=======================================
Performing Fits, Analyzing Outputs
=======================================

As shown in the previous sections, a simple fit can be performed with
the :func:`minimize` function.    For more sophisticated modeling,
the :class:`Minimizer` class can be used to gain a bit more control,
especially when using complicated constraints.


The :func:`minimize` function
===============================

The minimize function takes a function to minimize, a dictionary of
:class:`Parameter` , and several optional arguments.    See
:ref:`fit-func-label` for details on writing the function to minimize.

.. function:: minimize(function, params, args=None, kws=None, method='leastsq', **leastsq_kws)

   find values for the params so that the sum-of-squares of the returned array
   from function is minimized.

   :param function:  function to return fit residual.  See :ref:`fit-func-label` for details.
   :type  function:  callable.
   :param params:  a dictionary of Parameters.  Keywords must be strings
                   that match ``[a-z_][a-z0-9_]*`` and is not a python
                   reserved word.  Each value must be :class:`Parameter`.
   :type  params:  dict
   :param args:  arguments tuple to pass to the residual function as  positional arguments.
   :type  args:  tuple
   :param kws:   dictionary to pass to the residual function as keyword arguments.
   :type  kws:  dict
   :param method:  name of fitting method to use. See  :ref:`fit-engines-label` for details
   :type  method:  string
   :param leastsq_kws:  dictionary to pass to scipy.optimize.leastsq
   :type  leastsq_kws:  dict
   :return: Minimizer object, which can be used to inspect goodness-of-fit
            statistics, or to re-run fit.

   For backward compatibility, the keyword `engine` is retained as a synonym for `method`,
   but this should be considered deprecated.

   On output, the params will be updated with best-fit values and, where
   appropriate, estimated uncertainties and correlations.  See
   :ref:`fit-results-label` for further details.

..  _fit-func-label:

Writing a Fitting Function
===============================

An important component of a fit is writing a function to be minimized in
the least-squares sense.   Since this function will be called by other
routines, there are fairly stringent requirements for its call signature
and return value.   In principle, your function can be any python callable,
but it must look like this:

.. function:: func(params, *args, **kws):

   calculate objective residual to be minimized from parameters.

   :param params: parameters.
   :type  params:  dict
   :param args:  positional arguments.  Must match ``args`` argument to :func:`minimize`
   :param kws:   keyword arguments.  Must match ``kws`` argument to :func:`minimize`
   :return: residual array (generally data-model) to be minimized in the least-squares sense.
   :rtype: numpy array.  The length of this array cannot change between calls.


A common use for the positional and keyword arguments would be to pass in other
data needed to calculate the residual, including such things as the data array,
dependent variable, uncertainties in the data, and other data structures for the
model calculation.

The objective function should return the value to be minimized.  For the
Levenberg-Marquardt algorithm from :meth:`leastsq`, this returned value **must** be an
array, with a length greater than or equal to the number of fitting variables in the
model.  For the other methods, the return value can either be a scalar or an array.  If an
array is returned, the sum of squares of the array will be sent to the underlying fitting
method, effectively doing a least-squares optimization of the return values.


Since the function will be passed in a dictionary of :class:`Parameters`, it is advisable
to unpack these to get numerical values at the top of the function.  A simple example
would look like::

    def residual(pars, x, data=None, eps=None):
        # unpack parameters:
        #  extract .value attribute for each parameter
        amp = pars['amp'].value
        period = pars['period'].value
        shift = pars['shift'].value
        decay = pars['decay'].value

        if abs(shift) > pi/2:
            shift = shift - sign(shift)*pi

        if abs(period) < 1.e-10:
            period = sign(period)*1.e-10

        model = amp * sin(shift + x/period) * exp(-x*x*decay*decay)

        if data is None:
            return model
	if eps is None
            return (model - data)
        return (model - data)/eps

In this example, ``x`` is a positional (required) argument, while the ``data``
array is actually optional (so that the function returns the model calculation
if the data is neglected).   Also note that the model calculation will divide
``x`` by the varied value of the 'period' Parameter.  It might be wise to
make sure this parameter cannot be 0.   It would be possible to use the bounds
on the :class:`Parameter` to do this::

    params['period'] = Parameter(value=2, min=1.e-10)

but might be wiser to put this directly in the function with::

        if abs(period) < 1.e-10:
            period = sign(period)*1.e-10


..  _fit-engines-label:

Choosing Different Fitting Methods
===========================================

By default, the `Levenberg-Marquardt
<http://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm>`_ algorithm is
used for fitting.  While often criticized, including the fact it finds a
*local* minima, this approach has some distinct advantages.  These include
being fast, and well-behaved for most curve-fitting needs, and making it
easy to estimate uncertainties for and correlations between pairs of fit
variables, as discussed in :ref:`fit-results-label`.

Alternative algorithms can also be used. These include `simulated annealing
<http://en.wikipedia.org/wiki/Simulated_annealing>`_ which promises a
better ability to avoid local minima, and `BFGS
<http://en.wikipedia.org/wiki/Limited-memory_BFGS>`_, which is a
modification of the quasi-Newton method.

To select which of these algorithms to use, use the ``method`` keyword to the
:func:`minimize` function or use the corresponding method name from the
:class:`Minimizer` class as listed in the
:ref:`Table of Supported Fitting Methods <fit-engine-table>`.

.. _fit-engine-table:

 Table of Supported Fitting Methods:

 +-----------------------+--------------------+---------------------+-------------------------+
 | Fitting               | ``method`` arg to  | :class:`Minimizer`  | ``method`` arg to       |
 | Method                | :func:`minimize`   | method              | :meth:`scalar_minimize` |
 +=======================+====================+=====================+=========================+
 | Levenberg-Marquardt   |  ``leastsq``       | :meth:`leastsq`     |   Not available         |
 +-----------------------+--------------------+---------------------+-------------------------+
 | Nelder-Mead           |  ``nelder``        | :meth:`fmin`        | ``Nelder-Mead``         |
 +-----------------------+--------------------+---------------------+-------------------------+
 | L-BFGS-B              |  ``lbfgsb``        | :meth:`lbfgsb`      | ``L-BFGS-B``            |
 +-----------------------+--------------------+---------------------+-------------------------+
 | Simulated Annealing   |  ``anneal``        | :meth:`anneal`      | ``Anneal``              |
 +-----------------------+--------------------+---------------------+-------------------------+
 | Powell                |  ``powell``        |                     | ``Powell``              |
 +-----------------------+--------------------+---------------------+-------------------------+
 | Conjugate Gradient    |  ``cg``            |                     | ``CG``                  |
 +-----------------------+--------------------+---------------------+-------------------------+
 | Newtown-CG            |  ``newton``        |                     | ``Newton-CG``           |
 +-----------------------+--------------------+---------------------+-------------------------+
 | COBYLA                |  ``cobyla``        |                     |  ``COBYLA``             |
 +-----------------------+--------------------+---------------------+-------------------------+
 | Sequential Linear     |  ``slsqp``         |                     |  ``SLSQP``              |
 | Squares Programming   |                    |                     |                         |
 +-----------------------+--------------------+---------------------+-------------------------+


.. note::

   Use of :meth:`scipy.optimize.minimize` requires scipy 0.11 or higher.

.. note::

   The objective function for the Levenberg-Marquardt method **must**
   return an array, with more elements than variables.  All other methods
   can return either a scalar value or an array.


.. warning::

  The Levenberg-Marquardt method is *by far* the most tested fit method,
  and much of this documentation assumes that this is the method used.  For
  example, many of the fit statistics and estimates for uncertainties in
  parameters discussed in :ref:`fit-results-label` are done only for the
  ``leastsq`` method.

In particular, the simulated annealing method appears to not work
correctly.... understanding this is on the ToDo list.

..  _fit-results-label:

Goodness-of-Fit and estimated uncertainty and correlations
===================================================================

On a successful fit using the `leastsq` method, several goodness-of-fit
statistics and values related to the uncertainty in the fitted variables will be
calculated.  These are all encapsulated in the :class:`Minimizer` object for the
fit, as returned by :func:`minimize`.  The values related to the entire fit are
stored in attributes of the :class:`Minimizer` object, as shown in :ref:`Table
of Fit Results <goodfit-table>` while those related to each fitted variables are
stored as attributes of the corresponding :class:`Parameter`.


.. _goodfit-table:

 Table of Fit Results:  These values, including the standard Goodness-of-Fit statistics,
 are all attributes of the :class:`Minimizer` object returned by :func:`minimize`.

+----------------------+----------------------------------------------------------------------------+
| :class:`Minimizer`   | Description / Formula                                                      |
| Attribute            |                                                                            |
+======================+============================================================================+
|    nfev              | number of function evaluations                                             |
+----------------------+----------------------------------------------------------------------------+
|    success           | boolean (``True``/``False``) for whether fit succeeded.                    |
+----------------------+----------------------------------------------------------------------------+
|    errorbars         | boolean (``True``/``False``) for whether uncertainties were estimated.     |
+----------------------+----------------------------------------------------------------------------+
|    message           | message about fit success.                                                 |
+----------------------+----------------------------------------------------------------------------+
|    ier               | integer error value from scipy.optimize.leastsq                            |
+----------------------+----------------------------------------------------------------------------+
|    lmdif_message     | message from scipy.optimize.leastsq                                        |
+----------------------+----------------------------------------------------------------------------+
|    nvarys            | number of variables in fit  :math:`N_{\rm varys}`                          |
+----------------------+----------------------------------------------------------------------------+
|    ndata             | number of data points:  :math:`N`                                          |
+----------------------+----------------------------------------------------------------------------+
|    nfree `           | degrees of freedom in fit:  :math:`N - N_{\rm varys}`                      |
+----------------------+----------------------------------------------------------------------------+
|    residual          | residual array (return of :func:`func`:  :math:`{\rm Resid}`               |
+----------------------+----------------------------------------------------------------------------+
|    chisqr            | chi-square: :math:`\chi^2 = \sum_i^N [{\rm Resid}_i]^2`                    |
+----------------------+----------------------------------------------------------------------------+
|    redchi            | reduced chi-square: :math:`\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}` |
+----------------------+----------------------------------------------------------------------------+

Note that the calculation of chi-square and reduced chi-square assume that the
returned residual function is scaled properly to the uncertainties in the data.
For these statistics to be meaningful, the person writing the function to
be minimized must scale them properly.

After a fit using using the :meth:`leastsq` method has completed successfully,
standard errors for the fitted variables and correlations between pairs of
fitted variables are automatically calculated from the covariance matrix.
The standard error (estimated :math:`1\sigma` error-bar) go into the
:attr:`stderr` attribute of the Parameter.  The correlations with all other
variables will be put into the :attr:`correl` attribute of the Parameter --
a dictionary with keys for all other Parameters and values of the
corresponding correlation.

In some cases, it may not be possible to estimate the errors and
correlations.  For example, if a variable actually has no practical effect
on the fit, it will likely cause the covariance matrix to be singular,
making standard errors impossible to estimate.  Placing bounds on varied
Parameters makes it more likely that errors cannot be estimated, as being
near the maximum or minimum value makes the covariance matrix singular.  In
these cases, the :attr:`errorbars` attribute of the fit result
(:class:`Minimizer` object) will be ``False``.


..  _fit-minimizer-label:

Using the :class:`Minimizer` class
=======================================

For full control of the fitting process, you'll want to create a
:class:`Minimizer` object, or at least use the one returned from the
:func:`minimize` function.

.. class:: Minimizer(function, params, fcn_args=None, fcn_kws=None, iter_cb=None, scale_covar=True, **kws)

   creates a Minimizer, for fine-grain access to fitting methods and attributes.

   :param function:  objective function to return fit residual.  See :ref:`fit-func-label` for details.
   :type  function:  callable.
   :param params:  a dictionary of Parameters.  Keywords must be strings
                   that match ``[a-z_][a-z0-9_]*`` and is not a python
                   reserved word.  Each value must be :class:`Parameter`.
   :type  params:  dict
   :param fcn_args:  arguments tuple to pass to the residual function as  positional arguments.
   :type  fcn_args: tuple
   :param fcn_kws:  dictionary to pass to the residual function as keyword arguments.
   :type  fcn_kws:  dict
   :param iter_cb:  function to be called at each fit iteration
   :type  iter_cb:  callable or ``None``
   :param scale_covar:  flag for scaling covariance matrix and uncertainties to reduced chi-square (``leastsq`` only)
   :type  scale_cover:  boolean, default ``True``
   :param kws:      dictionary to pass as keywords to the underlying scipy.optimize method.
   :type  kws:      dict
   :return: Minimizer object, which can be used to inspect goodness-of-fit
            statistics, or to re-run fit.


The Minimizer object has a few public methods:

.. method:: leastsq(scale_covar=True, **kws)

   perform fit with Levenberg-Marquardt algorithm.  Keywords will be passed directly to
   `scipy.optimize.leastsq <http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.leastsq.html>`_.
   By default, numerical derivatives are used, and the following arguments are set:

    +------------------+----------------+------------------------------------------------------------+
    | :meth:`leastsq`  |  Default Value | Description                                                |
    | arg              |                |                                                            |
    +==================+================+============================================================+
    |   xtol           |  1.e-7         | Relative error in the approximate solution                 |
    +------------------+----------------+------------------------------------------------------------+
    |   ftol           |  1.e-7         | Relative error in the desired sum of squares               |
    +------------------+----------------+------------------------------------------------------------+
    |   maxfev         | 2000*(nvar+1)  | maximum number of function calls (nvar= # of variables)    |
    +------------------+----------------+------------------------------------------------------------+
    |   Dfun           | ``None``       | function to call for Jacobian calculation                  |
    +------------------+----------------+------------------------------------------------------------+


.. method:: anneal(**kws)

   perform fit with Simulated Annealing.  Keywords will be passed directly to
   `scipy.optimize.anneal <http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.anneal.html>`_.

    +------------------+----------------+------------------------------------------------------------+
    | :meth:`anneal`   |  Default Value | Description                                                |
    | arg              |                |                                                            |
    +==================+================+============================================================+
    |   schedule       | ``cauchy``     | annealing schedule                                         |
    +------------------+----------------+------------------------------------------------------------+
    |   maxiter        |  2000*(nvar+1) | maximum number of iterations                               |
    +------------------+----------------+------------------------------------------------------------+

    For me, this Simulated Annealing appears to never work.

.. method:: lbfgsb(**kws)

   perform fit with L-BFGS-B algorithm.  Keywords will be passed directly to
   `scipy.optimize.fmin_l_bfgs_b <http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin_l_bfgs_b.html>`_.


    +------------------+----------------+------------------------------------------------------------+
    | :meth:`lbfgsb`   |  Default Value | Description                                                |
    | arg              |                |                                                            |
    +==================+================+============================================================+
    |   factr          | 1000.0         |                                                            |
    +------------------+----------------+------------------------------------------------------------+
    |   approx_grad    |  ``True``      | calculate approximations of gradient                       |
    +------------------+----------------+------------------------------------------------------------+
    |   maxfun         | 2000*(nvar+1)  | maximum number of function calls (nvar= # of variables)    |
    +------------------+----------------+------------------------------------------------------------+

.. method:: fmin(**kws)

   perform fit with Nelder-Mead downhill simplex algorithm.  Keywords will be passed directly to
   `scipy.optimize.fmin <http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin.html>`_.

    +------------------+----------------+------------------------------------------------------------+
    | :meth:`fmin`     |  Default Value | Description                                                |
    | arg              |                |                                                            |
    +==================+================+============================================================+
    |   ftol           | 1.e-4          | function tolerance                                         |
    +------------------+----------------+------------------------------------------------------------+
    |   xtol           | 1.e-4          | parameter tolerance                                        |
    +------------------+----------------+------------------------------------------------------------+
    |   maxfun         | 5000*(nvar+1)  | maximum number of function calls (nvar= # of variables)    |
    +------------------+----------------+------------------------------------------------------------+


.. method:: scalar_minimize(method='Nelder-Mead', hess=None, tol=None, **kws)

   perform fit with any of the scalar minimization algorithms supported by
   `scipy.optimize.minimize <http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html>`_.

    +-------------------------+-----------------+-----------------------------------------------------+
    | :meth:`scalar_minimize` | Default Value   | Description                                         |
    | arg                     |                 |                                                     |
    +=========================+=================+=====================================================+
    |   method                | ``Nelder-Mead`` | fitting method                                      |
    +-------------------------+-----------------+-----------------------------------------------------+
    |   tol                   | 1.e-7           | fitting and parameter tolerance                     |
    +-------------------------+-----------------+-----------------------------------------------------+
    |   hess                  | None            | Hessian of objective function                       |
    +-------------------------+-----------------+-----------------------------------------------------+


.. method:: prepare_fit(**kws)

   prepares and initializes model and Parameters for subsequent
   fitting. This routine prepares the conversion of :class:`Parameters`
   into fit variables, organizes parameter bounds, and parses, checks and
   "compiles" constrain expressions.


   This is called directly by the fitting methods, and it is generally not
   necessary to call this function explicitly.  An exception is when you
   would like to call your function to minimize prior to running one of the
   minimization routines, for example, to calculate the initial residual
   function.  In that case, you might want to do something like::

      myfit = Minimizer(my_residual, params,  fcn_args=(x,), fcn_kws={'data':data})

      myfit.prepare_fit()
      init = my_residual(p_fit, x)
      pylab.plot(x, init, 'b--')

      myfit.leastsq()

   That is, this method should be called prior to your fitting function being called.


Getting and Printing Fit Reports
===========================================

.. function:: fit_report(params, modelpars=None, show_correl=True, min_correl=0.1)

   generate and return text of report of best-fit values, uncertainties,
   and correlations from fit.

   :param params:       Parameters from fit.
   :param modelpars:    Parameters with "Known Values" (optional, default None)
   :param show_correl:  whether to show list of sorted correlations [``True``]
   :param min_correl:   smallest correlation absolute value to show [0.1]


.. function:: report_fit(params, modelpars=None, show_correl=True, min_correl=0.1)

   print text of report from :func:`fit_report`.

An example fit with an error report::

    p_true = Parameters()
    p_true.add('amp', value=14.0)
    p_true.add('period', value=5.33)
    p_true.add('shift', value=0.123)
    p_true.add('decay', value=0.010)

    def residual(pars, x, data=None):
        amp = pars['amp'].value
        per = pars['period'].value
        shift = pars['shift'].value
        decay = pars['decay'].value

        if abs(shift) > pi/2:
            shift = shift - sign(shift)*pi
        model = amp*sin(shift + x/per) * exp(-x*x*decay*decay)
        if data is None:
            return model
        return (model - data)

    n = 2500
    xmin = 0.
    xmax = 250.0
    noise = random.normal(scale=0.7215, size=n)
    x     = linspace(xmin, xmax, n)
    data  = residual(p_true, x) + noise

    fit_params = Parameters()
    fit_params.add('amp', value=13.0)
    fit_params.add('period', value=2)
    fit_params.add('shift', value=0.0)
    fit_params.add('decay', value=0.02)

    out = minimize(residual, fit_params, args=(x,), kws={'data':data})

    fit = residual(fit_params, x)
    report_errors(fit_params)

would generate this report::

    [[Variables]]
         amp:        13.969724 +/- 0.050145 (0.36%) initial =  13.000000
         decay:      0.009990 +/- 0.000042 (0.42%) initial =  0.020000
         period:     5.331423 +/- 0.002788 (0.05%) initial =  2.000000
         shift:      0.125333 +/- 0.004938 (3.94%) initial =  0.000000
    [[Correlations]] (unreported correlations are <  0.100)
        C(period, shift)             =  0.800
        C(amp, decay)                =  0.576