Smoothing Spline ANOVA for Time-Dependent Spectral Analysis

In this paper, a locally stationary process is proposed using a Smooth Localized Complex Ex- ponential (SLEX) basis, whose spectrum is assumed to be smooth in both time and frequency. A smoothing Spline ANOVA (SS-ANOVA) is used to estimate and make inference on the time-varying log-spectrum. This approach allows the time and frequency domains to be modeled in an unified approach and jointly estimated. Because the SLEX basis is orthogonal and localized in both time and frequency, our method has good finite sample performance. It also allows for deriving desirable asymptotic properties. Inference procedures such as confidence intervals and hypothesis tests pro- posed 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 as for large data sets we only need to estimate the initial raw periodograms at a much coarser grid. We present simulation results and apply our method to an EEG data recorded during an epileptic seizure