Time-Series Analysis of
Workshop on Photometric Databases and
Data Analysis Techniques
92nd Meeting of the AAVSO
April 26, 2003
Matthew Templeton (AAVSO)
What is time-series analysis?
Applying mathematical and statistical
tests to data, to quantify and understand
the nature of time-varying phenomena
•Gain physical understanding of the system
•Be able to predict future behavior
Has relevance to fields far beyond
just astronomy and astrophysics!
Statistical time-series and
Arithmetic mean or average of a data set
Variance & standard deviation:
How much do the data vary about the mean?
• 1 sigma: 68% confidence level
• 3 sigma: 99.7% confidence level
Error Analysis of
Variable Star Data
Measurement of Mean and Variance are
not so simple!
•Mean varies: Linear trends? Fading?
•Variance is a combination of:
o Intrinsic scatter
o Systematic error (e.g. chart errors)
o Real variability!
Random errors always present in
your data, regardless of how high the
Be aware of non-random, systematic
trends (fading, chart errors, observer
Understand your data before you analyze it!
Methods of Time-Series
Use the right tool for the right job!
Fourier Analsysis: Basics
Fourier analysis attempts to fit a series
of sine curves with different periods,
amplitudes, and phases to a set of data.
Algorithms which do this perform
mathematical transforms from the
time “domain” to the period (or
f (time) F (period)
The Fourier Transform
For a given frequency (where =(1/period))
the Fourier transform is given by
F () = f(t) exp(i2t) dt
Recall Euler’s formula:
exp(ix) = cos(x) + isin(x)
Fourier Analysis: Basics 2
Your data place limits on:
• Period resolution
• Period range
If you have a short span of data, both the
period resolution and range will be lower
than if you have a longer span
Period Range & Sampling
Suppose you have a data set spanning
5000 days, with a sampling rate of 10/day.
What are the formal, optimal values of…
• P(max) = 5000 days (but 2500 is better)
• P(min) = 0.2 days (sort of…)
• dP = P2 / [5000 d] (d = n/(N), n=-N/2:N/2)
Effect of time span on FT
R CVn: P (gcvs) = 328.53 d
FTs can recover periods much shorter than
the sampling rate, but the transform will
suffer from aliasing!
Discrete Fourier Transform: the
classic algorithm (DFT)
Fast Fourier Transform: very good
for lots of evenly-spaced data (FFT)
Date-Compensated DFT: unevenly
sampled data with lots of gaps (TS)
Periodogram (Lomb-Scargle): similar
“Red noise” spectral measurements
Period, amplitude evolution
Light curve “shape” estimation via
Application: Light Curve
Shape of AW Per
m(t) = mean + aicos(it + i)
power spectrum as
a function of time
Many long period stars have changing
periods, including Miras with “stable”
pulsations (M, SR, RV, L)
“Mode switching” (e.g. Z Aurigae)
CVs can have transient periods (e.g.
WWZ is ideal for all of these!
of AAVSO Data
Long data strings are ideal,
particularly with no (or short) gaps
Be careful in selecting the window
width – the smaller the window, the
worse the period resolution (but the
larger the window, the worse the time
Wavelet Analysis: Z Aurigae
How to choose a window size?
Statistical Methods for
Correlation/Autocorrelation – how
does the star at time (t) differ from
the star at time (t+)?
Analysis of Variance/ANOVA – what
period foldings minimize the
variance of the dataset?
For a range of “periods” (), compare
each data point m(t) to a point m(t+)
The value of the correlation function at
each is a function of the average
difference between the points
If the data is variable with period ,
the autocorrelation function has a peak at
Excellent for stars with amplitude
variations, transient periods
Strictly periodic stars
Not good for multiperiodic stars
(unless Pn= n P1)
Autocorrelation: R Scuti
Many time-series analysis methods
Choose the method which best suits
your data and your analysis goals
Be aware of the limits (and
strengths!) of your data
Computer Programs for
•AAVSO: TS 1.1 & WWZ (now available for linux/unix)
•PERIOD98: designed for multiperiodic stars
•Statistics code index @ Penn State Astro Dept.
•Astrolab: autocorrelation (J. Percy, U. Toronto)