Peranso 3 User Guide

There is a long history and a wide variety in period-finding algorithms that can be used to study light curves of variable stars, exoplanets and asteroids. They can be divided into a number of types:

• Algorithms based on discrete Fourier transforms

These methods attempt to represent a set of observations with a series of trigonometric functions (sines and cosines, with different periods, amplitudes and phases). They are one of the oldest forms of time-series analysis and are also quite flexible. Fourier methods supported by Peranso are : Discrete Fourier Transform DFT (Deeming), Date Compensated Discrete Fourier Transform (Ferraz-Mello) DCDFT,  CLEANest (Foster) and Bloomfield. 

• Algorithms that model a light curve via a least-squares fit to a set of (orthogonal) basis functions

• The basis functions can be trigonometric, as in Lomb-Scargle, and Generalized Lomb-Scargle (Zechmeister & Kürster) and FALC (Harris), which fits a Fourier series of a selectable order.
• ANOVA (Schwarzenberg-Czerny) fits periodic orthogonal polynomials and evaluates the fit with an analysis of variance, making it sensitive to both sinusoidal and sharply non-sinusoidal signals.
• The basis functions can also be more complex, such as the wavelets in the Weighted Wavelet Z-Transform WWZ (Foster), which shows how periods change over time.

• Bayesian algorithms
  
  

These methods express the result as the probability that a signal with a given period is present in the data, which allows the relative probability of different peaks to be compared directly. Peranso implements the Bayesian Generalized Lomb-Scargle periodogram BGLS (Mortier et al. 2015). BGLS uses the magnitude errors of the observations and assumes a sinusoidal signal.

• Algorithms that minimize some measure of the dispersion of the time series data in phase space
  
  

The minimization can be done in various ways, including:

• Binned means: these look for the period that produces the least observational scatter around the mean light curve ("best phasing of the data"). Peranso implements PDM (Stellingswerf) and Jurkevich.
• Variance: as in Phase binned AoV, AoV Transit (Schwarzenberg-Czerny)
• Entropy: as in Conditional Entropy (Graham et al.). The best period gives the most ordered folded light curve, i.e. the lowest entropy.
• String lengths: these methods fold the observations on a series of trial periods, and at each trial period they compute the sum of the lengths of the line segments joining successive points (the string-length). Minima in a plot of string-length versus trial frequency indicate possible periods. Peranso includes Lafler-Kinman, Renson and Dworetsky
• Cross-cycle prediction: these methods fold the observations on a series of trial periods and predict each observation from its nearest phase neighbours, using only observations from other cycles. Like the string-length methods, they need no binning, but they compare each point with several neighbours instead of only the next one, and they take the magnitude errors into account. Peranso includes CCC.
  
  
  
• Algorithms based on statistical dependence

These methods make no assumption about the shape of the light curve. For each trial period they measure how strongly the brightness depends on phase, using the distance correlation concept of Székely. They perform particularly well on sparse data sets and on sawtooth-like light curves such as those of RR Lyrae stars and Cepheids. Peranso implements Phase Distance Correlation with errors PDC-E (Binnenfield et al.), which gives less weight to observations with large magnitude errors.

• Algorithms developed for specific types of variability

• Edge Enhanced Box-fitting Least Squares EEBLS (Kovács) analyses stellar photometric time series in search of periodic transits by exoplanets. It looks for signals characterized by a periodic alternation between two discrete levels, with much less time spent at the lower level.
• EASolver (Wils) determines the period of eclipsing Algol-type (EA) binaries from survey data with only a few observations in the faint state.
  
  
• New research in period finding continues to explore other approaches, such as Gaussian processes and neural networks

Methods using magnitude errors

Some period analysis methods can take the uncertainty of each observation into account. Instead of treating all observations equally, these methods give each observation a weight based on its magnitude error, usually 1/σ². Precise observations then have more influence on the result than noisy ones. This matters most when the data quality varies, for example when you combine observations from different observers or instruments, when the conditions change during a night, or when the star fades so that the errors grow towards the end of the run.

In the Period determination window, these methods are marked with ±

All other methods treat every observation as equally precise: DCDFT, DFT, CLEANest, Bloomfield, Lomb-Scargle, WWZ, PDM, Jurkewich, Dworetsky, Renson, Lafler-Kinman, Phase-binned AoV, AoV Transit, CE, EEBLS and EASolver. WWZ is called “weighted”, but its weights come from the wavelet window in time, not from the measurement errors. EASolver uses only the times of the faint-state observations, so magnitudes play no role at all.

Missing errors. Observations without a valid magnitude error (blank, zero or a placeholder value) get the median error of the other observations. If no observation in the data set has a valid error, as is often the case with visual estimates, the method automatically runs without weights.

When to use it. For ANOVA and PDC-E, the Use magnitude errors option lets you choose whether the errors are used. Enable it when your errors are realistic and the data quality clearly varies. If all observations have about the same error, the result hardly changes. If the errors are unrealistic, for example far too small for some observers, weighting can give those observations too much influence. Then it’s better to switch the option off. In case of doubt, run the analysis both ways: if both give the same period, the result is robust.

For PDC-E, switching the option off also makes the analysis much faster on large data sets.

Generalized LS, FALC, BGLS and CCC always use the errors when they are available. If you doubt the quality of the errors, compare their result with a method that doesn’t use them, such as PDM or Lomb-Scargle.