A Cepstral Model for Efficient Spectral Analysis of Covariate-dependent Time Series

This article introduces a novel and computationally fast model to study the association between covariates and power spectra of replicated time series. A random covariate-dependent Cram\'{e}r spectral representation and a semiparametric log-spectral model are used to quantify the association between the log-spectra and covariates. Each replicate-specific log-spectrum is represented by the cepstrum, inducing a cepstral-based multivariate linear model with the cepstral coefficients as the responses. By using only a small number of cepstral coefficients, the model parsimoniously captures frequency patterns of time series and saves a significant amount of computational time compared to existing methods. A two-stage estimation procedure is proposed. In the first stage, a Whittle likelihood-based approach is used to estimate the truncated replicate-specific cepstral coefficients. In the second stage, parameters of the cepstral-based multivariate linear model, and consequently the effect functions of covariates, are estimated. The model is flexible in the sense that it can accommodate various estimation methods for the multivariate linear model, depending on the application, domain knowledge, or characteristics of the covariates. Numerical studies confirm that the proposed method outperforms some existing methods despite its simplicity and shorter computational time. Supplementary materials for this article are available online.