On the Detection of Periodic Components in Observational and Experimental Data

Abstract

In this paper, the problem addressed is estimation of the false alarm probability, associated with maximum peaks on the periodograms, which are based on the least-squares fitting of the putative periodic signal. This class of periodograms include, for instance, the well-known Lomb-Scargle periodogram and multi-harmonic periodograms. The evaluation of the related distribution of the maximum periodogram values was performed using the so-called Rice method, based on the analysis of the number of up-crossings. This method allowed to obtain simple and closed approximations to the false alarm probability, associ- ated with maximum peaks on the periodograms. The comparison of these analytic approximations with results of Monte Carlo simulations shows that the precision of the analytic expressions should be suitable in the most practical cases. I. INTRODUCTION In many proctical problems in various reserch fields we of- ten deal with the task of detecting a periodic component, which our observational or experimental data can possibly contain. These input data may have various nature and structure, and the putative periodicity may have different physical sense (de- pending on the research field). For definiteness we will call the independent (one-dimensional) variable as time and denote it as t, though in practice its physical sense may differ from the physical time. The data array represent a sequence of discrete (also one-dimensional) measurements xi, taken at timings ti (which may be sampled arbitrarily, but are supposed to be known precisely). In addition, we usually have a sequence of quantities, characterising the measurement uncertainties. These quantities may be, for instance, the standard deviations of the measurement errors, σi, or the weights of observations wi ∝ 1/σ 2 i (so that the proportionality factor is ap riori unknown). These data can possibly contain a deterministic periodic signal of ap rioriunknown frequency f. The task is to detect this periodicity and (further) to estimate its parameters. One of the mostly popular mathematical tools solving this task is based on the periodogram approach. A periodogram is a function of the frequency of the putative periodic signal, which represents, in fact, some estimation of the power spectrum of the variable quantity being observed. High peaks on a periodogram indicate that the data likely contain a periodic signal of the respective frequency. However, the further prob- lem is to rigorously estimate the statistical significance of the observed periodogram peaks. It is this problem, which is addressed in this paper. To claim that the observed peak is statistically significant, we need first to evaluate the associated false alarm probability (hereafter FAP), i.e. the probability that a peak of the same or larger height could be inspired solely by measurement noise and then to check, whether it stays below some small tolerance level. The problem of evaluation of the FAP is directly connected with the problem of evaluation of the distribution function of the maximum periodogram value. This task is a non-trivial one, and in many publications this problem was considered (6), (7), (12)-(14), (17), (18), (20), (21). Recently, a significant progress was attained in this field by the author (2)-(5). In these papers, the approximations to the distribution function of the maximum are obtained using the so-called Rice method, which is based on the analysis of level up-crossings (1), (15). We present these results below.