Moment-generating function

Statistics

Physics

Statistical physics

Random variable

Mathematical optimization

Mathematical analysis

Probability distribution

Monte Carlo method

Probability density function

Polynomial chaos

Applied mathematics

Mathematics

Polynomial

Principle of maximum entropy

Computer Methods in Applied Mechanics and Engineering

Uncertainty propagation under multimodally distributed random variables, called multimodal distribution propagation for short, is a challenging problem due to the complicate probability density function of the random variables. In this paper, a general frame based on a new finite mixture model (λMM) constructed by derivative lambda probability density function and polynomial chaos expansion method is put forward to efficiently solve the multimodal distribution propagation problem. First, λMM with high accuracy and extensive applicability is proposed to represent the arbitrary unimodal distributions and multimodal distributions. Second, a pseudo EM method proved strictly is proposed to estimate the parameters of this λMM. Third, the statistical moments of the response by polynomial chaos expansion method is derived mathematically. Since the new mixture model can be decomposed into several unimodal probability distributions, the multimodal distribution propagation can be successfully converted into several unimodal distribution propagations, which further can be easily solved by polynomial chaos expansion method. Finally, the maximum entropy principle is adopted to evaluate the probability density function of the result of every unimodal distribution propagation, which is then superimposed into the final probability density function of the system response. The proposed frame only requires the first four statistical moments to evaluate the probability density function and avoid calculating the higher statistical moments, especially for the situation that the probability distribution of the system response is multimodal. Four examples are presented to verify the high accuracy and efficiency of the proposed general frame for uncertainty propagation.

A general frame for uncertainty propagation under multimodally distributed random variables