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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)

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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 Gain physical understanding of the system Be able to predict future behavior Has relevance to fields far beyond just astronomy and astrophysics!

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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: Intrinsic scatter Systematic error (e.g. chart errors) 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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**F () = f(t) exp(i2t) dt**

The Fourier Transform For a given frequency (where =(1/period)) the Fourier transform is given by F () = f(t) exp(i2t) dt Recall Euler’s 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 = P2 / [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 + aicos(it + 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 Pn= n P1)

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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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