__init__(self,
fcn,
xall=None,
functkw=None,
parinfo=None,
ftol=1.e-10,
xtol=1.e-10,
gtol=1.e-10,
damp=0.,
maxiter=200,
factor=100.,
nprint=1,
iterfunct='default',
iterkw=None,
nocovar=0,
fastnorm=0,
rescale=0,
autoderivative=1,
quiet=0,
diag=None,
epsfcn=None,
debug=0)
(Constructor)
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Inputs:
fcn:
The function to be minimized. The function should return the weighted
deviations between the model and the data, as described above.
xall:
An array of starting values for each of the parameters of the model.
The number of parameters should be fewer than the number of measurements.
This parameter is optional if the parinfo keyword is used (but see
parinfo). The parinfo keyword provides a mechanism to fix or constrain
individual parameters.
Keywords:
autoderivative:
If this is set, derivatives of the function will be computed
automatically via a finite differencing procedure. If not set, then
fcn must provide the (analytical) derivatives.
Default: set (=1)
NOTE: to supply your own analytical derivatives,
explicitly pass autoderivative=0
fastnorm:
Set this keyword to select a faster algorithm to compute sum-of-square
values internally. For systems with large numbers of data points, the
standard algorithm can become prohibitively slow because it cannot be
vectorized well. By setting this keyword, MPFIT will run faster, but
it will be more prone to floating point overflows and underflows. Thus, setting
this keyword may sacrifice some stability in the fitting process.
Default: clear (=0)
ftol:
A nonnegative input variable. Termination occurs when both the actual
and predicted relative reductions in the sum of squares are at most
ftol (and status is accordingly set to 1 or 3). Therefore, ftol
measures the relative error desired in the sum of squares.
Default: 1E-10
functkw:
A dictionary which contains the parameters to be passed to the
user-supplied function specified by fcn via the standard Python
keyword dictionary mechanism. This is the way you can pass additional
data to your user-supplied function without using global variables.
Consider the following example:
if functkw = {'xval':[1.,2.,3.], 'yval':[1.,4.,9.],
'errval':[1.,1.,1.] }
then the user supplied function should be declared like this:
def myfunct(p, fjac=None, xval=None, yval=None, errval=None):
Default: {} No extra parameters are passed to the user-supplied
function.
gtol:
A nonnegative input variable. Termination occurs when the cosine of
the angle between fvec and any column of the jacobian is at most gtol
in absolute value (and status is accordingly set to 4). Therefore,
gtol measures the orthogonality desired between the function vector
and the columns of the jacobian.
Default: 1e-10
iterkw:
The keyword arguments to be passed to iterfunct via the dictionary
keyword mechanism. This should be a dictionary and is similar in
operation to FUNCTKW.
Default: {} No arguments are passed.
iterfunct:
The name of a function to be called upon each NPRINT iteration of the
MPFIT routine. It should be declared in the following way:
def iterfunct(myfunct, p, iter, fnorm, functkw=None,
parinfo=None, quiet=0, dof=None, [iterkw keywords here])
# perform custom iteration update
iterfunct must accept all three keyword parameters (FUNCTKW, PARINFO
and QUIET).
myfunct: The user-supplied function to be minimized,
p: The current set of model parameters
iter: The iteration number
functkw: The arguments to be passed to myfunct.
fnorm: The chi-squared value.
quiet: Set when no textual output should be printed.
dof: The number of degrees of freedom, normally the number of points
less the number of free parameters.
See below for documentation of parinfo.
In implementation, iterfunct can perform updates to the terminal or
graphical user interface, to provide feedback while the fit proceeds.
If the fit is to be stopped for any reason, then iterfunct should return a
a status value between -15 and -1. Otherwise it should return None
(e.g. no return statement) or 0.
In principle, iterfunct should probably not modify the parameter values,
because it may interfere with the algorithm's stability. In practice it
is allowed.
Default: an internal routine is used to print the parameter values.
Set iterfunct=None if there is no user-defined routine and you don't
want the internal default routine be called.
maxiter:
The maximum number of iterations to perform. If the number is exceeded,
then the status value is set to 5 and MPFIT returns.
Default: 200 iterations
nocovar:
Set this keyword to prevent the calculation of the covariance matrix
before returning (see COVAR)
Default: clear (=0) The covariance matrix is returned
nprint:
The frequency with which iterfunct is called. A value of 1 indicates
that iterfunct is called with every iteration, while 2 indicates every
other iteration, etc. Note that several Levenberg-Marquardt attempts
can be made in a single iteration.
Default value: 1
parinfo
Provides a mechanism for more sophisticated constraints to be placed on
parameter values. When parinfo is not passed, then it is assumed that
all parameters are free and unconstrained. Values in parinfo are never
modified during a call to MPFIT.
See description above for the structure of PARINFO.
Default value: None All parameters are free and unconstrained.
quiet:
Set this keyword when no textual output should be printed by MPFIT
damp:
A scalar number, indicating the cut-off value of residuals where
"damping" will occur. Residuals with magnitudes greater than this
number will be replaced by their hyperbolic tangent. This partially
mitigates the so-called large residual problem inherent in
least-squares solvers (as for the test problem CURVI,
http://www.maxthis.com/curviex.htm).
A value of 0 indicates no damping.
Default: 0
Note: DAMP doesn't work with autoderivative=0
xtol:
A nonnegative input variable. Termination occurs when the relative error
between two consecutive iterates is at most xtol (and status is
accordingly set to 2 or 3). Therefore, xtol measures the relative error
desired in the approximate solution.
Default: 1E-10
Outputs:
Returns an object of type mpfit. The results are attributes of this class,
e.g. mpfit.status, mpfit.errmsg, mpfit.params, npfit.niter, mpfit.covar.
.status
An integer status code is returned. All values greater than zero can
represent success (however .status == 5 may indicate failure to
converge). It can have one of the following values:
-16
A parameter or function value has become infinite or an undefined
number. This is usually a consequence of numerical overflow in the
user's model function, which must be avoided.
-15 to -1
These are error codes that either MYFUNCT or iterfunct may return to
terminate the fitting process. Values from -15 to -1 are reserved
for the user functions and will not clash with MPFIT.
0 Improper input parameters.
1 Both actual and predicted relative reductions in the sum of squares
are at most ftol.
2 Relative error between two consecutive iterates is at most xtol
3 Conditions for status = 1 and status = 2 both hold.
4 The cosine of the angle between fvec and any column of the jacobian
is at most gtol in absolute value.
5 The maximum number of iterations has been reached.
6 ftol is too small. No further reduction in the sum of squares is
possible.
7 xtol is too small. No further improvement in the approximate solution
x is possible.
8 gtol is too small. fvec is orthogonal to the columns of the jacobian
to machine precision.
.fnorm
The value of the summed squared residuals for the returned parameter
values.
.covar
The covariance matrix for the set of parameters returned by MPFIT.
The matrix is NxN where N is the number of parameters. The square root
of the diagonal elements gives the formal 1-sigma statistical errors on
the parameters if errors were treated "properly" in fcn.
Parameter errors are also returned in .perror.
To compute the correlation matrix, pcor, use this example:
cov = mpfit.covar
pcor = cov * 0.
for i in range(n):
for j in range(n):
pcor[i,j] = cov[i,j]/sqrt(cov[i,i]*cov[j,j])
If nocovar is set or MPFIT terminated abnormally, then .covar is set to
a scalar with value None.
.errmsg
A string error or warning message is returned.
.nfev
The number of calls to MYFUNCT performed.
.niter
The number of iterations completed.
.perror
The formal 1-sigma errors in each parameter, computed from the
covariance matrix. If a parameter is held fixed, or if it touches a
boundary, then the error is reported as zero.
If the fit is unweighted (i.e. no errors were given, or the weights
were uniformly set to unity), then .perror will probably not represent
the true parameter uncertainties.
*If* you can assume that the true reduced chi-squared value is unity --
meaning that the fit is implicitly assumed to be of good quality --
then the estimated parameter uncertainties can be computed by scaling
.perror by the measured chi-squared value.
dof = len(x) - len(mpfit.params) # deg of freedom
# scaled uncertainties
pcerror = mpfit.perror * sqrt(mpfit.fnorm / dof)
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