Time Series: Cycles

This article discusses cyclical (or periodic) properties of stochastic processes. The spectrum (or spectral density function) is introduced as a method for characterizing the approximate periodic behavior of realizations from stochastic processes. The spectral representation is introduced as a way to decompose a stochastic process into its periodic components. Moving average filters change the cyclical properties of time series, and formulae characterizing these changes are developed. Specific moving average filters are discussed, and general formulae for extracting components associated with specific frequencies (bandpass filters) are presented. Spectral properties of ARMA process are derived. Methods for estimating spectra are outlined.