The Gaussian multiplicative approximation for state-space models

Applications such as structural health monitoring (SHM) often rely on the analysis of time-series using methods such as state-space models (SSM). In this paper, we propose an analytical method called the Gaussian multiplicative approximation (GMA) that is applicable to multiplicative state-space models that are often encountered in practical SHM applications. The method enables the analytical inference of the mean vector and the covariance matrix for the product of two hidden states in the transition and/or observation models using linear estimation theory and the online estimation of model parameters as hidden states. The potential of combining the GMA and Bayesian dynamic linear models (BDLM) is illustrated through the development of (1) a generic component called online autoregressive that can estimate both the state variable and the parameter together; (2) a generic component called trend multiplicative for multiplicative seasonality model to identify non-harmonic periodic pattern whose amplitude changes linearly with time; and (3) a generic component called double kernel regression to identify non-harmonic periodic pattern that involves the product of two periodic kernel regression components. The SHM-based case studies presented confirm that the GMA exceeds the performance of the existing nonlinear Kalman filter methods in terms of accuracy along with the computational cost.