Phase dispersion minimization

Phase dispersion minimization (PDM) is a data analysis technique that searches for periodic components of a time series data set. It is useful for data sets with gaps, non-sinusoidal variations, poor time coverage or other problems that would make Fourier techniques unusable. It was first developed by Stellingwerf in 1978 [ [http://adsabs.harvard.edu/abs/1978ApJ...224..953S "Period Determination Using Phase Dispersion Minimization", Stellingwerf, R.F., Astrophysical.J. v224, p953, 1978.] ] and has been widely used for astronomical and other types of periodic data analyses. The current version of this application is available for download [ [http://www.stellingwerf.com/rfs-bin/index.cgi?action=PageView&id=29 "PDM2 Application, Technical Manual, and test data sets", Stellingwerf, R. F., 2006.] ] .

Background

PDM is a variant of a standard astronomical technique called data folding. This involves guessing a trial period for the data, and cutting, or "folding" the data into multiple sub-series with a time duration equal to the trial period. The data is now plotted versus "phase", or a scale of 0->1, relative to the trial period. If the data is truly periodic with this period a clean functional variation, or "light curve", will emerge. If not the points will be randomly distributed in amplitude.

As early as 1926 Whittiker and Robinson ["The Calculus of Observations", Whittiker, E. T., Robinson, G. (London: Blackie and Son) 1926.] proposed an analysis technique of this type based on maximizing the amplitude of the mean curve. Another technique focusing on the variation of data at adjacent phases was proposed in 1964 by Lafler and Kinman [ [http://adsabs.harvard.edu/abs/1965ApJS...11..216L "An RR Lyrae Star Survey with Ihe Lick 20-INCH Astrograph II. The Calculation of RR Lyrae Periods by Electronic Computer", Lafler, J., Kinman, T. D. Astrophysical J., v11, p216, 1965.] ] . Both techniques had difficulties, particularly in estimating the significance of a possible solution.

PDM analysis

The PDM analysis tries to simplify the statistical properties of the analysis by dividing the folded data into a series of bins and computing the variance of the amplitude within each bin. These bin variances are then combined and compared to the overall variance of the data set. For a true period the ratio of the bin to the total variances will be small. For a false period the ratio will be approximately unity. A plot of this ratio versus trial period will usually indicate the best candidates for periodic components. Analyses of the statistical properties of this approach have been given in [ [http://adsabs.harvard.edu/abs/1985AJ.....90.2317L "A test of significance for periods derived using phase-dispersion-minimization techniques," Nemec & Nemec, Astronomical.J. v90, p2317, 1985.] ] and [ [http://adsabs.harvard.edu/abs/1997ApJ...489..941S "The Correct Probability Distribution for the Phase Dispersion Minimization Periodogram", Schwarzenberg-Czerny, A., Astrophysical J. v489, p941, 1997.] ] .

PDM2 updates

The original PDM technique has problems in two areas:
*1) The bin variance calculation is equivalent to a curve fit with step functions across each bin. This can introduce errors in the result if the underlying curve is non-symmetric, since errors toward the right side and left side of each bin will not exactly cancel. This low order error can be eliminated by replacing the step function by a linear fit drawn between bin means. Attempts to further increase accuracy by using more sophisticated curve fite (eg splines or polynomials) have been tried, but tend to fail if one or more bins are sparsely populated. This situation is encountered for most data sets at occasional trial periods due to aliasing, so we feel that a simple linear fit is the best choice in general.
*2) The original test of significance was based on an F test, which has been shown to be incorrect. The correct statistic is a beta distribution.These updates have been incorportated into a new formulation of PDM, called PDM2. See reference (2) for a detailed technical discussion, test cases, C source code, and a Windows application package.

References