Kalman Filtering And Smoothing

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

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### 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

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.