Version 1.2.7

# Bayesian Blocks

## Bayesian Blocks Functions

Based upon the algorithms and MATLAB routines of Jeffrey Scargle, these routines take an event list (i.e., an ordered list of the times of detected events), and create a piecewise-constant rate representation of the lightcurve. That is, the lightcurve is represented as a series of contiguous "blocks" within which the event rate is modeled as a constant. The determination of the location of the "change points" (i.e., the boundaries between the blocks) and the rates within the blocks is made via a Bayesian analysis (assuming Poisson statistics). A user-selectable parameter assigns the prior probability for the number of blocks, which is roughly equivalent to assigning a significance level threshhold for each detected block. (The simple `frequentist' interpretation of this parameter is best used for guiding, for example, Monte Carlo simulations to gauge the true significance level.)

A description of the algorithm can be found in:

```     "Studies in Astronomical Time Series Analysis. VI.
Optimum Partition of the Interval: Bayesian Blocks,
Histograms, and Triggers",  by J.D. Scargle et al., 2003 (in preparation)
```

Two routines must be called by the user. The event times are read by `sitar_make_data_cells`, which then creates the input (either as a series of event "cells", or bins for a binned-mode) for `sitar_global_optimum`. The latter outputs the locations of the statistically significant "change points" as well as the rates, counts, and associated errors within each block. These routines are described below.

1. sitar_make_data_cells ( tt, type, max_delt, frame, tstart, tstop )
Assign events to "cells" described by the cell population and size.
Run as: `isis> cell = sitar_make_data_cells(tt,type,max_delt,frame,tstart,tstop);`

Inputs:
• `tt : ` The times of the detected events
• `type : ` Describe cells as "events" (`type=1` or `2`), with the size (i.e., normalized duration) being the distance from halfway from the previous event to halfway to the next event (`type=1`), or from the current event to right before the subsequent event (`type=2`). (Note: all cell populations are greater than or equal to one, and the total number of cells is less than or equal to the total number of events.) Alternatively, events can be assigned to bins of uniform size (`type=3`). (For this case, cells can have zero population, and the total number of cells can greatly exceed the total number of events.)
• `max_delt : ` Events closer together than `max_delt` are grouped together in a single cell. (Note: this is for grouping of consecutive events, such that the resulting cell duration can be greater than `max_delt`.)
• `frame : ` The output cell sizes are in units of `frame`, or this is the bin size for `type=3`.
• ` tstart, tstop : ` The start and stop times for creating the output cells.
Outputs:
• `cell : ` A structure with fields:
`cell.pops` - an array of the number of events per cell;
`cell.size` - the size (i.e. duration) of a cell in units of `frame` or the mean counts per bin (derived from the total lightcurve) in binned mode (`type=3`);
`cell.lo_t,.hi_t` - the start and stop times of the cell;
`cell.dtcor` - an array, currently set to unity, for storing "dead time corrections" (or "efficiency" corrections) of the cells. Must be changed externally to this subroutine, and currently only used in `sitar_global_optimum` to deadtime correct the output rates of the resulting blocks.

2. sitar_global_optimum( cell, ncp_prior, ev_type [; first, mean_prior, rate_prior, rate_low=#, rate_high=#, alpha=#, beta=#] )
Find the maximum likelihood location of the change points.
Run as: `isis> results = sitar_global_optimum(cell,ncp_prior,type);`

Inputs:
• `cell : ` A structure, with cell.pops, cell.size, cell.lo_t cell.hi_t,cell.dtcor (see `sitar_make_data_cells`)
• `ncp_prior : ` Parameter for prior on number of 'blocks'
• ` type : ` Identical to type from sitar_make_data_cells
Optional Qualifiers:
• `first : ` If set, the code runs in 'trigger' mode, and returns upon the first sign of a change, so long as it is greater than 'first' cells from the beginning of the lightcurve
event mode priors (type=1 or 2)
• Default prior is that p_1, the probability of one or more photons in a frame, is uniformly distributed from 0->1.
• `alpha : ` >=0 implies p_1 prior is (1+alpha) (1-p_1)^alpha
• `mean_prior : ` if set, prior on *rate* is exp(-Lambda/Lambda_0), where Lambda_0 is the mean rate from the entire observation. Supercedes any choice on alpha.
• `rate_prior : ` if set, prior is chosen to be uniform between rates ranging from `rate_low` -- `rate_high`, with defaults of `rate_low` = 1/3 mean rate and `rate_high` = 3 times mean rate. Supercedes any choice of alpha or mean_prior.
binned mode priors and posteriors (type=3)
• `alpha, beta :` The prior on the bin rate (sort of) goes as (Lambda)^(alpha-1) Exp(-beta*Lambda), where Lambda=rate, and the default is alpha=beta=1
• `max_like : ` Use maximum likelihood, log(Pmax) = N log(N/T) -N, as the posterior test function. Overrides any choice of alpha & beta for binned mode.
Outputs:
• `results.cpt :` ```Array of change point locations for the maximum likelihood solution (indices are specific to the cell input);```
• `results.last, .best :` (Diagnostic purposes only) Arrays of the location of the last change point and the associated maximum log probability
• `results.cts : ` Counts in each block
• `results.rate : ` Rate in each block
• ` results.err : ` Poisson error for the block rate
• `results.lo_t,.hi_t : ` | Times of lower (>=) and upper (<) block boundaries.

This page was last updated May 2, 2017 by Michael Nowak. To comment on it or the material presented here, send email to mnowak@space.mit.edu.