「内積が見えると統計学も見える」第5回プログラマのための数学勉強会発表資料

ઢܗ୅਺͕ݟ͑Δͱ ౷ܭֶ΋ݟ͑Δ ಺ੵ ୈճϓϩάϥϚͷͨΊͷ਺ֶษڧձൃදࢿྉ ,FO`JDIJ.BUTVJ !LFONBUTV

ࣗݾ঺հ!LFONBUTV ɾ'BDFCPPLϖʔδ ɹIUUQTXXXGBDFCPPLDPNNBUTVLFOCPPL ɾ5XJUUFSΞΧ΢ϯτ ɹ!LFONBUTV ɾ2JJUBͰϒϩάΛॻ͍͍ͯ·͢ʢ౷ܭɺػցֶशɺ1ZUIPO౳ʣ ɹɹɹIUUQRJJUBDPNLFONBUTV ɹɹɹʢDPOUSJCVUJPOΛ௒͑·ͨ͠ʂʣ ɾझຯ ɹɹɹόϯυͰϕʔεΛ஄͍ͨΓ͍ͯ͠·͢ɻ
ɹɹɹओʹ౦ೆΞδΞ΁όοΫύοΫཱྀߦʹߦͬͨΓ͠·͢ ɹɹʢΧϯϘδΞɺϛϟϯϚʔɺόϯάϥσγϡɺ৽ᙜ΢Πάϧ࣏ࣗ۠FUDʣ ཱྀߦͷࣸਅIUUQNBUTVLFOKJNEPDPN TwitterΞΠίϯ

1ZUIPOλάͰ̍Ґʹ ͳΓ·ͨ͠ʂ 㱼ਓ

ࠓ೔ͷΞδΣϯμ ɾ಺ੵΛάϥϑΟΧϧʹཧղ͢Δ ɾ෼ࢄͱඪ४ภࠩͷ࿩ ɾ૬ؔ܎਺ͷ࿩ ɾճؼ෼ੳͷ࿩ ɾओ੒෼෼ੳͷ࿩ ͜͜·Ͱߦ͚ͳ͍͔΋ɾɾɾ

ຊൃදͷ಺༰͸ݸਓͷݟղͰ ͋Γɺॴଐ͢Δ૊৫ɾஂମͷ ެࣜݟղͰ͸͋Γ·ͤΜɻ

·ͨɺຊൃදͰ͸ཧղͷ͠΍͢͞Λ ༏ઌ͠ɺҰ෦ݫີͳઆ໌ͱͳͬͯ ͍ͳ͍෦෼͕͋Γ·͕͢ɺ ྃ͝ঝ͍ͩ͘͞ɻ

·ͣ͸ɺ಺ੵͷ࿩͔Βɻ

ͱ͢Δͱɺ಺ੵͱ͸ ϕΫτϧΛ n࣍ݩϕΫτϧ ఆٛ a = (a1, a2, · ·
· , an)T, b = (b1, b2, · · · , bn)T a, b a · b = a1b1 + · · · + anbn = n X i=1 aibi a b

ͱ͢Δͱɺ಺ੵͱ͸ ϕΫτϧΛ n࣍ݩϕΫτϧ ఆٛ a = (a1, a2, · ·
· , an)T, b = (b1, b2, · · · , bn)T a, b a · b = a1b1 + · · · + anbn = n X i=1 aibi n n 1 1 ɾ a b ϕΫτϧ͔ΒεΧϥʹม׵͢Δ ԿΒ͔ͷֻ͚ࢉతͳૢ࡞Ͱ͋Δ͜ͱ͸ Θ͔Δ͕ɺͦΕҎ্Α͘Θ͔Βͳ͍

ϕΫτϧ a = (a1, a2, · · · , an)T,
ͷ௕͞ ϊϧϜ ͸ɺ ͷΑ͏ʹɺࣗ਎ͱͷ಺ੵͷϧʔτͱͯ͠ දͤ·͢ a kak2 kak = p a1 · a1 + · · · + an · an = v u u t n X i=1 a2 i = p a · a

͜͜Ͱ༨ݭఆཧΛࢥ͍ग़͠·͢ɻ kb ak2 = kak2 + kbk2 2 kakkbk cos
✓

✓ kb ak2 = (b a) · (b a) = a · a + b · b 2a · b = kak2 + kbk2 2a · b ࠨลΛల։͢Δͱɾɾɾ

✓ kb ak2 = (b a) · (b a) = a · a + b · b 2a · b = kak2 + kbk2 2a · b ࠨลΛల։͢Δͱɾɾɾ ಉ͡ͳͷͰɺ

✓ kb ak2 = (b a) · (b a) = a · a + b · b 2a · b = kak2 + kbk2 2a · b ࠨลΛల։͢Δͱɾɾɾ ফڈՄೳʂ 2 a · b = 2 kakkbk cos ✓ ) a · b = kakkbk cos ✓

✓ kb ak2 = (b a) · (b a) = a · a + b · b 2a · b = kak2 + kbk2 2a · b ࠨลΛల։͢Δͱɾɾɾ 2 a · b = 2 kakkbk cos ✓ ) a · b = kakkbk cos ✓ a · b = kakkbk cos ✓

a · b = kakkbk cos ✓ ఆٛ̎ Αͬͯɺ΋͏Ұͭͷ಺ੵͷఆٛɺ ͱಉ஋Ͱ͋Δ͜ͱ͕Θ͔Γ·ͨ͠ɻ

͜͜Ͱ֯౓θ ͱ͸ ͜ΕͰ͢Ͷɻ

a · b = kakkbk cos ✓ Λ΋͏ͪΐͬͱݟ͑ΔܗͰߟ͑·͢ɻ

cos ✓ = kck krk DPTВͷఆٛ͸ Ͱ͢ɻ

) krk cos ✓ = kck cos ✓ = kck
krk ͱɺมܗͰ͖ΔͷͰɺ ɹccDcc͸൒ܘͷ௕͞ʹDPTВΛ͔͚ͨ΋ͷ ͱཧղͰ͖·͢ɻ

Ҏ্͔ΒɺϕΫτϧD͸ɺY࣠ʹਨ௚ํ޲ʹ্͔Β ϥΠτΛ౰ͯͨ࣌ͷ൒ܘϕΫτϧSͷӨʹͳΔ ෦෼ͱղऍ͕Ͱ͖·͢ɻ ͜ΕΛʮࣹӨʯ ͱݴͬͨΓ͠·͢ɻ ͜ͷ࣌ɺࣹӨͷ௕͞ ͸ Ͱ͢ɻ ) krk
cos ✓ = kck

a · b = kakkbk cos ✓ Ҏ্ͷٞ࿦Λ;·͑ͯɺ಺ੵΛཧղ͢Δ ϕΫτϧͷ௕͞ c
ϕΫτϧͷ௕͞ a a b θ c = kbk cos ✓

a b θ c = kbk cos ✓ Ҏ্ͷٞ࿦Λ;·͑ͯɺ಺ੵΛཧղ͢Δ a
· b = kakkbk cos ✓ ϕΫτϧͷ௕͞ c ϕΫτϧͷ௕͞ ͭ·Γɺಉ͡ํ޲Λ޲͘ ੒෼ʹ͋Θͤͯ͋͛ͯɺ ͦͷํ޲ͷ௕͞Λ ֻ͚ࢉͨ͠΋ͷʂ a

a b θ c = kbk cos ✓ ͜ͷֆͷ৔߹ɺBͷํ޲ʹ͋Θ͍ͤͯΔ Ҏ্ͷٞ࿦Λ;·͑ͯɺ಺ੵΛཧղ͢Δ
a · b = kakkbk cos ✓ ϕΫτϧͷ௕͞ c ϕΫτϧͷ௕͞ a ͭ·Γɺಉ͡ํ޲Λ޲͘ ੒෼ʹ͋Θͤͯ͋͛ͯɺ ͦͷํ޲ͷ௕͞Λ ֻ͚ࢉͨ͠΋ͷʂ

ͳͷͰɺ֯౓͕ਨ௚ͩͱɺࣹӨͨ͠ͷ௕͞ ͕ʹͳͬͯ͠·͏ͷͰɺ಺ੵ΋ʹͳΔɻ c ɹ ٯ΋વΓɻ ɹ ಺ੵ͕̌ͩͱਨ௚ͱݴ͑Δɻ

̎ͭͷϕΫτϧ ͷ ɹɹɹ௕͕̍ͩͬͨ͞৔߹ a b

a · b = kakkbk cos ✓ = cos ✓

a · b = kakkbk cos ✓ = cos ✓
಺ੵ͸DPTВ ͱͳΔɻ

͜ͷઅͷ·ͱΊ

͜ͷઅͷ·ͱΊ a · b = a1b1 + · · ·
+ anbn = n X i=1 aibi a · b = kakkbk cos ✓ a b θ c = kbk cos ✓ ಺ੵʹ͸̎ͭͷఆ͕ٛ͋Γɺ ࣹӨ͞ΕΔଆͷϕΫτϧͷ௕͕͞ͷ࣌͸ɺ಺ੵ͸ ࣹӨͷ௕͞Ͱ͋Δɻ ܭࢉ͢ΔͳΒͪ͜Β ҙຯ͕Θ͔Δͷ͸ͪ͜Β

͜͜Ͱ࣍ʹ౷ܭֶͷ࿩Λ

DBSTσʔληοτ ˠं͕૸Δ଎౓ͱɺϒϨʔΩΛ౿Μͩ࣌ʹࢭ·Δ ͜ͱ͕Ͱ͖Δ·Ͱͷڑ཭ͷσʔλ ͪΐͬͱݹ͍Ͱ͢ ώετάϥϜ ࢄ෍ਤ

෼ࢄɺඪ४ภࠩͱ͸ʁ ࢄΒ͹Γɿখ ෼ࢄɹɹɿখ ࢄΒ͹Γɿେ ෼ࢄɹɹɿେ ෼ࢄͱ͸ɺσʔλͷ ࢄΒ͹Γͷࢦඪɻ ্ͷάϥϑ͸ࢄΒ͹ Γ͕গͳ͘ɺ Լͷάϥϑ͸ࢄΒ͹
Γ͕େ͖͍ɻ Y͸େମʙͷൣғ Y͸େମʙͷൣғ ͜ΕΛɺ਺஋తͳ ࢦඪͰද͢ɻ

DBSTσʔληοτ TQFFEσʔλ = ¯ x = 1 n n X
i=1 xi ฏۉ = s 2 = 1 n n X i=1 ( xi ¯ x )2 ෼ࢄ ඪ४ภࠩ = s = v u u t 1 n n X i=1 ( xi ¯ x )2 ภࠩ

ඪ४ภࠩ = s = v u u t 1 n
n X i=1 ( xi ¯ x )2 DBSTσʔληοτ TQFFEσʔλ = ¯ x = 1 n n X i=1 xi ฏۉ = s 2 = 1 n n X i=1 ( xi ¯ x )2 ෼ࢄ ֤σʔλฏۉ͔ΒͷࠩΛͦ ΕͧΕ̎৐ͯ͠࿨Λͱͬͨ ΋ͷɻࢄΒ͹Γ۩߹ͷࢦඪ ภࠩ

෼ࢄɾඪ४ภࠩͷࢹ֮తΠϝʔδ = s 2 = 1 n n X
i=1 ( xi ¯ x )2 ภࠩ͸ɺ୯७ʹ଍͠ ͯ͠·͏ͱɺ௼Γ߹ͬ ͍ͯΔͷͰʹͳΔ ˠภࠩ৐͍ͯ͠ΔΦϨϯδͷਖ਼ํܗͷ໘ੵΛฏۉͨ͠ ΋ͷɻԼهͰ͸த৺ΛͣΒ͍ͯ͠Δ͕ɺ࠷খͱͳΔͷ͕ ฏۉ ͷҐஔ ¯ x ৐͢Δ https://goo.gl/6DROOA Ξχϝʔγϣϯɿ

෼ࢄɾඪ४ภࠩͷࢹ֮తΠϝʔδ ෼ࢄͷ··ͩͱɺ୯Ґ͕໘ੵʹͳͷͰݩͷσʔλͷ୯Ґ ͱ߹Θͳ͍ɻͳͷͰɺϧʔτΛͱͬͯݩͷ୯Ґʹ໭ͨ͠ ΋ͷ͕ඪ४ภࠩɻ = s = v u u
t 1 n n X i=1 ( xi ¯ x )2

෼ࢄɾඪ४ภࠩͷ΋͏ҰͭͷΠϝʔδ x = ( x1, · · · , xn)
x 0 = ( x1 ¯ x, · · · , xn ¯ x ) σʔλΛO࣍ݩϕΫτϧͱͯ͠ݟͯΈΔɻ x ฏۉ͔ΒͷภࠩͷϕΫτϧΛ ¯ x ͱ͢Δͱɺͦͷ࣌ɺͷ௕͞͸ x 0 k x 0k k x 0k = v u u t n X i=1 ( xi ¯ x )2 Ͱɺද͞ΕΔɻ

෼ࢄɾඪ४ภࠩͷ΋͏ҰͭͷΠϝʔδ Αͬͯ = s = v u u t 1
n n X i=1 ( xi ¯ x )2 = r 1 n v u u t n X i=1 ( xi ¯ x )2 = r 1 n k x 0k ͱͳΓɺඪ४ภࠩ͸ϕΫτϧͷ௕͞ͷ Ұछͱߟ͑ΒΕΔɻ x 0 k x 0k = v u u t n X i=1 ( xi ¯ x )2 x’ k x 0 k

ͪͳΈʹɺ &Yภࠩ஋ ໊લɹ ਺ֶɹ ภࠩ ඪ४ภࠩ Կݸ෼ʁ ˡºഒ ˡ
ాத ߴڮ ླ໦ ౉ล ਗ਼ਫ ໦ଜ ࢁຊ ฏۉ ඪ४ภࠩ ͜Ε͕ʮภࠩ஋ʯ

͜ͷઅͷ·ͱΊ

͜ͷઅͷ·ͱΊ σʔλΛO࣍ݩ্ͷ̍ຊͷϕΫτϧͱͯ͠ݟͯΈΔɻ ·ͨɺฏۉ͔ΒͷภࠩϕΫτϧΛԼهͷΑ͏ʹఆٛͨ͠ x s = r 1 n v
u u t n X i=1 ( xi ¯ x )2 = r 1 n k x 0k x 0 = ( x1 ¯ x, · · · , xn ¯ x ) ͜ͷΑ͏ʹ͢Δͱඪ४ภࠩ͸ɺ ͷΑ͏ʹϕΫτϧͷ௕͞ͷҰछͱଊ͑Δ͜ͱ͕Ͱ͖Δɻ

૬ؔ܎਺ʹ͍ͭͯ

૬ؔ܎਺ 9ͷ஋͕૿Ճ͢Δͱɺ:ͷ஋΋૿Ճ͢Δˠਖ਼ͷ૬͕ؔ͋Δ 9ͷ஋͕૿Ճ͢Δͱɺ:ͷ஋͕ݮগ͢Δˠෛͷ૬͕ؔ͋Δ 㱺͜ΕΛ਺஋Խͨ͠΋ͷʹʮ૬ؔ܎਺ʯ͕͋Δɻ r = P ( xi ¯
x )( yi ¯ y ) pP ( xi ¯ x )2 pP ( yi ¯ y )2 ૬ؔ܎਺

૬ؔ܎਺

૬ؔ܎਺͸ ׬શʹԣ࣠ͱॎ͕࣠ ґଘؔ܎ʹ͋ΓɺҰํ͕૿͑Δͱ ΋͏Ұํ΋૿͍͑ͯΔɻ ૬ؔ܎਺

૬ؔ܎਺͸ ΍͸Γɺ׬શʹԣ࣠ͱॎ͕࣠ ґଘؔ܎ʹ͋ΓɺҰํ͕૿͑Δͱ ΋͏Ұํ͕ݮ͍ͬͯΔɻ ૬ؔ܎਺

૬ؔ܎਺͸ ԣ࣠ͱॎ͕࣠શ͘ͳ͘ Ұํ͕૿͑ͯ΋΋͏Ұํ͸ ͦΕͱ͸ؔ܎ͳ͘஋͕ܾ·Δɻ ૬ؔ܎਺

૬ؔ܎਺ r = P ( xi ¯ x )( yi
¯ y ) pP ( xi ¯ x )2 pP ( yi ¯ y )2 ૬ؔ܎਺ ࣜΛΑ͘ݟͯΈΔͱɺ = x 0 · y 0 k x 0kk y 0k ͋Εɺ͜Εͬͯʁ

૬ؔ܎਺ r = P ( xi ¯ x )( yi
¯ y ) pP ( xi ¯ x )2 pP ( yi ¯ y )2 ૬ؔ܎਺ ࣜΛΑ͘ݟͯΈΔͱɺ = x 0 · y 0 k x 0kk y 0k = cos ✓ ಺ੵͷఆ͔ٛΒ DPTВͩʂ ͋Εɺ͜Εͬͯʁ θ x’ y’ σʔλ਺Oͷͱ͖ɺO࣍ݩͷ ߴ࣍ݩۭؒͷ̎ຊͷϕΫτϧ ͱݟͳ͢ ฏۉΛҾ͍ͯத৺Λଗ͑ͯ͋Δ͜ͱʹ஫ҙ

૬ؔ܎਺ Αͬͯɺσʔλ਺O࣍ݩ্ۭؒͷ̎ຊͷϕΫτϧͷؒͷ ֯౓Ͱ͋Δͱߟ͑ΒΕΔʂ https://goo.gl/bXgHnc Ξχϝʔγϣϯɿ = x 0 · y
0 k x 0kk y 0k = cos ✓

͜ͷઅͷ·ͱΊ

͜ͷઅͷ·ͱΊ r = P ( xi ¯ x )( yi
¯ y ) pP ( xi ¯ x )2 pP ( yi ¯ y )2 ૬ؔ܎਺ = x 0 · y 0 k x 0kk y 0k = cos ✓ ̎ͭͷσʔλؒͷ૬ؔؔ܎Λද͢ʮ૬ؔ܎਺ʯ͸ σʔλΛϕΫτϧͱͯ͠ଊ͑ΔͱɺO࣍ݩ্ۭؒͷ ̎ຊͷϕΫτϧͷؒͷ֯౓ͱଊ͑Δ͜ͱ͕Ͱ͖ͨɻ θ x’ y’

ճؼ෼ੳʹ͍ͭͯ

ճؼ෼ੳ ୯ճؼ෼ੳ DBSσʔληοτ ˠεϐʔυͱڑ཭ͷؒʹ௚ઢతͳؔ܎͕͋Γͦ͏ʹ ɹΈ͑Δɻ ڑ཭ ºεϐʔυ ↵

ˠεϐʔυͱڑ཭ͷؒʹ௚ઢతͳؔ܎͕͋Γͦ͏ʹ ɹΈ͑Δɻ ճؼ෼ੳ Ұ൪ྑ͍ઢΛબͿج४ͱͯ͠ ఺ͱઢͷؒͷ௕͞Λશ෦଍͠ ߹Θͤͨ΋ͷΛ࠷খ͢Δ͜ͱ Λߟ͑Δɻ ࢒ࠩ yi =
↵ + xi + ei ͜ΕΛ J ʹ͍ͭͯ ࿨ΛͱΔ

ճؼ෼ੳ yi = ↵ + xi + ei ei =
yi ( ↵ + xi) ˠ min S ( ↵, ) = min n X i=1 e 2 i = min n X i=1 { yi ( ↵ + xi)}2 ΑͬͯɺԼهͷ࠷খԽ໰୊ͱͳΔɻ ࠷খೋ৐๏ ↵ + xi yi ei

ճؼ෼ੳ ˠࠇ͍ઢ ࢒ࠩ ͕࠷খʹͳΔΑ͏ʹЋɺЌΛௐ੔͢Δ ЋΛม͑ͯΈΔ ЌΛม͑ͯΈΔ https://goo.gl/BDtIU0 https://goo.gl/jmrk07 Ξχϝʔγϣϯɿ Ξχϝʔγϣϯɿ

ճؼ෼ੳ @S(↵, ) @↵ = 0 @S(↵, ) @ =
0 Λղ͘͜ͱͰɺ4͕࠷খͱͳΔ ЋͱЌΛٻΊΒΕΔɻ

ճؼ෼ੳ લϖʔδͷܭࢉΛղ͘ͱɺ ˆ ↵ = ¯ y ˆ¯ x ͱͳΔɻ
ˆ = P ( xi ¯ x )( yi ¯ y ) P ( xi ¯ x )2

Λਂ۷Γͯ͠ΈΔ ˆ

ճؼ෼ੳ લड़ͷΑ͏ʹɺภࠩΛϓϥΠϜͰද͢ͱɺ ˆ = P ( xi ¯ x )(
yi ¯ y ) P ( xi ¯ x )2 = x 0 · y 0 k x 0k2 ಺ੵͱɺϕΫτϧͷ௕͞Ͱ දݱͰ͖Δʂ

ճؼ෼ੳ x’ y’ θ = x 0/k x 0k ޲͖͕x’ͱಉ͡Ͱ͋Δ୯ҐϕΫτϧ
ˆ = x 0 · y 0 k x 0k2 = x 0 k x 0k2 · y 0 ͜ͷ̎ͭͷϕΫτϧͷ಺ੵͱ ߟ͑ΒΕΔɻ

ճؼ෼ੳ ˆ = x 0 · y 0 k x
0k2 = x 0 k x 0k2 · y 0 ͜ͷ̎ͭͷϕΫτϧͷ಺ੵͱ ߟ͑ΒΕΔɻ x’ y’ θ = x 0/k x 0k ˆ = x 0 k x 0k2 · y 0 ͸x’ʹର͢Δy’ͷࣹӨͷ ௕͞Ͱ͋Δͱߟ͑ΒΕΔɻ ˆ = 1 k x 0k k x 0kk y 0k cos ✓ = k y 0k cos ✓ = 1 k x 0k x 0 k x 0k · y 0 ௕͞

͜ͷઅͷ·ͱΊ

͜ͷઅͷ·ͱΊ ͭͷσʔλͷؒʹઢܗؔ܎Λ౰ͯ͸ΊͯΈΔ෼ੳख๏ yi = ↵ + xi + ei ্هͷϞσϧʹରͯ͠ɺ࢒ࠩei
Λ࠷খʹ͢ΔЋɺЌΛ ࠷খೋ৐๏ͰٻΊͨɻ

͜ͷઅͷ·ͱΊ ܏͖ʹ૬౰͢Δ͸ϕΫτϧ x’ʹର͢ΔϕΫτϧ y’ ͷࣹӨͷ௕͞ͱͯ͠ଊ͑Δ͜ͱ͕Ͱ͖ͨɻ x’ y’ θ = x
0/k x 0k ˆ = x 0 k x 0k2 · y 0 ˆ

ओ੒෼෼ੳʹ͍ͭͯ

ྫɿ*SJT ΞϠϝ σʔληοτ ˠσʔλ෼ੳքͷ)FMMP8PSMEతͳσʔλͰ͢ ID sepal_length sepal_width petal_length petal_width target
0 5.1 3.5 1.4 0.2 0 1 4.9 3.0 1.4 0.2 0 2 4.7 3.2 1.3 0.2 0 3 4.6 3.1 1.5 0.2 0 4 5.0 3.6 1.4 0.2 0 5 5.4 3.9 1.7 0.4 0 6 4.6 3.4 1.4 0.3 0 7 5.0 3.4 1.5 0.2 0 8 4.4 2.9 1.4 0.2 0 9 4.9 3.1 1.5 0.1 0 ʜ http://blog.kaggle.com/2015/04/22/scikit-learn-video-3-machine-learning-ﬁrst-steps-with-the-iris-dataset/ ɹΞϠϝͷछผ TFUPTB WFSTJDPMPS WJSHJOJDB Ֆห ᣾ย

ࢄ෍ਤΛඳ͘ʹ͸ɺ̎࣍ݩʹ͠ͳͯ͘͸ͳΒͳ͍ɻ ӈਤ͸ɺ֤ྻͷ૊ Έ߹Θͤ͝ͱʹɺ ࢄ෍ਤΛॻ͍ͨྫ ྫɿ*SJT ΞϠϝ σʔληοτ

sepal_length sepal_width petal_length petal_width z1 z2 ͭͷಛ௃͔Βɺ ͳΔ΂͘৘ใΛ ଛͳΘͳ͍Α͏ʹ ͭʹݮΒͨ͠ɻ
[ [ ʮओ੒෼෼ੳʯͱ͍͏ explained: 0.9776 ྫɿ*SJT ΞϠϝ σʔληοτ ৘ใͷ໿͕ΩʔϓͰ͖͍ͯΔ

࣍ݩ͔Β࣍ݩʹམͱͨ࣌͠ͷΠϝʔδ ࣍ݩ͸ɺ΋͏૝૾Ͱ͖ͳ͍ੈք͕ͩɺ͜Εͱಉ༷ͳ͜ͱΛ ߦͬͯɺ࣍ݩ͔Β࣍ݩʹམͱͨ͠ͷ͕લϖʔδͷྫ ࣍ݩΛམͱ͢ ໘ΑΓ্ͷ఺͕ࠇ ໘ΑΓԼͷ఺͕άϨʔ σʔλ఺YJ ฏ໘ʹਨ௚ʹམͱ͢

sepal_length sepal_width petal_length petal_width z1 z2 ͭͷಛ௃͔Βɺ ͳΔ΂͘৘ใΛ ଛͳΘͳ͍Α͏ʹ ͭʹݮΒͨ͠ɻ
[ [ ʮओ੒෼෼ੳʯͱ͍͏ explained: 0.9776 ओ੒෼෼ੳ ଟ͘ͷಛੑΛ࣋ͭଟมྔσʔλΛɺগͳ͍ݸ਺Ͱ ૯߹తʹಛ௃Λද͢ྔʹ·ͱΊΔ͜ͱΛߦ͏෼ੳ ྫɿ*SJT ΞϠϝ σʔληοτ

ओ੒෼෼ੳͷ࢓૊Έ ΋͏গ͠୯७ͳσʔλͰ ID Math Japanese 1 20 9 2 4
2 3 12 15 4 5 10 5 10 6 6 8 11 7 1 4 8 15 9 9 5 6 ਺ֶɾࠃޠͷڭՊͷ੒੷σʔλ͔Β૯߹తͳֶྗͷ ࢦඪΛࢉग़ͯ͠Έ·͢ɻ㱺࣍ݩσʔλΛ࣍ݩʹམͱ͢

ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ ੺͍ઢ ௕̍͞ͷϕΫτϧ ΛԆ௕ͨ͠΋ͷʹਨ௚ʹ ઢΛམͱͨ͠ͱ͜ΖΛɺͦͷσʔλͷಛੑ஋ͱ͢Δɻ

ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ https://goo.gl/lAqbz5 Ξχϝʔγϣϯɿ ઢ্ʹམͪͨ఺ ੺͍఺ ͷ෼ࢄ͕Ұ൪େ͖͘ͳΔΑ͏ͳ ֯౓ΛબͿɻˡͦͷ֯౓ͷͱ͖ɺҰ൪σʔλΛઆ໌Ͱ͖͍ͯΔ ͱݴ͑Δɻ

xi = (xi1, xi2) ௕͞ kx i kcos
✓ θ z(1) a ಺ੵʂ ओ੒෼ͱͯ͠ͷ̍ͭ໨ͷ࣠Λͱ͢Δͱɺʹର͢Δ ओ੒෼͸ͱදͤΔɻ z(1) xi z(1)i = ( a · xi) a ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ = z(1)i = a · x i a = (a1, a2)T ৘ใଛࣦྔ

͜ͷ੺͍఺ͷ෼ࢄΛٻΊΔɻ ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ V(z(1) ) = 1 n n X
i=1 (z(1)i ¯ z(1) )2 ෼ࢄ = 1 n n X i=1 {( a · xi) ( a · ¯ x )}2 dதུd ͸࣠ͷ෼ࢄ ͸࣠ ͷڞ෼ࢄ sii sij ¯ z(1) z(1) = a2 1 s11 + 2a1a2s12 + a2 2 s22

͜ͷ੺͍఺ͷ෼ࢄΛٻΊΔɻ ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ V(z(1) ) = 1 n n X
i=1 (z(1)i ¯ z(1) )2 ෼ࢄ = 1 n n X i=1 {( a · xi) ( a · ¯ x )}2 dதུd ͸࣠ͷ෼ࢄ ͸࣠ ͷڞ෼ࢄ sii sij ¯ z(1) z(1) ෼ࢄΛ࠷େʹ͢ΔBͷ֯౓ ͭ·Γ ͜ΕΛ࠷େʹ͢ΔB BΛٻΊΔ = a2 1 s11 + 2a1a2s12 + a2 2 s22

ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ = a2 i s11 + 2a1a2s12 + a2
2 s22 ͨͩ͠ɺ ͸ɺϕΫτϧ͕௕͘ͳΕ͹ͳΔ΄Ͳେ͖͘ͳΔͷͰɺ ௕͞͸̍ a kak2 = 1 ͱ͍͏੍໿Λ͚ͭΔɻ Ҏ্ΑΓɾɾɾ

ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ ओ੒෼෼ੳ͸ҎԼͷ৚͖݅ͭ࠷େԽ໰୊ͱͯ͠ղ͘ɻ s.t. kak2 = 1 max a2 1
s11 + 2 a1a2s12 + a2 2 s22 ৚͖݅ͭ࠷దԽ໰୊ 㱺ʮϥάϥϯδϡͷະఆ৐਺๏ʯͰղ͘

s.t. kak2 = 1 max a2 1 s11 + 2
a1a2s12 + a2 2 s22 ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ F(a1, a2, ) = a2 1 s11 + 2a1a2s12 + a2 2 s22 (a2 1 + a2 2 1) @F @a1 = 2a1s11 + 2a2s12 2 a1 = 0 @F @a2 = 2a2s22 + 2a1s12 2 a2 = 0 @F @ = a2 1 + a2 2 1 = 0 ඍ෼ͯ͠ͱஔ͘

ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ લϖʔδͷࣜΛ੔ཧ͢Δͱɺ  s11 s12 s12 s22  a1
a2 =  a1 a2 ෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ͱͳ͍ͬͯΔʂ

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ

˞ΑΓ͜Ε͕ݴ͑Δ ෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ S =  s11 s12 s12 s22 =
 2.0 s12 s12 5.0 ྫ ෼ࢄ͕ɺԣ࣠ɺॎ࣠ͷ৔߹ ·ͨɺ෼ࢄͱڞ෼ࢄͷؔ܎͔Βɺڞ෼ࢄͷ্Լݶ͸ ͦΕͧΕͷඪ४ภࠩΛֻ͚ͨ΋ͷͱͳΔɻ |sij |  p siisjj |a · b| = |kakkbk cos ✓|  kakkbk ্هͷ਺஋ྫͷ৔߹ɺ ্Լݶ͸

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ ྫ ෼ࢄ͕ɺԣ࣠ɺॎ࣠ͷɺڞ෼ࢄͷ৔߹ S =  s11 s12 s12 s22
=  2.0 0 0 5.0

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ ྫ ෼ࢄ͕ɺԣ࣠ɺॎ࣠ͷɺڞ෼ࢄͷ৔߹
S =  s11 s12 s12 s22 =  2.0 1.0 1.0 5.0

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ ڞ෼ࢄͷ্ԼݶͷൣғͰΞχϝʔγϣϯͯ͠ΈΔɻ ԁप্ͷ੨͍఺ʹ ෼ࢄڞ෼ࢄߦྻΛ ֻ͚ͯઢܗม׵ͨ͠ ΋ͷ͕੺͍఺ ෼ࢄڞ෼ࢄߦྻ ྘ͱࠇͷઢ͕ɺ ݻ༗ϕΫτϧ https://goo.gl/WGWXWr
Ξχϝʔγϣϯɿ

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ ڞ෼ࢄͷ্ԼݶͷൣғͰΞχϝʔγϣϯͯ͠ΈΔɻ ෼ࢄڞ෼ࢄߦྻ https://goo.gl/rIzRCA Ξχϝʔγϣϯɿ ԁप্ͷ੨͍఺ʹ ෼ࢄڞ෼ࢄߦྻΛ ֻ͚ͯઢܗม׵ͨ͠ ΋ͷ͕੺͍఺ ྘ͱࠇͷઢ͕ɺ
ݻ༗ϕΫτϧ

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ ෼ࢄڞ෼ࢄߦྻΛϕΫτϧʹ͔͚ΔͱɺϕΫτϧ͕ճసͱ Ҿ͖Ԇ͹͠͞ΕΔɻ ݻ༗ϕΫτϧ͸ɺԼهͷପԁͷ৔߹ɺ௕͍ํͷϕΫτϧ͸௕࣠ํ޲ ୹͍ํͷϕΫτϧ͸୹࣠ํ޲Λࢦ͢Α͏ʹͳ͍ͬͯΔɻ ڞ෼ࢄ ڞ෼ࢄ ڞ෼ࢄ ্ݶ

෼ࢄڞ෼ࢄߦྻͷݻ༗஋໰୊ʁ ෼ࢄڞ෼ࢄߦྻΛϕΫτϧʹ͔͚ΔͱɺϕΫτϧ͕ճసͱ Ҿ͖Ԇ͹͠͞ΕΔɻ ݻ༗ϕΫτϧ͸ɺԼهͷପԁͷ৔߹ɺ௕͍ํͷϕΫτϧ͸௕࣠ํ޲ ୹͍ํͷϕΫτϧ͸୹࣠ํ޲Λࢦ͢Α͏ʹͳ͍ͬͯΔɻ ڞ෼ࢄ ڞ෼ࢄ ڞ෼ࢄ ্ݶ ݻ༗஋໰୊Λղ͍ͯɺ࣠ͷํ޲͕Θ͔Δͱ
ϕΫτϧBͷ޲͖͕ܾఆͰ͖Δɻ

ओ੒෼෼ੳͷ࢓૊Έ ࣍ݩͷྫ ϕΫτϧBͷ޲͖͕ܾ·ͬͨͷͰɺͦͷ্࣠ʹσʔλ Λམͱͯ͠ࢦඪͱͨ͠Γɺ෼ྨΛ࣮ߦͨ͠Γ͢Δɻ

͜ͷઅͷ·ͱΊ

͜ͷઅͷ·ͱΊ ࣍ݩΛ࡟ݮ͢Δख๏ͷ̍ͭͱͯ͠ओ੒෼෼ੳΛ঺հ͠ɺ ͦͷٻΊํʹ͸ɺ಺ੵΛཧղ͢ΔͱԿΛ͍ͯ͠Δॲཧ Ͱ͋Δ͔ɺΠϝʔδΛ͚ͭ΍͍͢͜ͱΛઆ໌ͨ͠ɻ ID sepal_length sepal_width petal_length petal_width target
0 5.1 3.5 1.4 0.2 0 1 4.9 3.0 1.4 0.2 0 2 4.7 3.2 1.3 0.2 0 3 4.6 3.1 1.5 0.2 0 4 5.0 3.6 1.4 0.2 0 5 5.4 3.9 1.7 0.4 0 ࣍ݩ͔Β ࣍ݩ΁

ࢀߟ ɾʲ౷ܭֶʳॳΊͯͷʮඪ४ภࠩʯʢ౷ܭֶʹ࠳ં͠ͳ͍ͨΊʹʣ 2JJUB IUUQRJJUBDPNLFONBUTVJUFNTFDBDCDF ɾʲ਺ֶʳݻ༗஋ɾݻ༗ϕΫτϧͱ͸Կ͔ΛՄࢹԽͯ͠ΈΔ 2JJUB IUUQRJJUBDPNLFONBUTVJUFNTBFDGDEB
ɾओ੒෼෼ੳ ɹɹIUUQXXXFPLBZBNBVBDKQdOBHBIBUBCTUBUSUBTZPQEG ɾࠓ೔ͷ1ZUIPOίʔυPO(JU)VC IUUQTHJUIVCDPNNBUTVLFO2JJUB@$POUFOUTCMPCNBTUFS NBUI@GPS@QSPHSBNNFSEFNP@NBUI@TUBUT@JQZOC

「内積が見えると統計学も見える」第5回 プログラマのための数学勉強会 発表資料

「内積が見えると統計学も見える」第5回 プログラマのための数学勉強会 発表資料

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「内積が見えると統計学も見える」第5回プログラマのための数学勉強会発表資料

「内積が見えると統計学も見える」第5回プログラマのための数学勉強会発表資料