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! Mean varies: Linear trends? Fading? Variance is a combination of: Intrinsic scatter Systematic error (e.g. chart errors) Real variability!

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!

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)

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)

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: 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 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) 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