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.

翻译：在非线性动力系统中，对于感兴趣量的非高斯分布进行精确表示对于估计、控制与决策至关重要，但当正向传播的代价高昂时，这一过程可能充满挑战。本文提出一种利用半非参数（SNP）密度（即Gallant-Nychka密度）来估计非线性动态演化状态概率密度函数的方法。SNP密度采用概率论中的Hermite多项式基函数来建模非高斯行为，并通过构造确保其在支撑集上处处为正。我们使用蒙特卡罗方法近似SNP系数极大似然估计中出现的期望积分，并引入一种凸松弛方法来生成有效的初始估计。该方法在混沌Lorenz系统的密度与分位数估计上进行了验证。结果表明，相较于原始蒙特卡罗采样，所提方法能够用显著更少的样本精确捕捉非高斯密度结构并计算分位数。