Time-Series Analysis of Astronomical Data

1
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)

2
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!
3
Discussion Outline

• Statistics
• Fourier Analysis
• Wavelet analysis
• Statistical time-series and autocorrelation
• Resources

4
PreliminariesElementary Statistics
Mean
Arithmetic mean or average of a data set
Variance standard deviation
How much do the data vary about the mean?
5
Example AveragingRandom Numbers

• 1 sigma 68 confidence level
• 3 sigma 99.7 confidence level

6
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

• Intrinsic scatter
• Systematic error (e.g. chart errors)
• Real variability!

7
Statistics Summary

• Random errors always present in your data,
  regardless of how high the quality
• Be aware of non-random, systematic trends
  (fading, chart errors, observer differences)

Understand your data before you analyze it!
8
Methods of Time-Series Analysis

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

Use the right tool for the right job!
9
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)
10
The Fourier Transform
For a given frequency ? (where ?(1/period)) the
Fourier transform is given by
F (?) ? f(t) exp(i2??t) dt
Recall Eulers formula exp(ix) cos(x) isin(x)
11
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
12
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/2N/2)

13
Effect of time span on FT
R CVn P (gcvs) 328.53 d
14
Nyquist frequency/aliasing
FTs can recover periods much shorter than the
sampling rate, but the transform will suffer from
aliasing!
15
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

16
Fourier TransformsApplications

• Multiperiodic data
• Red noise spectral measurements
• Period, amplitude evolution
• Light curve shape estimation via Fourier
  harmonics

17
Application Light Curve Shape of AW Per
m(t) mean ?aicos(?it ?i)
18
Wavelet Analysis

• Analyzing the power spectrum as a function of
  time
• Excellent for changing periods, mode switching

19
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!
20
Wavelet Analysisof 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!)

21
Wavelet Analysis Z Aurigae
How to choose a window size?
22
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?

23
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 ?
24
Autocorrelation Applications

• Excellent for stars with amplitude variations,
  transient periods
• Strictly periodic stars
• Not good for multiperiodic stars (unless Pn n P1)

25
Autocorrelation R Scuti
26
SUMMARY

• Many time-series analysis methods exist
• Choose the method which best suits your data and
  your analysis goals
• Be aware of the limits (and strengths!) of your
  data

27
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 _at_ Penn State Astro Dept.
• http//www.astro.psu.edu/statcodes/
• Astrolab autocorrelation (J. Percy, U. Toronto)
• http//www.astro.utoronto.ca/percy/analysis.html