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. State Space Model … x y A C C Independence Relationships:x 1 2 T A y C 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 Distribution
Unconditional mean of
is zero.
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 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. Alpha-gamma algorithm
RTS Smoother
Recurses directly on the filtered-and-smoothed estimates i.e.
Alpha-gamma algorithm    

19. (RTS) Forward pass:
Mean
Covariance

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

21. Identities:

22. Equations
Summary of update equations:

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

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

25. Covariance matrix can be written as:

26. We can define Inverse dynamics as:

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

28. Fusion Of Guassian Posterior Probability

29. Fusion Cont’d