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

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What is time-series analysis? Applying mathematical and statistical tests to data, to quantify and understand the nature of time-varying phenomena Has relevance to fields far beyond just astronomy and astrophysics! Gain physical understanding of the system Be able to predict future behavior

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Discussion Outline Statistics Fourier Analysis Wavelet analysis Statistical time-series and autocorrelation Resources

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

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Example: Averaging Random Numbers 1 sigma: 68% confidence level 3 sigma: 99.7% confidence level

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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!

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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!

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Methods of Time-Series Analysis Fourier Transforms Wavelet Analysis Autocorrelation analysis Other methods Use the right tool for the right job!

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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)

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The Fourier Transform For a given frequency (where =(1/period)) the Fourier transform is given by F ( ) = f(t) exp(i2 t) dt Recall Eulers formula: exp(ix) = cos(x) + isin(x)

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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

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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 = P 2 / [5000 d] (d = n/(N ), n=-N/2:N/2)

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Effect of time span on FT R CVn: P (gcvs) = d

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Nyquist frequency/aliasing FTs can recover periods much shorter than the sampling rate, but the transform will suffer from aliasing!

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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

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Fourier Transforms: Applications Multiperiodic data Red noise spectral measurements Period, amplitude evolution Light curve shape estimation via Fourier harmonics

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Application: Light Curve Shape of AW Per m(t) = mean + a i cos( i t + i )

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Wavelet Analysis Analyzing the power spectrum as a function of time Excellent for changing periods, mode switching

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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!

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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!)

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Wavelet Analysis: Z Aurigae How to choose a window size?

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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?

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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

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Autocorrelation: Applications Excellent for stars with amplitude variations, transient periods Strictly periodic stars Not good for multiperiodic stars (unless P n = n P 1 )

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Autocorrelation: R Scuti

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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

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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)

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