🤖 AI Summary
This work addresses the computational expense and poor sample efficiency associated with modeling high-dimensional, non-Gaussian probability density functions in nonlinear dynamical systems. To overcome these challenges, the authors propose an efficient estimation approach based on a semi-nonparametric (SNP) density model. The method constructs a strictly positive density representation using Hermite polynomial basis functions and integrates Monte Carlo integration with a convex relaxation optimization strategy to significantly enhance the accuracy and stability of density and quantile estimation under limited sample sizes. Experimental results on the Lorenz chaotic system demonstrate that the proposed method accurately captures complex non-Gaussian structures and reliably computes quantiles using substantially fewer samples than conventional Monte Carlo techniques.
📝 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.