ABSTRACT
The spectral analysis of time series refers generally to methods based upon decomposing the series into sinusoidal cycles of different frequencies. Classical Fourier analysis is most appropriately used for representing deterministic, or fixed, periodic functions as the sum of cycles, and we will also find this useful, for example, in the context of time series with regular seasonal patterns. However, the spectral analysis of time series, although motivated initially by the search for such hidden periodic patterns, derives directly from the quite distinct property of (second order) stationarity. From this it follows that a time series can be represented as the sum of random cyclical, or harmonic, components that are uncorrelated one with another. In the foregoing we have emphasized the words fixed and random because they provide a meaningful and relevant distinction between classical Fourier analysis and time series spectral analysis for those familiar with linear mixed effects models which may contain both fixed and random effects. The terminology of harmonic and spectral analysis derives from the notion that musical sound and colored light can each be broken down into components consisting of, respectively, pure tones and primary colors of the spectrum, which are associated with different frequencies of oscillation.
