1. Biointelligence Laboratory, Seoul National University
Ch 13. Sequential Data Pattern Recognition and Machine Learning, C. M. Bishop, 2006.
Summarized by
B.-W. Ku
Biointelligence Laboratory, Seoul National University

2. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
13.3. Linear Dynamical Systems
Inference in LDS
Learning in LDS
Extensions of LDS
Particle filters
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3. A Stochastic Linear Dynamical System
Fig
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4. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Inference in LDS
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5. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Inference Problem
Finding the marginal distributions for the latent variables conditional on the observation sequence.
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6. An Example Application: Tracking an Moving Object
Fig An illustration of a linear dynamical system being used to track a moving object.
Blue:Zn.
Green: Xn.
Red: The inferred.
One of the most important application of the Kalman filter.
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7. Mean and Variance of p(zn|x1, …, xn)
Kalman filter equations
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8. Interpretation of the Steps Involved
Fig
Kalman filter as a process of
Making successive predictions and then
Correcting the predictions using the new observations.
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9. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Derivation of Eq.13.89, 13.90, 13.94, 13.95
(13.85), (13.59)
forward recursion
(13.84)
(13.87)
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10. Mean and Variance of p(zn|x1, …, xn, xn+1, …, xN)
(13.84)
Kalman smoother equations.
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11. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Derivation of Eq ,
(13.62)
backward recursion
the forward-backward algorithm
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12. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Learning in LDS
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13. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Learning Problem
Determining the parameters θ = {A, Γ, C, Σ, μ0, V0} using the EM algorithm.
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14. Expectation of Log Likelihood Function
The complete data ({X, Z}) log likelihood function
(13.108)
The expectation of the log likelihood function with respect to p(Z | X, θ old)
(13.109)
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15. Maximizing the Expectation
Maximizing each term with respect to the parameters. (Section 2.3.4)
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16. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Evaluated θ new
(13.105)
(13.106)
(13.107)
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17. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Extensions of LDS
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.
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.
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18. (C) 2007, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Particle filters
Non-Gaussian emission density p(xn |zn)
 non-Gaussian p(zn | x1, …, xn)
 mathematically intractable integral
Sampling-importance-resampling
Fig A schematic illustration of the operation of particle filter.
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