Bootstrap Segmentation Analysis and Expectation Maximization

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Presentation transcript:

Bootstrap Segmentation Analysis and Expectation Maximization to Detect and Characterize Sources Jeffrey.D.Scargle@nasa.gov Space Science Division NASA Ames Research Center Diffuse Emission & LAT Source Catalog SLAC Workshop: May 23, 2005

Problems (Solutions) Algorithm provides no Error Estimate Bootstrap Errors Block Posterior Probabilities Point Spread: Overlapping Sources Expectation Maximization (EM) Point Spread: Function of Energy Maximum Likelihood Models?

One Dimensional1 Example: Swift GRB Data 1. But everything applies to 2D and higher!

Expectation Maximization (EM) Initialize: Find a good guess at the mixture model: How Many Sources? Locations of the Sources Source Parameters (size, spectra, …) Iterate: From the model: Divide the data into pieces that are relevant to each Source separately Re-determine the Source Locations and Parameters by fitting to the data pieces in 1 Repeat as needed

The “Maximization” Step with Point Data Maximize the Unbinned log-Likelihood: Log(L) =  i log[ a(ti) + b ] -  w(t) [ a(t) + b ] Where the model for each source (pulse) is: X(t) = [ a(t) + b ] a = source parameters (location, size, … ) b = local background constant w(t) is the EM partitioning, or weighting, function NB: the term  i log[ w(ti) ] in the full log-likelihood is a constant, irrelevant for model fitting.

Science!

Time Series of N Discrete Events Bootstrap Method: Time Series of N Discrete Events For many iterations: Randomly select N of the observed events with replacement Analyze this sample just as if it were real data Compute mean and variance of the bootstrap samples Bias = result for real data – bootstrap mean RMS error derived from bootstrap variance Caveat: The real data does not have the repeated events in bootstrap samples. I am not sure what effect this has.

Piecewise Constant Model (partitions the data space) Signal modeled as constant over each partition element (block).

Optimum Partitions in Higher Dimensions Blocks are collections of Voronoi cells (1D,2D,...) Relax condition that blocks be connected Cell location now irrelevant Order cells by volume Theorem: Optimum partition consists of blocks that are connected in this ordering Now can use the 1D algorithm, O(N2) Postprocessing: identify connected block fragments

Blocks Block: a set of data cells Two cases: Connected (can't break into distinct parts) Not constrained to be connected Model = set of blocks Fitness function: F( Model ) = sum over blocks F( Block )

Connected vs. Arbitrary Blocks

2D Synthetic Bootstrap Example: Raw Data

Local Mean & Variance of Area/Energy (idea due to Bill Atwood)