TY - GEN

T1 - A stochastic density matrix approach to approximation of probability distributions and its application to nonlinear systems

AU - Vladimirov, Igor G.

N1 - Publisher Copyright:
© 2015 IEEE.

PY - 2015/11/4

Y1 - 2015/11/4

N2 - This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonor-mal 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.

AB - This paper outlines an approach to the approximation of probability density functions by quadratic forms of weighted orthonor-mal 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.

UR - http://www.scopus.com/inward/record.url?scp=84964319967&partnerID=8YFLogxK

U2 - 10.1109/CCA.2015.7320758

DO - 10.1109/CCA.2015.7320758

M3 - Conference contribution

T3 - 2015 IEEE Conference on Control and Applications, CCA 2015 - Proceedings

SP - 1090

EP - 1095

BT - 2015 IEEE Conference on Control and Applications, CCA 2015 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - IEEE Conference on Control and Applications, CCA 2015

Y2 - 21 September 2015 through 23 September 2015

ER -