  Below you will find a list of programs which may be used to analyze variable star data or any kind of time series. Most variable star data consists of measurements of the brightness of the star at various times. Usually, the analysis of variable star data is aimed at obtaining possible periods of the star and is carried out by applying several different methods of time series analysis in conjunction with each other. One may employ these methods to confirm/deny periods obtained from other methods. A usual first step is to just look at the star's light curve. A light curve is a plot of brightness (magnitudes) versus time (Julian Date). Thus, by looking at the light curve, one can get a sense of the periodicity or irregularity of the star's variation including whether the variation is long term or short term. This can be difficult to see at first if it has been plotted at the wrong scale and so may require some trial and error to make to results more obvious. Assuming the signal is periodic, one may perform Fourier analysis on the time series. The method of Fourier analysis attempts to express the signal as a linear combination of sinusoids, each with a specific frequency and amplitude. The amplitude of a sinusoid at a given frequency shows the extent to which the signal is oscillating at that frequency. Thus, if the time series shows strong periodicity at a few periods, the amplitudes corresponding to the sine curves with those periods will be relatively large. The Fourier analysis program will output a graph of amplitude versus frequency. However, another measure of the presence of a frequency is power. Thus, the program may also output a plot of power versus frequency, commonly called a power spectrum. However, an important weakness in Fourier analysis results when regularly-spaced gaps are in the data (eg. when the star can no longer be seen at certain parts of the year). If this is the case, a sinusoid with the wrong period can be in phase with the oscillations of the signal where there is data, and out of phase where the data is missing, producing a good fit. Consequently, the power spectrum will show "fake" (normally called "alias") periods which are not the true periods: 1/Palias=1/Preal ± N/T where N is a whole number and T is the separation between regularly spaced points or the inherent periodicity of the times of observation. Assuming the signal is regularly mono-periodic, one way of testing a suspected period is by forming a phase diagram. This diagram is a plot of magnitude versus phase (between 0 and 1) relative to the suspected period. In producing a phase diagram, you take the time of each observation and subtract from it the time of the initial observation, then divide the result by the period in question and take the decimal part of this quotient. This will give a dimensionless number indicating at what fraction of a cycle the data point is. The periods associated with less scatter are more likely to be correct. Period98 This is a Fourier analysis program which allows you to perform a best fit (resulting in values of amplitude and phase of the sine curves being fitted) to your data with frequencies obtained from the power spectrum or frequencies which you specify. The program may be downloaded from: www.astro.univie.ac.at/~dsn/dsn/Period98/current (Note: Get the file P98inst.exe and run it. Then, follow the standard installation procedures.) TS1.2 This is an MS-DOS time series statistical program which allows you to plot the data and perform Fourier analysis. This program may be downloaded from the website of the American Association of Variable Star Observers (AAVSO). This page contains useful data analysis and data entry software you can download. http://www.aavso.org/data/software/ts.shtml One method of time series analysis which doesn't result in alias periods is self-correlation analysis. This form of analysis is suitable, not only to periodic data, but also to data which contains a substantial amount of irregularity and with irregularly spaced observations. However, it is not as helpful as Fourier analysis in determining the periods of multi-periodic stars. To learn about the self-correlation algorithm, see the Astrolab manual available as a reference, or see: Nyssa and Percy, "Autocorrelation Analysis of Variable Stars", International Amateur-Professional Photoelectric Photometry Communications, in press. Astrolab This program allows you to perform self-correlation analysis. This program was developed by students under the supervision of Prof. John R. Percy at the University of Toronto and can be downloaded here. This method can detect characteristic timescales, tau, in the data. This method determines the cycle- to- cycle behavior of the star, averaged over all the data. The measurements do not have to be equally spaced. For all pairs of measurements, the difference in magnitude and the difference in time are calculated. Delta mag is then plotted against delta time to some upper limit. This limit should be a few times greater than the expected timescales but less than the total time span of the data. The delta mags are binned in delta t so that, if possible, there are at least a few values in each bin; the delta mags in each bin are then averaged. The average delta mag will be a minimum at multiples of tau. Each minimum can be used to estimate tau. The height of the maxima is a measure of the average amplitude of the variability. If the variability were perfectly periodic and the magnitudes had no error, then the minima would fall to zero; in fact, the height of the minima is determined by the average error of the magnitudes and by the degree of irregularity. Download Astrolab Another form of time series analysis is wavelet analysis. This type of analysis is similar to Fourier analysis and also results in alias periods. Unlike Fourier analysis, wavelet analysis can give the frequency of a signal at a localized time. It does this by breaking down the data into sinusoids each of which are multiplied by a Gaussian function, whose width is proportional to the period of the sinusoid. The quality of the fit to such a function mostly depends on the signal's oscillations around the peak of the Gaussian. Thus, the quality of the fit determines to what extent the data is oscillating at the frequency of the sinusoid at the time of the peak of the Gaussian. Wavelet analysis is especially useful for stars which change their period, amplitude, or mode. WWZ11 This is an MS-DOS wavelet statistical program also available from the AAVSO. This program may contain a bug which causes it to crash in some situations, and possibly produce erroneous results. For more information on this and how to correct the bug (if it's still there), you can see: Redelmeier, "Wavelet Analysis of Seven Small-Amplitude Red Variable Stars", Journal of the AAVSO, to be submitted. www.aavso.org/cdata/software.stm Microsoft Excel is very useful for creating light curves and the output from the analysis programs can be entered directly into Excel. and easily graphed. The graphs shown in the examples section were all created using Excel. Self-Correlation Software is software that you can download and install in order to perform your own correlation analysis. (Courtesy of Akos Bakos) Download it here. (If download doesn't work, right click and choose "Save Target As...") For Akos Bakos' Data Analysis Methodology click here.(link opens in new window) Site created by JoAnne Hosick and Vince Velocci, and extended by Akos Bakos and Artur Chudolinski. Last updated July 19, 2004.