PDC-E
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The Phase Distance Correlation (PDC) periodogram was introduced by Zucker¹ and extended by Binnenfeld, Shahaf and Zucker² to take the magnitude errors of the observations into account (PDC-E). The method is based on the distance correlation concept of Székely et al.³, a statistical measure that is zero only if two variables are completely independent. PDC-E makes no assumption at all about the shape of the light curve. It is very well suited to sparse data sets, and it performs particularly well on sawtooth-like light curves such as those of RR Lyrae stars and Cepheids. For each trial frequency, PDC-E measures how strongly the brightness of the observations depends on their phase. For every pair of observations it calculates two distances:
If the trial period is not a true period, the brightness does not depend on phase and the PDC-E statistic will be close to 0. If the trial period is a true period, the PDC-E statistic reaches a local maximum. The maximum possible value is 1.
With Use magnitude errors enabled in the Period determination window, each observation keeps its own magnitude error. Observations with large errors then weigh less than accurate ones. If only some observations have an error, the missing errors are replaced by the median of the available ones. If no observation has an error, Peranso estimates a common error from the scatter between consecutive observations. With the option switched off, Peranso gives all observations the same error, estimated from the scatter between consecutive observations. All observations then weigh equally. Use this when the magnitude errors are missing, unreliable or about equal for all observations.
PDC-E compares every observation with every other observation. With magnitude errors enabled, its calculation time therefore grows with the square of the number of observations: a few seconds for a few hundred observations, but considerably longer for data sets with thousands of observations. With the option switched off, Peranso uses a much faster, mathematically identical algorithm whenever many observations share the same magnitude. This is the case for visual estimates (rounded to 0.1 mag) and for CCD magnitudes, which Peranso rounds to 0.001 mag. For large data sets, such as long-term visual light curves with thousands of observations, this can make PDC-E many times faster. PDC-E typically shows the true period P as the highest peak, with weaker peaks at its harmonics (P/2, P/3, ...).
PDC-E is strong at detecting whether a period is present, even in sparse or noisy data and for non-sinusoidal light curves. Its peaks are, however, usually broader than those of model-fitting methods such as Generalized LS, FALC or ANOVA. The top of a PDC-E peak can therefore lie slightly off the true period, especially for short data sets that cover only one or a few nights. Use PDC-E to find or confirm the period, then refine its value with a fitting method around the PDC-E peak. The longer the time span of the observations, the narrower the peaks become and the smaller this difference gets. The PDC-E dialog box is similar to the Lomb-Scargle dialog box. Unlike PDM, ANOVA or Conditional Entropy, PDC-E doesn’t need any binning parameters, so no additional parameters need to be entered. Prominent periods appear as peaks in the Period Window. (1) Zucker, S., 2018, Mon. Not. R. Astron. Soc., 474, L86 |