GLAST lunch on HBT

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Jeffrey D. Scargle Space Science and
Astrobiology Division NASA 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)
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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
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All of this will be in the Handbook of
Statistical Analysis of Event Data funded by
the NASA AISR Program MatLab Code Documentation E
xamples Tutorial Contributions welcome!
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Variable Source
Propagation To Observer
Photon Detection

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

• Random
• Scintillation,
• Dispersion, etc.?

• Random Detection
• of Photons (Poisson)

Correlations in source luminosity do not imply
correlations in time series data!
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The Wold - von Neumann Decomposition Theorem
X C R D
Moving Average Process
Any stationary process X can be represented as
the convolution of a constant pulse shape C and
a (white) random process R plus a linearly
deterministic process D.
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Time Series Data
Time-Tagged Events
Binned Event Times
Time-To-Spill
Point Measurements
Mixed Modes
Binning
Any Standard Variability Analysis Tool Bayesian
blocks, correlation, power spectra, structure
Fixed
Equi-Variance
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Area 1 / dt n / dt E / dt
dt
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Area 1 / dt n / dt E / dt
dt dt exposure
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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.
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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-11,
) ), prior, type ) ) ) if first gt 0
last(R) gt first Option trigger on first
significant block changepoints
last(R) return end end Now locate all
the changepoints index last( num_cells
) changepoints while index gt 1
changepoints index changepoints index
last( index - 1 ) end
Global optimum of exponentially large search
space in O(N2)!
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Cross- and Auto- Correlation Functions for
unevenly spaced data Edelson and Krolik The
Discrete Correlation Function a New Method for
Analyzing Unevenly Sampled Variability Data Ap.
J. 333 (1988) 646
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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 gt 0 index_tau lt 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 ) lt 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
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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
Data Contributions welcome! Automatic
variability analysis tools for High Energy
Pipelines
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Backup
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LAT
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LAT