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A state-space calculus for rational probability density functions and applications to non-Gaussian filtering

A state-space calculus for rational probability density functions and applications to non-Gaussian filtering
A state-space calculus for rational probability density functions and applications to non-Gaussian filtering

We propose what we believe to be a novel approach to performing calculations for rational density functions using state-space representations of the densities. By standard results from realization theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-real, which we call the density summand. The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Markov parameters. Some of the main contributions of the paper are explicit state-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular, state-space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.

Linear algebra, Non-Gaussian filtering, Probability theory, Rational functions, Realization theory for linear systems
0363-0129
724-740
Hanzon, Bernard
ec8a3e31-d488-4a69-8318-6ff08024dd7c
Ober, Raimund J.
31f4d47f-fb49-44f5-8ff6-87fc4aff3d36
Hanzon, Bernard
ec8a3e31-d488-4a69-8318-6ff08024dd7c
Ober, Raimund J.
31f4d47f-fb49-44f5-8ff6-87fc4aff3d36

Hanzon, Bernard and Ober, Raimund J. (2002) A state-space calculus for rational probability density functions and applications to non-Gaussian filtering. SIAM Journal on Control and Optimization, 40 (3), 724-740. (doi:10.1137/S036301299731610X).

Record type: Article

Abstract

We propose what we believe to be a novel approach to performing calculations for rational density functions using state-space representations of the densities. By standard results from realization theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-real, which we call the density summand. The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Markov parameters. Some of the main contributions of the paper are explicit state-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular, state-space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.

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More information

Published date: 2002
Keywords: Linear algebra, Non-Gaussian filtering, Probability theory, Rational functions, Realization theory for linear systems

Identifiers

Local EPrints ID: 424923
URI: http://eprints.soton.ac.uk/id/eprint/424923
ISSN: 0363-0129
PURE UUID: bb90051e-21b5-41a3-bc94-9c3118151e39
ORCID for Raimund J. Ober: ORCID iD orcid.org/0000-0002-1290-7430

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Date deposited: 05 Oct 2018 16:30
Last modified: 16 Mar 2024 04:37

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Contributors

Author: Bernard Hanzon
Author: Raimund J. Ober ORCID iD

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