Prml last yokkuns

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Prml last yokkuns

  1. 1. PRML 13.3 13.3.3 . @yokkuns . PRML 2010/09/11 @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 1 / 36
  2. 2. PRML 13.3 13.3.3 @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 2 / 36
  3. 3. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 3 / 36
  4. 4. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 4 / 36
  5. 5. ID : yokkuns : : Web http://twitter.com/yokkuns @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 5 / 36
  6. 6. - 1 R Tokyo.R http://groups.google.co.jp/group/r-study-tokyo @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 6 / 36
  7. 7. - 2 PRML http://groups.google.co.jp/group/grinning-math @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 7 / 36
  8. 8. - PRML 2010.09.11 RPML 13.3 13.3.3 9 R R II 2010.09.19 Tokyo.R#09 4 7 2010.09.25 +WEB Tokyo.Webmining#7 @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 8 / 36
  9. 9. - PRML 2010.09.11 RPML 13.3 13.3.3 9 R R II 2010.09.19 Tokyo.R#09 4 7 2010.09.25 +WEB Tokyo.Webmining#7 @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 9 / 36
  10. 10. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 10 / 36
  11. 11. @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 11 / 36
  12. 12. PCA @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 12 / 36
  13. 13. p(z n|z n−1 ) = N(z n| Az n−1 , γ) p(x n|z n) = N(x n|Cz n, σ) p(z1 ) = N(z1 |µ0 , P0 ) @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 13 / 36
  14. 14. - zn = Az n−1 + w n x n = Cz n + v n z 1 = µ0 + u w ∼ N(x|0, Γ) v ∼ N(v|0, Σ) u ∼ N(u|0, V0 ) @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 14 / 36
  15. 15. x1:n zn zn @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 15 / 36
  16. 16. zn α β @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 16 / 36
  17. 17. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 17 / 36
  18. 18. @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 18 / 36
  19. 19. p(z n|x1 , ..., x n) α(z n) = N(z n|µ n, V n) ˆ (13.59) ∫ c nα(z n) = ˆ p(x n|z n) α(z n−1 ) p(z n|z n−1 )dz n−1 ˆ @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 19 / 36
  20. 20. (13.75) (13.76) ∫ c nN(z n|µ n, V n) = N(x n|Cz n, Σ) N(z n| Az n−1 , Γ)N(z n−1 |µ n−1 , V n−1 )dz n−1 (2.115) ∫ N(z n| Az n−1 , Γ)N(z n−1 |µ n−1 , V n−1 )dz n−1 = N(z n| Aµ n−1 , AV n−1 AT + Γ) @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 20 / 36
  21. 21. (2.115) (2.116) P n−1 = AV n−1 AT + Γ µn = Aµ n−1 + K n(x n − C Aµ n−1 ) V n = (I − K nC)P n−1 K n = P n−1 CT (CP n−1 CT + Σ)−1 @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 21 / 36
  22. 22. @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 22 / 36
  23. 23. LDS β(z n) ˆ γ(z n) = α(z n)β(z n) ˆ ˆ γ(z n) γ(z n) = N(z n|µ n, V n) ˆ ˆ ∫ c n+1 β(z n) = ˆ β(z n+1 )p(x n+1 |z n+1 )p(z n+1 |z n)dz n+1 ˆ @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 23 / 36
  24. 24. µ n, V n ˆ ˆ J n = V n AT (P n)−1 µ n = µ n + J n(µ n+1 − Aµ n) ˆ ˆ Vˆn = V n + J n(V n+1 − P n)J T ˆ n µn Vn @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 24 / 36
  25. 25. EM 2 (13.65) ζ(z n−1 , z n) = (c n)−1 α(z n−1 )p(x n|z n)p(z n|z n−1 )β(z n) ˆ ˆ N(z n−1 |µ n−1 , V n−1 )N(z n|Az n−1 , Γ)N(x n|Cz n, Σ)N(z n|µ n, V n) ˆ ˆ = c nα(z n) ˆ (13.84) α(z n) ˆ ζ(z n−1 , z n) [µ n−1 , µ n]T ˆ ˆ zn z n−1 cov[z n−1 , z n] = ˆ J n−1 V n @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 25 / 36
  26. 26. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 26 / 36
  27. 27. LDS - θ = { A, Γ, C, Σ, µ0 , V0 } EM θ old p(Z|X, θ old ) (13.104) E[z n] = µn ˆ E[z n zT ] n−1 = J n−1 V n + µ nµT ˆ ˆ ˆ n−1 E[z n zT ] n = V n + µ nµT ˆ ˆ ˆn @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 27 / 36
  28. 28. LDS - (13.6) ∑ N ln p(X, Z|θ) = ln p(z1 |µ0 , V0 ) + ln p(z n|z n−1 , A, Γ) n=2 ∑ N + ln p(x n|z n, C, Σ) n=1 p(Z|X, θ old ) Q(θ, θ old ) = E Z|θ old [ln p(X, Z|θ)] θ = {A, Γ, C, Σ, µ0 , V0 } @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 28 / 36
  29. 29. LDS - µ0 new = E[z1 ] new V0 = E[z1 zT ] − E[z1 ]E[z1T ] 1  N  N −1 ∑    ∑      A new =    E[z n zT ]    E[z n−1 zT ]   n−1   n  n=2 n=2 1 ∑{ N Γ new = E[z n zT ] − A new E[z n−1 zT ] n n N−1 n=2 } −E[z n zT ]( A new )T + A new E[z n−1 zT ]( A new )T n−1 n−1  N  N −1 ∑    ∑      C new =    x n E[zT ]   E[z n zT ]   n   n  n=1 n=1 1 ∑{ N Σ new = x n xT − (C new )T E[z n]xT n n N n=1 } −x n E[zT ]C new + (C new )T E[z n zT ]C new n n @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 29 / 36
  30. 30. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 30 / 36
  31. 31. @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 31 / 36
  32. 32. 1 1 @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 32 / 36
  33. 33. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 33 / 36
  34. 34. 9/19( ) 9 R @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 34 / 36
  35. 35. AGENDA Linear Dynamical System LDS LDS LDS @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 35 / 36
  36. 36. @yokkuns (PRML ) PRML 13.3 13.3.3 2010/09/11 36 / 36

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