# Performing Fits, Analyzing Outputs¶

As shown in the previous chapter, a simple fit can be performed with the minimize() function. For more sophisticated modeling, the Minimizer class can be used to gain a bit more control, especially when using complicated constraints or comparing results from related fits.

Warning

Upgrading scripts from version 0.8.3 to 0.9.0? See Version 0.9.0 Release Notes

## The minimize() function¶

The minimize() function is a wrapper around Minimizer for running an optimization problem. It takes an objective function (the function that calculates the array to be minimized), a Parameters object, and several optional arguments. See Writing a Fitting Function for details on writing the objective.

minimize(function, params[, args=None[, kws=None[, method='leastsq'[, scale_covar=True[, iter_cb=None[, **fit_kws]]]]]])

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

Parameters: function (callable.) – function to return fit residual. See Writing a Fitting Function for details. params (Parameters.) – a Parameters dictionary. Keywords must be strings that match [a-z_][a-z0-9_]* and cannot be a python reserved word. Each value must be Parameter. args (tuple) – arguments tuple to pass to the residual function as positional arguments. kws (dict) – dictionary to pass to the residual function as keyword arguments. method (string (default leastsq)) – name of fitting method to use. See Choosing Different Fitting Methods for details scale_covar (bool (default True)) – whether to automatically scale covariance matrix (leastsq only) iter_cb (callable or None) – function to be called at each fit iteration. See Using a Iteration Callback Function for details. fit_kws (dict) – dictionary to pass to optimize.leastsq or optimize.minimize. MinimizerResult instance, which will contain the optimized parameter, and several goodness-of-fit statistics.

Changed in version 0.9.0: return value changed to MinimizerResult

On output, the params will be unchanged. The best-fit values, and where appropriate, estimated uncertainties and correlations, will all be contained in the returned MinimizerResult. See MinimizerResult – the optimization result for further details.

For clarity, it should be emphasized that this function is simply a wrapper around Minimizer that runs a single fit, implemented as:

fitter = Minimizer(fcn, params, fcn_args=args, fcn_kws=kws,
iter_cb=iter_cb, scale_covar=scale_covar, **fit_kws)
return fitter.minimize(method=method)


## Writing a Fitting Function¶

An important component of a fit is writing a function to be minimized – the objective function. 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:

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

calculate objective residual to be minimized from parameters.

Parameters: params (Parameters.) – parameters. args – positional arguments. Must match args argument to minimize() kws – keyword arguments. Must match kws argument to minimize() residual array (generally data-model) to be minimized in the least-squares sense. 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 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 Parameters, it is advisable to unpack these to get numerical values at the top of the function. A simple way to do this is with Parameters.valuesdict(), as with:

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

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

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

model = parvals['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 value of the ‘period’ Parameter. It might be wise to ensure this parameter cannot be 0. It would be possible to use the bounds on the Parameter to do this:

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


but putting this directly in the function with:

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


is also a reasonable approach. Similarly, one could place bounds on the decay parameter to take values only between -pi/2 and pi/2.

## Choosing Different Fitting Methods¶

By default, the Levenberg-Marquardt 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 MinimizerResult – the optimization result.

Alternative algorithms can also be used by providing the method keyword to the minimize() function or Minimizer.minimize() class as listed in the Table of Supported Fitting Methods.

Table of Supported Fitting Methods:

Fitting Method method arg to minimize() or Minimizer.minimize()
Levenberg-Marquardt leastsq
Nelder-Mead nelder
L-BFGS-B lbfgsb
Powell powell
Conjugate Gradient cg
Newton-CG newton
COBYLA cobyla
Truncated Newton tnc
Dogleg dogleg
Sequential Linear Squares Programming slsqp
Differential Evolution differential_evolution

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

Much of this documentation assumes that the Levenberg-Marquardt method is the method used. Many of the fit statistics and estimates for uncertainties in parameters discussed in MinimizerResult – the optimization result are done only for this method.

## MinimizerResult – the optimization result¶

class MinimizerResult(**kws)

New in version 0.9.0.

An optimization with minimize() or Minimizer.minimize() will return a MinimizerResult object. This is an otherwise plain container object (that is, with no methods of its own) that simply holds the results of the minimization. These results will include several pieces of informational data such as status and error messages, fit statistics, and the updated parameters themselves.

Importantly, the parameters passed in to Minimizer.minimize() will be not be changed. To to find the best-fit values, uncertainties and so on for each parameter, one must use the MinimizerResult.params attribute.

params

the Parameters actually used in the fit, with updated values, stderr and correl.

var_names

ordered list of variable parameter names used in optimization, and useful for understanding the the values in init_vals and covar.

covar

covariance matrix from minimization (leastsq only), with rows/columns using var_names.

init_vals

list of initial values for variable parameters using var_names.

nfev

number of function evaluations

success

boolean (True/False) for whether fit succeeded.

errorbars

boolean (True/False) for whether uncertainties were estimated.

message

ier

integer error value from optimize.leastsq (leastsq only).

lmdif_message

message from optimize.leastsq (leastsq only).

nvarys

number of variables in fit $$N_{\rm varys}$$

ndata

number of data points: $$N$$

nfree 

degrees of freedom in fit: $$N - N_{\rm varys}$$

residual

residual array, return value of func(): $${\rm Resid}$$

chisqr

chi-square: $$\chi^2 = \sum_i^N [{\rm Resid}_i]^2$$

redchi

reduced chi-square: $$\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}$$

aic

Akaike Information Criterion statistic (see below)

bic

Bayesian Information Criterion statistic (see below).

### Goodness-of-Fit Statistics¶

Table of Fit Results: These values, including the standard Goodness-of-Fit statistics, are all attributes of the MinimizerResult object returned by minimize() or Minimizer.minimize().
Attribute Name Description / Formula
nfev number of function evaluations
nvarys number of variables in fit $$N_{\rm varys}$$
ndata number of data points: $$N$$
nfree  degrees of freedom in fit: $$N - N_{\rm varys}$$
residual residual array, return value of func(): $${\rm Resid}$$
chisqr chi-square: $$\chi^2 = \sum_i^N [{\rm Resid}_i]^2$$
redchi reduced chi-square: $$\chi^2_{\nu}= {\chi^2} / {(N - N_{\rm varys})}$$
aic Akaike Information Criterion statistic (see below)
bic Bayesian Information Criterion statistic (see below)
var_names ordered list of variable parameter names used for init_vals and covar
covar covariance matrix (with rows/columns using var_names
init_vals list of initial values for variable parameters

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 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 $$1\sigma$$ error-bar) go into the stderr attribute of the Parameter. The correlations with all other variables will be put into the 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 errorbars attribute of the fit result (Minimizer object) will be False.

### Akaike and Bayesian Information Criteria¶

The MinimizerResult includes the traditional chi-square and reduced chi-square statistics:

$\begin{eqnarray*} \chi^2 &=& \sum_i^N r_i^2 \\ \chi^2_\nu &=& = \chi^2 / (N-N_{\rm varys}) \end{eqnarray*}$

where $$r$$ is the residual array returned by the objective function (likely to be (data-model)/uncertainty for data modeling usages), $$N$$ is the number of data points (ndata), and $$N_{\rm varys}$$ is number of variable parameters.

Also included are the Akaike Information Criterion, and Bayesian Information Criterion statistics, held in the aic and bic attributes, respectively. These give slightly different measures of the relative quality for a fit, trying to balance quality of fit with the number of variable parameters used in the fit. These are calculated as

$\begin{eqnarray*} {\rm aic} &=& N \ln(\chi^2/N) + 2 N_{\rm varys} \\ {\rm bic} &=& N \ln(\chi^2/N) + \ln(N) *N_{\rm varys} \\ \end{eqnarray*}$

When comparing fits with different numbers of varying parameters, one typically selects the model with lowest reduced chi-square, Akaike information criterion, and/or Bayesian information criterion. Generally, the Bayesian information criterion is considered the most conservative of these statistics.

## Using a Iteration Callback Function¶

An iteration callback function is a function to be called at each iteration, just after the objective function is called. The iteration callback allows user-supplied code to be run at each iteration, and can be used to abort a fit.

iter_cb(params, iter, resid, *args, **kws):

user-supplied function to be run at each iteration

Parameters: params (Parameters.) – parameters. iter (integer) – iteration number resid (ndarray) – residual array. args – positional arguments. Must match args argument to minimize() kws – keyword arguments. Must match kws argument to minimize() residual array (generally data-model) to be minimized in the least-squares sense. None for normal behavior, any value like True to abort fit.

Normally, the iteration callback would have no return value or return None. To abort a fit, have this function return a value that is True (including any non-zero integer). The fit will also abort if any exception is raised in the iteration callback. When a fit is aborted this way, the parameters will have the values from the last iteration. The fit statistics are not likely to be meaningful, and uncertainties will not be computed.

## Using the Minimizer class¶

For full control of the fitting process, you’ll want to create a Minimizer object.

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

creates a Minimizer, for more detailed access to fitting methods and attributes.

Parameters: function (callable.) – objective function to return fit residual. See Writing a Fitting Function for details. params (dict) – 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 Parameter. fcn_args (tuple) – arguments tuple to pass to the residual function as positional arguments. fcn_kws (dict) – dictionary to pass to the residual function as keyword arguments. iter_cb (callable or None) – function to be called at each fit iteration. See Using a Iteration Callback Function for details. scale_covar (bool (default True).) – flag for automatically scaling covariance matrix and uncertainties to reduced chi-square (leastsq only) nan_policy (str (default 'raise')) – Specifies action if userfcn (or a Jacobian) returns nan values. One of: ‘raise’ - a ValueError is raised ‘propagate’ - the values returned from userfcn are un-altered ‘omit’ - the non-finite values are filtered. kws (dict) – dictionary to pass as keywords to the underlying scipy.optimize method.

The Minimizer object has a few public methods:

minimize(method='leastsq', params=None, **kws)

perform fit using either leastsq() or scalar_minimize().

Parameters: method (str.) – name of fitting method. Must be one of the names in Table of Supported Fitting Methods params (Parameters or None) – a Parameters dictionary for starting values MinimizerResult object, containing updated parameters, fitting statistics, and information.

Changed in version 0.9.0: return value changed to MinimizerResult

Additional keywords are passed on to the correspond leastsq() or scalar_minimize() method.

leastsq(params=None, scale_covar=True, **kws)

perform fit with Levenberg-Marquardt algorithm. Keywords will be passed directly to optimize.leastsq. By default, numerical derivatives are used, and the following arguments are set:

leastsq() arg Default Value Description
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

Changed in version 0.9.0: return value changed to MinimizerResult

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

perform fit with any of the scalar minimization algorithms supported by optimize.minimize.

scalar_minimize() arg Default Value Description
method Nelder-Mead fitting method
tol 1.e-7 fitting and parameter tolerance
hess None Hessian of objective function

Changed in version 0.9.0: return value changed to MinimizerResult

prepare_fit(**kws)

prepares and initializes model and Parameters for subsequent fitting. This routine prepares the conversion of Parameters into fit variables, organizes parameter bounds, and parses, “compiles” and checks constrain expressions. The method also creates and returns a new instance of a MinimizerResult object that contains the copy of the Parameters that will actually be varied in the fit.

This method is called directly by the fitting methods, and it is generally not necessary to call this function explicitly.

Changed in version 0.9.0: return value changed to MinimizerResult

emcee(params=None, steps=1000, nwalkers=100, burn=0, thin=1, ntemps=1, pos=None, reuse_sampler=False, workers=1, float_behavior='posterior', is_weighted=True, seed=None)

Bayesian sampling of the posterior distribution for the parameters using the emcee Markov Chain Monte Carlo package. The method assumes that the prior is Uniform. You need to have emcee installed to use this method.

Parameters: params (Parameters or None) – a Parameters dictionary for starting values steps (int) – How many samples you would like to draw from the posterior distribution for each of the walkers? nwalkers (int) – Should be set so $$nwalkers >> nvarys$$, where nvarys are the number of parameters being varied during the fit. “Walkers are the members of the ensemble. They are almost like separate Metropolis-Hastings chains but, of course, the proposal distribution for a given walker depends on the positions of all the other walkers in the ensemble.” - from [1]. burn (int) – Discard this many samples from the start of the sampling regime. thin (int) – Only accept 1 in every thin samples. ntemps (int) – If ntemps > 1 perform a Parallel Tempering. pos (np.ndarray) – Specify the initial positions for the sampler. If ntemps == 1 then pos.shape should be (nwalkers, nvarys). Otherwise, (ntemps, nwalkers, nvarys). You can also initialise using a previous chain that had the same ntemps, nwalkers and nvarys. reuse_sampler (bool) – If you have already run emcee() on a given Minimizer object then it possesses an internal sampler attribute. You can continue to draw from the same sampler (retaining the chain history) if you set this option to True. Otherwise a new sampler is created. The nwalkers, ntemps and params keywords are ignored with this option. Important: the Parameters used to create the sampler must not change in-between calls to emcee(). Alteration of Parameters would include changed min, max, vary and expr attributes. This may happen, for example, if you use an altered Parameters object and call the minimize() method in-between calls to emcee() . workers (int or Pool-like) – For parallelization of sampling. It can be any Pool-like object with a map method that follows the same calling sequence as the built-in map function. If int is given as the argument, then a multiprocessing-based pool is spawned internally with the corresponding number of parallel processes. ‘mpi4py’-based parallelization and ‘joblib’-based parallelization pools can also be used here. Note: because of multiprocessing overhead it may only be worth parallelising if the objective function is expensive to calculate, or if there are a large number of objective evaluations per step (ntemps * nwalkers * nvarys). float_behavior (str) – Specifies the meaning of the objective function if it returns a float. One of: ‘posterior’ - the objective function returns a log-posterior probability‘chi2’ - the objective function is returning $$\chi^2$$. See Notes for further details. is_weighted (bool) – Has your objective function been weighted by measurement uncertainties? If is_weighted is True then your objective function is assumed to return residuals that have been divided by the true measurement uncertainty (data - model) / sigma. If is_weighted is False then the objective function is assumed to return unweighted residuals, data - model. In this case emcee will employ a positive measurement uncertainty during the sampling. This measurement uncertainty will be present in the output params and output chain with the name __lnsigma. A side effect of this is that you cannot use this parameter name yourself. Important this parameter only has any effect if your objective function returns an array. If your objective function returns a float, then this parameter is ignored. See Notes for more details. seed (int or np.random.RandomState) – If seed is an int, a new np.random.RandomState instance is used, seeded with seed. If seed is already a np.random.RandomState instance, then that np.random.RandomState instance is used. Specify seed for repeatable sampling. MinimizerResult object containing updated params, statistics, etc. The MinimizerResult also contains the chain, flatchain and lnprob attributes. The chain and flatchain attributes contain the samples and have the shape (nwalkers, (steps - burn) // thin, nvarys) or (ntemps, nwalkers, (steps - burn) // thin, nvarys), depending on whether Parallel tempering was used or not. nvarys is the number of parameters that are allowed to vary. The flatchain attribute is a pandas.DataFrame of the flattened chain, chain.reshape(-1, nvarys). To access flattened chain values for a particular parameter use result.flatchain[parname]. The lnprob attribute contains the log probability for each sample in chain. The sample with the highest probability corresponds to the maximum likelihood estimate.

This method samples the posterior distribution of the parameters using Markov Chain Monte Carlo. To do so it needs to calculate the log-posterior probability of the model parameters, F, given the data, D, $$\ln p(F_{true} | D)$$. This ‘posterior probability’ is calculated as:

$\ln p(F_{true} | D) \propto \ln p(D | F_{true}) + \ln p(F_{true})$

where $$\ln p(D | F_{true})$$ is the ‘log-likelihood’ and $$\ln p(F_{true})$$ is the ‘log-prior’. The default log-prior encodes prior information already known about the model. This method assumes that the log-prior probability is -np.inf (impossible) if the one of the parameters is outside its limits. The log-prior probability term is zero if all the parameters are inside their bounds (known as a uniform prior). The log-likelihood function is given by [1]:

$\ln p(D|F_{true}) = -\frac{1}{2}\sum_n \left[\frac{\left(g_n(F_{true}) - D_n \right)^2}{s_n^2}+\ln (2\pi s_n^2)\right]$

The first summand in the square brackets represents the residual for a given datapoint ($$g$$ being the generative model) . This term represents $$\chi^2$$ when summed over all datapoints. Ideally the objective function used to create lmfit.Minimizer should return the log-posterior probability, $$\ln p(F_{true} | D)$$. However, since the in-built log-prior term is zero, the objective function can also just return the log-likelihood, unless you wish to create a non-uniform prior.

If a float value is returned by the objective function then this value is assumed by default to be the log-posterior probability, i.e. float_behavior is ‘posterior’. If your objective function returns $$\chi^2$$, then you should use a value of ‘chi2’ for float_behavior. emcee will then multiply your $$\chi^2$$ value by -0.5 to obtain the posterior probability.

However, the default behaviour of many objective functions is to return a vector of (possibly weighted) residuals. Therefore, if your objective function returns a vector, res, then the vector is assumed to contain the residuals. If is_weighted is True then your residuals are assumed to be correctly weighted by the standard deviation of the data points (res = (data - model) / sigma) and the log-likelihood (and log-posterior probability) is calculated as: -0.5 * np.sum(res **2). This ignores the second summand in the square brackets. Consequently, in order to calculate a fully correct log-posterior probability value your objective function should return a single value. If is_weighted is False then the data uncertainty, $$s_n$$, will be treated as a nuisance parameter and will be marginalised out. This is achieved by employing a strictly positive uncertainty (homoscedasticity) for each data point, $$s_n=exp(\_\_lnsigma)$$. __lnsigma will be present in MinimizerResult.params, as well as Minimizer.chain, nvarys will also be increased by one.

## emcee() - calculating the posterior probability distribution of parameters¶

emcee() can be used to obtain the posterior probability distribution of parameters, given a set of experimental data. An example problem is a double exponential decay. A small amount of Gaussian noise is also added in:

>>> import numpy as np
>>> import lmfit
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(1, 10, 250)
>>> np.random.seed(0)
>>> y = 3.0 * np.exp(-x / 2) - 5.0 * np.exp(-(x - 0.1) / 10.) + 0.1 * np.random.randn(len(x))
>>> plt.plot(x, y)
>>> plt.show()


Create a Parameter set for the initial guesses:

>>> p = lmfit.Parameters()
>>> p.add_many(('a1', 4.), ('a2', 4.), ('t1', 3.), ('t2', 3., True))

>>> def residual(p):
...     v = p.valuesdict()
...     return v['a1'] * np.exp(-x / v['t1']) + v['a2'] * np.exp(-(x - 0.1) / v['t2']) - y


Solving with minimize() gives the Maximum Likelihood solution.:

>>> mi = lmfit.minimize(residual, p, method='Nelder')
>>> lmfit.printfuncs.report_fit(mi.params, min_correl=0.5)
[[Variables]]
a1:   2.98623688 (init= 4)
a2:  -4.33525596 (init= 4)
t1:   1.30993185 (init= 3)
t2:   11.8240752 (init= 3)
[[Correlations]] (unreported correlations are <  0.500)
>>> plt.plot(x, y)
>>> plt.plot(x, residual(mi.params) + y, 'r')
>>> plt.show()


However, this doesn’t give a probability distribution for the parameters. Furthermore, we wish to deal with the data uncertainty. This is called marginalisation of a nuisance parameter. emcee requires a function that returns the log-posterior probability. The log-posterior probability is a sum of the log-prior probability and log-likelihood functions. The log-prior probability is assumed to be zero if all the parameters are within their bounds and -np.inf if any of the parameters are outside their bounds.:

>>> # add a noise parameter

>>> # This is the log-likelihood probability for the sampling. We're going to estimate the
>>> # size of the uncertainties on the data as well.
>>> def lnprob(p):
...    resid = residual(p)
...    s = p['f']
...    resid *= 1 / s
...    resid *= resid
...    resid += np.log(2 * np.pi * s**2)
...    return -0.5 * np.sum(resid)


Now we have to set up the minimizer and do the sampling.:

>>> mini = lmfit.Minimizer(lnprob, mi.params)
>>> res = mini.emcee(burn=300, steps=600, thin=10, params=mi.params)


Lets have a look at those posterior distributions for the parameters. This requires installation of the corner package.:

>>> import corner
>>> corner.corner(res.flatchain, labels=res.var_names, truths=list(res.params.valuesdict().values()))


The values reported in the MinimizerResult are the medians of the probability distributions and a 1 sigma quantile, estimated as half the difference between the 15.8 and 84.2 percentiles. The median value is not necessarily the same as the Maximum Likelihood Estimate. We’ll get that as well. You can see that we recovered the right uncertainty level on the data.:

>>> print("median of posterior probability distribution")
>>> print('------------------------------------------')
>>> lmfit.report_fit(res.params)
median of posterior probability distribution
------------------------------------------
[[Variables]]
a1:   3.00975345 +/- 0.151034 (5.02%) (init= 2.986237)
a2:  -4.35419204 +/- 0.127505 (2.93%) (init=-4.335256)
t1:   1.32726415 +/- 0.142995 (10.77%) (init= 1.309932)
t2:   11.7911935 +/- 0.495583 (4.20%) (init= 11.82408)
f:    0.09805494 +/- 0.004256 (4.34%) (init= 1)
[[Correlations]] (unreported correlations are <  0.100)
C(a2, t2)                    =  0.981
C(a2, t1)                    = -0.927
C(t1, t2)                    = -0.880
C(a1, t1)                    = -0.519
C(a1, a2)                    =  0.195
C(a1, t2)                    =  0.146

>>> # find the maximum likelihood solution
>>> highest_prob = np.argmax(res.lnprob)
>>> hp_loc = np.unravel_index(highest_prob, res.lnprob.shape)
>>> mle_soln = res.chain[hp_loc]
>>> for i, par in enumerate(p):
...     p[par].value = mle_soln[i]

>>> print("\nMaximum likelihood Estimation")
>>> print('-----------------------------')
>>> print(p)
Maximum likelihood Estimation
-----------------------------
Parameters([('a1', <Parameter 'a1', 2.9943337359308981, bounds=[-inf:inf]>),
('a2', <Parameter 'a2', -4.3364489105166593, bounds=[-inf:inf]>),
('t1', <Parameter 't1', 1.3124544105342462, bounds=[-inf:inf]>),
('t2', <Parameter 't2', 11.80612160586597, bounds=[-inf:inf]>)])

>>> # Finally lets work out a 1 and 2-sigma error estimate for 't1'
>>> quantiles = np.percentile(res.flatchain['t1'], [2.28, 15.9, 50, 84.2, 97.7])
>>> print("2 sigma spread", 0.5 * (quantiles[-1] - quantiles[0]))


## Getting and Printing Fit Reports¶

fit_report(result, modelpars=None, show_correl=True, min_correl=0.1)

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

Parameters: result – MinimizerResult object as returned by minimize(). modelpars – Parameters with “Known Values” (optional, default None) show_correl – whether to show list of sorted correlations [True] min_correl – smallest correlation absolute value to show [0.1]

If the first argument is a Parameters object, goodness-of-fit statistics will not be included.

report_fit(result, modelpars=None, show_correl=True, min_correl=0.1)

print text of report from fit_report().

An example fit with report would be

#!/usr/bin/env python
#<examples/doc_withreport.py>

from __future__ import print_function
from lmfit import Parameters, minimize, fit_report
from numpy import random, linspace, pi, exp, sin, sign

p_true = Parameters()

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

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 = 1001
xmin = 0.
xmax = 250.0

random.seed(0)

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

fit_params = Parameters()

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

print(fit_report(out))

#<end of examples/doc_withreport.py>


which would write out:

[[Fit Statistics]]
# function evals   = 85
# data points      = 1001
# variables        = 4
chi-square         = 498.812
reduced chi-square = 0.500
Akaike info crit   = -689.223
Bayesian info crit = -669.587
[[Variables]]
amp:      13.9121944 +/- 0.141202 (1.01%) (init= 13)
period:   5.48507044 +/- 0.026664 (0.49%) (init= 2)
shift:    0.16203676 +/- 0.014056 (8.67%) (init= 0)
decay:    0.03264538 +/- 0.000380 (1.16%) (init= 0.02)
[[Correlations]] (unreported correlations are <  0.100)
C(period, shift)             =  0.797
C(amp, decay)                =  0.582
C(amp, shift)                = -0.297
C(amp, period)               = -0.243
C(shift, decay)              = -0.182
C(period, decay)             = -0.150