A stochastic density matrix approach to approximation of probability
distributions and its application to nonlinear systems
I. Vladimirov. (2015)cite arxiv:1506.04775Comment: 12 pages, 3 figures. Submitted to the IEEE Multi-Conference on Systems and Control, 21-23 September 2015, Sydney, Australia.
Abstract
This paper outlines an approach to the approximation of probability density
functions by quadratic forms of weighted orthonormal basis functions with
positive semi-definite Hermitian matrices of unit trace. Such matrices are
called stochastic density matrices in order to reflect an analogy with the
quantum mechanical density matrices. The SDM approximation of a PDF satisfies
the normalization condition and is nonnegative everywhere in contrast to the
truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic
structure, such as the Hermite polynomial and Fourier bases, the SDM
approximation can be chosen so as to satisfy given moment specifications and
can be optimized using a quadratic proximity criterion. We apply the SDM
approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion
processes governed by nonlinear stochastic differential equations. This leads
to an ordinary differential equation for the SDM dynamics of the approximating
PDF. As an example, we consider the Smoluchowski SDE on a multidimensional
torus.
Description
A stochastic density matrix approach to approximation of probability
distributions and its application to nonlinear systems
cite arxiv:1506.04775Comment: 12 pages, 3 figures. Submitted to the IEEE Multi-Conference on Systems and Control, 21-23 September 2015, Sydney, Australia
%0 Generic
%1 vladimirov2015stochastic
%A Vladimirov, Igor G.
%D 2015
%K distributions statistics
%T A stochastic density matrix approach to approximation of probability
distributions and its application to nonlinear systems
%U http://arxiv.org/abs/1506.04775
%X This paper outlines an approach to the approximation of probability density
functions by quadratic forms of weighted orthonormal basis functions with
positive semi-definite Hermitian matrices of unit trace. Such matrices are
called stochastic density matrices in order to reflect an analogy with the
quantum mechanical density matrices. The SDM approximation of a PDF satisfies
the normalization condition and is nonnegative everywhere in contrast to the
truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic
structure, such as the Hermite polynomial and Fourier bases, the SDM
approximation can be chosen so as to satisfy given moment specifications and
can be optimized using a quadratic proximity criterion. We apply the SDM
approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion
processes governed by nonlinear stochastic differential equations. This leads
to an ordinary differential equation for the SDM dynamics of the approximating
PDF. As an example, we consider the Smoluchowski SDE on a multidimensional
torus.
@misc{vladimirov2015stochastic,
abstract = {This paper outlines an approach to the approximation of probability density
functions by quadratic forms of weighted orthonormal basis functions with
positive semi-definite Hermitian matrices of unit trace. Such matrices are
called stochastic density matrices in order to reflect an analogy with the
quantum mechanical density matrices. The SDM approximation of a PDF satisfies
the normalization condition and is nonnegative everywhere in contrast to the
truncated Gram-Charlier and Edgeworth expansions. For bases with an algebraic
structure, such as the Hermite polynomial and Fourier bases, the SDM
approximation can be chosen so as to satisfy given moment specifications and
can be optimized using a quadratic proximity criterion. We apply the SDM
approach to the Fokker-Planck-Kolmogorov PDF dynamics of Markov diffusion
processes governed by nonlinear stochastic differential equations. This leads
to an ordinary differential equation for the SDM dynamics of the approximating
PDF. As an example, we consider the Smoluchowski SDE on a multidimensional
torus.},
added-at = {2015-06-17T20:08:17.000+0200},
author = {Vladimirov, Igor G.},
biburl = {https://www.bibsonomy.org/bibtex/2c4714f02cced34464d36aec2a02e65e9/shabbychef},
description = {A stochastic density matrix approach to approximation of probability
distributions and its application to nonlinear systems},
interhash = {540063b7c9caebe6ef795a83cf2c33cb},
intrahash = {c4714f02cced34464d36aec2a02e65e9},
keywords = {distributions statistics},
note = {cite arxiv:1506.04775Comment: 12 pages, 3 figures. Submitted to the IEEE Multi-Conference on Systems and Control, 21-23 September 2015, Sydney, Australia},
timestamp = {2015-06-17T20:08:17.000+0200},
title = {A stochastic density matrix approach to approximation of probability
distributions and its application to nonlinear systems},
url = {http://arxiv.org/abs/1506.04775},
year = 2015
}