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 • Mixture Models -> Hidden Markov Model • Factor Analysis -> Kalman Filter
Application Applications of Kalman filter are endless! • Control theory • Tracking • Computer vision • Navigation and guidance system
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.
Parameterization Transition From one node to another:
Unconditional mean of is zero. Unconditional Distribution • Unconditional covariance is:
Inference • Calculation of the posterior probability of the states given an output sequence • Two Classes of Problems: • Filtering • Smoothing
Filtering Problem is to calculate the mean vector and Covariance matrix. Notations:
Filtering Cont’d Time update: Measurement update: Time Update step:
Equations Mean Covariance Using the equations 13.26 and 13.27
Equations Summary of the update equations
Kalman Gain Matrix Update Equation:
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,
Information Filter (Inverse Covariance Filter) Conversion of moment parameters to canonical parameters: … Eqn. 13.5 Canonical parameters of the distribution of
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)
RTS Smoother • Recurses directly on the filtered-and-smoothed estimates i.e. • Alpha-gamma algorithm
(RTS) Forward pass: Mean Covariance
conditioned on Estimate the probability of Backward filtering pass:
Equations Summary of update equations:
Alpha-beta algorithm Two-Filter smoother Forward Pass: Backward Pass: Naive approach to invert the dynamics which does not work is:
We can invert the arrow between Cont’d Covariance Matrix is: Which is backward Lyapunov equation.
Summary: Forward dynamics: Backward dynamics: Last issue is to fuse the two filter estimates.