Time Series in the Frequency Domain
Tucker McElroy, Dimitris N. Politis · 2019
This chapter considers the representation of time series structure in terms of frequencies. It defines the spectral density function and examines its relation to the spectral decomposition of a (large) autocovariance matrix. The discussion leads to inverse autocovariances, which are well defined when the spectral density is invertible, and to the notion of partial autocorrelation. The chapter introduces the spectral density on which the Fourier representation of time series is based. It also introduces the notion of inverse autocovariances, and explores the operation of whitening a time series. In many applications, such as computing the Gaussian likelihood or forecasting, it is necessary to compute the spectral decomposition of a Toeplitz matrix of autocovariances. The spectral density is the Fourier series associated with the autocovariance function, and summarizes information about periodic components within a time series.