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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

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Presentation Transcript
outline
Outline
  • Introduction
  • State Space Model
  • Parameterization
  • Inference
    • Filtering
    • Smoothing
introduction
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
Application

Applications of Kalman filter are endless!

  • Control theory
  • Tracking
  • Computer vision
  • Navigation and guidance system
state space model

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
Parameterization

Transition From one node to another:

unconditional distribution

Unconditional mean of

is zero.

Unconditional Distribution
  • Unconditional covariance is:
inference
Inference
  • Calculation of the posterior probability of the states given an output sequence
  • Two Classes of Problems:
  • Filtering
  • Smoothing
filtering
Filtering

Problem is to calculate the mean vector and Covariance matrix.

Notations:

filtering cont d
Filtering Cont’d

Time update:

Measurement update:

Time Update step:

equations
Equations

Mean

Covariance

Using the equations 13.26 and 13.27

equations13
Equations

Summary of the update equations

slide14

Kalman Gain Matrix

Update Equation:

interpretation and relation to lms
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
Information Filter (Inverse Covariance Filter)

Conversion of moment parameters to canonical parameters:

… Eqn. 13.5

Canonical parameters of the distribution of

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

rts forward pass
(RTS) Forward pass:

Mean

Covariance

backward filtering pass

conditioned on

Estimate the probability of

Backward filtering pass:

equations22
Equations

Summary of update equations:

two filter smoother

Alpha-beta algorithm

Two-Filter smoother

 Forward Pass:

 Backward Pass:

Naive approach to invert the dynamics which does not work is:

cont d

  

  

We can invert the arrow between

Cont’d

Covariance Matrix is:

Which is backward Lyapunov equation.

summary
Summary:

Forward dynamics:

Backward dynamics:

Last issue is to fuse the two filter estimates.