Methods of Periodicity Analysis: Relationship between the Rayleigh Analysis and a Maximum Likelihood Method
Abstract
For periodicity analysis of occurrence rates of discrete events, one can use the maximum likelihood method or the 'Rayleigh analysis'. In a maximum likelihood analysis using a sinusoidal distribution, one tries various values of amplitude A and phase angle Theta(0) of the distribution function. We show that these two methods are essentially equivalent to one another in spite of their different mathematical origins. Using the Rayleigh analysis, therefore, we can simply calculate A and Theta(0) which maximize the likelihood. Using the cumulative nature of the logarithmic likelihood, we can identify time intervals during which the periodicity is in operation. When a periodicity operates only in certain time intervals, it is important to identify these intervals. Mathematically, the above technique is applicable only to discrete events. However, with slight modifications we can apply this technique to general cases  periodicity analysis of measurements of continuously varying quantities.
 Publication:

The Astrophysical Journal
 Pub Date:
 October 1992
 DOI:
 10.1086/171816
 Bibcode:
 1992ApJ...397..584B
 Keywords:

 Maximum Likelihood Estimates;
 Periodic Variations;
 Rayleigh Equations;
 Solar Flares;
 Solar Physics;
 Distribution Functions;
 Power Spectra;
 Solar Cycles;
 Astrophysics;
 METHODS: ANALYTICAL;
 SUN: ACTIVITY