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Kalman Filtering And Smoothing

Kalman Filtering And Smoothing. Jayashri. Outline . Introduction State Space Model Parameterization Inference Filtering Smoothing. Introduction. Two Categories of Latent variable Models Discrete Latent variable -> Mixture Models Continuous Latent Variable-> Factor Analysis Models

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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

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