1. Data Analysis Do it yourself!

2. 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!

3. 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 …

4. 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

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

6. Average and sample standard deviation Average Sample standard deviation

7. 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

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

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

10. Visual Inspection

11. Lets take a look

12. Fourier Analysis

13. 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

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

15. 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

16. 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!!!

17. 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

18. 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)

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

20. 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.

21. 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!

22. 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

23. Aliasing

24. Aliasing: UZ Hya

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

26. 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

27. DCDFT Much better estimates of period, amplitude

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

31. Fourier transform (CLEANEST) of TU Cas

32. Wavelet Analysis

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

34. 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

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

36. Lets take a look

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