The statistical approach to the analysis of time-series

The problems of statistics are broadly classified into problems of specification and problems of inference, and a brief recapitulation is given of some standard methods in statistics, based on the use of the probability p (S/H) of the data S on the specification H (or on the use of the equivalent likelihood function). The general problems of specification and inference for time-series are then also briefly surveyed. To conclude Part I, the relation is examined between the information (entropy) concept used in communication theory, associated with specification, and Fisher´s information concept used in statistics, associated with inference. In Part II some detailed methods of analysis are described with special reference to stationary time-series. The first method is concerned with the analysis of probability chains (in which the variable X can assume only a finite number of values or ´states´, and the time t is discrete). The next section deals with autoregressive and autocorrelation analysis, for series defined either for discrete or continuous time, including proper allowance for sampling fluctuations; in particular, least-squares estimation of unknown coefficients in linear autogressive representations, and Quenouille´s goodness of fit test for the correlogram, are illustrated. Harmonic or periodogram analysis is theoretically equivalent to autocorrelation analysis, but in the case of time-series with continuous spectra is valueless in practice without some smoothing device, owing to the peculiar distributional properties of the observed periodogram; one such arithmetical device is described in Section 7. Finally the precise use of the likelihood function (when available) is illustrated by reference to two different theoretical series giving rise to the same autocorrelation function.