.. note::
    :class: sphx-glr-download-link-note

    Click :ref:`here <sphx_glr_download_examples_example_brute.py>` to download the full example code
.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_example_brute.py:


Global minimization using the ``brute`` method (a.k.a. grid search)
===================================================================

This notebook shows a simple example of using ``lmfit.minimize.brute`` that
uses the method with the same name from ``scipy.optimize``.

The method computes the function’s value at each point of a multidimensional
grid of points, to find the global minimum of the function. It behaves
identically to ``scipy.optimize.brute`` in case finite bounds are given on
all varying parameters, but will also deal with non-bounded parameters
(see below).


.. code-block:: default

    import copy

    import matplotlib.pyplot as plt
    import numpy as np

    from lmfit import Minimizer, Parameters, fit_report








Let's start with the example given in the documentation of SciPy:

"We illustrate the use of brute to seek the global minimum of a function of
two variables that is given as the sum of a positive-definite quadratic and
two deep “Gaussian-shaped” craters. Specifically, define the objective
function f as the sum of three other functions, ``f = f1 + f2 + f3``. We
suppose each of these has a signature ``(z, *params), where z = (x, y)``,
and params and the functions are as defined below."

First, we create a set of Parameters where all variables except ``x`` and
``y`` are given fixed values.


.. code-block:: default

    params = Parameters()
    params.add_many(
            ('a', 2, False),
            ('b', 3, False),
            ('c', 7, False),
            ('d', 8, False),
            ('e', 9, False),
            ('f', 10, False),
            ('g', 44, False),
            ('h', -1, False),
            ('i', 2, False),
            ('j', 26, False),
            ('k', 1, False),
            ('l', -2, False),
            ('scale', 0.5, False),
            ('x', 0.0, True),
            ('y', 0.0, True))








Second, create the three functions and the objective function:


.. code-block:: default



    def f1(p):
        par = p.valuesdict()
        return (par['a'] * par['x']**2 + par['b'] * par['x'] * par['y'] +
                par['c'] * par['y']**2 + par['d']*par['x'] + par['e']*par['y'] + par['f'])


    def f2(p):
        par = p.valuesdict()
        return (-1.0*par['g']*np.exp(-((par['x']-par['h'])**2 +
                (par['y']-par['i'])**2) / par['scale']))


    def f3(p):
        par = p.valuesdict()
        return (-1.0*par['j']*np.exp(-((par['x']-par['k'])**2 +
                (par['y']-par['l'])**2) / par['scale']))


    def f(params):
        return f1(params) + f2(params) + f3(params)









Just as in the documentation we will do a grid search between ``-4`` and
``4`` and use a stepsize of ``0.25``. The bounds can be set as usual with
the ``min`` and ``max`` attributes, and the stepsize is set using
``brute_step``.


.. code-block:: default

    params['x'].set(min=-4, max=4, brute_step=0.25)
    params['y'].set(min=-4, max=4, brute_step=0.25)








Performing the actual grid search is done with:


.. code-block:: default

    fitter = Minimizer(f, params)
    result = fitter.minimize(method='brute')








, which will increment ``x`` and ``y`` between ``-4`` in increments of
``0.25`` until ``4`` (not inclusive).


.. code-block:: default

    grid_x, grid_y = [np.unique(par.ravel()) for par in result.brute_grid]
    print(grid_x)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    [-4.   -3.75 -3.5  -3.25 -3.   -2.75 -2.5  -2.25 -2.   -1.75 -1.5  -1.25
     -1.   -0.75 -0.5  -0.25  0.    0.25  0.5   0.75  1.    1.25  1.5   1.75
      2.    2.25  2.5   2.75  3.    3.25  3.5   3.75]




The objective function is evaluated on this grid, and the raw output from
``scipy.optimize.brute`` is stored in the MinimizerResult as
``brute_<parname>`` attributes. These attributes are:

``result.brute_x0`` -- A 1-D array containing the coordinates of a point at
which the objective function had its minimum value.


.. code-block:: default

    print(result.brute_x0)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    [-1.    1.75]




``result.brute_fval`` -- Function value at the point x0.


.. code-block:: default

    print(result.brute_fval)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    -2.8923637137222027




``result.brute_grid`` -- Representation of the evaluation grid. It has the
same length as x0.


.. code-block:: default

    print(result.brute_grid)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    [[[-4.   -4.   -4.   ... -4.   -4.   -4.  ]
      [-3.75 -3.75 -3.75 ... -3.75 -3.75 -3.75]
      [-3.5  -3.5  -3.5  ... -3.5  -3.5  -3.5 ]
      ...
      [ 3.25  3.25  3.25 ...  3.25  3.25  3.25]
      [ 3.5   3.5   3.5  ...  3.5   3.5   3.5 ]
      [ 3.75  3.75  3.75 ...  3.75  3.75  3.75]]

     [[-4.   -3.75 -3.5  ...  3.25  3.5   3.75]
      [-4.   -3.75 -3.5  ...  3.25  3.5   3.75]
      [-4.   -3.75 -3.5  ...  3.25  3.5   3.75]
      ...
      [-4.   -3.75 -3.5  ...  3.25  3.5   3.75]
      [-4.   -3.75 -3.5  ...  3.25  3.5   3.75]
      [-4.   -3.75 -3.5  ...  3.25  3.5   3.75]]]




``result.brute_Jout`` -- Function values at each point of the evaluation
grid, i.e., Jout = func(\*grid).


.. code-block:: default

    print(result.brute_Jout)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    [[134.         119.6875     106.25       ...  74.18749997  85.24999999
       97.1875    ]
     [129.125      115.         101.75       ...  74.74999948  85.99999987
       98.12499997]
     [124.5        110.5625      97.5        ...  75.5624928   86.99999818
       99.31249964]
     ...
     [ 94.12499965  85.24999772  77.24998843 ... 192.         208.5
      225.875     ]
     [ 96.49999997  87.81249979  79.99999892 ... 199.8125     216.5
      234.0625    ]
     [ 99.125       90.62499998  82.99999992 ... 207.875      224.75
      242.5       ]]




**Reassuringly, the obtained results are indentical to using the method in
SciPy directly!**

Example 2: fit of a decaying sine wave

In this example, will explain some of the options ot the algorithm.

We start off by generating some synthetic data with noise for a decaying
sine wave, define an objective function and create a Parameter set.


.. code-block:: default

    x = np.linspace(0, 15, 301)
    np.random.seed(7)
    noise = np.random.normal(size=x.size, scale=0.2)
    data = (5. * np.sin(2*x - 0.1) * np.exp(-x*x*0.025) + noise)
    plt.plot(x, data, 'b')


    def fcn2min(params, x, data):
        """Model decaying sine wave, subtract data."""
        amp = params['amp']
        shift = params['shift']
        omega = params['omega']
        decay = params['decay']
        model = amp * np.sin(x*omega + shift) * np.exp(-x*x*decay)
        return model - data


    # create a set of Parameters
    params = Parameters()
    params.add('amp', value=7, min=2.5)
    params.add('decay', value=0.05)
    params.add('shift', value=0.0, min=-np.pi/2., max=np.pi/2)
    params.add('omega', value=3, max=5)




.. image:: /examples/images/sphx_glr_example_brute_001.png
    :class: sphx-glr-single-img





In contrast to the implementation in SciPy (as shown in the first example),
varying parameters do not need to have finite bounds in lmfit. However, in
that case they **do** need the ``brute_step`` attribute specified, so let's
do that:


.. code-block:: default

    params['amp'].set(brute_step=0.25)
    params['decay'].set(brute_step=0.005)
    params['omega'].set(brute_step=0.25)








Our initial parameter set is now defined as shown below and this will
determine how the grid is set-up.


.. code-block:: default

    params.pretty_print()





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    Name      Value      Min      Max   Stderr     Vary     Expr Brute_Step
    amp           7      2.5      inf     None     True     None     0.25
    decay      0.05     -inf      inf     None     True     None    0.005
    omega         3     -inf        5     None     True     None     0.25
    shift         0   -1.571    1.571     None     True     None     None




First, we initialize a Minimizer and perform the grid search:


.. code-block:: default

    fitter = Minimizer(fcn2min, params, fcn_args=(x, data))
    result_brute = fitter.minimize(method='brute', Ns=25, keep=25)








We used two new parameters here: ``Ns`` and ``keep``. The parameter ``Ns``
determines the \'number of grid points along the axes\' similarly to its usage
in SciPy. Together with ``brute_step``, ``min`` and ``max`` for a Parameter
it will dictate how the grid is set-up:

**(1)** finite bounds are specified ("SciPy implementation"): uses
``brute_step`` if present (in the example above) or uses ``Ns`` to generate
the grid. The latter scenario that interpolates ``Ns`` points from ``min``
to ``max`` (inclusive), is here shown for the parameter ``shift``:


.. code-block:: default

    par_name = 'shift'
    indx_shift = result_brute.var_names.index(par_name)
    grid_shift = np.unique(result_brute.brute_grid[indx_shift].ravel())
    print("parameter = {}\nnumber of steps = {}\ngrid = {}".format(par_name,
          len(grid_shift), grid_shift))





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    parameter = shift
    number of steps = 25
    grid = [-1.57079633 -1.43989663 -1.30899694 -1.17809725 -1.04719755 -0.91629786
     -0.78539816 -0.65449847 -0.52359878 -0.39269908 -0.26179939 -0.13089969
      0.          0.13089969  0.26179939  0.39269908  0.52359878  0.65449847
      0.78539816  0.91629786  1.04719755  1.17809725  1.30899694  1.43989663
      1.57079633]




If finite bounds are not set for a certain parameter then the user **must**
specify ``brute_step`` - three more scenarios are considered here:

**(2)** lower bound (min) and brute_step are specified:
range = (min, min + Ns * brute_step, brute_step)


.. code-block:: default

    par_name = 'amp'
    indx_shift = result_brute.var_names.index(par_name)
    grid_shift = np.unique(result_brute.brute_grid[indx_shift].ravel())
    print("parameter = {}\nnumber of steps = {}\ngrid = {}".format(par_name, len(grid_shift), grid_shift))





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    parameter = amp
    number of steps = 25
    grid = [2.5  2.75 3.   3.25 3.5  3.75 4.   4.25 4.5  4.75 5.   5.25 5.5  5.75
     6.   6.25 6.5  6.75 7.   7.25 7.5  7.75 8.   8.25 8.5 ]




**(3)** upper bound (max) and brute_step are specified:
range = (max - Ns * brute_step, max, brute_step)


.. code-block:: default

    par_name = 'omega'
    indx_shift = result_brute.var_names.index(par_name)
    grid_shift = np.unique(result_brute.brute_grid[indx_shift].ravel())
    print("parameter = {}\nnumber of steps = {}\ngrid = {}".format(par_name, len(grid_shift), grid_shift))





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    parameter = omega
    number of steps = 25
    grid = [-1.25 -1.   -0.75 -0.5  -0.25  0.    0.25  0.5   0.75  1.    1.25  1.5
      1.75  2.    2.25  2.5   2.75  3.    3.25  3.5   3.75  4.    4.25  4.5
      4.75]




**(4)** numerical value (value) and brute_step are specified:
range = (value - (Ns//2) * brute_step, value + (Ns//2) * brute_step, brute_step)


.. code-block:: default

    par_name = 'decay'
    indx_shift = result_brute.var_names.index(par_name)
    grid_shift = np.unique(result_brute.brute_grid[indx_shift].ravel())
    print("parameter = {}\nnumber of steps = {}\ngrid = {}".format(par_name, len(grid_shift), grid_shift))





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    parameter = decay
    number of steps = 24
    grid = [-1.00000000e-02 -5.00000000e-03  5.20417043e-18  5.00000000e-03
      1.00000000e-02  1.50000000e-02  2.00000000e-02  2.50000000e-02
      3.00000000e-02  3.50000000e-02  4.00000000e-02  4.50000000e-02
      5.00000000e-02  5.50000000e-02  6.00000000e-02  6.50000000e-02
      7.00000000e-02  7.50000000e-02  8.00000000e-02  8.50000000e-02
      9.00000000e-02  9.50000000e-02  1.00000000e-01  1.05000000e-01]




The ``MinimizerResult`` contains all the usual best-fit parameters and
fitting statistics. For example, the optimal solution from the grid search
is given below together with a plot:


.. code-block:: default

    print(fit_report(result_brute))
    plt.plot(x, data, 'b')
    plt.plot(x, data + fcn2min(result_brute.params, x, data), 'r--')




.. image:: /examples/images/sphx_glr_example_brute_002.png
    :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    [[Fit Statistics]]
        # fitting method   = brute
        # function evals   = 375000
        # data points      = 301
        # variables        = 4
        chi-square         = 11.9353671
        reduced chi-square = 0.04018642
        Akaike info crit   = -963.508878
        Bayesian info crit = -948.680437
    ##  Warning: uncertainties could not be estimated:
    [[Variables]]
        amp:    5.00000000 (init = 7)
        decay:  0.02500000 (init = 0.05)
        shift: -0.13089969 (init = 0)
        omega:  2.00000000 (init = 3)

    [<matplotlib.lines.Line2D object at 0x1a1e071410>]



We can see that this fit is already very good, which is what we should expect
since our ``brute`` force grid is sampled rather finely and encompasses the
"correct" values.

In a more realistic, complicated example the ``brute`` method will be used
to get reasonable values for the parameters and perform another minimization
(e.g., using ``leastsq``) using those as starting values. That is where the
`keep`` parameter comes into play: it determines the "number of best
candidates from the brute force method that are stored in the ``candidates``
attribute". In the example above we store the best-ranking 25 solutions (the
default value is ``50`` and storing all the grid points can be accomplished
by choosing ``all``). The ``candidates`` attribute contains the parameters
and ``chisqr`` from the brute force method as a namedtuple,
``(‘Candidate’, [‘params’, ‘score’])``, sorted on the (lowest) ``chisqr``
value. To access the values for a particular candidate one can use
``result.candidate[#].params`` or ``result.candidate[#].score``, where a
lower # represents a better candidate. The ``show_candidates(#)`` uses the
``pretty_print()`` method to show a specific candidate-# or all candidates
when no number is specified.

The optimal fit is, as usual, stored in the ``MinimizerResult.params``
attribute and is, therefore, identical to ``result_brute.show_candidates(1)``.


.. code-block:: default

    result_brute.show_candidates(1)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none


    Candidate #1, chisqr = 11.935
    Name      Value      Min      Max   Stderr     Vary     Expr Brute_Step
    amp           5      2.5      inf     None     True     None     0.25
    decay     0.025     -inf      inf     None     True     None    0.005
    omega         2     -inf        5     None     True     None     0.25
    shift   -0.1309   -1.571    1.571     None     True     None     None




In this case, the next-best scoring candidate has already a ``chisqr`` that
increased quite a bit:


.. code-block:: default

    result_brute.show_candidates(2)





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none


    Candidate #2, chisqr = 13.994
    Name      Value      Min      Max   Stderr     Vary     Expr Brute_Step
    amp        4.75      2.5      inf     None     True     None     0.25
    decay     0.025     -inf      inf     None     True     None    0.005
    omega         2     -inf        5     None     True     None     0.25
    shift   -0.1309   -1.571    1.571     None     True     None     None




and is, therefore, probably not so likely... However, as said above, in most
cases you'll want to do another minimization using the solutions from the
``brute`` method as starting values. That can be easily accomplished as
shown in the code below, where we now perform a ``leastsq`` minimization
starting from the top-25 solutions and accept the solution if the ``chisqr``
is lower than the previously 'optimal' solution:


.. code-block:: default

    best_result = copy.deepcopy(result_brute)

    for candidate in result_brute.candidates:
        trial = fitter.minimize(method='leastsq', params=candidate.params)
        if trial.chisqr < best_result.chisqr:
            best_result = trial








From the ``leastsq`` minimization we obtain the following parameters for the
most optimal result:


.. code-block:: default

    print(fit_report(best_result))





.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none

    [[Fit Statistics]]
        # fitting method   = leastsq
        # function evals   = 21
        # data points      = 301
        # variables        = 4
        chi-square         = 10.8653514
        reduced chi-square = 0.03658367
        Akaike info crit   = -991.780924
        Bayesian info crit = -976.952483
    [[Variables]]
        amp:    5.00323088 +/- 0.03805940 (0.76%) (init = 5.25)
        decay:  0.02563850 +/- 4.4571e-04 (1.74%) (init = 0.025)
        shift: -0.09162988 +/- 0.00978382 (10.68%) (init = 0)
        omega:  1.99611629 +/- 0.00316225 (0.16%) (init = 2)
    [[Correlations]] (unreported correlations are < 0.100)
        C(shift, omega) = -0.785
        C(amp, decay)   =  0.584
        C(amp, shift)   = -0.121




As expected the parameters have not changed significantly as they were
already very close to the "real" values, which can also be appreciated from
the plots below.


.. code-block:: default

    plt.plot(x, data, 'b')
    plt.plot(x, data + fcn2min(result_brute.params, x, data), 'r--',
             label='brute')
    plt.plot(x, data + fcn2min(best_result.params, x, data), 'g--',
             label='brute followed by leastsq')
    plt.legend()





.. image:: /examples/images/sphx_glr_example_brute_003.png
    :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 Out:

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1a1e8649d0>



Finally, the results from the ``brute`` force grid-search can be visualized
using the rather lengthy Python function below (which might get incorporated
in lmfit at some point).


.. code-block:: default

    def plot_results_brute(result, best_vals=True, varlabels=None,
                           output=None):
        """Visualize the result of the brute force grid search.

        The output file will display the chi-square value per parameter and contour
        plots for all combination of two parameters.

        Inspired by the `corner` package (https://github.com/dfm/corner.py).

        Parameters
        ----------
        result : :class:`~lmfit.minimizer.MinimizerResult`
            Contains the results from the :meth:`brute` method.

        best_vals : bool, optional
            Whether to show the best values from the grid search (default is True).

        varlabels : list, optional
            If None (default), use `result.var_names` as axis labels, otherwise
            use the names specified in `varlabels`.

        output : str, optional
            Name of the output PDF file (default is 'None')
        """
        from matplotlib.colors import LogNorm

        npars = len(result.var_names)
        fig, axes = plt.subplots(npars, npars)

        if not varlabels:
            varlabels = result.var_names
        if best_vals and isinstance(best_vals, bool):
            best_vals = result.params

        for i, par1 in enumerate(result.var_names):
            for j, par2 in enumerate(result.var_names):

                # parameter vs chi2 in case of only one parameter
                if npars == 1:
                    axes.plot(result.brute_grid, result.brute_Jout, 'o', ms=3)
                    axes.set_ylabel(r'$\chi^{2}$')
                    axes.set_xlabel(varlabels[i])
                    if best_vals:
                        axes.axvline(best_vals[par1].value, ls='dashed', color='r')

                # parameter vs chi2 profile on top
                elif i == j and j < npars-1:
                    if i == 0:
                        axes[0, 0].axis('off')
                    ax = axes[i, j+1]
                    red_axis = tuple([a for a in range(npars) if a != i])
                    ax.plot(np.unique(result.brute_grid[i]),
                            np.minimum.reduce(result.brute_Jout, axis=red_axis),
                            'o', ms=3)
                    ax.set_ylabel(r'$\chi^{2}$')
                    ax.yaxis.set_label_position("right")
                    ax.yaxis.set_ticks_position('right')
                    ax.set_xticks([])
                    if best_vals:
                        ax.axvline(best_vals[par1].value, ls='dashed', color='r')

                # parameter vs chi2 profile on the left
                elif j == 0 and i > 0:
                    ax = axes[i, j]
                    red_axis = tuple([a for a in range(npars) if a != i])
                    ax.plot(np.minimum.reduce(result.brute_Jout, axis=red_axis),
                            np.unique(result.brute_grid[i]), 'o', ms=3)
                    ax.invert_xaxis()
                    ax.set_ylabel(varlabels[i])
                    if i != npars-1:
                        ax.set_xticks([])
                    elif i == npars-1:
                        ax.set_xlabel(r'$\chi^{2}$')
                    if best_vals:
                        ax.axhline(best_vals[par1].value, ls='dashed', color='r')

                # contour plots for all combinations of two parameters
                elif j > i:
                    ax = axes[j, i+1]
                    red_axis = tuple([a for a in range(npars) if a != i and a != j])
                    X, Y = np.meshgrid(np.unique(result.brute_grid[i]),
                                       np.unique(result.brute_grid[j]))
                    lvls1 = np.linspace(result.brute_Jout.min(),
                                        np.median(result.brute_Jout)/2.0, 7, dtype='int')
                    lvls2 = np.linspace(np.median(result.brute_Jout)/2.0,
                                        np.median(result.brute_Jout), 3, dtype='int')
                    lvls = np.unique(np.concatenate((lvls1, lvls2)))
                    ax.contourf(X.T, Y.T, np.minimum.reduce(result.brute_Jout, axis=red_axis),
                                lvls, norm=LogNorm())
                    ax.set_yticks([])
                    if best_vals:
                        ax.axvline(best_vals[par1].value, ls='dashed', color='r')
                        ax.axhline(best_vals[par2].value, ls='dashed', color='r')
                        ax.plot(best_vals[par1].value, best_vals[par2].value, 'rs', ms=3)
                    if j != npars-1:
                        ax.set_xticks([])
                    elif j == npars-1:
                        ax.set_xlabel(varlabels[i])
                    if j - i >= 2:
                        axes[i, j].axis('off')

        if output is not None:
            plt.savefig(output)









and finally, to generated the figure:


.. code-block:: default

    plot_results_brute(result_brute, best_vals=True, varlabels=None)



.. image:: /examples/images/sphx_glr_example_brute_004.png
    :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  33.866 seconds)


.. _sphx_glr_download_examples_example_brute.py:


.. only :: html

 .. container:: sphx-glr-footer
    :class: sphx-glr-footer-example



  .. container:: sphx-glr-download

     :download:`Download Python source code: example_brute.py <example_brute.py>`



  .. container:: sphx-glr-download

     :download:`Download Jupyter notebook: example_brute.ipynb <example_brute.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_