Rhythms and autocorrelation analysis

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Abstract
The time domain approach to the analysis of time‐series data is based on the concept of autocorrelation which arises whenever the same variable is observed sequentially over time, on the same individual. The parameter function is the theoretical autocorrelation function that may be estimated by the sample autocorrelogram. While the periodic pattern of a time series is reproduced by the autocorrelation function estimate where the lags of maximum autocorrelation correspond to peaks and troughs in the series, the ability of autocorrelograms to provide information on the respective period of different rhythm components is addressed. Firstly, the autocorrelation concept and the theoretical autocorrelation function are defined for univariate time series; related forms are considered and their properties reviewed, including the multidimensional and multivariate cases. Secondly, focusing on univariate time series, the methods of statistical inference available in autocorrelation analysis are outlined, and their underlying assumptions are discussed. Thirdly, the specific problem of periodic time series is addressed and the application of autocorrelation analysis to the study of rhythms is presented. Practical guidelines are given for the definition of the maximum lag in an autocorrelogram in relation with the length of the series and the number and period of rhythm components, for the series length required to detect a periodicity for a given signal‐to‐noise ratio in a power analysis, and for the problems of data unequally spaced in time and of noninteger periods, among others.