大數據天文學 時間序列分析 Michael Ting-Chang Yang 楊庭彰 1 / 70
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(cont.) 7 / 70
South Oscillation El Nino 8 / 70
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: time : quantity : period : amplitude : scale : shape : ordered data 10 / 70
variation/variability 11 / 70
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(cont.) 13 / 70
RR Lyrae 14 / 70
SS Cyg 15 / 70
Kepler transit 16 / 70
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Chan-Kao Chang et al. (2014) 18 / 70
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data reduction barycenter correction etc 22 / 70
: Barycenter Correction 23 / 70
: Barycenter Correction (cont.) 24 / 70
Fourier Transformation F(ξ) = f(x)e 2πixξ dx Discrete Fourier Transformation (DFT): (discrete Fourier transoformation) (DFT) 25 / 70
Fast Fourier Transformation A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse. : Cooley Tukey FFT algorithm (Numerical Recipes, Ch. 12) : 2 n 2 n 2 n 26 / 70
Light Curve: Single period 27 / 70
Fourier Power Spectrum 28 / 70
Light Curve: Two periods 29 / 70
Fourier Power Spectrum 30 / 70
Light Curve: Two periods with small trend 31 / 70
Fourier Power Spectrum 32 / 70
Light Curve: Two periods with big trend 33 / 70
Fourier Power Spectrum 34 / 70
Light Curve: step function 35 / 70
Fourier Power Spectrum 36 / 70
FFT : Reduce the complexity from O(n 2 ) to O(n log n) 10 8 FFT DFT : 2 n trend, gap power spectrum 37 / 70
Fourier Decomposition Technique Use fitting with different components of Fourier series to decompose the Fourier components Ngeow et al. (2013) 38 / 70
Gibbs Effect in Fourier Transformation 39 / 70
(observation gap) (uncertain trend of the raw data) 40 / 70
/ Fitting / 41 / 70
Lomb s Paper 42 / 70
Scargle s Paper 43 / 70
Lomb-Scargle Periodogram Define a time delay τ tan 2ωτ = j sin 2ωt j j cos 2ωt j The basic function will be orthogonal The power at frequency ω will be: P x (ω) = 1 2 [ j X j cos ω(t j τ)] 2 j cos 2 ω(t j τ) Then, generate a periodogram for investigation + [ j X j sin ω(t j τ)] 2 j sin 2 ω(t j τ) 44 / 70
The advantages / disadvantages of LS periodogram Advantages: Orthogonal set in each testing frequencies: avoid power leakage Good for the unevenly spaced data Disadvantages: Unweightted fitting Trend Originally designed for mono-frequency: power leakage if multiple frequencies are close enough 45 / 70
Fourier Analysis 46 / 70
Phase Based Methods Epoch folding: χ 2 maximazing Phase Dispersion Minimization (PDM) Analysis of Variance (AoV/ANOVA) 47 / 70
Epoch Folding Search trial period (folded light curve) χ 2 trial period χ 2 48 / 70
Epoch Folding Search (cont.) 49 / 70
Phase Dispersion Minimization 50 / 70
Light Curve: two sinsoidal function 51 / 70
Periodogram 52 / 70
Analysis of Variance (AoV, ANOVA) Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences among group means and their associated procedures, developed by statistician and evolutionary biologist Ronald Fisher. 53 / 70
Light Curve: two sinsoidal function 54 / 70
Periodogram 55 / 70
Periodogram / Spectrogram Periodogram: Time frequency Spectrogram: Time frequency power 56 / 70
: Dynamical Power Spectrum - : HHT : Wavelet 57 / 70
Dynamical Power Spectrum dynamical power spectrum 58 / 70
Hilbert-Huang Transformation 59 / 70
Comparison of Spectrograms 60 / 70
Data Quality Error of the measurement Time span sampling frequency Observation gap Alias Better data: Data with smaller errors Continuous data: less gaps Longer time span of the data 61 / 70
一個物理系統不同時間尺度 緻密雙星系統 62 / 70
: Fourier Analysis : Phase based analysis 63 / 70
Phase Diagram of Astronomical Objects 64 / 70
Some Coding Resources Python: numpy.fft scipy.fftpack scipy.signal.lombscargle pynfftls astroml.time_series.lomb_scargle C: FFTW R: lomb GUI: period04 https://www.univie.ac.at/tops/period04/ Matlab: plomb, fft 65 / 70
Kepler http://kepler.nasa.gov/ PTF http://www.ptf.caltech.edu/ CSS/CRTS http://crts.caltech.edu/ ASAS http://www.astrouw.edu.pl/asas/ Pan-STARRS http://pan-starrs.ifa.hawaii.edu/public/ LSST http://www.lsst.org/ GAIA http://sci.esa.int/gaia/ 66 / 70
It s the Golden Era of Astronomy 67 / 70