Ch 13 Sequential Data Pattern Recognition and Machine

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Ch 13. Sequential Data Pattern Recognition and Machine Learning, C. M. Bishop, 2006. Summarized

Ch 13. Sequential Data Pattern Recognition and Machine Learning, C. M. Bishop, 2006. Summarized by B. -W. Ku Biointelligence Laboratory, Seoul National University http: //bi. snu. ac. kr/ 1

Contents l 13. 3. Linear Dynamical Systems ¨ 13. 3. 1 Inference in LDS

Contents l 13. 3. Linear Dynamical Systems ¨ 13. 3. 1 Inference in LDS ¨ 13. 3. 2 Learning in LDS ¨ 13. 3. 3 Extensions of LDS ¨ 13. 3. 4 Particle filters (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 2

A Stochastic Linear Dynamical System Fig. 13. 15 (C) 2007, SNU Biointelligence Lab, http:

A Stochastic Linear Dynamical System Fig. 13. 15 (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 3

13. 3. 1 Inference in LDS (C) 2007, SNU Biointelligence Lab, http: //bi. snu.

13. 3. 1 Inference in LDS (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 4

Inference Problem l Finding the marginal distributions for the latent variables conditional on the

Inference Problem l Finding the marginal distributions for the latent variables conditional on the observation sequence. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 5

An Example Application: Tracking an Moving Object Fig. 13. 22 An illustration of a

An Example Application: Tracking an Moving Object Fig. 13. 22 An illustration of a linear dynamical system being used to track a moving object. Blue: Zn. Green: Xn. Red: The inferred. l One of the most important application of the Kalman filter. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 6

Mean and Variance of p(zn|x 1, …, xn) l Kalman filter equations (C) 2007,

Mean and Variance of p(zn|x 1, …, xn) l Kalman filter equations (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 7

Interpretation of the Steps Involved Fig. 13. 21 l Kalman filter as a process

Interpretation of the Steps Involved Fig. 13. 21 l Kalman filter as a process of ¨ Making successive predictions and then ¨ Correcting the predictions using the new observations. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 8

Derivation of Eq. 13. 89, 13. 90, 13. 94, 13. 95 (13. 85), (13.

Derivation of Eq. 13. 89, 13. 90, 13. 94, 13. 95 (13. 85), (13. 59) forward recursion (13. 84) (13. 87) (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 9

Mean and Variance of p(zn|x 1, …, xn+1, …, x. N) (13. 84) l

Mean and Variance of p(zn|x 1, …, xn+1, …, x. N) (13. 84) l Kalman smoother equations. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 10

Derivation of Eq. 13. 100, 13. 101 (13. 62) backward recursion the forward-backward algorithm

Derivation of Eq. 13. 100, 13. 101 (13. 62) backward recursion the forward-backward algorithm (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 11

13. 3. 2 Learning in LDS (C) 2007, SNU Biointelligence Lab, http: //bi. snu.

13. 3. 2 Learning in LDS (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 12

Learning Problem l Determining the parameters θ = {A, Γ, C, Σ, μ 0,

Learning Problem l Determining the parameters θ = {A, Γ, C, Σ, μ 0, V 0} using the EM algorithm. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 13

Expectation of Log Likelihood Function l The complete data ({X, Z}) log likelihood function

Expectation of Log Likelihood Function l The complete data ({X, Z}) log likelihood function (13. 108) l The expectation of the log likelihood function with respect to p(Z | X, θ old) (13. 109) (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 14

Maximizing the Expectation l Maximizing each term with respect to the parameters. (Section 2.

Maximizing the Expectation l Maximizing each term with respect to the parameters. (Section 2. 3. 4) (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 15

Evaluated θ new (13. 105) (13. 106) (C) 2007, SNU Biointelligence Lab, http: //bi.

Evaluated θ new (13. 105) (13. 106) (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ (13. 107) 16

13. 3. 3 Extensions of LDS l Problem: Beyond the linear-Gaussian assumption. ¨ Considerable

13. 3. 3 Extensions of LDS l Problem: Beyond the linear-Gaussian assumption. ¨ Considerable interest in extending the basic linear dynamical system in order to increase its capabilities. ¨ Gaussian p(zn |xn) – A significant limitation. l Some extensions ¨ Gaussian mixture p(zn). ¨ Gaussian mixture p(xn |zn) – Impractical. ¨ The extended Kalman filter. ¨ The switching state space model / the switching hidden Markov model. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 17

13. 3. 4 Particle filters l Non-Gaussian emission density p(xn |zn) non-Gaussian p(zn |

13. 3. 4 Particle filters l Non-Gaussian emission density p(xn |zn) non-Gaussian p(zn | x 1, …, xn) mathematically intractable integral l Sampling-importanceresampling Fig. 13. 21 A schematic illustration of the operation of particle filter. (C) 2007, SNU Biointelligence Lab, http: //bi. snu. ac. kr/ 18