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.

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