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```									Time-Series Analysis of
Astronomical Data
Workshop on Photometric Databases and
Data Analysis Techniques

92nd Meeting of the AAVSO
Tucson, Arizona
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!
Discussion Outline
 Statistics
 Fourier Analysis
 Wavelet analysis
 Statistical time-series and
autocorrelation
 Resources
Preliminaries:
Elementary Statistics
Mean:
Arithmetic mean or average of a data set

Variance & standard deviation:
How much do the data vary about the mean?
Example: Averaging
Random Numbers

• 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!
•Variance is a combination of:
o Intrinsic scatter
o Systematic error (e.g. chart errors)
o Real variability!
Statistics: Summary
 Random errors always present in
your data, regardless of how high the
quality
 Be aware of non-random, systematic
differences)

Understand your data before you analyze it!
Methods of Time-Series
Analysis

 Fourier Transforms
 Wavelet Analysis
 Autocorrelation analysis
 Other methods

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
frequency) domain.
f (time)  F (period)
The Fourier Transform

For a given frequency  (where =(1/period))
the Fourier transform is given by

F () =  f(t) exp(i2t) dt

Recall Euler’s formula:
exp(ix) = cos(x) + isin(x)
Fourier Analysis: Basics 2

• 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
Nyquist frequency/aliasing
FTs can recover periods much shorter than
the sampling rate, but the transform will
suffer from aliasing!
Fourier Algorithms
 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
to DFT
Fourier Transforms:
Applications

 Multiperiodic data
 “Red noise” spectral measurements
 Period, amplitude evolution
 Light curve “shape” estimation via
Fourier harmonics
Application: Light Curve
Shape of AW Per
m(t) = mean + aicos(it + i)
Wavelet Analysis

   Analyzing the
power spectrum as
a function of time

   Excellent for
changing periods,
“mode switching”
Wavelet Analysis:
Applications
 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.
superhumps)

WWZ is ideal for all of these!
Wavelet Analysis
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
resolution!)
Wavelet Analysis: Z Aurigae
How to choose a window size?
Statistical Methods for
Time-Series Analysis
   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?
Autocorrelation
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 
Autocorrelation: Applications
 Excellent for stars with amplitude
variations, transient periods
 Strictly periodic stars
 Not good for multiperiodic stars
(unless Pn= n P1)
Autocorrelation: R Scuti
SUMMARY
 Many time-series analysis methods
exist
 Choose the method which best suits
 Be aware of the limits (and
Computer Programs for
Time-Series Analysis

•AAVSO: TS 1.1 & WWZ (now available for linux/unix)
http://www.aavso.org/data/software/

•PERIOD98: designed for multiperiodic stars
http://www.univie.ac.at/tops/Period04/

•Statistics code index @ Penn State Astro Dept.
http://www.astro.psu.edu/statcodes/

•Astrolab: autocorrelation (J. Percy, U. Toronto)
http://www.astro.utoronto.ca/~percy/analysis.html

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