Module confidence

Returns the probalitiy for two given parameter sets. nfix is the number of fixed parameters.

Calculates the confidence interval for parameters
from the given minimizer.

The parameter for which the ci is calculated will be varied, while
the remaining parameters are re-optimized for minimizing chi-square.
The resulting chi-square is used  to calculate the probability with
a given statistic e.g. F-statistic. This function uses a 1d-rootfinder
from scipy to find the values resulting in the searched confidence
region.

Parameters
----------
minimizer : Minimizer
    The minimizer to use, should be already fitted via leastsq.
p_names : list, optional
    Names of the parameters for which the ci is calculated. If None,
    the ci is calculated for every parameter.
sigmas : list, optional
    The probabilities (1-alpha) to find. Default is 1,2 and 3-sigma.
trace : bool, optional
    Defaults to False, if true, each result of a probability calculation
    is saved along with the parameter. This can be used to plot so
    called "profile traces".

Returns
-------
output : dict
    A dict, which contains a list of (sigma, vals)-tuples for each name.
trace_dict : dict
    Only if trace is set true. Is a dict, the key is the parameter which
    was fixed.The values are again a dict with the names as keys, but with
    an additional key 'prob'. Each contains an array of the corresponding
    values.

See also
--------
conf_interval2d

Other Parameters
----------------
maxiter : int
    Maximum of iteration to find an upper limit.
prob_func : ``None`` or callable
    Function to calculate the probability from the optimized chi-square.
    Default (``None``) uses built-in f_compare (F test).
verbose: bool
    print extra debuggin information. Default is ``False``.


Examples
--------

>>> from lmfit.printfuncs import *
>>> mini=minimize(some_func, params)
>>> mini.leastsq()
True
>>> report_errors(params)
... #report
>>> ci=conf_interval(mini)
>>> report_ci(ci)
... #report

Now with quantiles for the sigmas and using the trace.

>>> ci, trace=conf_interval(mini, sigmas=(0.25,0.5,0.75,0.999),trace=True)
>>> fixed=trace['para1']['para1']
>>> free=trace['para1']['not_para1']
>>> prob=trace['para1']['prob']

This makes it possible to plot the dependence between free and fixed.

Calculates confidence regions for two fixed parameters.

The method is explained in *conf_interval*: here we are fixing
two parameters.

Parameters
----------
minimizer : minimizer
    The minimizer to use, should be already fitted via leastsq.
x_name : string
    The name of the parameter which will be the x direction.
y_name : string
    The name of the parameter which will be the y direction.
nx, ny : ints, optional
    Number of points.
limits : tuple: optional
    Should have the form ((x_upper, x_lower),(y_upper, y_lower)). If not
    given, the default is 5 std-errs in each direction.

Returns
-------
x : (nx)-array
    x-coordinates
y : (ny)-array
    y-coordinates
grid : (nx,ny)-array
    grid contains the calculated probabilities.

Examples
--------

>>> from lmfit.printfuncs import *
>>> mini=minimize(some_func, params)
>>> mini.leastsq()
True
>>> x,y,gr=conf_interval2d('para1','para2')
>>> plt.contour(x,y,gr)

Other Parameters
----------------
prob_func : ``None`` or callable
    Function to calculate the probability from the optimized chi-square.
    Default (``None``) uses built-in f_compare (F test).