Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines

In many practical problems, time series are realizations of nonstationary random processes. These processes can often be modeled as processes with slowly changing dynamics or as piecewise stationary processes. In these cases, various approaches to estimating the time-varying spectral density have been proposed. Our approach in this article is to estimate the log of the Dahlhaus local spectrum using a Bayesian mixture of splines. The basic idea of our approach is to first partition the data into small sections. We then assume that the log spectral density of the evolutionary process in any given partition is a mixture of individual log spectra. We use a mixture of smoothing splines model with time varying mixing weights to estimate the evolutionary log spectrum. The mixture model is fit using Markov chain Monte Carlo techniques that yield estimates of the log spectra of the individual subsections. In addition to an estimate of the local log spectral density, the method yields pointwise credible intervals. We use a reversible jump step to automatically determine the number of different spectral components.