Presentation on theme: "Time-Series Analysis of Astronomical Data"— Presentation transcript:
1 Time-Series Analysis of Astronomical Data Workshop on Photometric Databases andData Analysis Techniques92nd Meeting of the AAVSOTucson, ArizonaApril 26, 2003Matthew Templeton (AAVSO)
2 What is time-series analysis? Applying mathematical and statistical tests to data, to quantify and understand the nature of time-varying phenomenaGain physical understanding of the systemBe able to predict future behaviorHas relevance to fields far beyond just astronomy and astrophysics!
6 Error Analysis of Variable Star Data Measurement of Mean and Variance arenot so simple!Mean varies: Linear trends? Fading?Variance is a combination of:Intrinsic scatterSystematic error (e.g. chart errors)Real variability!
7 Statistics: SummaryRandom errors always present in your data, regardless of how high the qualityBe 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 TransformsWavelet AnalysisAutocorrelation analysisOther methodsUse the right tool for the right job!
9 Fourier Analsysis: Basics Fourier analysis attempts to fit a seriesof sine curves with different periods,amplitudes, and phases to a set of data.Algorithms which do this performmathematical transforms from thetime “domain” to the period (orfrequency) domain.f (time) F (period)
10 F () = f(t) exp(i2t) dt The Fourier TransformFor a given frequency (where =(1/period))the Fourier transform is given byF () = f(t) exp(i2t) dtRecall Euler’s formula:exp(ix) = cos(x) + isin(x)
11 Fourier Analysis: Basics 2 Your data place limits on:Period resolutionPeriod rangeIf you have a short span of data, both theperiod resolution and range will be lowerthan if you have a longer span
12 Period Range & Sampling Suppose you have a data set spanning5000 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)
14 Nyquist frequency/aliasing FTs can recover periods much shorter thanthe sampling rate, but the transform willsuffer from aliasing!
15 Fourier AlgorithmsDiscrete 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
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) gapsBe 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 ateach is a function of the averagedifference between the pointsIf the data is variable with period ,the autocorrelation function has a peak at
24 Autocorrelation: Applications Excellent for stars with amplitude variations, transient periodsStrictly periodic starsNot good for multiperiodic stars (unless Pn= n P1)
26 SUMMARY Many time-series analysis methods exist Choose the method which best suits your data and your analysis goalsBe 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 starsStatistics code Penn State Astro Dept.Astrolab: autocorrelation (J. Percy, U. Toronto)