Methods for Cross-Analyzing Radio and ray Time Series Data Fermi Marries Jansky
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1 Methods for Cross-Analyzing Radio and ray Time Series Data Fermi Marries Jansky Jeff Scargle NASA Ames Research Center Fermi Gamma Ray Space Telescope Special Thanks to Jim Chiang, Jay Norris, Brad Jackson, Roger Blandford, Tony Readhead, Walter Max Moerbeck, Joey Richards, Vasiliki Pavlidou, Anne Lahteenmaki
2 Prac>cal Time Series Analysis Methods Data Issues Light Curve Representa>ons (Data Cells) ScaHer Plots Correla>on Func>ons (Edelson and Krolik) Spectra Amplitude (Power) Phase Wavelet Transform (Scalogram) Wavelet Power (Scalegram) Structure Func>ons Time Scale/Time Frequency Analysis Cau>ons: sta>onarity, nonlinearity, correla>ons,
3 Data Issues FmJ 2010 Transform event data to (x,t) data cells Sampling: study interval distribu>on hist( diff( t ) ) Get and understand error distribu>on Random Systema>c
4 Light Curve Representa>on FmJ 2010 Gamma Ray Event data: A cell for each photon ( 1/dt n, dt n ) Independent events? Flux Measurements: Data cell trivial: ( x n, t n ) Error distribu>on? Op>mal blocks (piecewise constant)
5 Height = 1 / dt n / dt E / dt dt
6 Area = 1 / dt n / dt E / dt dt = dt exposure
7 Preliminary
8 Preliminary
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10 ScaHer Plots Auto Self dependency Introduce lag: X(t+τ) vs X(t) Interpolate as needed Preliminary FmJ 2010 Cross Joint Probability Distribu>on Introduce lag: X(t+τ) vs Y(t) Interpolate X,Y to same >mes Show Time Sequence!?
11 Fm Edelson and Krolik: The Discrete Correla>on Func>on: a New Method for Analyzing Unevenly Sampled Variability Data, Ap. J. 333, 1988, 646 star>ng point for all else! J2 01 0
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15 Preliminary
16 Time Frequency/Time Scale Analysis Transform to a new view of the 0me series informa0on. A Reality in joint >me & frequency (or scale) representa>on Atomic decomposi>on Time frequency atoms Over complete representa>ons Op>mal Basis Pursuit (Mallat), etc. Uncertainty Principle: T F resolu>on tradeoff Non sta>onary processes Flares Trends & Modula>ons Sta>s>cal change points Instantaneous Frequency Local vs. Global structure Interference (cross terms in bi linear representa>on) Time Frequency/Time Scale Analysis (Temps Fréquence) Patrick Flandrin hhp://perso.ens lyon.fr/patrick.flandrin/publis.html; A Wavelet tour of Signal Processing (Une Explora>on des Signaux en OndeleHes) Stéphane Mallat
17 Preliminary
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21 Mul> taper Analysis (Thomson 1982) Tapers (windows) reduce sidelobe leakage = bias Incomplete use of data loss of informa>on Mul>tapers recover this informa>on Leakage minimiza>on = eigenvalue problem Eigenfunc>ons: efficient window func>ons Eigenvalues measure effec>veness determine how many terms to include Spectral Analysis for Physical Applica3ons: Mul3taper and Conven3onal Univariate Techniques, Don Percival and Andrew Walden (1993)
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23 Preliminary
24 Preliminary Mul> tapered
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27 Sta>onarity vs. Non Sta>onarity Formal defini>on requires infinite amount of data Local sta>onarity depends on scale Construct sta>onarity measure S[ x(t) ] E.g. variance of TF distribu>on vs. >me marginal Any such measure has sta>s>cal fluctua>ons Simulate surrogate data: scramble Fourier phase Construct distribu>on of S( surrogate data ) Tes3ng Sta3onarity with Time Frequency Surrogates, Jun Xiao, Pierre Borgnat, and Patrick Flandrin
28 Fm From: Flandrin & Borgnat Revisi>ng and tes>ng sta>onarity, 2008 interpreted as sta>onary or nonsta>onary depending on the observa>on scale TL: nonsta>onary TR: sta>onary (periodic) BL: nonsta>onary BR: sta>onary (homogeneous texture) J2 01 0
29 behavior Func1on Domain Range Auto Cross Physical Interp Bayesian blk. Light Curve Time Flux mul>var. BB Flares, events etc. ScaHer Plot Flux 1 Flux 2 Dependency (not just cor.) Correla>on Lag <X 2 > <XY> Correlated behavior/lags Spectrum Frequency Power Periodicity 1/f noise Phase Shizs, lags Structure Lag <X 2 > <XY> Correlated behavior/lags Scalogram Scale/Time Power Dynamic behavior Scalegram Scale Power 1/f noise QPOs Distribu>on Time/scale/ frequency Power Dynamic
30 Prac>cal Sugges>ons Study distribu>on of sample intervals dt n = t n+1 t n Never subtract mean of >me series Edelson and Krolik CF is the source of all other analysis Use self terms in E&K CF to assess observa>onal errors Don t confuse source noise with observa>onal noise (doubly stochas>c, or Cox processes) Source correla>on does not imply correlated errors H0: All AGN are iden>cal stochas>c dynamical systems Any sta>onary random process is exactly shot noise (random pulses; the Wold Decomposi>on Theorem) Linearity is a physical property, not a >me series one Do not bin data (unless absolutely necessary)
31 Backup Slides
32 Variable Source Propaga>on To Observer Photon Detec>on Luminosity: random or determinis>c Photon Emission Independent Random Process (Poisson) Random Scin>lla>on, Dispersion, etc.? Random Detec>on of Photons (Poisson) Correla>ons in source luminosity do not imply correla>ons in >me series data!
33 Fm J2 Preliminary 01 0
34 All of this will be in the Handbook of Sta3s3cal Analysis of Event Data funded by the NASA AISR Program MatLab Code Documenta>on Examples Tutorial Contribu>ons welcome!
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