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Time-Series Analysis of Astronomical Data

Time-Series Analysis of Astronomical Data. Workshop on Photometric Databases and Data Analysis Techniques 92 nd Meeting of the AAVSO Tucson, Arizona April 26, 2003. Matthew Templeton (AAVSO). What is time-series analysis?.

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Time-Series Analysis of Astronomical Data

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  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: 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 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) = 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 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 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 @ 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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