  This article introduces a flexible and adaptive nonparametric method for
estimating the association between multiple covariates and power spectra of
multiple time series. The proposed approach uses a Bayesian sum of trees model
to capture complex dependencies and interactions between covariates and the
power spectrum, which are often observed in studies of biomedical time series.
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 sparse estimation of covariate effects and efficient
variable selection. By averaging over the posterior distribution of trees, the
proposed method can recover both smooth and abrupt changes in the power
spectrum across multiple covariates. Empirical performance is evaluated via
simulations to demonstrate the proposed method's ability to accurately recover
complex relationships and interactions. The proposed methodology 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.
