A summary of many of the new techniques developed in the last two decades for spectrum analysis of discrete time series is presented in this tutorial. An examination of the underlying time series model assumed by each technique serves as the common basis for understanding the differences among the various spectrum analysis approaches. Techniques discussed include the classical periodogram, classical Blackman-Tukey, autoregressive (maximum entropy), moving average, autotegressive-moving average, maximum likelihood, Prony, and Pisarenko methods. A summary table in the text provides a concise overview for all methods, including key references and appropriate equations for computation of each spectral estimate.

Spectrum analysis&#8212;A modern perspective

Group representations in probability and statistics

The subject of this paper is the higher-order spectra or polyspectra of multivariate stationary time series. The intent is to derive (i) certain mathematical properties of polyspectra, (ii) estimates of polyspectra based on an observed stretch of time series, (iii) certain statistical properties of the proposed estimates and (iv) several applications of the results obtained.

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An Introduction to Polyspectra

(1994). Engineering Applications of Correlation and Spectral Analysis. Technometrics: Vol. 36, No. 2, pp. 220-221.

Engineering Applications of Correlation and Spectral Analysis

The temporal characteristics of rainfall variability in Africa are examined using departure series for 84 regions of the continent and five larger-scale zones. The forms of non-randomness which are investigated include linear trends, persistence and quasi-periodic fluctuations. No long-term trends in African rainfall are evident. In some sectors, most noteably along the Benguela coast and equatorial Africa, rainfall anomalies tend to persist over several months and interseasonal correlations are also high. Spectral analysis revealed significant quasi-periodicities clustered in four bands at 2.2–2.4, 2.6–2.8, 3.3–3.8 and 5.0–6.3 years. These are common throughout equatorial and southern Africa, but only weakly evident in northern Africa. A cross-spectral analysis with the Southern Oscillation suggests a strong influence on rainfall variability in southern Africa and parts of the equatorial region and minimal influence in northern Africa. Coherence with the Southern Oscillation is particularly strong in the QBO range of 2.2–2.4 years. In general, the co-spectra suggest an inverse relationship between the SOI and rainfall in equatorial regions (i.e., higher rainfall during low-index years) and a positive relationship with rainfall in most other regions where an influence can be demonstrated.

The quasi-periodic behavior of rainfall variability in Africa and its relationship to the southern oscillation

The object of this paper is to present a simplified account of the motivation behind the spectral analysis of time-series and to give a heuristic discussion of the statistical problems which arise. It is directed mainly at statisticians with little experience of the theory and applications of spectral analysis. It is of necessity short and omits a great deal of important detail. A much lengthier exposition for non-statisticians is in the course of preparation.

G. M. Jenkins

Papers

General Considerations in the Analysis of Spectra