The Search for Periodic Components in Observational Data

This review is devoted to the problem of searching for periodicities in observational data using periodograms based on the general statistical likelihood ratio test and special variants of it, including the classical Lomb-Scargle periodogram. Primary emphasis is on the problem of estimating the statistical significance of the periodicities detected in this manner. We assume that a universal solution of this problem exists involving an effective method in which the periodogram is regarded as a random process (or random field), while an approximation for the required "false alarm probability" is constructed by a generalized Rice method. Besides a unified method for determining the expected noise levels (or significance levels) of these periodograms, we also examine some important special cases with different models of periodic signals (linear and nonlinear). The false alarm probability associated with an observed signal is approximated in most cases by a formula of the type , where z is the observed maximum readout in the periodogram and P is an algebraic polynomial with coefficients that depend on the conditions of the problem. We also examine the problem of separating composite signals with several frequencies from noise. In this case correct analysis of the data requires the use of so-called multifrequency periodograms based on models of signals containing several periodic components. We show that a complete solution to this problem requires construction of 2(n)-1 such periodograms, where n is the total number of possible frequencies. Finally, we describe some program packages that we have developed which make it easier to perform practical frequency analysis of series using this theory.