Technical notes on GSFC entry for 1B.1 challenge.

We have returned results for challenges 1B.1.1a-c, 1B.1.2, and
partial results for 1B.1.4. We looked briefly at 1B.1.3 but were
unable to quickly dig any signal out of the noise.

Our approach centered on an effort to apply the XSPEC program
(xspec.gsfc.nasa.gov), the standard tool used in X-ray astronomy to fit
models to energy spectra. We operated entirely in frequency space so
first FFT'd the MLDC data and wrote the results out in the standard
file formats used by XSPEC. The tool to do this includes an option to
only select a limited frequency range.

XSPEC provides a scriptable, interactive environment for matching
models to data, estimating best-fit parameters, and generating
confidence regions. The model we used was based on the
Cornish-Littenberg fast binary code (N. J. Cornish & T. B. Littenberg,
Phys. Rev. D 76, 083006 (2007)) which we modified to output the FFT of
the A and E TDI variables. All our analysis was then performed
simultaneously on the Real and Imaginary parts of the A and E FFTs.

Three features of XSPEC were used extensively :

1) Local minimization. XSPEC uses a tuned version of the
   Levenberg-Marquardt (L-M) optimization algorithm to start at an 
   initial guess for the parameters and move to a local minimum of 
   the fit statistic. This is very efficient but does not work well 
   for problems with many local minima.

2) Metropolis-Hastings Markov Chain Monte Carlo. XSPEC has extensive
   facilities for running MCMC chains using the Metropolis-Hastings
   algorithm. Multi-dimensional Normal and Cauchy proposal
   distributions are available. Annealed chains can also be created. 
   The chains are written out as FITS files which can then be analyzed 
   in detail using tools such as fv (fv.gsfc.nasa.gov).

3) Confidence regions. While XSPEC generates the Fisher matrix
   information after an L-M fit this is often not a good estimator for
   the confidence regions. To generate 1-D confidence ranges we used
   one of two methods. Following an L-M fit the XSPEC "error" command
   tries different values of the parameter of interest. For each value
   a separate fit is performed and the statistic evaluated. A search
   algorithm then zeroes in on the parameter values which give a
   previously specified change in the statistic value from that at the
   best fit (for 90% confidence ranges this difference is
   2.605). Alternatively, if an MCMC chain has been created then this
   is analyzed to find the minimum width range which contains the
   required fraction of the posterior probability density
   function. (note that the first method is Frequentist and the second
   Bayesian although this difference is usually elided).


Challenges 1B.1.1a-c

For these three cases we performed an initial search for the signal by
using a 13 bin sliding box and calculating excessive variance within
the box. We found that L-M fits had a tendency to get stuck in local minima
however by stepping through frequency values and doing an L-M fit at
each frequency for the other parameters we were able to determine the
global minima. This required about 400 frequency steps taking a few
minutes on a 2.4 GHz MacBook Pro. We estimated confidence regions both
based on L-M fitting and on a 50,000 step MCMC chain starting at the
best-fit parameters. For the MCMC proposal distribution we used a
Cauchy distribution with covariance set from the Fisher matrix divided
by a constant scaling factor (usually ~8) set to get a chain with 0.75
repeat fraction. The MCMC chains showed that the Psi and Phase
parameters are highly correlated (with a 2:1 periodicity) and the Iota
parameter has a double horned PDF with horns separated by pi/2.
(Note:In submittng the information in the web form, the fit information for 
these challenges was attached to the entry for 1B.1.4.)

Challenge 1B.1.2

We used the L-M method exclusively for this challenge. We started with 
the specified values for the frequency and sky position. We held the
sky position fixed but found that we had to allow the frequency to vary 
by up to 1e-7 Hz to give the best fit. For two pairs of sources, 12-13
and 17-18, the binaries were close in frequency so we fit for both 
simultaneously. 90% confidence ranges on all parameters were evaluated
using the error command. Once scripted the entire analysis runs in a
few minutes.

Challenge 1B.1.4

We did not attempt a full solution but experimented with applications
of the XSPEC MCMC routines in two frequency regions.

In the 3.01 < f < 3.012 mHz region which shows very high S/N we
performed a long (5 million step) MCMC run with two sources. This
found a best-fit but with large-enough residuals for us to speculate
that there are one or more additional sources in this region. We did
not explore this further since it is clearly not an optimum method. 

For the second region, 3.014 < f < 1.015 mHz, we interactively
experimented with a series of MCMC chains including models with 2 to 6
binaries. These were punctuated by occasional L-M fits. In all there
were ~15 chains with a total of \lesssim 200,000 chain steps,
requiring ~1hour of computation.  In the end, our best fit result
included 4 binaries, but the residuals were, apparently, not
exhausted.  There was not time to explore further.