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fit
fit data
Find the best fit model parameters for the current data by minimizing the current statistic.
| Syntax: | fit | <fit method parameters> |
|---|
The arguments to fit depend on the fitting method currently in use. See the method command for details (and for the usage of the USE_CHAIN_RULE option in the user's startup Xspec.init file). Output from the fit command also depends on the fitting method currently in use.
Using the Levenberg-Marquardt algorithm, the parameters accepted are the maximum <number of iterations> before the user is prompted, the <critical delta>, which is the (absolute, not fractional) change in the statistic between iterations less than which the fit is deemed to have converged, and <critical beta>.
The <critical beta> provides an optional second stopping criterion, and it refers to the |beta|/N value reported during a Levenberg-Marquardt fit. This is the norm of the vector derivatives of the statistic with respect to the parameters divided by the number of parameters. At the best fit this should be zero, and so provides another measure of how well the fit is converging. <critical beta> is set to a negative value by default, which renders it inactive.
Including the string delay as an argument to fit turns on delayed gratification. It is turned off by nodelay. Delayed gratification modifies the way the damping parameter is set and has been shown in many cases to speed up convergence. The default is nodelay.
If <number of iterations>, <critical delta>, <critical beta>, delay, or nodelay is entered through the fit command, it also becomes the future default value for the currently loaded fit method (ie. Levenberg-Marquardt).
Examples:
XSPEC> fit // Fit with the default number of iterations and critical delta // chi-squared. XSPEC> fit 60 // Fit with 60 as the number of iterations. XSPEC> fit 50 1.e-3 // Fit with 1.e-3 as the critical delta. XSPEC> fit 50 1.e-3 20. // Same fit, but will now use |beta|/N = 20.0 as another stopping // criterion in addition to that of the critical delta. XSPEC> fit delay // Same fit, but will now use delayed gratification.
Global optimization (the global keyword)
| Syntax: | fit | global [<maxgen>[<popsize>]] |
|---|
A plain fit is a local optimizer: from the current parameter values it descends to the nearest minimum of the statistic. On a multi-modal or degenerate parameter space — common when fitting several physically distinct models to the same data, or a model with near-degenerate components — it can converge to a local minimum that is far from the global best fit, with the result depending on the starting point.
Including the keyword global (any three-or-more-character prefix of global) prefixes the fit with a derivative-free global search. XSPEC runs a Differential Evolution sweep over the soft-limit box of every thawed parameter, moves to the best point it finds, and then polishes that point with the current method (Levenberg–Marquardt, migrad, or simplex). Because the search uses only statistic evaluations and never derivatives, it works for the entire model library, including local models that provide no analytic gradients. The final statistic, parameter errors, and covariance are produced by the ordinary polish, so they are identical in form to those of a normal fit.
The search box is the soft (“min”/“max”) limit range of each parameter,
which is always finite, so fit global never refuses to
start. A positive parameter whose range spans more than four decades (a
column density or normalisation, say) is searched in 
so the
wide scale does not swamp the others; a parameter with a soft minimum of
zero and a very wide range draws a non-fatal warning, since its scale
cannot be inferred. The optional <maxgen>sets the number of
Differential Evolution generations (default 200) and <popsize>the
population size (default automatic,
for 
thawed parameters); these are the only tunable controls.
fit global is never-regress: if the global search
plus polish does not improve on the statistic of the current fit, the
original parameters are restored and a message is printed, so running it on
an already-good fit cannot make it worse. The run is reproducible under
xset seed, and the Differential Evolution likelihood batch can be
spread over 
processes with parallel global <N>.
After the fit, any other distinct minima within
(about 
for one degree of freedom) of the best fit are
reported as competitive basins, with their
and
parameter values. These flag genuine degeneracies that a single best-fit
number hides; the suggestion is to test them as distinct models with
compare. The basin list is also available in PyXspec as
Fit.globalBasins after Fit.globalFit().
Example:
XSPEC> fit global
... Differential Evolution search, then a normal polish ...
fit global: 1 additional competitive basin(s) within delta-stat < 9
of the best fit (stat = 243.95):
basin dStat stat parameters(in fit order)
2 3.1846 247.14 2.0014, 0.00039876, ...
(test these as distinct models with `compare`.)
XSPEC> fit global 300 40
// Run a longer global search: 300 generations, population 40.