math_29

Transcript

1. 29 ϕΫτϧҎ্ʹߴີ౓ͳॻ͖ํ ౷ܭֶ͕࠷ڧͷֶ໰Ͱ͋Δɹ਺ֶฤ
2. લճͷτϐοΫ ૬ؔ܎਺ΛϕΫτϧͰදݱ
3. ࠓճͷτϐοΫ • ୯ճؼ෼ੳΛϕΫτϧͰදݱ • ॏճؼ෼ੳΛϕΫτϧͰදݱ
4. 2ষͷσʔλΛϕΫτϧͷܗࣜͰද͢ Ӧۀ๚໰ճ਺:ɹx = (x1 , x2 , ⋯, xn ) ܖ໿݅਺ɹɹ:ɹy = (y1 , y2 , ⋯, yn ) yi = a + bxi Ͱදͨ͠ͱ͖ͷͣΕ ϵi = yi − (a + bxi )ΑΓ ͣΕɹɹɹɹ:ɹϵ = (ϵ1 , ϵ2 , ⋯, ϵn ) ⃗ y = a + b ⃗ x + ⃗ ϵ a = (a, a, ⋯, a) ⃗ y = ⃗ a + b ⃗ x + ⃗ ϵ ৚݅ΛϕΫτϧܗࣜͰද͢ͱ ϕΫτϧ+εΧϥ͸Ͱ͖ͳ͍ͷͰ ࠷খೋ৐๏ n ∑ i=1 ϵ2 i = ϵ ⋅ ϵ = ∥ϵ∥2
5. ୯ճؼ෼ੳɿ ⃗ y = ⃗ a + b ⃗ x + ⃗ ϵ ୯ճؼ෼ੳͷߟ͑ํΛΑΓγϯϓϧʹද͢͜ͱ͕Ͱ͖ͨ ࠷খೋ৐๏ɿ n ∑ i=1 ϵ2 i = ϵ ⋅ ϵ = ∥ϵ∥2
6. ౷ܭֶɾػցֶशͷࠜຊతͳߟ͑ํ ʮxͷ஋ʹΑͬͯyͷ஋͕Ͳ͏ͳΔ͔Λઆ໌Ͱ͖ΔΑ͏ͳ਺ֶϞσϧΛ ߟ͑ɺཧ࿦্ͷyͱ࣮ࡍͷyΛ͍ۙ΋ͷʹ͢Δʯ
7. ྫ୊1 • આ໌ม਺͕ෳ਺͋Δ৔߹ɺͲͷΑ͏ʹ෼ੳ͢Ε͹Α͍͔ʁ x1 = (x11 , x21 , ⋯, xn1 ) x2 = (x12 , x22 , ⋯, xn2 ) xk = (x1k , x2k , ⋯, xnk ) ·ͣɺxΛఆٛ͠௚͢ nਓ kݸͷ આ໌ม਺ ॏճؼ෼ੳͷϕΫτϧදݱɿy = β0 + β1 x1 + β2 x2 + ⋯ + βk xk + ϵ
8. ྫ୊1 ΑΓγϯϓϧʹද͢ͱ x1 = (x11 , x21 , ⋯, xn1 ) x2 = (x12 , x22 , ⋯, xn2 ) xk = (x1k , x2k , ⋯, xnk ) X = x11 x12 ⋯ x1k x21 x22 ⋯ x2k x31 x32 ⋯ x3k ⋮ ⋮ ⋯ ⋮ xn1 xn2 ⋯ xnk X = (x1 , x2 , ⋯, xk ) x1 = x11 x21 x31 ⋮ xn1 x2 = x12 x22 x32 ⋮ xn2 xk = x1k x2k x3k ⋮ xnk ϵ = ϵ1 ϵ2 ϵ3 ⋮ ϵn y = y1 y2 y3 ⋮ yn β = β1 β2 β3 ⋮ βn ॏճؼ෼ੳͷϕΫτϧදݱɿy = Xβ + ϵ ϕΫτϧΛ·ͱΊΔ ྻϕΫτϧͰද͢
9. ߦྻͷֻ͚ࢉ ʮNߦOྻͷߦྻʯʹʮOߦ̍ྻͷߦྻʯΛֻ͚Δͱ
   ʮNߦྻͷߦྻʯ͕Ͱ͖͕͋Δ ͠ΓͱΓͷΑ͏ͳϧʔϧ͕ଘࡏ͓ͯ͠Γɺ
   ࠨଆͷྻ਺ͱӈଆͷߦ਺͕Ұக͍ͯ͠ͳ͚Ε͹ֻ͚ࢉ͸ग़དྷͳ͍
10. xΛఆٛ͠௚͢ y1 y2 y3 ⋮ yn = 1 x11 x12 ⋯ x1k 1 x21 x22 ⋯ x2k 1 x31 x32 ⋯ x3k ⋮ ⋮ ⋯ ⋮ 1 xn1 xn2 ⋯ xnk β1 β2 β3 ⋮ βk + ϵ1 ϵ2 ϵ3 ⋮ ϵn y = Xβ + ϵ x = 1 x11 x12 ⋯ x1k 1 x21 x22 ⋯ x2k 1 x31 x32 ⋯ x3k ⋮ ⋮ ⋯ ⋮ 1 xn1 xn2 ⋯ xnk
11. ୯ճؼ෼ੳ y1 y2 y3 ⋮ yn = 1 x1 1 x2 1 x3 1 ⋮ 1 xn (a b) + ϵ1 ϵ2 ϵ3 ⋮ ϵn y = Xβ + ϵ
12. ·ͱΊ ߦྻΛ࢖͏͜ͱͰγϯϓϧʹද͢͜ͱ͕Ͱ͖Δ