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Kalman Filtering And Smoothing

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1. Kalman Filtering And Smoothing Jayashri

2. Outline • Introduction • State Space Model • Parameterization • Inference • Filtering • Smoothing

3. Introduction • Two Categories of Latent variable Models • Discrete Latent variable -> Mixture Models • Continuous Latent Variable-> Factor Analysis Models • Mixture Models -> Hidden Markov Model • Factor Analysis -> Kalman Filter

4. Application Applications of Kalman filter are endless! • Control theory • Tracking • Computer vision • Navigation and guidance system

5. A A … C C C x y y y y x x x 2 T 0 0 1 2 T 1 State Space Model C • Independence Relationships: • Given the state at one moment in the time, the states in the future are conditionally independent of those in the past. • The observation of the output nodes fails to separate any of the state nodes.

6. Parameterization Transition From one node to another:   

7. Unconditional mean of is zero. Unconditional Distribution • Unconditional covariance is:

8. Inference • Calculation of the posterior probability of the states given an output sequence • Two Classes of Problems: • Filtering • Smoothing

9. Filtering Problem is to calculate the mean vector and Covariance matrix. Notations:

10. Filtering Cont’d Time update: Measurement update: Time Update step:

11. Measurement Update step:

12. Equations Mean Covariance Using the equations 13.26 and 13.27

13. Equations Summary of the update equations

14. Kalman Gain Matrix Update Equation:

15. Interpretation and Relation to LMS The update equation can be written as, • Matrix A is identity matrix and noise term w is zero • Matrix C be replaced by the Update equation becomes,

16. Information Filter (Inverse Covariance Filter) Conversion of moment parameters to canonical parameters: … Eqn. 13.5 Canonical parameters of the distribution of

17. Smoothing • Estimation of state x at time t given the data up to time t and later time T • Rauch-Tung-Striebel (RTS) smoother (alpha-gamma algorithm) • Two-filter smoother (alpha-beta algorithm)

18. RTS Smoother • Recurses directly on the filtered-and-smoothed estimates i.e. • Alpha-gamma algorithm    

19. (RTS) Forward pass: Mean Covariance

20. conditioned on Estimate the probability of Backward filtering pass:  

21. Identities:

22. Equations Summary of update equations:

23. Alpha-beta algorithm Two-Filter smoother  Forward Pass:  Backward Pass: Naive approach to invert the dynamics which does not work is:

24.       We can invert the arrow between Cont’d Covariance Matrix is: Which is backward Lyapunov equation.

25. Covariance matrix can be written as:

26. We can define Inverse dynamics as:

27. Summary: Forward dynamics: Backward dynamics: Last issue is to fuse the two filter estimates.

28. Fusion Of Guassian Posterior Probability

29. Fusion Cont’d