2 $ # 8 % % 0S  # % 0 ( &   2 0 " & ( 0 ! # " % 2   ! 2 % ! ( 5 ¦  0 % 2 ) 2 !  5 0 # S  $ 1 "2 $ ( & ! # $  5 0 # % $ # % 0 ( ! 5 0   # ( % 2 H $  9 " 5 0  4  5 # $  % 2 " ¤ 0 " % # & !  ! % $ # $  # # & % %  ( #  ! 4 !  G # % 0 ! " ( # 0 5 5 # % 0 ( "2 ) 0S # 2 " % 0 " !   42 # % 2  4 # & % %  ( ( # 0 H % $ ! $ # " 2 $ # % !   2 0 @0 ! " ( % 7 0 9 (  ' " & 4 0  ( & ! " % % # 2 % ! $  G # & % %  ( # ( 2 0 " 0  !2 $ ( & ! ( " 4 !  # & % %  ( # ( % 2 " 0 4 !  G# % 0 ! " 0  " B && & ¦ 0 ! % ) 4R H  !   ! ( $  # ! ! $   # 4 R  H  !  ¤ # $  # # 5 ( # " 0 "  $ # & ! #  " 2 $ # % ! # 2  2 " $S ( # 2 $  9   2 0 " " 2 "  # $   ( # 4 R H  !  ¤# $  # # ! $ 2# $  % 4 R H  !   ! #2 @2 ( ( " 4 !  "% 4 % % 2 "2 " ! 0   B  & ¦ # $ 1 2 " # 2 " 0 " # $  # $ 1 2 " # 0 '5  " # # (  ' " 4 #  ! $   # 0 5  " # S 5 4 4  5 9 " % 0 " # % 2 $ 1 0 '5 F %  2 " $   # % $ ! 2 % ! $  G (  H 0  0 5 " # 2 5 ¤   5 $ 1 # !   9 (  ' " & 4 0  ( # %  2 " 0 ! & " 2S ( 4 $ 4 2 8 0 4 ! H 4  % % $ ! 8 7 ( " 4 !  # % !  H ( %  2 " 0 # 2  2 " $S  B  & ¦ (  '" 4 0 ( "% 4  ¤ $  ! & ( $ ( # !   # " A H S ( # %2  4 "2 $ # % ! ! & ( 2 # %  5 ( 2 # %2 0 " % 0 " " 4 !  9 # & % %  ( # ( G 2 " 0 " % # & !  !  H 4 # % ¤ # $  # % $ ! % %  2 " 5  & # ( " 4 !  ! 2  " 0 &  0 ) 0 % %    2 " % 0 ' 5 &S  B ¥  ¦ ¢ ¢ ¢ ¥ 9 4 2 ¦ ( % 0 ) ! $ H % % 2 §¤ # " % 0 @ 2 $ # #  "%  # ! ( 0 5 5 # % 0 ( # & #22 " $ " % @ $  # # $  #  # % 2 " $  # #  9 "% 4 " R ! 5 %  5 # $ T B 4 ' " 2 !  )  0S  F 2 % ! $  G # & % %  ( # ( ! $ " 5 $ ! " # 0  (   2 0 " 0  ! 2 $ ( & ! B  % 2 " $ 5 & 8 S  "% 0 ( %  # & %2 4 0 8 # & % %  ( ( ! H 4  %  !2 $ ( & ! B ¦ & ¦ # % %  ( # (  H 4 # % S  ( # ! $  5 ! 0  ( ! H 4  %  ! #2 42 %2 4 B ¥ # & ) 0 #2 @% ! "  "% 4  0 ) & " % @ $  # % 2 " $  # # ! " $ 0S Q B # " $ H 2 ! " " 0S ( " # " A H S ( ! H 4  % $ ( ! 2 0 & % 2  %  ¥ 0 G ( " % ( %  & ( 2 $ 1 # (  ' " & 4 # ( !    @ & ( ( "# % 2 " $  # ! $ 2 4 0  9 "% 4 $ 12  U § B & "2 8   4  5 ! $  F ! R2 $ 52 " ! 0  %  2 " % " " 0 % $ ! " !   " % @ 2  ( # &     @ & ( ) % 2 ! " # $  5 ( # (  ' " & 4 #  9 # " $ H 2 ! " " 0S ( $  # " A H S ( ! H 4  % ( % 0 ! ) % $ " % 0 % " %  5 # & % %  ( ( # # 0 H # ( F  H 0 5 2    0 ! "  ! $  T            ¢    ©     ¦  ¨  ¢  © ¨ ¤¦ ¡ ¡ ¡ P (k, l) = 0 |Ck ∩ Cl | N P (k, l) log £¢ ¡ K ¤¤ Nk N " P (k) = k =1 l=1 I (P, P ) = 0 K P (k, l) P (k)P (l) V I (P, P 0 ) = E (P ) + E (P 0 ) − 2 × I (P, P 0 ) ¤ "# $ 5 % $  & % %  ( ) %2 ! "% [email protected] $# 0  "# $ ( % " " 0 ) %2 ! "# $ 5  " ! " % %  2 " 0 4 !  G % £S ( %  2 " 0 2 ! 0 £ ( ! $ # 4 0  9 ) % 2 ! " # $  5 % $ F & 2 5  # # 0 ¡ £ ¡ ¦£ ¢ ¤ ¡ ¢ P0 P ¢ £ ¡ £ ¤ ¦ ¢ ¡£ P E (P ) ¢¡ ¤¨ ¥  #2  G ( & 8 7 ! H 4  % %2 0 " ! 5 % $ ! ! & "2 B & & & & ¦ # % %  ( # ( G 2 " 0 " % # !  ! %    2 " % 0 '5 % $ ! % %  2 " 5  # B ¥ # "% [email protected] $# #  " %  # (  ' " & 4 0  ( #  0 2 5 %2 !  #  0 " & # C B # ( ¨  ( & 4 # 5 5 @ 0 & " 2 4 2 8 !  ! $  ( %  2 " 5 %  G % # " A H  # (  H 4 # % S  # & 2 5  # # 0 " %  # #  $ 1 8 $ 0 9 # ! " # $ 5 # ( # G2 " 0 " % # & !  ! # " A H  9 # ( ¨  ( & 4 (  H 4 # % % $ % "#2# %  5 % 2 " $  # % $ 9 # & '5 ! ' 5 ! # ! " # $  5 ( ! H 4  %  & % %  ( " % 0 "  B ! 2  " 0 &  0 ) 0 %    2 " % 0 '5 &S  ! $ # $ 1 2 # % 2 0 9 (  ' " & 4 0  ( # %  2 " 0 ! & " 2S ( ! H 4  %  ! % !  H ( " % 0 " " 4 !  # ! " R 4 0 ! 0  # " % ! & 3 2 ( ! $ # & # 0 H $ 1 2 " # 0 ' 5  " # '5 ! '5 ! % $ ! # 2  2 " $S ( " % #    !  9 3 ¢ ¦  ¦ C  &    0 4 R " # U # ! $  # % 0 ( 9 ¦ ¢ ¢ ¥ 9 % 0 § ( % 0 ) ¢¤ ( # ! $ " $ 0 #  9 # & % %  ( ( # # 0 H # ) ! 0  8 $ 0 5 0 G ! 2 0 G ! $  T  K K ¥ ¢ ¢ ¤£ £ ¢       ¨  ©    ¦ ¢¦ ¡ © ¤¢    ¨ ¨ B  0 H  ) ) %2 ! 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" # $  5 $ 0 @ $  % 4 4  5 " % 0 # # 2 # 2  ' 5 %  0 " &S  F ! 5 % 4 4  5 ! " B  && & ¦ " % 4  S  ! $   ! " # $  5 $ 0 @ $  % % $ ! ! 5 $  £ (¢ ¦ " % 0 ! $  5 ! " # $  5  ! # 2 @ 2 ( £ 5¢ ¦ # " % 0 " # 2 8 # ! " # $  5 8 $ ( ! % %  2 # $ G £ H¢ && & ¦ " % 0 ! $  5 ! " # $  5  # % 0 ( " % 4  S  ! ! ) " % 2 £ 0¢ B ¥ B ¦ %  2 " 5 # 0  # % 0 ( 2 % 7 & ( £ U " 2  2 "  U !  ) " 0 ¢   ! $ # 4 0  # 2 4 2 "   2 $ 1 2 $ 5 ! % % 2 " 5  & # 9 " % 0 ! $  5 ! "# $ 5  ! $# #  H 0 52   0 # ! $ " 0 ! &   ! " 0 $ 1 #  2 4 ! 0  B ! H ! 0S  ( % 2 5 0 ! 0  ( ! ! ! 0 4 & ( B ¥  ¦  " % 0 @ 2 $ # 0  " # 2 ' 5 ! 0 ! & 2 ' 0  # % 0 ( " % 4 &  &  @ $  % % $S ( %  2 " 0 ! ) & " % 2S  " % 0 ! & ) (  ' " & 4 0 C B 4 ' " 2 !  )  0S  F # & " % # & !  " %  # # & % %  ( # 5  $ 1  # % 0 ( ! ( ! S  ( " % 0 ( %  & ( ( % !  # 2 0 4 # 2  G  $ # % $S $ 1 # & % %  ( #  ! % % 0 5 # % ( " 4 !  2 $ 1 5 9  0 " % 4 & ! 5 % 2 " #  £ B " % 0 ( % 5 # ( $ 1 2 ' 5 ! 0 ! & 2 ' ) % 2 ! 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"% 4 "!G "% 2# ( # " A H  #  $ 1 # " # % 2 # % 42 ( # (  H 4 # % ¤# $  # % $ F &25 ## 0 # " A H  # (  H 4 # % ¤# $  # % $ 4 4  5 2 % 7& ( "# ( ¨ ¥ ¦  ¢  ¥ ¨ ¦ ¢ ¤  %  B !2 ! 5 & ( #  ! $   # !2 0 # # 5 & % # % 2# % 42 ( ! H 4  %  # ! $ " 0 #22 " $ # ! " R 4 0 ! 0  ( % U  4 ! H 4  %  " # & '5 ! '5 ! # ! " # $  5 ( 8 $ ( " # ( ¨  ( & 4 ¤ ¦ ! 0  $ 1 2 " # 0 '5  " # ' 5 ! '5 ! % $ ! $ # & # 0 H " # ¦ ¢ ¢ ¢ ¥ 9 B  0 "  0 ¨ ! 0 ) ) ¦¤ # % 0 ( & "% # & !  & " & 0 2 $ 1 " % 0 ( % 5 # 0 ) %2 ! "# $ 5 5 0  # H $# ( (  ' " & 4 ! R2 4 !  0 C Sk Sk Dk Ck Dk K L  ¤¡ ¤  ¥ ¢ ¤£ £  ¡    ©   ¦ ¡  ¡ ¢ ¢ ¦ ¤ ¦ ¢ §¢ ¨ ¤¢ ¡ ) £ ¡ B # ! "# $ 5 #  ! 4 !  G ! $   & #22 " $ & " & 0 # " A H S ( 4 $ 4 2 % 2 4 % $  $ 1  # % 0 ( 5 0  # ¤ # $  #  ! % %  2 " 5  & # ( " # " % #    !  # ! $ " $ 0 #  $ 1 $ 1 2 " # 2 ! $ 'S  9 0  5 ! $   ! ! & ( 2 # %  5 F 5 0  # ¤ # $  #  ! 2 # 2  '5 !$  " I B # % 0 ( % 2 "2 " ! 0  0  ! $ " 5 3 ! $   # 5 0 # ¤# $  # #  # % 0 ( # % 2 "2 " ! 0  #  ! # 2  2 " $S (  H 2 # #   " #  2 9 4 ' " 2 !  )  0S  ( # 5 % 0 4 !  G !  #  ! !  2  & 4 0 ! $   9 % 7 % I B # "% ( & 5 & !  # (  '"& 4 #  ! 0  & #22 " $ 2 %  "  %  4 ( 2 5 %2 !  4  4  %  # 9 ! "# $ 5 ( # $  %  % # 0  0 ! ( % 2 "%  5 % # !  0 9 ! "# $ 5 ( # 0  " % 2 "%  5 % 5 0 # ¤# $  # % $ 2# ! 0  B # ! " # $ 5 # ( F # 5 0 # ¤# $  # #  # $  " 2# "% % % 2"%  5 # % 0 ( # $ 5 %2 # % 2# % 42 ( & % %  ( 5 0  # ¤ # $  # % $ %  % $  ! $ ) 2 " # @ % 2S ( ! ( 2 5 & ( ! $  T 72 ! & @ 2 9 # %  2 # % 42 ( F B & % ! 5 %  5 5 0  # ¤# $  #  # % 0 ( # " % # & !  # ! " # $ 5 #  ! 72 " % ( 2 ! $   ¢ ¦  3 ¨ Q # 22 " $  2 9   2 ! ) % $S ( ( 2 0S  F & 2 ( $ " & 5 0  # ¤ # $  #  ! % %  2 " 2 " ! 0  ( $ 2  $ 0 # 2 0  B # %  2 # % 4 2 ( F # 5 0 # ¤# $  # # ( # % 0 ( # % 2 "2 " ! 0  # ( F ! # # 0  ! $   # % 2# % 42 ( F # 5 0 # ¤# $  # #&  # % 0 ( # & $ " 5 3 # %  2 "2 " ! 0  #  # 22 " $ # 2 $  9 " % 4&4 0 ( %  & ( %2 &%  2 # % 42 ( $ 1 0 '5 ! $# # % %  ( #  ! % % 2 "2 " ! 0  ! 0  5 % 4 4  5 2 # " % ( 5 & !  # (  ' " 4 8 $ 0 $ 12 "% (2 " % 0 ( % 5 # 0 '5  !  0 % $ " % 0 # 2  2 " $ % 9 ) % 2 ! " # $  5 5 0  # H $ # $ ( # 0 5 $ 0 B  %  2 " 5 # 0  # % 0 ( & " % # & !  ¢ ¦  3 ¨ Q (   5 ( % " & ¦ § § 9 B  0 " ) % 2  2 0 ¦¤ # % 0 ( & " % # & !  (  ' " & 4 0 C Sm−1 Sm−1 ⊂ Sm Sm Sm−1 ⊂ Sm (m − 1) Sm−1 ⊂ Sm Sm Sm Sm m m (m + 1)  § £ § ¥ ££ ¨£ B (  '"& 4 0 ! ! &  & 5 5 0S ( % 7 0 & #    !  " % 4  0 ) & " #  R   0 ! 0  " % 4 " 2 0 ! " %  B % $ 4 4  5 % # %  2 # % 4 2 ( F # 5 0 # ¤# $  # ( %  #2 0 %2 H 4  5 " $  " #2 0 4 9 % $ 4 4  5 "%  2 $ 1 # % 2# % 42 ( (m − 2) ¢¢ (m − 1) ¥ £ ¤ £ ¥ ¡ ££ ¦ £ ¡ ¢ £ ¡ £ ¤ ¦ ¢ ¡ £ £ £ ¥ ¡¦ ¡ ¢ Ck  ! " # $ 5  ! $   Distkd − Yk σk d  Zkd = % 2# % 42 ( 0  ( 5 % %2 " !  0  ( ! $ " 0 52 ( %2 9 M −1 M d=1 (Distkd − Yk )2 Distkd # ! $ 0@ # ( ( !0 ( %0 "# % 2"02 @& ( " dist(xid , µkd ) |Lk | −k → µ % 2# % 42 ( 0  ! $ # F Lk  M d=1 Distkd i∈Lk  % % U 4 #  !  $50 5 (    H 4 # % S  ! &!5 ( # " A H  # ( % % U 4 5 % 0 " # 2 ( 0  − → µk ¥ ¡ ¦ 0 "%0 "% # & !  ! (¨ ¤  ¤  !  $50 5  !  $50 5 "  # ! " $ 0 #  # $  " " & % %  ( ( ¨  ( & 4 % $ ! "% # 5 % 0 "#2 ( # ( 4 $ 42 %2 4  9 !  $ 5 0 5 # "% [email protected] $# #  0 " & #  # %  2 # % 4 2 ( # ( ' 5 ! '5 ! 0  " % 0 % ! 5 %   − → µk "2 $# 2 5 ¤  5 9 # (¨  ( & 4 8 $0 # &25 ## 0 |xid − µkd | |Sk |   d∈Sk  Zkd σk = σk d M Yk = Yk Distkd = Distkd →µ Lk = {i ∈ D|dist(−i , −k ) ≤ δk } x→ Lk →µ δk = min{l=k} dist(−l , −k ) µ→ δk Sk → dist(−i , Ck ) = x & " A  !  " # ! "# $ 5   $ 1  # % 0 ( 5 0 # ¤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¤  5 # !  0 9 " % 0 ! $  5 % 2 " $  # 0  $ 1 "% %2 " !  # $  4 4  5 & ! & (2# %  5 "# % 2 " $  # % $ 2 3 B # " A H  # (  H 4 # % S  # % 0 ( " % 4 ! 2  " 0 &  0 # 2 # 2  '5 # ( ¨  ( & 4 # ! " $ 0S ( ! 0  # " A H S ( # %2  4  # &2 5  # # 0 "%  # # $ 1 8 $ 0 # ( ¨  ( & 4 #  " % 0 ¥ 0   4 ! % "2 $ ! "# %  5 "# 2 5 ¤   B & # ¤    !  " # "% 0 ! $  5 % 2 " $  # 0  ( %2 #2  @ % 2 " $  # % $ 9 #  4 " ! 2 4 !  % $ # % 0 Q B % 2 "2 " ! 0  0  ! $ 0@ & B  & & ¦ # ( ¨  ( 4 8 $ 0 # 2 5  # # 0 # %  2 # % 4 2 ( #  ! '5 ! ' 5 ! B ( % 2 " $  #  @ $  % % $ ! ! & (2 # %  5 B ¥ # (¨ (& 4 Ck K −i → x ¦  # (¨  ( & 4 # 5 F # &25 ## 0 # %  2 # % 4 2 ( # ( #  H 4 # % ¤ # $  # #  $ 1 2 # % 2 0 9 # & ' 5 ! '5 ! # ! " # $  5 #  " % 0 " % # & !  ! # ( ¨  ( ¤& 4 # (  H 4 # % S  ! #2 42 "   ! $   #  0 " & #2  ! " ! ! & "2 F "#2# %  5 # 0 '  ( %  5 # 0 C K ¢ £¤ £¤ £¢ £ ¢¡ £¤ ¡ ¢¢ £ ££ £ ¦¡ §§ B # ! "# $ 5 8 $ 0 # " A H  # ( % 2 " 0 2 5  # # 0 B   #  0 2 5 %2 !  # # 0 '  #2  ! " % ¦ # ! " # $ 5 # ( % 2 " 0 4 !  G B ) 0 % %  2 " % 0 '5 & B ¥ "2 $ # % ¦ #2 @2 ( # (  ' " & 4 0 C B # ! "# $ 5 #  "% 0 ! 0  & #  0 42 %2 4 5 % 0 "#2 ( 0  9 Dmindist B ¦ ! 2 % " %  5 " % @ 2  ( # ! " # $  5 #  $ 1 # " A H S ( 4 $ 4 2 % 2 4 ! H 4  %  9 B¥ (  ' " & 4 0  F # !2 0 # # 5 & % " %  # # ! " R 4 0 ! 0  8 $ Q B " A H S  ( $ 1 2 " # 2 ! & " 5 0 ! 0 5 4 4  5 & ! & ( 2 # %  5 " # %  2 # % 42 ( 0  # !   0 9 ( # %2  4 ( # " % 0 " # 2 ( " %  # # %2 # 2  @ # '5  !  # $  # # ( & 2# 9 # & % %  ( % 2# % 42 ( % $ " " A H  % $ ! $   " % [email protected] $ #  " # 2 5 %2 !  C B "2 $ (  ! "%2 " % 4  0 ) & " # 9 # % 2 # 2  @ # '5  !  # $   # ( %  2 " 0 # 2  2 " $S  ! $ # & # 0 H 9 § ¤ ¥ ¨ ¦¥ ¤ ¦ ¥ ¦ ¤ ¥  ¥  &    0 (  ' " & 4  @ $  % % $ 9 # " A H  8 $ 0 # &2 5  # # 0 # % 2 # % 42 ( # ( % 2 " 5  & # 0  ! $ " 5 3 ! $  T B ( # $  ( # "% 0 " #2 ( "%  # # " A H  8 $ ( #   $ 1 #  ! $ # # %  2 # % 4 2 ( ( ! H 4  %  2 # % 2 0 " R ! 5 % 0 " # 2 ( " "  B § ( '5  !  & 8 7 " 9 " A H S  F # & 2 5  # # 0 # %  2 # % 4 2 ( # (  H 4 # % S  "  % & ( G2 "2 #    & ! % $ " # Cminsize   k α −i → x Si → → dod (−i → −j ) = |Si | − |{d ∈ Si ∩ Sj | |xid − xjd | ≤ }| x x →→ → → → → dod (−i , −j ) = max{dod (−i → −j ), dod (−j → −i )} xx x x x x B # %  2 # % 4 2 ( # $ 1  $ 1 " % 4  $ # ! $ # # '5  !  # R ! " $ 1 # %  2 # % 4 2 ( # # $ ! H 4  % ( ! $ # # '5  !  " % 2  # # " A H  8 $ ( $ 1 $ 1 2 " # 2 ! & " 5 0 ! 0 5 # $   " #  2 9 & " 2  0 % ¤  2 # % 42 ( ( % 0 ! ) ( 5 0  # % $ # % 0 ( $ 1 # R ' "   U 'S  ! $ # & # 0 H " # 2 5 ¤   B "2 $ (  ! " %2 & " # 9 ¥ ¤ ¡ ¨ ¦¥  ¥ ¨ ¤ ¥ ¥  ¡ ¢ ¤ ¦ ¥ ¦ ¤¥ ¥  ! $   "  % 9 5 % 0 " #2 ( ( ! $ # 4  @ $  % %  B # %2 # 2  @ # '5  !  & # 0 H $ 1 2 % '5 " % $ " % # 22 " $ # 2 9 8 $ 2 4  ( %   # ! !  5 ! $  2 $ 1 # %  2 # % 42 ( S  !  H2 5 ! $   #2 0 4 9 # ! "# $ 5 #  "% 0 "% # & !  ! # ( ¨  ( & 4 ( $ 1 2 " # 0 '5 ! % $ ! $ # " % 4  0 ) & " % # 0 H # # ! $ " $ 0 #  9 ¦ § § 9 C ( % 0   ¤ # % 0 Q ¤ dod # $  #  ! $# # ( H4 #% ¤  " # '5 ! ' 5 K £0 £ 0 ££ ) £ →µ ∆k = min{l=k} dist(−l , −k ) µ→ B # & % %  ( #  # % 0 ( "2 $ ! H $ ( "% 0 " & 4 4  5 # & ! & (2# %  5 " %  # # R !  0 ¤2 5 & " % # & !  ! $# 4 0 F ! $ 2!&  $# "# # (¨  ( & 4 #  # $  " F 5 %0 "#2 ( 0  "%  ( # " A H  #  #2 $ T B # ! "# $ 5 8 $ 0 " % 0 % " ! 0   0 # " A H  #  # $  " ( % 2 " 5 %  G % #2 0 4 9 # ( ¨  ( & 4 # ( % 2 " 5 %  G % " % 4 $ 12 % $ # $  %  % 9 # & $ 0@ & & ! "%  # # ! "# $ 5 8 $ 0 # &2 5  # # 0 # %  2 # % 4 2 ( #  9 ( !  H 0S ( " $  § B & % 4 " # ¨ ¤ ¥  ¥ ¤ ¡ ( # 0 '  ! R 2 % ! ( % $ 9 % 7 % I B # "%0 ( %  # !!  5 # ! "# $ 5 8 $ 0 # "% 0 ( %  # ! !  5 ! "# $ 5 # % 2 # % 42 ( #  ! 2 5  # # 0 " 9 "% 0 "# ! 0 42 %2 4 #  #2 $  9 $ 1 0 '5 ! $   0 4 2 % 2 4 8 $ ( #  ! % %  2 " 5  & # 9 " % 0 # # 2  ! 5 ! ( !  ! 0  #  ! 2!" #2 $  − → µk ∆k K × (L − 2) Ck  Ck d Zkd ¥ § £ ¤ £ ¥ ¡ ££ ¦ £ ¡ ¢ £ ¡ £ ¤ ¦ ¢ ¡ £ £ £ ¥ ¡¦ ¡ ¢  B   4 8 !0  5 @0 " % 4 $ ) ! 0 $ 1 0 '5 ! $ # " % 0 # # 2  ! 5 %  "  %  4 %  2 " 5 %  G % $ "# P 9 ! 0  2 % 7 & ( " # & " A  !  ! " # $  5 % $S ( & " 2  0 $ 1 0 C B B ¦ B ¦ $ 1 !2 ( ¤F ¤" # S 5 9 # % ( ¤ "# B¥ $1  " " P   $  5 % $ !0  2 % 7& ( "# & # % 0 ( ! $ $ ) %  ( # % ( ¤ & " A  !  ! "# $ 5 % $ 9 " % % ( "%0"  B # &25 ## 0 "# $ 5 #  ! 0  ! " 5  # ! F 4 $ 42 %2 4 ! '5 ! '5 ( " # # & " A  !  # ! " # $  5 & "% # & !  & "22 H2# #   ! " $ 0 %  µ(a, b) = a(1/β )b µ(Nk , |Sk |) µ : R×R → R µ(0, 0) = 0 ∀ d ∈ S \Sk , max{i∈Dk } xid − min{i∈Dk } xid > w ∀ d ∈ Sk , max{i∈Dk } xid − min{i∈Dk } xid ≤ w α Nk ≥ α × N Sk ⊆ S α  Dk 0≤α≤1 w≥0 (Dk , Sk ) Dk ⊆ D w D # %  2 # % 4 2 ( # # ! $ # 4 $ 4 2 8 0 4 ! $ ) ! 0  % $S $ 1 2 # % 2 0 # ! & "2# % ( % $ ! $# # 0 H # 2 $ 1 ! R "2 ! 5 % $ ! #2 42 "   F # ( %  2 " 0 4 !  G 0  ! $   ¦ § § 9 B  0 " 5 $ 2   5  ! T¤ # % 0 ( ¢ ( ¨  ( & 4  "%  ( ! " # $ 5 $ 0 &2 5  # # 0 "# B '5  !  # $   " # D £ ££ )) £ ( " A H  $ 1 0 '5 9 % 7 % I k =1 Q= Nk × |Sk |  K &@! # %5 "# " % [email protected] $ # ! R "2 ! 5  " % 0 #2 42 "   # ( ¨  ( & 4 (  H 4 # % S  #2 $  9 # " A H  # ( % 2 "2 % 7& ( (   0 @ ! " % 2S  ( & F ¥ ( ! 2 ! 0 @ " % 0 # 2 0 G % # 2  G & " $ 5 & 8 # !   0 " # (  ' " & 4 " "  B £ & $"5 3 ) 0 % %    2 " % 0 ' 5 &S  ( " % 0 ( %  & (¢ # " A H S ( ! H 4  % % 2 0 " ! 5 % $S ( # % 2  4 F # & 2 5  # # 0 " %  # 2 $ 1 8 $ 5 ! 42 !   $ # " # ! " $ 0 # ( # % $ #  # '5  !  # ( ¨  ( & 4 8 $ 0 @ $  % #  !  $  ! ) ! B & & F # &2 5  # # 0 " %  # 2 $ 1 #  5 "%  # ¦ # % ! 5 %  5 # ( ¨  ( 4 # ( & ¢ ( # $  # & @ ! # %  5 # %  2 # % 4 2 ( #  " 9 # & 2 5  # # 0 " %  # ! $  2 $ 1 # " A H S ( ! H 4  %  ! 0  & ! & ( ( % % U 4 0  # !  0 "# % 2# $ G 0  ( " % 0 " $# & ! ( ¨  ( & 4  ¤ %   8 $ 0 2 "2 %2 # ( ¨  ( & 4 #  → → dod (−k → −l ) < Dmindist µ µ " % [email protected] $ # & "2 42 8 !  ( %  2 "2 ( %  5 0  " % 0 72 ! & @ " # (¨ (& 4 #  !  $!) ! B  & ¦ "2 $ ! H $ ( " % 0 " 4 4  5 # & 72 " % ( 2 # !   0 " %  # ( ¨  ( & 4 % $ 5 $ 0 F # & % )2 # # 0 ! "  " % @ $  % 2 $ 1 # " A H  #  − → µk − → µl → → µ x dod (−k → −i ) = 0  "% [email protected] $# % 2 "2 ( %  5 0  F # '5  !  # $  #  " %  # # 2 " %  ( # ( ¨  ( & 4 8 $ 0 ( # " A H  #  ! % )2# # 0 B & ¦ § ¤ ¥ ¨ ¦¥ ¤ ¦ ¥ ¦ ¤ ¥  ¥  ( (  ' " 4 ( ¨  ( & 4 $ 1 0 '5 F # & 2 5  # # 0 # %  2 # % 4 2 ( # (  H 4 # % S  ! % 2 4 ! " & ( B ¥ −i → x D O 0 !0  ( − → µk  # "% [email protected] $# #  0 " & #  "2 $# % "2 $ # # ! " # $ 5 # ( % 2 " 0 4 !  G 0 C B # ( ¨  ( & 4 # ( H 4 # % S  " # & %%( # (  H 4 # % S  # & & ! 5 " %  # #  H 4 # % 8 $ ( 9 ) 0 % %    2 " % 0 '5 &S ( # 0 '  ! R 2 4 !  0  # % 0 Q  ¢ £¤ £¤ £¢ £ ¢¡ £¤ ¡ ¢¢ £ O D ££ £ ¦¡ § M D (Ci , Cj ) = 1 |Si ∪ Sj | d∈Si ∪Sj (1 − M (Ci , Cj , d))  & @  &   ! " "# ! $  0@ 0 # 2 # # ! " # $  5 8 $ ( ! % %  2 # $ G ( 0 ! '5   4 2 $ 1 9 " 2 $ (  ! " % 2 5 %  ( " # ! $ # 4   @ $  % %  B " % ! & 32 ( # R ! " " # # & 2 5  # # 0 # %  2 # % 42 ( ( ! H 4  %  " %  ( # ! " # $ 5 8 $ ( ! " % $  9 # " % ! & 32 ( # R ! " # 2 0 " ( # ! " # $ 5 8 $ ( ! " % # %  2 # $ G # ( # & $ " 5 3 " %  # $ 1 # !   ! @2 ! ! 0 " % @ $  ¨ ¤ ¥  ¥¥  § ¡ ¦ ¥ ¤ ¡ ¢ ¨ ¢  ( # 0 5 8 $ ( 9 ! $ # 4 " " 5 " % 0 # 2  2 " $ % 9 " % 0 ( %   B # &25 ## 0 ! "% % 2# $G 0  ( " 0 " $# & !  5 @0  ¤ Cj "9 " Ci Sk # % 2 # % 42 ( # # (  H 4 # % S  Ck d∈Sk R(Ck , d) RSim(Ci , Cj ) = # % 0 ( & #22 " $ "% 4  0 ) & " # 52 ( %2 "  B % $S ( # $ # # ( $ 0 " # ! $  0 @ 0  " %  ( #   5 # % 2 # % 42 ( # ( % 2 " 5  & # 0  !2 0 G ! $    # ! " # $ 5 & ! "% & "2 ! 0 2 42# (  $ 5 0 5  $&#&12%24"%$0(% U!"(2 $"#%#$%50"G%#$ (! $5"%#$%!A20"!!&0(!2##%(2$515F22(#$%"2#%"%%202""!! 5 2 σ 2 (Ck , d) σ 2 (D, d)  ¥ R(Ck , d) = 1 − 0 H  ) " 0 5  ( !0 ( %0 "# % 2"02 @& ( 0 !"% "!   0 !  4 4  5 2 % 7 & ( ( !  H 0S ( " # ! " # $  5 % $ ! $   %  2 # % 4 2 ( % $S ( 5 % % 2 " !  ( 5 2 ( % 2 %  B ! $ " 0 # 2  2 " $S  ! 0  # ! " R 4 0 ! 0  ( %  2 " 0 5 7 2 5 &  # 0  ! " 2 @ &S ( % 7 0 $ 1 2 4 0 % U ( %  ¥ 0 G ( # & " # $ A 0 " %  ! # 2 $ 1 #  2 $ # # ( ! # 2  2 " $S ( " % #    !  ¦  § § 9 B  0 "  2 ¡¤ ( # ! $ " $ 0 #  9 £B 5 " 9 # ! " # $  5 # ( # " % % 2 " !  # %  2 # % 4 2 ( ( # % U  4 ! H 4  %  3  C  §  T 9 & " 2 # % ( (  2 $ # % $ " 2 # # 5 & % I  ¤ £ C ¢  R (  4 $ ( # ! " R 4 0 ! 0  # % 2 0 " ! 5 ! 8 7 ( ! $ " 0 # 2  2 " $S  F % 2  #  " % # # 2 0  # " % 0 " # 2 8 # 4 ' " 2 !  )  0 #  $ 1 # !   ¦ d Ck ¢¨ ¢ £ £ ¨)£ B # & % ! $  " ! " % 4  0 % 7 " %  ! # 9 ! '5 $ 0 @ '5 # " % @ $  2 $ 1 9 % 2 " 5 %  G 0  " % 0 #2 42 "   # ! "# $ 5 # C B & $ %2 42 (   ! " 0 # ! "# $ 5 # ( & "2 0 $ 1 ( ! $# 4 0  $  9 "%2 " " 0 & " & 0 # ! "# $ 5 ( ! H 4  % %2 0 " ! 5 % $ $  9 &  $  ! ) & " & 0 # " A ¤ H S ( ) 0 " % 5 ! $   %2 0 " ! 5 % $ # %  ¥ 0 G # ! $ 2 # $  ( 2 % 7 & ( ! "  " $  "  ! ! 0S ( ! R "2 ! 5 C B # % ( ¤ " #  2S # & @ ! # %  5 " # ! " # $  5  " 9 ! $ # ! $ ) ! 0  ( " !"% 5 ( H $ 5 !  U 'S  # % 0 ( # $ % " %  5 ( # " A H  #  "%0 % % 2"5 &# % ! $  A F #2 4 "# #2 $ T µ w  Dk − → µk Sk α D Sk = {d | |xid − µkd | ≤ w ∀ i ∈ Dk }  & & ! 5 "2 $# % "# & " 5 # ! "# 4 $ 42 80 4 ! $ ) ! 0  ( ! R "2 ! 5   $ 1  ! $ # # % 2# % 42 ( # (  H 4 # % S C B # &2 5  # # 0 " %  # 2 $ # " A H S (  H 4 # % % $ " % 2 0 ! ) 4 4  5 " % 4 ! 2  " 0 &  0 2 # 2  '5 " # " AH % $ ( !  H 0S Q B $ 1 2 " # 0 '5  " # " # # ! " # $  5 # ( %  2 " 0 4 !  G 0  ! $   & #    !  (  ' " & 4 0 C − → µk Dk § £ ¤ £ ¥ ¡ ££ ¦ £ ¡ ¢ £ ¡ £ ¤ ¦ ¢ ¡ £ £ £ ¥ ¡¦ ¡ ¢  ! " $ 0S  F %  2 " 0 ! & " 2 % $S ( ! " # $  5 ( %  2 " 0 " 5 3 0S ( ) % 0 ' 5 % " A H  % $ 5 $ 0 # $   $ 1 ! 2 ( ¤ F ¤ " # S 5 9 5 % ) ! @ %  5 " 2 0 U  2S $ 1 5 FS $ 1 # $ A # 2 $  B  ¦ 50 # S  ( %  2 # % 4 2 ( $ 1 0 '5 ! $ # ! " # $  5 $ 1 0 '5 ( # (2   #  ! #2 0 2 "2 %2 B & & F& & & & & ¦ # % % 2 " 5  # A ( 8 $ 5 (  H2# #   # % )2  # $  #  "%  # # % % 2 " 5  # # " % [email protected] $ # #  #2 $  9 " % 4 !2  " 0 & 0 & !2 " "# ! 2 4 !   8 $ 0 2 "2 %2 # ( ¨  ! " % 5 5 @0 ! ! ! 0 4 & ( B ¥ d Ck   √ wkd = 1/ M K  " 2 $ # 4 4  5 ( R 5  !  4 ' " 2 !  )  0S  " I B 5 0  # S  ( # %  2 # % 4 2 ( #  ! $ # # ( 2   ( ! $ " 5 @ % $ F & 2 5  # # 0 9 # " A H  # (  H 4 # % ¤ # $  # "2 $ (  ! " %2 5 %  ( " # ( ( ¤ ¦ ¢ ¥ ¨ ¦ ¢¤  ( %  2 "  % %  B ! " # $  5 ( ( ¨  ! " % 5 $ 1 0 '5 F 5 0  # S  ( # %  2 # % 4 2 ( # (  H 4 # % S  ! $ # # ( 2   ( ! $ " 5 @ % $ 2 5  # # 0 " 9 # % 0 4 ¤ ¦ 4 ' " 2 !  )  0S  ! $ # 0  5 ! $   # 0 H #  ¦ C B  H 4 # % ¤# $  # % $ ! % % 2 " 5  & # % S ( $ 2  $ 0 9 & ! & ( 2 # %  5 ! " # $  5  ! $   5 % 0 " !   4 2 %  # " % 0 # 2 5 & !  # ( 2   % $ ! " # $  5 $ 1 0 '5 ( %  2 # % 4 2 ( $ 1 0 '5 F ! 2 5  # # 0 0 @  ¦ C &    0 " ¦ § § 9 B  0 " 2 %  5 2 % 4  Q¤ # % 0 ( & #    !  (  ' " & 4 0  9 ! $ " 0 # 2  2 " $S  ( " ! 0  0  ( # ! " R 4 0 ! 0  ( %  2 " 0 5 7 2 5 &  # 0  ! " 2 @ &S ( % 7 ¦ ¤  " Rmin = Rmin − 1 M −1 R(Ck , a) × M D (Ci , Cj ) < Rmin ∀ (Ci , Cj , Ck ) |Sk | d∈Sk  ¢  ££ £)£ Smin = Smin − 1  #2 $ # #  ! $ %2 42 ( B $ |Sk | < Smin  H 2 # #   " 2  # % %  2 # $ G % $ 5 $ 0 # $   $ 1 5 FS $ 1 # $ A & "2 ! 0 2 42 # ! $  "% 0 #2 42 80 4 £ 9 ¢ " # ! "# $ 5 8 $ ( #  ! % % 2# $ G B ¥ " % 2 " " 0 " 2  #  H 2 5 # ! " # $  5 ( ! H 4  % % $S $ 1 5 FS $ 1 # $ A # & ! & " 2 #  0 " & 8 $ ( % % %  2 " 5 %  G 4 ' " 2 !  )  0S  9 " # 2 $ # 8 $ ( #  & # 2 0 2 "2 %2 !2  @0 # R !  ¦ B£ $ 1 ! "  % F¢ # & 2 5  # # 0 # %  2 # % 4 2 ( # %2  4 $ 0 ( R # #   F "% 0 " $# & ! ! "# $ 5  2# ! % % 2# $ G F # & #2 !  " $ 0 " %  # " # ! "# $ 5 8 $ ( " I B ! $ 2 ! &  $# " # 5 % %2 " !  ( 52 ( %2 %  # 2 # ! " # $ 5 % $ F &2 5  # # 0 # !  0 " # % 2 # % 42 ( %  B # %  2 # % 4 2 ( # 5 ( 5 % % 2 " !  ( 5 2 ( % 2S  ( 4 $ 4 2 % 2 4 ! $  0 @ 0  " & ¦ # ! " # $ 5 8 $ 0 # 2 5  # # 0 # % 2# % 42 ( ( 4 $ 42 %2 4 ! H 4  %  $ 1 2 '5 ! 0 ! & 2 ' " # & # 2  2 " $ 4 ' " 2 !  )  0S C Sk ⊆ Si ∪ Sj Rmin = 1 RSim Ck Cj  # $ 12 4 0 % U ( #2 $ # 8 $ ( ! $# & # 0 H " % 0 ( % 5 # 0 (µ(Ci , d) − µ(Ck , d))2 + σ 2 (Ci , d) σ 2 (D, d) R(Cj , d|Ck ) ≤ 0 £ ¢¡ £¤ $ ¡ ¢¢ £ ££ %  %2# 2# R(Ci , d|Ck ) ≤ 0 £ ¦¡ 0 ¢ min(R(Ci ,d|Ck ),R(Cj ,d|Ck )) max(R(Ci ,d|Ck ),R(Cj ,d|Ck )) £¤ £¤ £¢ Ci  C j ⇒ Ck Ci Smin = M Smin Rmin Smin Rmin  R(Ci , d|Ck ) = 1 − M (Ci , Cj , d) = § B #  & ! # " A H S ( ! H 4  %  ! ( % 2 " " 0 ! $   # & " $  A 0 " %  # " $ 0 G  2S $ 1 # G 2 " 5 7 # " A H S ( " % 0 " $ 0 # !   0 9 #  & ! # " A H S ( $ 1 # G 2 " 5 7 # " A H S ( # % 2  4 0 U  2 P  5 0  # S  ( %  2 ) & ! % $ # % 0 ( @ 2 ! ! 0 (  ' " & 4 0  $ 1 # !   # 2 $ T B #  & ! # " A H S ( " # 2 8  2S $ 1 # & " $  A 0 5 %  ( " %  # # " A H  # ( %  2 " ¤ 2 ! 5 # ( ( 5&0  # S & # % 0 ( " % 4 & 4¥!  G2 % $ # 2 " ! 0  & ! # G2 " 5 7 # " A H S ( " %&0 " $ 0 9 " % 4  0 2 "2 % £ B 2 ( $ " " % 0 ! $  5 %  0  # % 0 ( # G 2 " 5 7 # " A H S ( $ 1 #  ! # " A H S ( $ 1 " % 0 " $ 0 # % 2  4 $ 0 # ! $  A$  " " 2 0 U  2S $ 1 " !  # % ! 2 0 G ( " % #    !  # ! $ " $ 0 #  9 0  5 ! $  T B %  2 # 2 5 & ( ( ! H ! 0S  (  H 2 # #   $ 1 " % % 2 " !  2 # # $ 0 %  2 " 0 & ! 5 % $ ! $   " % 4 @ 2 " 0 ) & % # & # # 0  5 # G 2 " ¤ 5 7 # " A H S ( " $  A 0S  !  ¤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§C  &    0 (  ' " & 4 0  (  0 ! & % & ) & ( 2S C B ! $ " 0 # 2  2 " $S  ! 0   H 2 # % ' & !  4  5 " 0 " $# & ! % $ 2 " !  # % !2 % ! $  G ( " 9 # !   !  "%  # ! $  2 $ 1 # % 2# % 42 ( ( #  H 4 # % ¤# $  # # ( # & 4 !  G # ! " # $  5 8 $ 0 ! 2 5  # # 0S ( 2 # % 2 0 " 4 !  2 $ 1 9 & #    !  & " & 0 ) % 2 ! " # $  5 $ 0 %  2 # 2 5 & ( ( # ! H ! 0 #  ! "  0 ( 0S ( " % 0 " " 4 !   0 % 2 ) 2 !  $ 1 2 % ' 5 " % $ 9 ¦ § § § 9 B  0 " $ 2 C¤ # % 0 Q ¥ ¥ ¨ £  −= → µk Nk i∈Dk −i → x  # ! "# $ 5 # ( # ( ¨  ! "% 5 #  ! $  A F !"" 4 " →→ Dk = {i ∈ D|k = ArgM inl LW (−l , −i )} µx  # ! "# $ 5 8 $ 0 # &2 5  # # 0 # " A H  #  ! $  A F ! " " 4 & F &87 " 9 ¦ "% 4  0 " % 42 !  8 ¢ # (2   # ( F 5 % 0 " !   4 2S ( # % 2  4 $  # $   ! % %  ( ( " % 0 " " 4 !  M j =1 exp(−h Ykd × 2 × Ykj ) ! $  A F #2 4 0 # %0 ( wkd exp(−h × Ykd ) − xid )2 %  2 # % 4 2 ( $ 1 0 '5 ! $ # Nk i∈Dk (µkd − → µk d  ¡ ££ ) £   h wkd = Ykd = ( ¨  ! " % 5 ( # ! "# $ 5 # ( # (2   #  ! $  A F ! " " 4  d=1 →→ LW (−l , −i ) = µx wld × (µld − xid )2 M →→ Dk = {i ∈ D|k = ArgM inl LW (−l , −i )} µx ¤ 2 8 !   # (2   ( 52 ! " 0 4 0  ( " % 0 ( %  & ( & "2 4 ( %  2 "  % 0  9 ' 5  !  # $    " #  2 " %  ( ( ¨  ! " % 5 $ 0 " A H  $ 1 0 '5 ! 2 5  # # 0 W § £ ¤ £ ¥ ¡ ££ ¦ £ ¡  ¢ £ ¡ £ ¤ ¦ ¢ ¡ £ £ £ ¥ ¡¦ ¡ ¢ |xid − xjd| < δ ∀ d ∈ Sk  F ! $ 2 ! & G %2 5 % 0 " #2 ( % $ "%  ( # % 2# % 42 ( # (  H 4 # % S  ! $ # # ! $  0@ # ! $  2# ! $# ! "# $ 5 ¤ % $ "% 4 !  G " # " A H  8 $ ( 9 # " $ H 2 ! " " 0S (  H 4 # % % $ & % %  ( " % 0 "  B ! "# $ 5 ¤ % $ "% 4  0 ) & "2  # $ 1  " "#2 8 S %  H 4 # % ¤ # $  # % $ 5 $ 0S $ 1 ! 7 2 ! & @ 5 %  ( " % @ 2  ( # ! "# $ 5 8 $ 0 # &2 5  # # 0 # % 2# % 42 ( ( #  H 4 # % # C B # % 2# % 42 ( ( 4 $ 42 %2 4 ! H 4  % % $ ! 0  # ! "# $ 5 #  ! #2 ! & " 5 0 ! 0 5 ( G2 " 5 A H  4 4  5 % %  ( # " 9 " % 0 ( % 5 # ( " " # 2 %2 4 ! " & ( " # (  ' " & 4 0 C B # ! " # $ 5 ¤ # ( % 2"0 4!  G 0 ! $   # & !& (2# %  5 # % 2# % 42 ( (  0 42 %2 4 ! H 4  %  " # " A H S ( 4 $ 42 %2 4 ! H 4  %  9 2 $ #  (  ' " & 4 0  F # ! 2 0 # # 5 & % " %  # # ! " R 4 0 ! 0  # 2  !§ B ! " # $ 5 ¤ % $ 2 # # $ 0 "# !$  $5 %  δ Sk Sk ⊃ Sk Sk (Dk , Sk ) Sk δ −i → x Sk −j → x δ Ck δ  mind mins δ (Dk , Sk ) Sk ⊆ Sk " X Dk ⊆ Dk ( 5 @0 52 ! " 0 4 ¤# $  # " $  " 9 2 # %2 ¦ B 52 ! " 0 4 ¤# $  # " $  " ! $   2# ! " # $ 5 ¤ % $ # !  0 4 !  G (Dk , Sk ) δ 9 (Dk , Sk ) pScore(X ) ≤ δ δ≥0 δ pScore  2×2 = |(xia − xib ) − (xja − xjb )| xia xib xja xjb "# "%0 ( %  # !!  5 52!"0 4 0 ( 9 " # & % %  ( " % 0 "  B # " A H S (  H 4 # % ¤ # $  # % $S ( ¥  ¤ ¥  ( £ ¦ 0  ! 2 % 7 & ( ! $   " 2 $ (  ! " % 2 5 %  ( " # ! " # $  5 ¤ (  R (  4 $ 0 @ $  % %  B # %  2 # % 4 2 ( (  H 4 # % ¤ # $  # % $ ! $ # ¦ ¥ ¤ ¡ ¤ ¥ ¨ # 4  4 #  " %  2 $ 1 # " A H  #  !  $  ! ) ! ( # !   0 " # ! ( 0 5 5 # % 0 ( G 2 " 5 A H S C B # ! $  0 @ # 5 ( $ 1 2 " % ( 2 %  2 " $   @ & % $ ! $ # # 2 0 4 # %  2 # % 4 2 ( # % 2 0 " ! 5 ! $ # # '5  !  # ! $  0 @ # ( ! 0  # & # 2 ! & " 5 0 ! 0 5 # 0  " %  # % #  $  ! ) # % 2 0 " ! 5 9 G2 " 0 !  H 0   5 ) 0 ! " 7 % $  # $ 1 2 " 0 4 !  G %2 ¤  2 H # & % %  ( ( # # 0 H # ( # % 0 (   4 8 ! 0  4 4  5 9 # 0 5 # % 2 0 " ! 5 # % 0 ( $ 1 " % " 0 " # %  5 # ! $ " $ 0 #  9 ¦ § § 9 B  0 " ) % 0 ¤ # % 0 ( 9 % 7 % I i, j ∈ Dk a, b ∈ Sk pScore ¤ δ #    © ¦ ¢ ¡£ £ &' δ d s |s|/|d| ) £ B ! 0  2 % 7 & (   0 @ ! " % 2S  ( @ 2 " 0  ! ! $ $ ) %   0  F " % 4  % % 2 " !    !  & $ 5 0 5 5 %  ( "# (  H 4 # % ¤# $  # $ 0 # &2 5  # # 0 @2 " 0 ) & % # # 0  5 0  ( # ! H 4 4 ( ! H 4  % C B £ # 0 5 ! "  % # % 0 ( 8 $ ( F  0 ) &¢ # # # 0  5 (  0 "  " ! H 4  %  " 9 # %0 ( # # 05 0 ( # ! H 4 4 ( % 2"!    !  0 "# P 9 " 9 ! $# % 2 "2 " ! 0  0  " % 0 # #2 % 7& ( ( #  H 4 # % ¤# $ # #  9 "% 4 %  ! $   & % %  2 " 5  & # %  2 # % 4 2 ( 0  9 # & % ! 5 %  5 # " A H  # (  H 4 # % S  5 @0 s s c pi × log(pi ) i=1 pi Entropie(E ) = i E c d ¤ % 2 "2 " ! 0  d s∈subset(d) |Es | Entropie(Es ) |E |  − d E IG(E, d) = Entropie(E ) − & % %  2 " 5  & # %  2 # % 4 2 ( 0  F " !    0 ! ! 0  & & ! 5 %  2 " 2 " ! 0  ¤ # $  # $ 1 0 ' 5 # % 0 ( # # 0  5 $ 1 0 '5 ( # ! H 4 4 ( ! H 4  %  $ 1 " % 0 " 2 # # 5 & % % ! $ # 4 " " 5 9 ! H ! 0S  ( %  2 " 5 $ ! " # %  5 0  ( # !   & # 2  2 " $ ¤ ¦ ¥ ¨ ¡   ¦ ¤ ¥ ¤ ¥ ¡ $ (  $ 5  0 5  # % 0 ( " % 4   4 2 # " 2 0 G " # 2 5  B ¥( ¤ ¦ ' ¡ ¤ ¡ # &  $ 5  0 5 ! "  " % @ 2  ( #  2 9 " % 4 $ 1 2 # U '  # & " $  A 0 # 0  " % 0 " &S % # G2 " 5 7 # " A H  # C B # & % %  ( #  # % 0 ( ! "#2 8 " $  2 $ 1 "2 $ ! H  !  H 2 5 ( " 4 !  2 $ 1 5 9 & % 4 " # S % %  2 " 5 0 % $ 5 $ 0 9 #  & ! # " A H S ( $ 1 # G 2 " 5 7 # " A H S ( # $  0 U 2  $ 1 0  # % 0 ( 5 0 # S  ( % 2 ) & ! % $ # % 0 ( @2 ! ! 0 (  ' " & 4 0  !2 0 ! " %  5 $ 0 2 3 ¢ £¤ £¤ £¢ ¤ £ ¢¡ £¤ ¡ ¢¢ £ ££ £ ¦¡ § ¤  & F ! ( % " 0@ # & % % 2"5 &# # % 2# ¦  H2# #   # %2  4  ! % % 2 " 5  # % ¤ % 42 ( #  ! $ # # ! $  0 @ ! " % # 5 % 0 " # 2 ( #  ! # 2 42 %2 4 9 # " A H S ( 8 7 ! H 4  % % $ ! $   #&#   # % # # ( # %0 ( " %  @ # " %2 0 ! " %  5 # 5 $ 1 "2 0 G $ ( " % 2 @ 4 R  H  !  $ ( & " $ 5 £2 ( 0  # 2 0  B ( ! 0 # 0 ' $ ( "2 $ ! G  " % 2  # # " A H  # 5 ! " % # 5 % 0 "#2 ( #  H2 0 G #  $ 1  H 0 H  !  $  5 %  ( "# S 5 " 9 # & % % 2 " 5  & # # % 2 # % 42 ( #  ! $# "% 0 " !   42 " !    $ # % $ $ 12 ( %2 ! "# $ 5 % $ # % 0 ( " % 0 " !   42 # ! H 4 4 ( ! H 4  % % $ " I B "  5 %  5 4  4 $ 0 ! 2 % " ! 0   0S ( " !  G & " 2  2 H 0 H  !  % $ " %  #  2S $ 1 5 %  ( " 9 % 2# % 42 ( ( % 0 ! ) ( 5 0 # ¤# $  # % $ # % 0 ( # !2 0 2 42# "%  # ! "# $ 5 $ ( # ! H 4 4 #  $ 1 $ 12 ( %2 # &2 5  # # 0 # % 2 # % 42 ( ( "% 0 " !   42 ! H 4  % %  B # ! $  0@ ( #  0@ ! "%2 # "2 "  ( ! $ # " % ( !  5 5 0S # ! " # $  5 5 ( # ! H 4 4 #  $ 1 " % $ 1 2 ( % 2 # & 2 5  # # 0 # %  2 # % 4 2 ( # # ! $ # ! " # $  5 4  4 % $S ( # ! H 4 4 # ( # ! $  0 @ #  ! " % # 5 % 0 " # 2 ( #  H 2 0 G ( 9 " % 4 @ 2 " 2 $ " % £ B # ! "# $ 5 # ( # ! H 4 4 ( ! H 4  %  ! #2 42 80 4 " B  9 # ! "# $ 5 8 $ 0 # &2 5  # # 0 # % 2# % 42 ( ( ! H 4  %  ! #2 42 80 4 B 9 # &2 5  # # 0 # % 2 # % 42 ( # # ! $ # & % %  ( ! " # $  5 % $S ( # ! H 4 4 # ( # ! $  0 @ #  ! " % # 5 % 0 " # 2 ( #  ! # 2 4 2 % 2 4 B ¥ ! " % # 2 4  !  4  5 %  H % $ ! @ $  ! " ( " 2 0 G % " # ) % 2 ! " # $  5 5 0  # H $ # $ ( G 2 " 5 A H S C   ¤ ¡ ¢ ¢ ¢¦ ¢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£2 ( 0  ! " $ § B # " 0 ( 2 ( % 0 5 # " % ! & 32 ( #  2 4 ! 0  2 ! "  !2 0 G F " % 0 @ ! # " # !"R 4 ¤0 ! 0  #  9 # ! "# $ 5 ¤ # (  0 % 7 % 2 " 0 4 !  G 0  ! $   # 0 H ( # !  0 " % @ ! # #  H 4 # % #  B & "% 4 & ! 5 %2 "# 2 $ 1 F ! $ 2 ! &  $ # "2 0 " & 2 0 " 0 # 2 # & E5  " # " # " % ( & 5 & !  N mind O(N 2 × M log M + M 2 × N log N ) M mins δ  0 42 %2 4 % !  H 0  "# S 5 " 9 mins 5 % $ 1 & # ¤# $  # 0  9 %  %2 3 B & "% 4 & ! 5 %2 " #  0 42 80 4 % !  H 0  # !  0 9 F ! $ 2 ! & G %2 5 % 0 "#2 ( % $ "%  ! "  % G 0  ( # & "2 4 & ! " 8 8 $ 0 # ! $  0@ #  2 3 B 5 % $1&# 0 ! $# &50  ¤& ( 5 %  ( " # 9  0 42 80 4 % !  H % $ "  0 42 %2 4 % !  H % $ ! 0  2 % 7& ( 9 ! "  % G %  sj +1 − si > δ " " sj − si ≤ δ sj − si−1 > δ}  " δ −−− →→→ S ( xi , xj , S ) −−− →→→ S ( xi , xj , Sk ) = {si ...sj |j > i 5 % $ 1 & # " " 5 (  0 42 80 4 2 0 " ( 5 % $ 1 & # ¤# $  # 0  # % 0 ( # "% # & !  # % 2 # % 42 ( # (  H 4 # % S  F # !  0 ( %  # ! !  5 # % 2# % 42 ( # (  0 42 80 4  H 4 # % S C Sk −−− →→→ S ( xi , xj , S ) = s1 ...sM  " % 0 # # 2  ! 5 ! ( ! S  # % 0 ( # & ) % 0 ! ! $ # # ! $  0 @ # ! $  ( # 5 % ! & 32 ( # (  H 4 # % S  F " % 0 ( %   # ! !  5 5 % $ 1 & # 0  #  4 " ! 2 4 !  % $ # % 0 ( 4 !  G %  9" # " A H  8 $ ( F # &2 5  # # 0 # % 2# % 42 ( # (  0 42 80 4  H 4 # % S  !  H2 5 ! $  T −j → x  S −i → x Sk § £¥ ¡¡ B  H 4 # % ¤ # $  # % $ ! % %  2 " 5  & # % F ! 2  @ 0S ( ! " 2 @ &S ( % 7 0 %  2 # % 4 2 ( $ 1 0 '5 ( 5 % 0 " !   4 2S  ! ! & ( %   ( 8 2  '5  ! 2 0 G 2 $  0 @  ¦ C % 7 % ¦ ! $ " 0 # 2  2 " $S  ( " ! 0  0  ( 2 !  2 !  0 5 % 0 # # 2 0 % %  5 0  ( ! 2 ! & $ 1 ! ( ! " 2 @ &S ( % 7 0 # $ 1 2 4 0 % U ( #  2 $ # # ( ! # 2  2 " $ 2 $  0 @ T  ¦ § ¦ £  " # ! R " 2 ! 5¢ # ! " # $  5 # 5 ( # ! H 4 4 ( ! H 4  %  " # ! " # $  5 8 $ 0 # & 2 5  # # 0 # %  2 # % 4 2 ( ( ! H 4  %  " % # 2 4 2 8 0 4 2 $ 1 # ! " # $  5 #  ' 5 ! '5 # 2 $  9 # ! " ¤# $ 5 # ( %  2F" 0 4 !  G 0  ! $   ! " 5 F # ! F 4 $ 42 %2 4 & "2 # % ( ( ! R "2 ! 5 ! 2 4 !   ! 8 7 ( ! $ " 0 # 2  2 " $S  ( % 0 4 ( 2 9  § Q "% 0 $ 1 ¦ ! R "2 ! 5 ! 2 4 !   "% #2 42 %2 4 2 $ 1 # % 2# % 42 ( #  '5 ! ' 5 # 2 $  9 # ! " # $  5 8 $ 0 # & 2 5  # # 0 # %  2 # % 4 2 ( ( ! H 4  %  ! 8 7 ( ! $ " 0 # 2  2 " $S  F ( % 0 4 ( 9 2 $ 9 3  C  §  T ¦ ! R "2 ! 5 ( %  5 # $ 0 "% 0 ( %  # ! !  5 9 # ! "# $ 5 8 $ 0 # &2 5  # # 0 # % 2# ¤ % 4 2 ( ( ! H 4  %  " % # 2 4 2 8 0 4 2 $ 1 # ! " # $  5 #  # !   0 " % '5 ! '5 " 9 # ! " # $  5 # " % ! & 3 2 ( #  ! 0  ! " 5  # ! F 4 $ 4 2 % 2 4 & " 2 # % ( 0  ( ! R " 2 ! 5 ! 2 4 !   ! 8 7 ( ! $ " 0 # 2  2 " $S  F % 2  #  "% # #2 0  9 I  ¤ £ C   " 9 # % 2# % 42 ( #  ! $# # "% 0 ( % 5 # 0 ) %2 ! "# $ 5 5 0 # H $# ( # (  ' " ¤ & 4 #  # " % ! & 3 2 ( # $ 1 2 % '5 " # ( ! $ # " % # 0 H # # & " % # & !  # %  @0 # $  % $ 1 ) % 2 ! " # $  5 5 0  # H $ # ( # (  ' " & 4 # " % ! & 3 2 ( #  9 %  2 " 0 # 2 4 2 "  S ( 4 R  H  !   2 5 £ 2 ( 5 F 5 0§ B # & 2 5  # # 0 # %  2 # % 4 2 ( ( ! H 4  %  ! # 2 4 2 % 2 4 F ! ( % " 0 @ # & ! & ( 2 # %  5 # " A H S ( ! H 4  %  ! # 2 4 2 8 0 4 9 " % 4 # ! @ % 2 " 9 # " A H S ( $  $ 1 ! ! & ( 2 # %  5 % F ! ( % " 0@ # &2 5  # # 0 # % 2 # % 42 ( ( ! H 4  %  ! #2 42 80 4 9 # &2 5  # # 0 # % 2 # % 42 ( # # ! $ # ! " # $  5 4  4 % $S ( # ! H 4 4 ! " % 5 % 0 " # 2 ( ( & 8 7 4 $ 4 2 8 0 4 ! $  0 @ % $ ! $   " & F ! ( % " 0@ # & % % 2 " 5  & # # " A H  ¦  H2# #   # %2  4  ! % % 2 " 5  # % # ( # ! $  0@ ! "% # 5 % 0 " #2 ( #  ! #2 42 %2 4 9 # % 2# % 42 ( ( 8 7 ! H 4  % % $ ! $   ¢ £¤ £¤ £¢    £ ¢¡ £¤ ¡ ¢¢ £ ££ £ ¦¡ ¨ § § ¢ ¦ 4 $ 42 %2 4 "2  # # % 0 ( 2 # 2  '5 " # P " "% ( F ( # ##05 % 2 "2 " ! 0  0  P  {vj +1 , ..., vn } [i, n − l + 1] I ({vi , ..., vj }) + $1 5 F %¥0G (  0 " &S  F &  $ 5  0 5 & " & 0 i ∈ [1, n − l + 1] ¤& 5 & !  l−1 l−1 !$  W (Plj +1 ) −1 Plj +1 −1 j Pli = ({vi , ..., vj }, Plj +1 ) −1 i P1 = {vi , ..., vn } l ∈ [2, k − 1]  # % 2 "2 " ! 0  #   $ 5 0 5 %  9 !$  ¦ !$  #  % " % 0 @ 2 $ # %  ¥ 0 G 0  ( " 2 ! 5 & ( ! "  " $  4 ' " 2 !  )  0S  ( " % 4  $  ! & ( C B # ##05 % ( #  0 42 "   # % 2 "2 " ! 0  ( "2 $# % $ 5 % ! ! $ 5 & ! ! 0  !  F " # 2 # %  5 ! ' # 2 § ( 4 ' " 2 !  )  0S  9 $ 1 2 4 0 % U ( %  2 " 0 4 4 0 ! )  !  0  ( & ! 2  # % £  W (Pl ) = {vi , ..., vn } l I (ci )  l ci !$  9 2 0 " ( # ! "# $ 5 # ( # 2 " ! %2 # ( 4 4  # 0  vm Mi − mi i ∈ [1, l] # %0 ( # $% "%  5 9 l ¤ $50 5 i ∈ [1, n] Pli i=1 W (Pl ) Pl % 2 "2 " ! 0  0  " % % § Gi = Mi m=mi  ! " # $ 5 $ ( & "2 @0 ! ) ( !"% 5  "# Gi P m=mi I (ci ) = (vm − Gi )2 Mi  ci = {vmi , ..., vMi } !0  & % %  ( # ! 0 "# ! " # $  5 % $S ( 2 " ! % 2S C B ! $ # 4 $ 42 80 4 " 4 $ 4 2 % 2 4 # " % 0 7 2 " % ( 2S (   $  5 % $ ! 0  ! " # $  5 % $ 7 2 " % ( 2 % § B # & & ! 5 # ! "# $ 5 # ( £ ¢ # ! " # $ 5 ¤0 ! "%2 # 2 " ! %2 # ( £ ¢ 4 4  # 0  ( #2 @ ¤F ¤#2 @ 9 & 8 7 # ! " # $ 5 ( ! H 4  % % $ ! $   9  H 4 # % " 5 (  0 42 "   % 2 "2 " ! 0  0  ! @ $  ! " ( " 4 !  ¦ 0 ¨ ¢ ¥ 9 B  0 " U 0 ( 2 Q¤ # % 0 ( & #    !  " % 4 % %  2 " 2 " ! 0  ( 4 ' " 2 !  )  0S C B #  H 4 # % ¤ # $  # % ! % % 2 "2 " ! 0  F # ! $  0@ ( 9 2 0 " ( 9 & % %  ( !   H 4 # % % $ "2  3 E ci mi W Mi I k k E = {v1 , ..., vn } n ©  ©  ¡     ¨    © ¢      ©   §¦ ¢ ¤ ¤¤ ¡ ¢¢ &  ' #    $ '  &  ¢ § ¡§ B  ¡ ¦ ! $   ¦ ¢9 ¨9 ¥ 9 §¤  H 4 # % S  ( ! ' # 2 § ( " % 4 % %  2 " 2 " ! 0 T  ¥B  B A@ 9 [0]+ [1]+ [8,9] => [0] + [1] + [8,9] [0,1,8] + [9] => [0,1] + [8,9] i=3 i=4 [0,1,8,9] l=1 i=2 [1,8,9] [8,9] [9] => [0,1,8,9] B "9  H 4 # % S  ! $   9 4 ' "2 !  ) 0 " 5 ( % 2 " $ 5 & 8 S (   4 8 % $ ! "%  4 ¥B  ! $ ) 7 0 C ! #2 42 %2 4 F %  ¥ 0 G ( # % 0 ( 2 # 2  '5 " # P  k=3 B [0,1] + [8]+ [9] l=3 [8]+ [9] [1,8] + [9] [1]+ [8,9] l=2 [0,1] + [8,9] [0]+ [1,8,9] i=1 E = [0, 1, 8, 9] i I ({v1 , ..., vi }) + W (Pk+1 ) −1 [1, n − k + 1] i i 1 Pk = ({v1 , ..., vi }, Pk+1 ) −1 % 2 "2 " ! 0  0  "2 $ ! "# %  5 %  9 ¤ £ ¢ ¡ ¢ ¡ £ ¢ ¤ ¡ ¢ k  0 " &S  F 9 % 7 %  §¥ ¢¡ ¥¥ B 5 @0 9 # & % %  ( # ( "   4  5  H 4 # % S  " 9 # &@! # H # & % % ( # (  H 4 # % S  9 # & '5 0 5 & % %  ( # (  H 4 # % S  # ! 0 4!  G  H 4 # % S C B %  %2# 9 % 2 " $ H2 ! "#2 ( 0  ! 0  & ! & % & ) & " & 0 " A H S  2 # #   % § B  R (  4 $ ( # & '5 0 5 ! " R 4 0 ! 0  # ( 4 4  5 # $ @ # !  0 " %  # # ! "# $ 5 8 $ 0 # " A H  # ( # 5 % 0 % " ! 0   0 # C Dy = {yi |1 ≤ i ≤ θk # " # −i → zik = 1 x − |1 ≤ i ≤ N } → zik = 0 Dz = { zi → Dx = {−i |1 ≤ i ≤ N } x →→ N} yi = (−i , −i ) xz k =1 πk = 1 K →→ θk = (−k , − , F reqsk , πk ) µ σk θ = (θ1 , ..., θ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© # 4 ' "2 !  ) 0 #  # % 0 ( # %  #22 " $ B ! "2  0 '5  # % 0 ( # & " % # & !  # $  % $ 1 # %  2 " 0 $ 1 & #  $ 1 2 !  & ' " ! R 2 % 0 4 ( # %  72 " # $ A # $  % 9 %  2 " 5 # " " 5 # % 0 Q πk d K σkd θk µkd −− − − −→ F reqskd k ∈ [1, K ] d θ ¥¤ ¤ ¤ ¤    ©   ¨  £ §¦ £ ¤ ¡¡ ¢¡ ¢ ¡ ¤   $ '  &  ¢ § ¡§ Q(θ ) = Ez {log P (Dy |θ )|Dx , θ } N K − → = i=1 k =1 Ez {zik |Dx , θ } × log P ( xi |θk ) + log P (Dy |θ ) = = N i=1 N i=1 K k =1 zik K k =1 zik P (Dy |θ ) = N i=1 → × log(πk × P (−i |θk )) x → × log P (−i |θk ) + N x i=1 K k =1 Ez {zik |Dx , θ } K k =1 zik × log πk × log πk i=1 k =1 N K → (πk × P (−i |θk ))zik x →→ P (yi |θ ) = P (−i , −i |θ ) xz → k = πk × P (−i |θk ) x K → = (πk × P (−i |θk ))zik x k =1 $1  " !$  i=1 zik = 1 P (yi |θ )  0 % 9 #" AH # ( P (Dy |θ ) = N 5 % 0 ( %  & ( % 2S ( # R ' "   U 'S  & % %  ( " % 0 "  Q(θ ) = Ez {log P (Dy |θ )|Dx , θ }  ! 42 " # F "% 2 @ ! 2 $ 1 5 9  R (  4 5 # $  #  0 42 "   # & '5 0 5 # & % %  ( # (  H 4 # % S  ! '5 ! ' 5 F " $ % %  5 R (  4  ! #    $# F " % 2 @ !  I 4 ' " 2 !  )  0S  ( %  2 " 0 4 2 " # IS (  0 " & ! R 2 4 !  0  9 " % 4   4 !  G # $  T B # & '5 0 5 # ! " R 4 0 ! 0  8 $ 0 @ $  % # ( % 2"5 %  G %  R (  4 $ ( # ! " R 4 0 ! 0  #  ! $  A F ! " " 4 £ %  2 " 0 # 2 4 2 8 0 ¢   0 " & B R (  4 $( #"%0! $5 # !"R 40!0  # ( ¦ % 2"5 %  G %  R (  4 $ ( # & ' 5 0 5 # ! " R 4 0 ! 0  #  ! $  0 @ & £ %  2 " 0 4 2 " # I¢ I  0 " & B ¥  0 " & $ 1 0 '5 F  R (  4  ! # 2 4 2 "  S ( " % 0 " " 4 !  # # 0 '  8 $ ( ( %  2 # # 5 5 $ # % $ % " # 2 # %  5 2 $ 1 9 § 4 ' " 2 !  ) %  2 " 0 # 2 4 2 "  S ( 4 R  H  !  5 ! ( $  # & ! ! $   # %  # 2  2 " $ # $  % $ 1 (  ' " & 4 0 C θ    θ Dz Dz θ θ Dz ¤ 0S  " # i=1 log L(θ |Dy ) = log P (yi |θ ) N L(θ |Dy ) B 4 ' "2 ! 0 )  %  # ! #2 42 80 4 F " $ [email protected] $ 1 & ! # 2 4 2 8 0 4 " 9 " 2 $ (  !  % $S ( $ 2  $ 0 # 4 ! " ( 4 4  # % $ ! ! & ( 2 # %  5 ( " 4 !  %  2 " 5 %  G " " 5 ( 4 ' " 2 ! 0 )   $ ( %  2 " 0 # 2  2 " $S C L(θ |Dy ) = P (Dy |θ ) = i=1 P (yi |θ )  N 0 % 9 # !" $0 # ( # % $ #  " % 4 4 0 ( %  & ( % 2 # & ! & % & ) & " & " %  # " A H  #  $ 1 # R ' "   U 'S  " 2 0 G % S  2 3 B ¥ ¤ ¡ ¤ ¦ &" % 9 # "R  4  5 # & % %  (  ¡ ¢ ¤ ¦ ¥¨ ¡ ¥¨ ¦¥ S (  ! 0  % § B  R (  4 $ ( 5 % 0  H 4 # 2 0 ! @ 0  ! # 2 4 2 8 0 4 ( # !   0 " # G 2 " 5 A H S C ¢  ¥ ¦ ¥ ¡ ' ¥ ¤  ¢ ¥ ¡  8$0 "!  0! !0  £¡¢ ¡ ¡ £ ¢ ¤ ¡ ¢ L(θ |Dy ) 2 Dy θ ¥ ¢¡ → P (−i |θk ) = x d∈Sk P (xid |θkd ) B #   0 2 4  %2 " $ 4 " # $ 12 ! & 4 $% # % 2# % 42 ( #  ! $ # % % 2# # $ 0 ) P  # %  !& (2# %  5 # $  % $ 1 # 0 5  # %0 ( # !"R 40 !0  # ( ! $  A F N N i=1 z ik £ B Q¢ πk = 2 !  ) & " 0 5 # % 2 # % 42 ( #  ! $# & #    $# "# % 2 " $ H2 ! " #2 ( 0  # 2 4 0  "2 $ # % # %  U£ 5 % Q k =1 i=1 z ik = N λ= K N k =1 λ πk = 1 = K k =1 K N i=1 z ik λ πk = ∂Q = ∂ πk N i=1 z ik i=1 N z ik −λ=0 πk B # !"R 40!0  # ( ! $ A F # 2 4 0  # % 0 ( % 6 0 ! " % 0  5 $ 1 5 ( !  H 0S ( # %  U  £ B   $ % " #  R (  4 $ ( # $ % %  5 % 2 # ! " R 4 8 $ 0 " !    0 ! ! 0  & @ 2 ! & ( 0 # $ 1 # !   " % 2 " " 0 " # %  2 " 5 %  G " " 5 ( 4 $ 4 2 "  S C πk ¤0 !0  i=1 k =1 Q (θ ) = N K → z ik × log P (−i |θk ) + x i=1 k =1 z ik × log(πk ) + λ × (1 − K N k =1 πk ) K  ) %0!)0 C % 2 " 5 %  G  @ $  % 0  5 %  ( "2 $ ! " # %  5 %  9 0  5 ( ! $ " 0 52 2 " $ 4  "% 0 #22 " $ 9 "% [email protected] $# !$T B $ 1 " % 2 0 ! " %  5 0  # $  # %  2 " 5 %  G 0  ! # 2 4 2 8 0 4 F ' 5 ! '5 %  " 9 # ! " # $  5 8 $ 0 # " A H  # ( 5 % 0 % " ! 0   0S ( # & " 2  2 H 0 H  !  #  # $ % %  5 #    $ # %  & ! & ( 2 # %  5 " # # ! @ % 2 # R ' "   U 'S  9  I 4 ' " 2 !  )  0S  ( %  2 " 0 # 2 4 2 8 0  ( # 0 '  ( %  5 # 0  # % 0 Q B ¥ B Q %  2 " 0 $ 1 &S  %   # 9 $ % %  5 & #    $ # " % 0 ! $  5  R (  4  & % %  ( " % 0 " & # & "22 H 0 H  !  # 5 ! 4 2 " # F 5 %  ( " # 2 # %  5  I 4 ' " 2 !  )  0S  ( %  2 " 0 4 2 " # IS ( # 0 '  0  9 " % 4  0 % 2 § Q K k =1 πk  Q =1 θ £ λ z ik z ik z ik = Ez {zik |Dx , θ } = 0 × P (zik = 0|Dx , θ ) + 1 × P (zik = 1|Dx , θ ) → = P (zik = 1|−i , θ ) x −i |θk ) → π k ×P ( x =P − → j π j × P ( xi | θ j ) ¥B Q¢ ¥ ¦ £ ¢ ¡¤ ¥ ¤ ¦ £ ¢ ¢ ¡ £ ¢ ¤ ¡ ¢ ¥ ¢¡ 2# F reqskd(xid ) xid −µkd σkd log F reqskd (xid ) d F "!  0! !0  Q d µkd B 2 2# 1 2  2 !  ) & " 0 5 − log P (xid |θkd ) 2# √1 2πσkd d $ 12!& 4 $% log d∈Sk d  2 !  ) & " 0 5 2# √1 e 2πσkd $ 12!& 4 $% xid −µkd σkd 1 −2 → log P (−i |θk ) = x log P (xid |θkd ) = £¡¢ ¡ P (xid |θkd) = 2 % 2 " 5 %  G 0  ( & @2 ! & ( 0  !  $ 5 0 5 ! 0  # %  ¥ % 4 4   1 ∂ → log P (−i |θk ) = 2 (xid − µkd ) x ∂µkd σkd N 1 1 z ik 2 (xid − µkd) = 2 σkd σkd i=1 N N i=1 z ik × (xid − µkd ) = 0 N i=1 z ik × µkd N i=1 z ik × xid N i=1 z ik 5 % Q µkd = F "!  0! !0  ( & @2 ! & ( 0  "2 $# % # %  $ 5 0  Q i=1 σkd B 5 % Q z ik × xid = £ B Q¢ ∂Q = ∂ µkd 1 1 ∂ =− log √ ∂σkd σkd 2πσkd 1 xid − µkd ∂ − ∂σkd 2 σkd ∂Q = ∂ σkd N i=1 N z ik i=1 N z ik i=1 z ik − 2 = (xid − µkd )2 3 σkd 1 (xid − µkd)2 + 3 σkd σkd (xid − µkd )2 −1 2 σkd (xid − µkd )2 = 2 σkd =0 N z ik i=1 =0 5 % Q B B Q " B Q 9 B Q 9 B Q # %  2 " 0 $ 1 & #  %   #  R (  4 $ ( # ! 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This note was uploaded on 11/29/2010 for the course DEC 123 taught by Professor Fr during the Spring '10 term at ENS Cachan.

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