time-series analysis of astronomical data
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Time-Series Analysis of Astronomical Data. Workshop on Photometric Databases and Data Analysis Techniques 92 nd Meeting of the AAVSO Tucson, Arizona April 26, 2003. Matthew Templeton (AAVSO). What is time-series analysis?. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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Discussion OutlineDiscussion Outline
Statistics Fourier AnalysisWavelet analysisStatistical time-series and
autocorrelationResources
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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?
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Example: AveragingExample: AveragingRandom NumbersRandom Numbers
• 1 sigma: 68% confidence level• 3 sigma: 99.7% confidence level
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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!
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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!
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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!
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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)
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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)
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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
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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)
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Effect of time span on FTEffect of time span on FT
R CVn: P (gcvs) = 328.53 d
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Nyquist frequency/aliasingNyquist frequency/aliasing
FTs can recover periods much shorter thanthe sampling rate, but the transform willsuffer from aliasing!
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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
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Fourier Transforms:Fourier Transforms:ApplicationsApplications
Multiperiodic data“Red noise” spectral measurementsPeriod, amplitude evolutionLight curve “shape” estimation via
Fourier harmonics
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Application: Light Curve Application: Light Curve Shape of AW PerShape of AW Per
m(t) = mean + aicos(it + i)
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Wavelet AnalysisWavelet Analysis
Analyzing the power spectrum as a function of time
Excellent for changing periods, “mode switching”
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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!
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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!)
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Wavelet Analysis: Z AurigaeWavelet Analysis: Z Aurigae
How to choose a window size?
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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?
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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
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Autocorrelation: ApplicationsAutocorrelation: Applications
Excellent for stars with amplitude variations, transient periods
Strictly periodic starsNot good for multiperiodic stars
(unless Pn= n P1)
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Autocorrelation: R ScutiAutocorrelation: R Scuti
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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
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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