Adaptive Bayesian Covariate Dependent Spectral Analysis of Multiple Time Series



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The frequency-domain properties of time series have been found to contain valuable information in many studies. It is often the case that biomedical time series are collected from multiple participants in conjunction with multiple covariates in order to analyze the association between the characteristics of biological processes and various clinical and behavioral outcomes. In this work, we propose flexible and adaptive nonparametric Bayesian methods to estimate the association between multiple covariates and the power spectrum of multiple time series. For stationary time series data, we introduce a Bayesian sum of trees model to capture complex dependencies and interactions between covariates and the power spectrum. Local power spectra corresponding to terminal nodes within trees are estimated nonparametrically using Bayesian penalized linear splines. The trees are considered to be random and fit using a Bayesian backfitting Markov chain Monte Carlo (MCMC) algorithm that sequentially considers tree modifications via reversible-jump MCMC techniques. For high-dimensional covariates, a sparsity-inducing Dirichlet hyperprior on tree splitting proportions is considered, which provides a sparse estimation of covariate effects and efficient variable selection. For nonstationary time series, Voronoi tessellation is used as the partition model for the partition of both time and covariates spaces. The tessellation is adaptively updated via the reversible-jump MCMC technique. The Bayesian penalized linear splines model is used to estimate the local power spectra within each disjoint region of the tessellation. Empirical performance is evaluated via simulations to demonstrate the proposed methods' ability to accurately recover complex relationships and interactions. The Bayesian sum of trees model is used to study gait maturation in young children by evaluating age-related changes in power spectra of stride interval time series in the presence of other covariates.



Bayesian backfitting, Gait variability, Multiple time series, Reversible jump Markov chain Monte Carlo, Spectrum analysis, Whittle likelihood