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ロジスティック回帰 Part 1 - 基礎編
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Kan Nishida
September 19, 2019
Science
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ロジスティック回帰 Part 1 - 基礎編
Kan Nishida
September 19, 2019
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Transcript
ϩδεςΟοΫճؼ Part 1 جૅฤ Exploratory Seminar #19
EXPLORATORY
3 εϐʔΧʔ ా צҰ CEO EXPLORATORY ུྺ 2016ɺσʔλαΠΤϯεͷຽओԽͷͨΊɺExploratory, Inc Λ
্ཱͪ͛Δɻ Exploratory, Inc.ͰCEOΛΊΔ͔ͨΘΒɺσʔλαΠΤϯεɾ ϒʔτΩϟϯϓɾτϨʔχϯάͳͲΛ௨ͯ͠γϦίϯόϨʔͰ ߦΘΕ͍ͯΔ࠷ઌͷσʔλαΠΤϯεͷීٴͱڭҭʹऔΓ Ήɻ ถΦϥΫϧຊࣾͰɺ16ʹΘͨΓσʔλαΠΤϯεͷ։ൃνʔ ϜΛ͍ɺػցֶशɺϏοάɾσʔλɺϏδωεɾΠϯςϦδΣ ϯεɺσʔλϕʔεʹؔ͢Δଟ͘ͷΛੈʹૹΓग़ͨ͠ɻ @KanAugust
Vision ΑΓΑ͍ҙࢥܾఆΛ͢ΔͨΊʹ σʔλΛ͏͜ͱ͕ͨΓલʹͳΔ
Mission σʔλαΠΤϯεͷຽओԽ
6 ୈ̏ͷ σʔλαΠΤϯεɺAIɺػցֶश౷ܭֶऀɺ։ൃऀͷͨΊ͚ͩͷͷͰ͋Γ·ͤΜɻ σʔλʹڵຯͷ͋ΔਓͳΒ୭͕ੈքͰ࠷ઌͷΞϧΰϦζϜΛͬͯ ϏδωεσʔλΛ؆୯ʹੳͰ͖Δ͖Ͱ͢ɻ Exploratory͕ͦ͏ͨ͠ੈքΛՄೳʹ͠·͢ɻ
ୈ1ͷ ୈ̎ͷ ୈ̏ͷ ϓϥΠϕʔτ(ߴ͍/ݹ͍) Φʔϓϯɾιʔε(ແྉ/࠷ઌ) UI & ϓϩάϥϛϯά ϓϩάϥϛϯά 2016
2000 1976 ϚωλΠθʔγϣϯ ίϞσΟςΟԽ ຽओԽ ౷ܭֶऀ σʔλαΠΤϯςΟετ Exploratory ΞϧΰϦζϜ Ϣʔβʔɾ ମݧ πʔϧ Φʔϓϯɾιʔε(ແྉ/࠷ઌ) UI & ࣗಈԽ ϏδωεɾϢʔβʔ ςʔϚ σʔλαΠΤϯεͷຽओԽ
質問 ExploratoryɹϞμϯˍγϯϓϧ UI 伝える データアクセス データ ラングリング 可視化 アナリティクス 統計/機械学習
ϩδεςΟοΫճؼ Part 1 جૅฤ Exploratory Seminar #19
質問 伝える データアクセス データ ラングリング 可視化 アナリティクス 統計/機械学習
ઢܗճؼͷ෮श
USͷͪΌΜσʔλ
ͷྸ͍͔ͭ͘ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ
ͷྸ vs. ͷྸ
ઢܗճؼ
Father_Age = 0.866 * Mother_Age + 6.28 ย ʢ͖ʣ
ڵຯͷର ΧςΰϦʔ/ೋ߲ 17 ΧςΰϦʔ/ଟ߲
ڵຯͷର ΧςΰϦʔ/ೋ߲ 18 ΧςΰϦʔ/ଟ߲
͜ͷϢʔβʔίϯόʔτ͢Δ͔ʁ ͜ͷऔҾෆਖ਼͔ʁ ͜ͷैۀһΊΔ͔ʁ ͜ͷͪΌΜະख़ࣇͰੜ·ΕΔ͔ʁ ೋ߲ͷ࣭
20 ͷΛೋ߲ͷ
21 ͪΌΜͷ৷ظؒ
22 premature = gestation week < 37 ੜ·Εͯ͘ΔͪΌΜະख़ࣇ͔ʁ TRUE FALSE
Numeric Binary ͷΛೋ߲ྨͷ
͕35ࡀΑΓ্͔Ͳ͏͔ɺͷྸΛͱʹ༧ଌ͍ͨ͠ɻ
ճؼͷΞϧΰϦζϜΛͬͯೋ߲ͷΛղܾͰ͖ͳ͍͔ʁ
YES YOU CAN!
͔͠͠ɺͪΐͬͱ͕ඞཁͰɺੲͷਓۤ͠Μͩɻ
͓͔͛ͰɺࢲୡϩδεςΟοΫճؼͱͯ͠͏͚ͩɻ
ͲΜͳ͜ͱͬͯΔͷ͔ͪΐͬͱ͍ͯΈ·͠ΐ͏ɻ
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ ϩδεςΟοΫճؼͷΈ
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ ϩδεςΟοΫճؼͷΈ
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ 3. ༧ଌ͞ΕΔ֬ΛͱʹTRUE͔FALSEͷϥϕϧ͚Λͯ͠Ξτϓοτ͢Δ ϩδεςΟοΫճؼͷΈ
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ 3. ༧ଌ͞ΕΔ֬ΛͱʹTRUE͔FALSEͷϥϕϧ͚Λͯ͠Ξτϓοτ͢Δ 1. ֬ 2. Φοζ
3. ϩάɾΦοζ 4. ֬ ϩδεςΟοΫճؼͷΈ
Step by Step
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ 3. ༧ଌ͞ΕΔ֬ΛͱʹTRUE͔FALSEͷϥϕϧ͚Λͯ͠Ξτϓοτ͢Δ 1. ֬ 2. Φοζ
3. ϩάɾΦοζ 4. ֬ ϩδεςΟοΫճؼͷΈ
Binary Numeric TRUE or FALSE 1 or 0
None
None
֬ 100% 0% ͷྸ
ͷ༧ଌͳͷͰઢܗճؼʹ͔͚ͯΈΑ͏ʂ
ઢܗճؼͷϞσϧ
P(Father > 35) = a * Mother_Age + b P():
֬ΛٻΊΔؔ
None
Pr(Father > 35) = 0.039 * Mother_Age -0.85
ઢܗճؼͷϞσϧ Pr(Father > 35) = 0.039 * Mother_Age -0.85
͓͞Μ͕35ࡀͷͱ͖ͷ֬ʁ
Pr(Father > 35) = 0.039 * 35 - 0.85 =
0.515 51.5% ͷ֬Ͱ35ࡀΑΓ্ɻ Pr(Father > 35) = 0.039 * Mother_Age -0.85 ͷྸ: 35
35ࡀ 51.5%
͓͞Μ͕20ࡀͷͱ͖ͷ֬ʁ
Pr(Father > 35) = 0.039 * 20 - 0.85 =
-0.07 Pr(Father > 35) = 0.039 * Mother_Age -0.85 ϚΠφε 7% ͷ֬Ͱ35ࡀΑΓ্ɻ ͷྸ: 20
20ࡀ -7%
ϚΠφεͷ֬ʁʁʁ
ϚΠφεͷ֬ڹ͖͓͠Ζ͍͕ɺ ࣮ࡍʹશ͘ҙຯΛͳ͞ͳ͍ɻ
͜ͷลΓʹ͘Δਓͨͪͷઆ໌͕͏·͘Ͱ͖ͳ͍ɻ
Ͱɺσʔλ͜ͷลʹ͔ͬ͠Γ͋ΔͷͰɺͪΌΜͱઆ໌Ͱ͖ΔϞ σϧ͕΄͍͠ɻ
ઢܗճؼ0͔Β1·Ͱͷ͔͠औΒͳ͍֬Λ༧ଌ͢ΔͨΊʹ ͋·Γద͍ͯ͠ͳ͍ɻ
্ݶɺԼݶΛ͚ͭΔͷͲ͏ͩʁ 0% 100%
ͱ͍͏͜ͱɺ͕21ࡀΑΓए͔ͬͨΒ0ˋͱ͍͏͜ͱɻ 0% 100%
0% 100% ͔͠͠ɺ࣮ࡍʹ1(ͷྸ͕35ΑΓ্ʣͷਓୡ͍Δɻ
0% 100% ͓͍͠ɺ͏Ұඞཁʂ
σʔλͷมʂ
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ 3. ༧ଌ͞ΕΔ֬ΛͱʹTRUE͔FALSEͷϥϕϧ͚Λͯ͠Ξτϓοτ͢Δ 1. ֬ 2. Φοζ
3. ϩάɾΦοζ 4. ֬ ϩδεςΟοΫճؼͷΈ
ͲΜͳม͕Ͱ͖Δ͔ʁ ֬ͩͱ0͔Β1ͷؒͱ͍͏ͷൣғʹ੍ݶ͕͋Δͷ͕ͩɻ ճؼͷϞσϧΛ͏ʹ࿈ଓͰʹ੍ݶ͕ͳ͍΄͏͕͍͍ɻ [- Infinity - infinity] ੍ݶͷͳ͍࿈ଓ [0 -
1] range ੍ݶͷ͋Δൣғ
Logit (Logistic Unit) ϩδοτؔ
Logit (Logistic Unit) Log of Odds ϩάɾΦοζ
66 Φοζ(Odds) ى͖͏Δೋͭͷ݁Ռͷ֬ͷൺ
67 Φοζ Φοζ = TRUEͷ֬ / FALSEͷ֬
68 ͕35ࡀΑΓ্Ͱ͋ΔΦοζ Φοζ = TRUEͷ֬ / FALSEͷ֬ ͕35ࡀΑΓ্ͷ͕֬10% ͕35ࡀΑΓ্ͷ͕֬90% 0.1111…
= 10 / 90
֬ vs. Φοζ P(Father > 35) = 0.2 P(TRUE) =
P(Father > 35) = 0.2 P(FALSE): 1 - P(Father > 35) = 0.8 Φοζ = P(TRUE) / P(FALSE) = 0.2 / 0.8 = 0.25
֬ vs. Φοζ P(Father > 35) = 0.75 P(TRUE) =
P(Father > 35) = 0.75 P(FALSE): 1 - P(Father > 35) = 0.25 Φοζ = P(TRUE) / P(FALSE) = 0.75 / 0.25 = 3
Pr(Father > 35) = 0 0 / (1 - 0)
= 0 ֬ Φοζ Pr(Father > 35) = 0.5 0.5 / (1 - 0.5) = 1 Pr(Father > 35) = 0.9 0.9 / (1 - 0.9) = 9 Pr(Father > 35) = 0.999 0.999 / (1 - 0.999) = 999 Pr(Father > 35) = 1 1 / (1 - 1) = ແݶ ม
Probability can only range from 0 to 1, Odds can
be 0 up to any positive number. But, we still have a problem. We want the variable that can range from any negative number to any positive number.
֬ 0 1 Φοζ 0 1 ແݶ
֬ 0 1 Φοζ 0 1 ແݶ ແݶ -ແݶ 0
ཧ
ϩάɾΦοζ log( P(y) 1 - P(y) ) ΦοζʹϩάΛ͔͚Δ
None
֬ 0 1 Φοζ 0 1 ແݶ ແݶ -ແݶ 0
ϩάɾΦοζ log( Odds( P(y) )) Odds( P(y) )
ϩδοτؔ log( Odds( P(y) )) = Logit(P(y)) ֬ΛϩάɾΦοζʹมͯ͘͠ΕΔؔ
• ֬0͔Β1ͷؒͷൣғͷ͚ͩΛؚΉɻ • Φοζ0͔Βϓϥεແݶେʹଓ͘ͷൣғͷ͚ͩΛؚΉɻ • ϩάɾΦοζϚΠφεແݶେ͔ΒϓϥεແݶେͷؒͷͲΜͳ ͰऔΓ͏Δɻ
ϩάɾΦοζͲΜͳͰͱΕΔɻ ͱ͍͏͜ͱɺ࿈ଓΛ༧ଌ͢ΔͨΊͷճؼͷΞϧΰϦζϜ ͕͑Δʂ
ϩδεςΟοΫճؼ खݩʹ͋Δ༧ଌมΛݩʹͯ͠ɺڵຯͷରͰ͋Δೋ߲มͷ ϩάɾΦοζΛ༧ଌ͢ΔͨΊͷϞσϧΛ࡞ΔͨΊͷΞϧΰϦζ Ϝɻ
ઢܗճؼ ϩδεςΟοΫճؼ Logit( P(Father > 35) ) = a *
Mother_Age + b Father_Age = a * Mother_Age + b
ϩδεςΟοΫճؼΛྲྀ͠ݟͯΈΔʂ
None
None
Logit( Pr(Father > 35) ) = 0.029 * Mother_Age -
0.1012
͕40ࡀͷ࣌ɺ͕35ࡀҎ্Ͱ͋Δ֬ʁ
Logit( P(Father > 35) ) = 0.29 * Mother_Age -
10.12 Logit( P(Father > 35) ) = 0.29 * 40 - 10.12 = 1.48 ͕35ࡀҎ্Ͱ͋ΔϩάɾΦοζ1.48ɻ ???
ϩάɾΦοζͷཧղࢲͷײΛ͍͑ͯΔɻɻɻ
ϩάɾΦοζΛͻͬ͘Γฦͯ֬͠ ʹ͍ͨ͠ɻ
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ 3. ༧ଌ͞ΕΔ֬ΛͱʹTRUE͔FALSEͷϥϕϧ͚Λͯ͠Ξτϓοτ͢Δ 1. ֬ 2. Φοζ
3. ϩάɾΦοζ 4. ֬ ϩδεςΟοΫճؼͷΈ
Logit(P(Father > 35)) = 0.03 * Mother_Age - 0.85 P(Father
> 35) = Logit (0.03 * Mother_Age - 0.85) -1 ٯؔ
Logit(Pr(Father > 35)) = 0.03 * Mother_Age - 0.85 Pr(Father
> 35) = (0.03 * Mother_Age - 0.85) Logistic
P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12)
͕40ࡀͷ࣌ɺ͕35ࡀҎ্Ͱ͋Δ֬ʁ
P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12) P(Father
> 35) = Logistic(0.29 * 40 - 10.12) = Logistic(1.48) = 0.8145 ͕40ࡀͷ࣌ɺ͕35ࡀҎ্Ͱ͋Δ֬81%ɻ
͕20ࡀͷ࣌ɺ͕35ࡀҎ্Ͱ͋Δ֬ʁ
P(Father > 35) = Logistic(0.29 * Mother_Age - 10.12) P(Father
> 35) = Logistic(0.29 * 20 - 10.12) = Logistic(-4.32) = 0.01312 ͕20ࡀͷ࣌ɺ͕35ࡀҎ্Ͱ͋Δ֬1.3%ɻ
͜ͷϩδεςΟοΫճؼͷϞσϧͲΜͳ͔Μ͡ͷͷʁ
100 ઢܗճؼͷϞσϧ vs ϩδεςΟοΫճؼͷϞσϧ
ઢܗճؼͷ߹ΛݟͯΈΔɻ
102 Father_Age = a * Mother_Age + b ʢ͖ʣ ย
ઢܗճؼͷϞσϧʢܭࢉࣜʣ
103 ʢ͖ʣ ย
104 Father_Age = 0.87 * Mother_Age + 6.28 ʢ͖ʣ ย
ઢܗճؼͷϞσϧʢܭࢉࣜʣ
None
ͷྸ ͷྸ
ͷྸ ͷྸ
How about this Logistic Regression Model?
109 ͕35Ҏ্ͷ֬ = logistic(a * Mother_Age + b) ʢ͖ʣ ย
ͱยΛௐઅ͢Δ͜ͱͰ࣮σʔλͱ Ϛον͢ΔΑ͏ͳۂઢ͕ඳ͚Δɻ
110 ϩδεςΟοΫճؼͷϞσϧ
111 ͕35Ҏ্ͷ֬ = logistic(0.29 * Mother_Age - 10.12) ย
None
֬ (Father > 35) ͷྸ
ϩδεςΟοΫۂઢ
֬ (Father > 35) ͷྸ
࣮σʔλ Ϟσϧ (ϩδεςΟοΫۂઢ) ͱͷσʔλͷฏۉΛऔΔͱɻɻɻ
ͱ͜ΖͰɺຊʹΓ͔ͨͬͨͷೋ߲ͷ֬Ͱͳ ͯ͘ɺʢTRUE/FALSEʣͷ༧ଌɻ ͷྸΛͱʹɺ͕̏̑ࡀҎ্ͳͷ͔Ͳ͏ ͔ΛΓ͍ͨɻ
ͱ͜ΖͰɺຊʹΓ͔ͨͬͨͷೋ߲ͷ֬Ͱͳ ͯ͘ɺʢTRUE/FALSEʣͷ༧ଌɻ ͷྸΛͱʹɺ͕̏̑ࡀҎ্ͳͷ͔Ͳ͏ ͔ΛΓ͍ͨɻ TRUE or FALSE ?
1. TRUE/FALSEΛ̌/̍ͱ͍͏ʹͯ֬͠Λ༧ଌ͢Δʹ͢Δ 2. ͦͷ··ͩͱ͍͔͕ͭ͋͘ΔͷͰσʔλΛՃ͢Δ 3. ༧ଌ͞ΕΔ֬ΛͱʹTRUE͔FALSEͷϥϕϧ͚Λͯ͠Ξτϓοτ͢Δ 1. ֬ 2. Φοζ
3. ϩάɾΦοζ 4. ֬ ϩδεςΟοΫճؼͷΈ
120 100% 0% 30 35 40 ͷྸ ͷྸ͕ 35ࡀҎ্Ͱ͋Δ ֬
֬Ͱͳ͘TRUE͔FALSEʹྨ͍ͨ͠ɻ
121 30 35 40 ͷྸ TRUE FALSE ͷྸ͕ 35ࡀҎ্ 100%
0% ֬Ͱͳ͘TRUE͔FALSEʹྨ͍ͨ͠ɻ
122 30 35 40 ͷྸ 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE
͖͍͠Λઃ͚Δɻྫ͑ɺ͕֬50%Λڥʹ͢Δɻ
123 30 35 40 ͷྸ 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE
TRUE FALSE ͜ͷۂઢΛ͏͜ͱͰྨ͢Δ͜ͱ͕Ͱ͖ΔΑ͏ʹͳͬͨɻ
124 30 35 40 ͷྸ 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE
TRUE FALSE ͜ͷۂઢΛ͏͜ͱͰྨ͢Δ͜ͱ͕Ͱ͖ΔΑ͏ʹͳͬͨɻ
125 30 35 40 ͷྸ 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE
TRUE FALSE ࣮ࡍʹɺͷྸ͕35ࡀҎ্ͩͬͨσʔλʹͯΊͯΈΔ
126 30 35 40 ͷྸ 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE
TRUE FALSE ͍ͨͬͯΔ ͍ͨͬͯͳ͍
127 30 35 40 ͷྸ 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE
TRUE FALSE ࣮ࡍʹɺͷྸ͕35ࡀҎ্͡Όͳ͔ͬͨσʔλʹͯΊͯΈΔ
128 30 35 40 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE TRUE
FALSE ͍ͨͬͯΔ ͍ͨͬͯͳ͍
129 30 35 40 50% ͷྸ͕ 35ࡀҎ্ TRUE FALSE TRUE
FALSE ͍ͨͬͯͳ͍ ͍ͨͬͯͳ͍
130 = 12/14 = 0.857 (85.7%) ͜ͷۂઢϞσϧΛ͏ͱɺ 85.7%ͷਫ਼Ͱ༧ଌͰ͖Δɻ
131 ༧ଌม͕1͚ͭͩͷ߹ΛΈ͖ͯͨɻ
132 ༧ଌม͕ෳͷ߹ʁ
133 ͕35Ҏ্ͷ֬ = logistic(0.3 * Mother_Age - 10) ย
༧ଌม͕Mother_Age͚ͩͷ߹
134 ͕35Ҏ্ͷ֬ = logistic(0.3 * Mother_Age + 1.2 * Mother_Japanese
- 10) ༧ଌม͕Mother_AgeͱMother_Japaneseͷ ̎ͭͷ߹
Q & A
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