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. Preliminaries: Elementary Statistics
Mean:
Arithmetic mean or average of a data set
Variance & standard deviation:
How much do the data vary about the mean?

5. Example: Averaging Random 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. F () =  f(t) exp(i2t) dt
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)

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/2:N/2)

13. Effect of time span on FT
R CVn: P (gcvs) = 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 Transforms: Applications
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 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!)

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)
PERIOD98: designed for multiperiodic stars
Statistics code Penn State Astro Dept.
Astrolab: autocorrelation (J. Percy, U. Toronto)