sherpa> PROJECTION [<dataset range> | ALLSETS] [ <arg_1> , ... ]
where <dataset range> = #, or more generally #:#,#:#,..., such that #
specifies a dataset number, and #:# represents an inclusive range of
datasets; one may specify multiple inclusive ranges by separating them
with commas. The default is to estimate limits using data from all
appropriate datasets.
The command-line arguments <arg_n> may be:
PROJECTION Command Arguments
| <sherpa_modelname>.{<paramname> | <#>} |
A specified model component parameter (e.g., GAUSS.pos). |
| <modelname>.{<paramname> | <#>} |
A specified model component parameter (e.g., g.pos). |
The user may configure PROJECTION via
the Sherpa state object structure proj.
The current values of the fields of this structure may be
displayed using the command print(sherpa.proj),
or using the more verbose Sherpa/S-Lang module
function list_proj().
The structure fields are:
proj Structure Fields
| fast |
If 1, use a fast optimization algorithm
(LEVENBERG-MARQUARDT
or
SIMPLEX)
regardless of the current
METHOD.
If 0, use the current
METHOD. |
| sigma |
Specifies the number of sigma
(i.e., the change in statistic).
|
Field values may be set using directly, e.g.,
sherpa> sherpa.proj.sigma = 2.6
NOTE: strict checking of value inputs is not done,
i.e., the user can errantly change arrays to scalars,
etc. To restore the default settings of the structure
at any time, use the Sherpa/S-Lang module function
restore_proj().
Confidence interval bounds are determined for each selected parameter
in turn. A given parameter's value is varied along a grid of values
while the values of all the other nominally thawed parameters are
allowed to float to new best-fit values (compare to UNCERTAINTY, where the values
of all the other nominally thawed parameters remain fixed to their
best-fit values). This method of estimating confidence interval
bounds gives truly accurate results only in special cases (see caveats
below).
Because PROJECTION estimates confidence intervals for
each parameter independently, the relationship between
sigma and the change in statistic value delta_S can be particularly simple:
sigma = the square root of delta_S for statistics
sampled from the chi-square distribution
and for the Cash statistic, and is approximately equal to
the square root of (2 * delta_S) for fits based on
the general log-likelihood (log(L)).
Confidence Intervals for the projection command
| 68.3% | 1.0 | 1.00 | 0.50 |
| 90.0% | 1.6 | 2.71 | 1.36 |
| 95.5% | 2.0 | 4.00 | 2.00 |
| 99.0% | 2.6 | 6.63 | 3.32 |
| 99.7% | 3.0 | 9.00 | 4.50 |
An estimated confidence interval is accurate if and only if:
-
the chi-square or log(L) surface in parameter space is
approximately shaped like a multi-dimensional paraboloid, and
-
the best-fit point is sufficiently far from parameter space boundaries.
One may determine if these conditions hold, for example, by plotting
the fit statistic as a function of each parameter's values (the curve
should approximate a parabola) and by examining contour plots of the
fit statistics made by varying the values of two parameters at a time
(the contours should be elliptical, and parameter space boundaries
should be no closer than approximately
3-sigma from the best-fit point).
The INTERVAL-PROJECTION
and
REGION-PROJECTION
commands can be used for these checks.
If either of these conditions does not hold, then the output from
PROJECTION may be meaningless except to give an idea of
the scale of the confidence intervals. To accurately
determine the confidence intervals, one would have to reparameterize
the model, or use Monte Carlo simulations or Bayesian methods.
The user may retrieve the upper- and lower-bound estimates
using the Sherpa/S-Lang module function
get_proj.
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