Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Jeffrey D. Scargle

Space Science and Astrobiology DivisionNASA Ames Research Center

Fermi Gamma Ray Space Telescope

Special thanks:Jim Chiang, Jay Norris, and Greg Madejski, …

Applied Information Systems Research Program (NASA)Center for Applied Mathematics, Computation and Statistics (SJSU)

Institute for Pure and Applied Mathematics (UCLA)

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Bin-free Algorithms for Estimation of …

Light Curve Analysis (Bayesian Blocks) Auto- and Cross-

Correlation Functions Fourier Power Spectra (amplitude and phase) Wavelet Power Structure Functions

Energy-Dependent Time Lags (An Algorithm for Detecting Quantum Gravity Photon Dispersion in Gamma-Ray Bursts : DisCan. 2008 ApJ 673 972-980)

… from Energy- and Time-Tagged Photon Data… with Variable Exposure and Gaps

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

All of this will be in the

Handbook of Statistical Analysis of Event Data

… funded by the NASA AISR Program

MatLab CodeDocumentationExamplesTutorial

Contributions welcome!

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Variable Source

PropagationTo Observer

PhotonDetection

Luminosity: random or deterministic Photon Emission Independent Random Process (Poisson)

Random Detection of Photons (Poisson)

Correlations in source luminosity do not imply correlations in time series data!

Random Scintillation, Dispersion, etc.?

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

X = C * R + D

Any stationary process X can be represented as the convolution of a constant pulse shape C and

a (white) random process Rplus a linearly deterministic process D.

The Wold - von Neumann Decomposition Theorem

Moving Average Process

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Time Series Data

Binning

Time-Tagged Events Binned Event Times Time-To-Spill

Mixed Modes

Point Measurements

Fixed Equi-Variance

Any Standard Variability Analysis Tool:

Bayesian blocks, correlation, power spectra, structure

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

dt

Area = 1 / dtn / dtE / dt

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

dt’ = dt × exposure

Area = 1 / dt’n / dt’E / dt’

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Bayesian Blocks Piecewise-constant Model of Time Series Data

Optimum Partition of Interval, Maximizing Fitness Of Step Function Model

Segmentation of Interval into Blocks, Representing Data as Constant In the Blocks -- within Statistical Fluctuations

Histogram in Unequal Bins -- not Fixed A Priori but determined by Data

Studies in Astronomical Time Series Analysis. V. Bayesian Blocks, a New Method to Analyze Structure in Photon Counting Data, Ap. J. 504 (1998) 405.

An Algorithm for the Optimal Partitioning of Data on an Interval," IEEE Signal Processing Letters, 12 (2005) 105-108.

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

The optimizer is based on a dynamic programming concept of Richard Bellman

best = [ ]; last = [ ];for R = 1: num_cells [ best(R), last(R) ] = max( [0 best] + ... reverse( log_post( cumsum( data_cells(R:-1:1, :) ), prior, type ) ) );

if first > 0 & last(R) > first % Option: trigger on first significant block changepoints = last(R); return endend

% Now locate all the changepointsindex = last( num_cells );changepoints = [];while index > 1 changepoints = [ index changepoints ]; index = last( index - 1 );end

Global optimum of exponentially large search space in O(N2)!

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Cross- and Auto- Correlation Functions for unevenly spaced data

Edelson and Krolik:

The Discrete Correlation Function: a New Method for Analyzing Unevenly Sampled Variability DataAp. J. 333 (1988) 646

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

for id_2 = 1: num_xx_2 xx_2_this = xx_2( id_2 ); tt_2_this = tt_2( id_2 ); tt_lag = tt_2_this - tt_1 - tau_min; % time lags relative to this point index_tau = ceil( ( tt_lag / tau_bin_size ) + eps );

% The index of this array refers to the inputs tt and xx; % the values of the array are indices for the output variables % sf cd nv that are a function of tau. % Eliminate values of index_tau outside the chosen tau range: ii_tau_good = find( index_tau > 0 & index_tau <= tau_num ); index_tau_use = index_tau( ii_tau_good );

if ~isempty( index_tau_use ) % There are almost always duplicate values of index_tau; % mark and count the sets of unique index values ("clusters") ii_jump = find( diff( index_tau_use ) < 0 ); % cluster edges num_clust = length( ii_jump ) + 1; % number of clusters for id_clust = 1: num_clust % get index range for each cluster if id_clust == 1 ii_1 = 1; else ii_1 = ii_jump( id_clust - 1 ) + 1; end

if id_clust == num_clust ii_2 = length( index_tau_use ); else ii_2 = ii_jump( id_clust ); end ii_lag = index_tau_use( ii_1 ); % first of duplicates values is ok xx_arg = xx_1( ii_tau_good( ii_1 ): ii_tau_good( ii_2 ) ); sum_xx_arg = xx_2_this .* sum( xx_arg ); vec = ones( size( xx_arg ) ); cf( ii_lag ) = cf( ii_lag ) + sum_xx_arg; % correlation and structure fcn sf( ii_lag ) = sf( ii_lag ) + sum( ( xx_2_this * vec - xx_arg ) .^ 2 ); nv( ii_lag ) = nv( ii_lag ) + ii_2 - ii_1 + 1; err_1( ii_lag ) = err_1( ii_lag ) + sum_xx_arg .^ 2; err_2( ii_lag ) = err_2( ii_lag ) + std( xx_2_this * xx_arg ); end % for id_clust

end

end % for id_2

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Summary:A variety of new and standard time series analysis tools can be implemented for time- and/or energy tagged data.

Future:Many applications to TeV and other photon data.

Handbook of Statistical Analysis of Event DataContributions welcome!

Automatic variability analysis tools for High Energy Pipelines:

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

Backup

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

LAT

Statistical Analysis of High-Energy Astronomical Time Series

Jeff Scargle NASA Ames – Fermi Gamma Ray Space Telescope

LAT