Data-Efficient Non-Gaussian Semi-Nonparametric Density Estimation for Nonlinear Dynamical Systems

TL;DR

This paper introduces a semi-nonparametric density estimation method using Hermite polynomial bases for nonlinear dynamical systems, enabling accurate non-Gaussian density representation with fewer samples.

Contribution

The paper develops a novel SNP density estimation approach with Monte Carlo approximation and convex relaxation, improving efficiency in modeling non-Gaussian distributions in nonlinear systems.

Findings

Accurately captures non-Gaussian density structures in the Lorenz system.

Computes quantiles effectively with significantly fewer samples than traditional Monte Carlo.

Demonstrates improved efficiency and accuracy in density estimation for chaotic systems.

Abstract

Accurate representation of non-Gaussian distributions of quantities of interest in nonlinear dynamical systems is critical for estimation, control, and decision-making, but can be challenging when forward propagations are expensive to carry out. This paper presents an approach for estimating probability density functions of states evolving under nonlinear dynamics using Seminonparametric (SNP), or Gallant-Nychka, densities. SNP densities employ a probabilists' Hermite polynomial basis to model non-Gaussian behavior and are positive everywhere on the support by construction. We use Monte Carlo to approximate the expectation integrals that arise in the maximum likelihood estimation of SNP coefficients, and introduce a convex relaxation to generate effective initial estimates. The method is demonstrated on density and quantile estimation for the chaotic Lorenz system. The results demonstrate that the proposed method can accurately capture non-Gaussian density structure and compute quantiles using significantly fewer samples than raw Monte Carlo sampling.