A Survey of Spectral Analysis

A wide variety of applications of spectral analysis have been reported in the literature since spectral estimation methods were introduced by M. S. Bartlett and J. W. Tukey about 15 years ago. In no sense, however, can it be said that spectral analysis is widely used or even understood by statisticians and many of the applications of the technique have in fact been made by physicists and engineers. It is suggested that there are two reasons for this: (1) The genuine difficulties which statisticians (as opposed to physicists and engineers) face in thinking in terms of frequency concepts. (2) The highly mathematical nature of papers written on spectral analysis. This undue emphasis on mathematical work has led many statisticians to believe that spectral analysis is very difficult to apply. This is not the case-in fact the important ideas in spectral analysis are no more difficult than those involved in estimating a probability density function by means of a histogram. In this paper we shall try to present, using the minimum of mathematics, all those ideas in spectral analysis which are necessary in order to be able to apply the technique. In the last resort the only way to understand spectral analysis is to use it and so where possible the main ideas have been illustrated by means of examples. Two forms of spectral analysis are discussed in detail, namely, (1) spectral analysis of a single time-series to be referred to as auto-spectra; (2) spectral analysis of pairs of time-series to be referred to as crossspectra. However other forms of spectral analysis are mentioned briefly in section 7. Cross-spectral analysis is useful in two contexts: