Model Selection for Strongly Nonlinear Systems

Abstract

The selection of the optimal model, among a class of feasible models, which correctly identifies the various sources of nonlinearities through the bias and variance trade-off, is of paramount importance for a robust inference. The model selection method critically relies on the accuracy and efficiency of the parameter estimation algorithm. To this end, a general Bayesian inference framework has been developed by the authors in order to estimate the parameters of nonlinear dynamical systems using a Markov Chain Monte Carlo (MCMC) sampling technique. The MCMC algorithm involves a state estimation problem performed using the Extended Kalman Filter (EKF). Due to assimilation of dense observational data, the conditional probability density functions of the state become weakly non-Gaussian despite strong nonlinearity, leading to effective performance of EKF. However, the performance of this filter degrades when the conditional probability density functions of the state become strongly non-Gaussian due to nonlinearities in the model and measurements, in addition to the sparsity of observational data. These nonlinear effects may introduce strongly non-Gaussian features in the system leading to difficulties in the parameter estimation and model selection. In this paper, a robust parameter estimation and model selection algorithm is reported which can handle strong non-Gaussian effects in the response. In particular, we extend our previous work on the Bayesian parameter estimation to tackle the strongly non-Gaussian state estimation problem with an efficient particle filter that exploits the estimates of the Ensemble Kalman Filter (EnKF) as the proposal distribution. Consequently, the efficiency and accuracy of the Bayesian model selection method is investigated for a strongly non-Gaussian system using the improved Bayesian parameter estimation algorithm.