Chapter 1 - The Methods of Time-Series Analysis

Abstract

The chapter presents methods designed for the purpose of analyzing series of statistical observations taken at regular intervals in time. The methods have a wide range of applications, such as astronomy, meteorology, seismology, oceanography, communications engineering and signal processing, the control of continuous process plants, neurology and electroencephalography, and economics. The methods apply to stationary or nonevolutionary time series. Such series manifest statistical properties which are invariant throughout time, so that the behavior during one epoch is the same as it would be during any other. There are two distinct yet broadly equivalent modes of time-series analysis which may be pursued. On the one hand are the time-domain methods that have their origin in the classical theory of correlation. Such methods deal preponderantly with the autocovariance functions and the cross-covariance functions of the series, and they lead inevitably towards the construction of structural or parametric models of the autoregressive moving-average type for single series and of the transfer-function type for two or more causally related series. On the other hand are the frequency-domain methods of spectral analysis. These are based on an extension of the methods of Fourier analysis which originate in the idea that, over a finite interval, any analytic function can be approximated, to whatever degree of accuracy is desired, by taking a weighted sum of sine and cosine functions of harmonically increasing frequencies. Furthermore, this chapter discusses a simple technique as smoothing the periodogram that should provide a theoretical resolution to the problems encountered in attempts to detect the hidden periodicities in economic and astronomical data.