Time-Series Analysis of Time-Series Analysis of Astronomical DataAstronomical Data

Matthew Templeton (AAVSO)

Workshop on Photometric Databases and Data Analysis Techniques

92nd Meeting of the AAVSOTucson, Arizona

April 26, 2003

What is time-series analysis?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

Discussion OutlineDiscussion Outline

Statistics Fourier AnalysisWavelet analysisStatistical time-series and

autocorrelationResources

Preliminaries:Preliminaries:Elementary StatisticsElementary Statistics

Mean:

Arithmetic mean or average of a data set

Variance & standard deviation:

How much do the data vary about the mean?

Example: AveragingExample: AveragingRandom NumbersRandom Numbers

• 1 sigma: 68% confidence level• 3 sigma: 99.7% confidence level

Error Analysis of Error Analysis of Variable Star DataVariable Star Data

Measurement of Mean and Variance arenot so simple!

•Mean varies: Linear trends? Fading?•Variance is a combination of:

o Intrinsic scattero Systematic error (e.g. chart errors)o Real variability!

Statistics: SummaryStatistics: 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 Methods of Time-Series AnalysisAnalysis

Fourier Transforms Wavelet Analysis Autocorrelation analysis Other methods

Use the right tool for the right job!

Fourier Analsysis: BasicsFourier 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)

The Fourier TransformThe 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)

Fourier Analysis: Basics 2Fourier Analysis: Basics 2

Your data place limits on:

• Period resolution• Period range

If you have a short span of data, both theperiod resolution and range will be lowerthan if you have a longer span

Period Range & SamplingPeriod 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 FTEffect of time span on FT

R CVn: P (gcvs) = 328.53 d

Nyquist frequency/aliasingNyquist frequency/aliasing

FTs can recover periods much shorter thanthe sampling rate, but the transform willsuffer from aliasing!

Fourier AlgorithmsFourier 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:Fourier Transforms:ApplicationsApplications

Multiperiodic data“Red noise” spectral measurementsPeriod, amplitude evolutionLight curve “shape” estimation via

Fourier harmonics

Application: Light Curve Application: Light Curve Shape of AW PerShape of AW Per

m(t) = mean + aicos(it + i)

Wavelet AnalysisWavelet Analysis

Analyzing the power spectrum as a function of time

Excellent for changing periods, “mode switching”

Wavelet Analysis: Wavelet Analysis: ApplicationsApplications

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 AnalysisWavelet Analysisof AAVSO Dataof 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 AurigaeWavelet Analysis: Z Aurigae

How to choose a window size?

Statistical Methods for Statistical Methods for Time-Series AnalysisTime-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?

AutocorrelationAutocorrelation

For a range of “periods” (), compareeach data point m(t) to a point m(t+)

The value of the correlation function ateach 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: ApplicationsAutocorrelation: Applications

Excellent for stars with amplitude variations, transient periods

Strictly periodic starsNot good for multiperiodic stars

(unless Pn= n P1)

Autocorrelation: R ScutiAutocorrelation: R Scuti

SUMMARYSUMMARY

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 Computer Programs for Time-Series AnalysisTime-Series Analysis

•AAVSO: TS 1.1 & WWZ (now available for linux/unix)http://www.aavso.org/data/software/

•PERIOD98: designed for multiperiodic starshttp://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