Receiver Operator Characteristic Curve

はじめての『はじめてのパターン認識』第3章
A first introduction of ROC curve (Section 3.2).

Transcript

1. ୈষ
   ϕΠζͷࣝผنଇ ड৴ऀಈ࡞ಛੑۂઢ ͸͡Ίͯͷύλʔϯೝࣝྠಡձ ޻౻঑  1
2. 065-*/& ϕΠζͷࣝผنଇ ࠞಉߦྻ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ষ຤໰୊ ·ͱΊ ຕ ຕ ຕ ຕ ຕ !2
3. ϕΠζͷࣝผنଇ ٤Ԏ͍ͯ͠Δਓ͕͍ͨʜ පؾͩͬͨʜ Cd P(Cd |x) = p(x|Cd )P(Cd ) p(x) ݈߁ͩͬͨ Ch P(Ch |x) = p(x|Ch )P(Ch ) p(x) ࣝผΫϥε = argmax p(x|Ci )P(Ci ) i x !3
4. ϕΠζͷࣝผنଇ ࣝผΫϥε = argmax p(x|Ci )P(Ci ) i ϵ(x) = min [P(Ci |x)] ΫϥεΛޡΔ֬཰ʢස౓ʣ i පؾͩͬͨʜ Cd P(Cd |x) = p(x|Cd )P(Cd ) p(x) ݈߁ͩͬͨ Ch P(Ch |x) = p(x|Ch )P(Ch ) p(x) !4
5. ϕΠζͷࣝผنଇ පؾͩͬͨʜ Cd ݈߁ͩͬͨ Ch ຊ౰͸݈߁ͩͬͨͷʹ
   පؾͩͱݴͬͯ͠·ͬͨ ຊ౰͸පؾͩͬͨͷʹ
   ݈߁ͩͱݴͬͯ͠·ͬͨ ِӄੑ ِཅੑ ࣝผڥք !5 ϵ(x) = min [P(Ci |x)] ΫϥεΛޡΔ֬཰ʢස౓ʣ i පؾͰ͋Δ͜ͱΛཅੑͱ͢Δͱ
6. පؾͩͬͨʜ Cd ݈߁ͩͬͨ Ch ϕΠζͷࣝผنଇ ࣝผΫϥε = argmax p(x|Ci )P(Ci ) i ࠞಉߦྻ ࣝผͷੑೳΛͲ͏ධՁ͢Δ͔ !6
7. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
   51 ِӄੑ
   '/ 1
   
   51
   
   
   '/ O ِཅੑ
   '1 ਅӄੑ
   5' /
   
   '1
   
   5/ ࣝผΫϥε ਅͷΫϥε ͜ΕΒͷ஋Λ࢖ͬͯࣝผੑೳΛධՁ͢Δ !7
8. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
   51 ِӄੑ
   '/ 1
   
   51
   
   
   '/ O ِཅੑ
   '1 ਅӄੑ
   5' /
   
   '1
   
   5/ ࣝผΫϥε ਅͷΫϥε ِཅੑ཰GBMTF
   QPTJUJWF
   SBUF FP N = FP FP + TN ӄੑͷ΋ͷΛཅੑͱ൑அׂͨ͠߹ !8
9. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
   51 ِӄੑ
   '/ 1
   
   51
   
   
   '/ O ِཅੑ
   '1 ਅӄੑ
   5' /
   
   '1
   
   5/ ࣝผΫϥε ਅͷΫϥε ਅཅੑ཰USVF
   QPTJUJWF
   SBUF TP P = TP TP + FN ཅੑͷ΋ͷΛਖ਼͘͠ཅੑͱ൑அׂͨ͠߹ !9
10. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
    51 ِӄੑ
    '/ 1
    
    51
    
    
    '/ O ِཅੑ
    '1 ਅӄੑ
    5' /
    
    '1
    
    5/ ࣝผΫϥε ਅͷΫϥε ద߹཰QSFDJTJPO TP TP + FP ཅੑͱ൑அͨ͠΋ͷͷ͏ͪຊ౰ʹཅੑͰ͋Δׂ߹ !10
11. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
    51 ِӄੑ
    '/ 1
    
    51
    
    
    '/ O ِཅੑ
    '1 ਅӄੑ
    5' /
    
    '1
    
    5/ ࣝผΫϥε ਅͷΫϥε ࠶ݱ཰SFDBMM TP P = TP TP + FN ໢ཏੑʹؔ͢Δࢦඪ
    શཅੑͷ͏ͪݕग़Ͱ͖ͨ΋ͷͷׂ߹ !11
12. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
    51 ِӄੑ
    '/ 1
    
    51
    
    
    '/ O ِཅੑ
    '1 ਅӄੑ
    5' /
    
    '1
    
    5/ ࣝผΫϥε ਅͷΫϥε ਖ਼֬౓BDDVSBDZ TP + TN P + N = TP + TN TP + FP + TN + FN ਖ਼ࣝ͘͠ผׂͨ͠߹ !12
13. ࠞಉߦྻ Q O ߦ࿨ Q ਅཅੑ
    51 ِӄੑ
    '/ 1
    
    51
    
    
    '/ O ِཅੑ
    '1 ਅӄੑ
    5' /
    
    '1
    
    5/ ࣝผΫϥε ਅͷΫϥε '஋'NFBTVSF 2 1 precision + 1 recall ద߹཰ͱ࠶ݱ཰ͷௐ࿨ฏۉ !13
14. ࠞಉߦྻ ِཅੑ཰GBMTF
    QPTJUJWF
    SBUF FP N = FP FP + TN ӄੑͷ΋ͷΛཅੑͱ൑அׂͨ͠߹ ਅཅੑ཰USVF
    QPTJUJWF
    SBUF TP P = TP TP + FN ཅੑͷ΋ͷΛਖ਼͘͠ཅੑͱ൑அׂͨ͠߹ ద߹཰QSFDJTJPO TP TP + FP ཅੑͱ൑அͨ͠΋ͷͷ͏ͪ
    ຊ౰ʹཅੑͰ͋Δׂ߹ ࠶ݱ཰SFDBMM TP P = TP TP + FN ໢ཏੑʹؔ͢Δࢦඪ
    શཅੑͷ͏ͪݕग़Ͱ͖ͨ΋ͷͷׂ߹ ਖ਼֬౓BDDVSBDZ TP + TN P + N = TP + TN TP + FP + TN + FN ਖ਼ࣝ͘͠ผׂͨ͠߹ '஋'NFBTVSF 2 1 precision + 1 recall ద߹཰ͱ࠶ݱ཰ͷௐ࿨ฏۉ !14
15. ࠞಉߦྻ ྫ୊ ద߹཰ͱ࠶ݱ཰͸ͳͥτϨʔυΦϑͷؔ܎ʹ͋Δͷ͔ʁ ໢ཏੑΛ্͛Α͏ͱཅੑͱड͚ೖΕΔ΋ͷΛ૿΍͢ͱ
    ِཅੑͱ൑அ͢Δ΋ͷ΋૿Ճ͠ɼ݁Ռͱͯ͠ద߹཰͕Լ͕ΔͨΊ ద߹཰QSFDJTJPO TP TP + FP ཅੑͱ൑அͨ͠΋ͷͷ͏ͪ
    ຊ౰ʹཅੑͰ͋Δׂ߹ ࠶ݱ཰SFDBMM TP P = TP TP + FN ໢ཏੑʹؔ͢Δࢦඪ
    શཅੑͷ͏ͪݕग़Ͱ͖ͨ΋ͷͷׂ߹ !15
16. ِཅੑ཰GBMTF
    QPTJUJWF
    SBUF FP N = FP FP + TN ӄੑͷ΋ͷΛཅੑͱ൑அׂͨ͠߹ ਅཅੑ཰USVF
    QPTJUJWF
    SBUF TP P = TP TP + FN ཅੑͷ΋ͷΛਖ਼͘͠ཅੑͱ൑அׂͨ͠߹ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ පؾͷਓ͔݈߁ͳਓͳͲͷαϯϓϧ਺ʹେ͖ͳภΓ͕͋Δ৔߹Ͱ΋
    ཅੑӄੑͷ΋ͷ͚ͩͰ࡞ΒΕ͍ͯΔͷͰɼͦΕͧΕӄੑཅੑͷӨڹΛड͚ͳ͍ 30$ۂઢ !16
17. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ཰
     ਅཅੑ཰
     # " $ ࣝผڥք Cp* Cn* ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ཅੑʢࠨʣଆͷ
    ֬཰෼෍ʹ஫໨͢Δ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅ !17
18. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ཰
     ਅཅੑ཰
     # " $ Cp* Cn* ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ ࣝผڥք ཅੑʢࠨʣଆͷ
    ֬཰෼෍ʹ஫໨͢Δ !18 ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅
19. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ِཅੑ཰
     ਅཅੑ཰
     # " $ Cp* Cn* ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ ࣝผڥք ཅੑʢࠨʣଆͷ
    ֬཰෼෍ʹ஫໨͢Δ !19 ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅
20. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣͱਅཅੑ཰ɾِཅੑ཰ (0,0) (0.1,0.7) (1.0,1.0) ཅੑʢࠨʣଆͷ
    ֬཰෼෍ʹ஫໨͢Δ ʢԣɼॎʣʢِཅੑ཰ɼਅཅੑ཰ʣ !20 ਅཅੑ཰ɿ੺ ِཅੑ཰ɿ੨ ୈछͷޡΓ ୈछͷޡΓͷ༨ࣄ৅
21. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 4 ΔE = 3 !21
22. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 2 ΔE = 1 !22
23. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ΔE = 0 30$͔Βݴ͑Δ͜ͱ͸Կ͔ʁ !23
24. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ͷॏͳΓ͕খ͍͞΄Ͳ 30$ͷԼ෦෼ͷ໘ੵ͕େ͖͘ͳ͍ͬͯΔ !24
25. 30$ͷԼ෦෼ͷ໘ੵɿ"6$
    "SFB
    6OEFS
    UIF
    $VSWF ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ׬ᘳʹࣝผͰ͖Δɿ AUC = 1 શࣝ͘ผͰ͖ͳ͍ɿ AUC = 0.5 0.5 ≦ AUC ≦ 1.0 !25
26. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ "6$Λେ͖͘͢ΔͨΊʹ͸ ྑ͍෼཭ੑͷ֬཰෼෍Λ࣋ͭಛ௃ྔͷ૊Έ߹Θͤ !26
27. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣͱਅཅੑ཰ɾِཅੑ཰ (0,0) (0.1,0.7) (1.0,1.0) ʢԣɼॎʣʢِཅੑ཰ɼਅཅੑ཰ʣ ͋͘·Ͱ΋
    ֬཰෼෍ͷੑೳධՁ Ͳͷ஋ʹ஫໨͢Ε͹ྑ͍ʁ 30$ !27
28. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ͋͘·Ͱ΋
    ֬཰෼෍ͷੑೳධՁ # " $ Ͳͷ஋ʹ஫໨͢Ε͹ྑ͍ʁ ͲͷࣝผڥքΛબ΂͹ྑ͍ʁ 30$ ཅੑͷ֬཰෼෍ͱӄੑͷ֬཰෼෍ !28
29. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ # " $ ࣝผڥք Cp* Cn* ِཅੑ཰ ਅཅੑ཰ ཅੑͷྖҬʹ͓͚Δଛࣦ ଛࣦ
    ਅཅੑ཰ʹ༝དྷ͢Δଛࣦ ଛࣦ
    ِཅੑ཰ʹ༝དྷ͢Δଛࣦ r(C1 |x) = Σ2 k=1 L1k P(Ck |x) !29
30. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ཅੑͷྖҬʹ͓͚Δଛࣦ ଛࣦ
    ਅཅੑ཰ʹ༝དྷ͢Δଛࣦ ଛࣦ
    ِཅੑ཰ʹ༝དྷ͢Δଛࣦ r(C1 |x) = Σ2 k=1 L1k P(Ck |x) r(C1 |x) = L11 P(C1 |x) r(C1 |x) = L12 P(C2 |x) ਖ਼ࣝ͘͠ผͨ͠ͷʹଛࣦʁ - Λʹͯ͠د༩Λখ͘͢͞Δ r(C1 |x) = L12 P(C1 |x) ཅੑͷྖҬʹ͓͚Δ
    ࠷ऴతͳଛࣦ ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ !30
31. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ ӄੑྖҬʹ͓͚Δଛࣦ΋ಉ༷ʹॲཧ͢Δͱɼଛࣦ͸྆ऀͷظ଴஋ΛͱΔ͜ͱͰ r = E[r(x)] = ∫ R1 +R2 min[L12 P(C2 |x), L21 P(C1 |x)]p(x)dx = ∫ R1 L12 p(x|C2 )P(C2 )dx + ∫ R2 L21 p(x|C1 )P(C1 )dx ΫϥεΛཅੑʢQ ʣɼΫϥεΛӄੑʢO ʣͱ͢Ε͹ r = ∫ p* Lp*n* p(x|Cn* )P(Cn* )dx + ∫ n* Ln*p* p(x|Cp* )P(Cp* )dx = Lp*n* P(Cn* )∫ p* p(x|Cn* )dx + Ln*p* P(Cp* )∫ n* p(x|Cp* )dx = Lp*n* P(Cn* )ϵ2 + Ln*p* P(Cp* )ϵ1 ୈछͷޡΓ ୈछͷޡΓ ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ !31
32. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ r = Lp*n* P(Cn* )ϵ2 + Ln*p* P(Cp* )ϵ1 30$ʹ͋Δɼਅཅੑ཰Џ ͱِཅੑ཰Џ ͰࣜΛॻ͖௚͢ͱ࣍ࣜͷΑ͏ʹͳΔɽ 1 − ϵ1 = Lp*n* P(Cn* ) Ln*p* P(Cp* ) ϵ2 + (1 − r Ln*p* P(Cp* ) ) = αϵ2 + h(r) ͜ͷࣜ͸1 $Q ͱ1 $O ͓Αͼଛࣦ-Q O ͱ-O Q ʹΑͬͯЏ ͷ܏͖͕ܾ·Γɼ
    ੾ยΛSͷؔ਺ͱͨ͠௚ઢͷํఔࣜͱଊ͑Δ͜ͱ͕Ͱ͖Δɽ ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ Ͱ͸ɼ༩͑ΒΕͨࣜͷதͰ࠷΋ద੾ͳࣜ͸ͲͷΑ͏ͳܗʹͳΔͩΖ͏͔ʁ
    ৚݅͸ҎԼͷ௨ΓͰ͋Δɽ min r = max h(r) ௚ઢ͕30$ͱަ఺Λ࣋ͭ ʜଛࣦ࠷খ ʜ͍࣋ͬͯΔσʔλͷ৚݅ͷൣғ !32
33. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 1 − ϵ1 = αϵ2 + h(r) ଛࣦ͕࠷΋খ͘͞ͳΔΑ͏ʹࣝผڥքΛܾΊΔ min r = max h(r) ௚ઢ͕30$ͱަ఺Λ࣋ͭ ʜଛࣦ࠷খ ʜ͍࣋ͬͯΔσʔλͷ৚݅ͷൣғ ৚݅ ੾ยI S খ
    
    ଛࣦSେ ଛࣦ࠷খ͔ͭ30$ͱަ఺Λ࣋ͭ఺ (0.1,0.7) ͍͍ͩͨ ͘Β͍ʁ ਅཅੑ཰ͱِཅੑ཰ͷׂ߹͕
    ఺ͷ஋ʹͳΔΑ͏ʹࣝผڥքΛఆΊΔͱ
    ࠷΋ޡΓΛখͨࣝ͘͞͠ผ͕ՄೳʹͳΔ !33
34. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ 30$͸Ϋϥεͷ෼෍͕Θ͔Βͳͯ͘΋ඳ͘͜ͱ͕Ͱ͖Δɽ ਅͷΫϥεͱσʔλͷείΞ      Q O  ᮢ஋ͷείΞ4Ҏ্ͷείΞΛ࣋ͭσʔλΛQ ͱࣝผ͢Δɽ
    ͜ΕΛɼᮢ஋ΛมԽͤͯ͞܁Γସ͑͢ɽ !34
35. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ    Q   O  30$͸Ϋϥεͷ෼෍͕Θ͔Βͳͯ͘΋ඳ͘͜ͱ͕Ͱ͖Δɽ ᮢ஋4 ਅཅੑ཰ ِཅੑ཰ 㱣
          
                    —㱣
      !35
36. ड৴ऀಈ࡞ಛੑۂઢʢ30$ʣ 30$άϥϑ    Q   O  30$͸Ϋϥεͷ෼෍͕Θ͔Βͳͯ͘΋ඳ͘͜ͱ͕Ͱ͖Δɽ !36
37. # " $ Cp* Cn* ِཅੑ཰ ਅཅੑ཰ ষ຤໰୊  30$ۂઢΛඳ͍ͨͱ͜ΖɼϥϯμϜࣝผΛද͢
    ౓ͷ௚ઢͷԼଆʹདྷͯ͠·ͬͨɽͲ͏͢Ε͹ྑ͍͔ɽ ্ਤͷਅཅੑ཰ͱِཅੑ཰ͷେ͖͕͞ٯస͍ͯ͠Δྫͱଊ͑ΒΕΔͷͰɼ
    ୯७ʹཅੑͱӄੑͷΫϥεΛٯస͢Ε͹͍͍
    ͋Δ͍͸ɼ྆ํͷείΞʹϚΠφεΛ͔͚Δ͔ᮢ஋ͷεΩϟϯΛٯ͔Βߦ͏ͱ͔ !37
38. ·ͱΊ ੾ยI S খ
    
    ଛࣦSେ ෼ྨͷධՁ͸ࠞಉߦྻ͔ΒಘΒΕΔ
    ͍͔ͭ͘ͷࢦඪͰߦ͑Δ ࠞಉߦྻΛ༩͑Δ֬཰෼෍ͷධՁ͸
    30$ͷԼ෦໘ੵ "6$ ʹΑͬͯධՁ͢Δ ֬཰෼෍ͷ࠷΋ྑ͍ࣝผڥք͸
    30$ͱϕΠζͷࣝผنଇͷ
    ଛࣦ͕࠷খʹͳΔ఺ΛબͿ͜ͱͰಘΒΕΔ 30$͸֬཰෼෍͕Θ͔Βͳͯ͘΋
    ඳ͘͜ͱ͕Ͱ͖Δ !38
39. ଛࣦߦྻ ଛࣦʹؔ͢Δࣜʹग़ͯ͘Δ-ʹ͍ͭͯ r(Ci |x) = ΣK k=1 Lik P(Ck |x) ,ͷ࣌ɼS͸࠷΋؆୯ͳ-Λ࢖ͬͯ࣍ͷΑ͏ʹॻ͖දΘͤΔɽ Lkj p(Ck |x) = (1 − Ikj )p(Ck |x) = {( 1 1 1 1) − ( 1 0 0 1)} ( p(C1 |x) p(C2 |x)) = ( 0 1 1 0) ( p(C1 |x) p(C2 |x)) = ( p(C2 |x) p(C1 |x)) ͜ͷதͰখ͍͞ํΛऔΔ࣌͸ҎԼͷΑ͏ʹॻ͚Δ min ( p(C2 |x) p(C1 |x)) !39