Remarks on the theory, computation and application of the spectral analysis of series of events

Abstract

The concepts of a covariance density function and spectrum of a continuous-time series of events (point stochastic process) are clarified by relating them to the covariance sequence and the spectrum of a discrete-time series of events (sequence of binary-valued random variables). It is also shown that there is a well-developed spectral theory for point process which uses the theory of distributions. The periodogram for an observed continuous-time series of events is then derived in two ways, and it is also shown to be related to the maximum likelihood estimates of the phase and amplitude of the intensity function in a non-homogeneous Poisson process. Estimates of the spectrum obtained by smoothing the periodogram are discussed and it is shown that severe computation-time problem involved in estimating the spectrum of a series of events can be obviated by using time-average smoothing of the estimated spectrum. This technique is also well suited to trend analysis of non-stationary series of events. Some tentative ideas on the filtering of point processes are given. In line with the expository and exploratory nature of the paper, a spectral analysis of R-R intervals in an electrocardiogram is used to illustrate the ideas and point up unsolved problems. © 1970.