Smoothing Spline ANOVA for Time-Dependent Spectral Analysis

In this article we propose a smoothing spline ANOVA model (SS-ANOVA) to estimate and to make inference on the time-varying log-spectrum of a locally stationary process. The time-varying spectrum is assumed to be smooth in both time and frequency. This assumption essentially turns a time-frequency spectral estimation problem into a 2-dimensional surface estimation problem. A smooth localized complex exponential (SLEX) basis is used to calculate the initial periodograms, and a SS-ANOVA is fitted to the log-periodograms. This approach allows the time and frequency domains to be modeled in a unified approach and jointly estimated. Inference procedures, such as confidence intervals, and hypothesis tests proposed for the SS-ANOVA can be adopted for the time-varying spectrum. Because of the smoothness assumption of the underlying spectrum, once we have the estimates on a time-frequency grid, we can calculate the estimate at any given time and frequency. This leads to a high computational efficiency, because for large datasets we need only estimate the initial raw periodograms at a much coarser grid. We study a penalized least squares estimator and a penalized Whittle likelihood estimator. The penalized Whittle likelihood estimator has smaller mean squared errors, whereas inference based on the penalized least squares method can adopt existing results. We present simulation results and apply our method to electroencephalogram data recorded during an epileptic seizure.