Parameter-space Searches in Forward Fitting

Tests of 1- and 2-dimensional searches through parameter space for various tilted elliptical
Gaussian models are shown here. (function cmp_gcount_count,det_index,map_ctr,dx,dy)

  1. Simulated count rate profiles for 100,000 c/s/SC from a Gaussian
    source superposed on model profiles using the same phase_map_ctr and
    Gaussian source.
  2. Contours of the Cash statistic for model Gaussian profile
    and Simulated profile.

    1000 cts/s/SC, widths=[0.1,0.1], SIM_XY=[31,31]

    Note that the Chi-square statistic (dotted) has multiple extrema,
    displaced from the Cash statistic minimum. This is typical behavior
    for count rates of this order. The 0.5,0.5 arcsec displacement of the
    Cash statistic minimum from (0,0) may be due to pixelization in the
    source simulation (hsi_image), although one of the signs appears to be

  3. Total Cash statistic for model Gaussian of guessed
    widths 1-100 arcsec, simulated width=8x8 arcsec.

    The Cash statistics for the fine subcollimators are
    underweighted in this test, but note that the
    minimum lies near the true width of 8. The Cash statistic
    should be weighted in proportion to the number of bins for a
    proper measure of goodness of fit.

  4. The Cash statistics computed separately for each detector as a
    function of guessed size (1-100 arcsec), for the 8x8 arcsec
    simulation of the previous figure. There are 3 things to note:
    • The finest subcollimators over resolve the source and give little size information;
    • The coarsest subcollimators put only an upper limit on source size
    • Only the subcollimator with resolution ~3 x source width shows a significant minimum
  5. Cash statistic for simulated 4x8 arcsec Gaussian as function of
    guessed tilt angle. As before, there are 3 things to note:
    • The finest subcollimators over resolve the source;
    • The coarsest subcollimators under resolve, and give no tilt information;
    • Only the subcollimators with angular resolutions from ~ 2 to ~5 times the widths give a minimum near the correct value of tilt (45 degrees).
    This again indicates that the parameter search must "know" about under- and over - resolution.

Last modified May 22, 2000.