A Bayesian Method for the Detection of a Periodic Signal of Unknown Shape and Period

Abstract

We present a new method for the detection and measurement of a periodic signal in a data set when we have no prior knowledge of the existence of such a signal or of its characteristics. It is applicable to data consisting of the locations or times of individual events. To address the detection problem, we use Bayes’ theorem to compare a constant rate model for the signal to models with periodic structure. The periodic models describe the signal plus background rate as a stepwise distribution in m bins per period, for various values of m. The Bayesian posterior probability for a periodic model contains a term which quantifies Ockham’s razor, penalizing successively more complicated periodic models for their greater complexity even though they are assigned equal prior probabilities. The calculation thus balances model simplicity with goodness-of-fit, allowing us to determine both whether there is evidence for a periodic signal, and the optimum number of bins for describing the structure in the data. Unlike the results of traditional “frequentist” calculations, the outcome of the Bayesian calculation does not depend on the number of periods examined, but only on the range examined. Once a signal is detected, we again use Bayes’ theorem to estimate the frequency of the signal. The probability density for the frequency is inversely proportional to the multiplicity of the binned events and is thus maximized for the frequency leading to the binned event distribution with minimum combinatorial entropy. The method is capable of handling gaps in the data due to intermittent observing or dead time.