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).

import copy

from matplotlib.colors import LogNorm
import matplotlib.pyplot as plt
import numpy as np

from lmfit import Minimizer, create_params, 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. 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.

params = create_params(a=dict(value=2, vary=False),
                       b=dict(value=3, vary=False),
                       c=dict(value=7, vary=False),
                       d=dict(value=8, vary=False),
                       e=dict(value=9, vary=False),
                       f=dict(value=10, vary=False),
                       g=dict(value=44, vary=False),
                       h=dict(value=-1, vary=False),
                       i=dict(value=2, vary=False),
                       j=dict(value=26, vary=False),
                       k=dict(value=1, vary=False),
                       l=dict(value=-2, vary=False),
                       scale=dict(value=0.5, vary=False),
                       x=dict(value=0.0, vary=True, min=-4, max=4, brute_step=0.25),
                       y=dict(value=0.0, vary=True, min=-4, max=4, brute_step=0.25))

Second, create the three functions and the objective function:

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)

Performing the actual grid search is done with:

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).

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

[-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.

print(result.brute_x0)

[-1.    1.75]

result.brute_fval – Function value at the point x0.

print(result.brute_fval)

-2.8923637137222027

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

print(result.brute_grid)

[[[-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).

print(result.brute_Jout)

[[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 identical to using the method in SciPy directly!

Example 2: fit of a decaying sine wave

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

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

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, 'o')
plt.show()


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

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, if a parameter does not have finite bounds, then it does need a brute_step attribute specified:

params = create_params(amp=dict(value=7, min=2.5, brute_step=0.25),
                       decay=dict(value=0.05, brute_step=0.005),
                       shift=dict(value=0.0, min=-np.pi/2., max=np.pi/2),
                       omega=dict(value=3, max=5, brute_step=0.25))

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

params.pretty_print()

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:

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

print(fit_report(result_brute))

[[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)

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:

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

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)

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

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)

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

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)

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

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:

print(fit_report(result_brute))

[[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)

plt.plot(x, data, 'o')
plt.plot(x, data + fcn2min(result_brute.params, x, data), '--')
plt.show()

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).

result_brute.show_candidates(1)

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:

result_brute.show_candidates(2)

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:

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:

print(fit_report(best_result))

[[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:    5.00323085 +/- 0.03805940 (0.76%) (init = 5)
    decay:  0.02563850 +/- 4.4572e-04 (1.74%) (init = 0.03)
    shift: -0.09162987 +/- 0.00978382 (10.68%) (init = 0)
    omega:  1.99611629 +/- 0.00316225 (0.16%) (init = 2)
[[Correlations]] (unreported correlations are < 0.100)
    C(shift, omega) = -0.7855
    C(amp, decay)   = +0.5838
    C(amp, shift)   = -0.1208

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.

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

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).

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')
    """
    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([])
                else:
                    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 not in (i, 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([])
                else:
                    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:

plot_results_brute(result_brute, best_vals=True, varlabels=None)
plt.show()