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Data Analysis Do it yourself!

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What to do with your data? Report it to professionals (e.g., AAVSO) –Excellent! A real service to science; dont neglect this Publish observations (e.g., JAAVSO) Analyze it – yourself!

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But … Im not a mathematician –Let the computer do the math Im not a programmer –Get programs from the net (often free) I dont know how to use or interpret them –Neither do the pros! –Practice, practice, practice …

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Time Series Analysis A time series is a set of data pairs t is the time, x is the data value Usually, times are assumed error-free Data = Signal + Error x can be anthing, e.g. brightness of variables star, time of eclipse, eggs/day from a laying hen

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Basic properties of data x Actual Mean = = expected value Standard deviation = expected rms difference from mean Estimated Average = estimated Sample standard deviation = estimated

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Average and sample standard deviation Average Sample standard deviation

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Method #1: worlds best Eye + Brain: Look at the data! Plot x as a function of t: Explore! Scientific name: Visual Inspection Worlds best – but not infallible Programs: –TShttp://www.aavso.org –MAGPLOThttp://www.aavso.org

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Method #2: Fourier Analysis Period analysis and curve-fitting Powerful, well-understood, popular Programs –TShttp://www.aavso.org –PerAnSohttp://www.peranso.com

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Method #3: Wavelet Analysis Time-frequency analysis Old versions bad, new version good Programs: –WWZhttp://www.aavso.org –WinWWZhttp://www.aavso.org

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Visual Inspection

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Lets take a look

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Fourier Analysis

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Fourier analysis for period search Match the data to sine/cosine waves = frequency Period = Amplitude = A = size of fluctuation Obvious choice is period; mathematically sound choice is frequency

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Null Hypothesis (important!) Null hypothesis: no time variation at all So = constant So, Quite important! Often neglected. Even the pros often forget this.

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Is it real? Fit produces a test statistic under the null hypothesis Is usually /degree of freedom (d) Linear: is significant (not just by accident) at 95% confidence 95% confidence means 5% false-alarm probability

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Meaning of significance Significance does not mean the signal is linear, sinusoidal, periodic, etc. It only means the null hypothesis is incorrect, i.e., the signal is not constant Important!!!

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Pre-whitening If you find a significant fit, then subtract the estimated signal, leaving residuals Analyze the residuals for more structure This process is called pre-whitening

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How to choose frequency? Test all reasonable values, get a strength of fit for each. Common is chi-square per degree of freedom (but there are many) Plot frequency.vs. fit – the Fourier transform (aka periodogram, aka power spectrum)

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Fourier decomposition Any periodic function of period P (frequency ) can be expressed as a Fourier series:

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Fundamental + harmonics For a pure sinusoid, expect response at frequency For a general periodic signal at a given frequency, expect a fundamental component at, as well as harmonics at frequencies etc.

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Lots of Fourier methods FFT: fast Fourier transform –Not just fast: its wicked fast –Requires even time spacing –Requires N=integer power of 2 –Beware! DFT: discrete Fourier transform –Applies to any time sampling, but incorrect results for highly uneven (as in astronomy!) –Beware!

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Problems from uneven time sampling Aliasing: false peaks, often from a periodic data density Aliases at Common in astronomy: data density have a period P = 1 yr = 365.2422 d, so Solution: pre-whitening

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Aliasing

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Aliasing: UZ Hya

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Problems from uneven time sampling Mis-calculation of frequency (slightly) and amplitude (greatly); sabotages prewhitening

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Solution: better Fourier methods (for astronomy) Lomb-Scargle modified periodogram –Improvement over FFT, DFT CLEAN spectrum –Bigger improvement DCDFT: date-compensated discrete Fourier transform (this is the one you want) CLEANEST spectrum: DCDFT-like for multiple frequencies

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DCDFT Much better estimates of period, amplitude

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Lets take a look Peranso (uses DCDFT and CLEANEST) Available from CBA Belgium –http://www.peranso.com

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Fourier transform (CLEANEST) of TU Cas

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Wavelet Analysis

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Wavelets Fit sine/cosine-like functions of brief duration Shift them through time Gives a time-frequency analysis

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Problems Same old same old: uneven time spacing, especially variable data density, invalidate the results But: even worse than Fourier Essentially useless for most astronomical data

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Wavelet methods DWT: discrete wavelet transform –Just not right for unevenly sampled data (astronomy!) Solution: WWZ = weighted wavelet Z-transform

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Lets take a look

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Data Analysis Do it yourself Use your eyes and brain Healthy skepticism tamino_9@hotmail.com Enjoy!

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