Functional mixed effects wavelet estimation for spectra of replicated time series

Abstract

Motivated by spectral analysis of replicated brain signal time series, we propose a functional mixed effects approach to model replicate-specific spectral densities as random curves varying about a deterministic population-mean spectrum. In contrast to existing work, we do not assume the replicate-specific spectral curves to be independent, i.e. there may exist explicit correlation between different replicates in the population. By projecting the replicate-specific curves onto an orthonormal wavelet basis, estimation and prediction is carried out under an equivalent linear mixed effects model in the wavelet coefficient domain. To cope with potentially very localized features of the spectral curves, we develop estimators and predictors based on a combination of generalized least squares estimation and nonlinear wavelet thresholding, including asymptotic confidence sets for the population-mean curve. We derive $L_{2}$-risk bounds for the nonlinear wavelet estimator of the population-mean curve–a result that reflects the influence of correlation between different curves in the replicate population– and consistency of the estimators of the inter- and intra-curve correlation structure in an appropriate sparseness class of functions. To illustrate the proposed functional mixed effects model and our estimation and prediction procedures, we present several simulated time series data examples and we analyze a motivating brain signal dataset recorded during an associative learning experiment.