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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
 Resources
      Elementary Statistics
  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!
  •Mean varies: Linear trends? Fading?
  •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
  Be aware of non-random, systematic
   trends (fading, chart errors, observer

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

 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
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
    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:

 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:
 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!
           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
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?
  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
 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
       Time-Series Analysis

•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)

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