A Bayesian blind source separation by embedding Gaussian process prior and applications on structural health monitoring

Abstract

This study aims to explore the Blind Source Separation (BSS) of heterogeneous monitoring data in the context of the Bayesian inference and its application on structural health monitoring (SHM) problems. In comparison with conventional Independent Component Analysis (ICA) and Second-Order Statistics (SOS) techniques, the Bayesian blind source separation approach can explicitly account for and quantify the measurement error and uncertainty, and do not suffer from the underdetermined problem. More importantly, it directly gives rise to the (posterior) probability density distributions of the source signals within a flexible framework, which greatly facilitate the reliability assessment in relation to SHM. To accommodate the temporal non-i.i.d. source signals that are prevalent in real-world problem, the Gaussian Process (GP) is introduced to define the prior distribution over the unknown source. With the likelihood function in conjunction with priors for source, mixing matrix and noise, the joint posterior probability of the three categories of unknown parameters is derived by the Bayes' theorem in this GP BSS framework. To enhance the universal performance of the proposed GP BSS framework when dealing with complicated SHM problems, an investigation of the GP's kernel combinability is further conducted systematically in this study. A noisy Squared-Exponential (SE) kernel, a multiplied combination of kernels, and a flexibly additive combination of kernels are developed and verified with the intention to improve accuracy in the BSS problem. The Gibbs-within-Metropolis sampler, which can be viewed as an enhanced Markov Chain Monte Carlo (MCMC) algorithm, is pursued to numerically obtain the probabilistic characteristics of the sources, mixing matrix, and noise, respectively. As a fast alternative to the MCMC, the sampler in terms of the variational Bayesian (VB) learning is also explored. The performance of the two methods is first evaluated by numerical simulation. Afterwards, the methods are applied to separate the experimental structural responses of a scaled frame structure under uncorrelated external excitations. The waveforms of the separated sources are found to be highly identical to the real excitations, and the corresponding column elements of the mixing matrix indicate the locations of the relevant external excitations. Results from both the numerical and experimental studies show the accuracy of the proposed method, in comparison with the ICA-based and SOS-based methods. In recognizing that many SHM-oriented signals often perform the difficult non-stationarity in that the source signal intermittently appears or disappears, the proposed GP BSS method is further extended for broader applications for the nonstationary source separation. A novel time-varying kernel function is developed in the GP framework to describe such discontinuous sources statistically. A sampler based on the sequential Metropolis-Hasting (sequential M-H) and the Gibbs algorithm is encoded to approximate the unknown parameters. Simulation tests prove the fidelity of this nonstationary GP BSS method even when the number of sources is unknown or misestimated. Two real SHM-oriented applications of the proposed GP BSS method are presented. The first one is the multi-source analysis of the strain data collected from the Tsing Ma Bridge (TMB). The effects of the daily circle of temperature, the railway and highway traffic on the TMB are extracted and decomposed separately based on the proposed method. It can be found that after removal of the separated components, better assessments could be achieved based on a damage detection procedure developed by the Bayesian Dynamic Linear Model (BDLM). The second SHM-based application of the GP BSS method is the identification of the wheel local defect of the high-speed train (HST). Such localized defects can generate influential effect on the rail track only when the defective wheel rolls over the monitored rail. This kind of intermittent pattern can be captured by the proposed nonstationary GP BSS method. It is proven that the GP BSS can efficiently identify the incipient defects of HST wheels.