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 sumofsquares of the array returned fromfunction
is minimized.Parameters:  function (callable.) – function to return fit residual. See Writing a Fitting Function for details.
 params (
Parameters
.) – aParameters
dictionary. Keywords must be strings that match[az_][az09_]*
and cannot be a python reserved word. Each value must beParameter
.  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.
Returns: MinimizerResult
instance, which will contain the optimized parameter, and several goodnessoffit statistics.
Changed in version 0.9.0: return value changed to MinimizerResult
On output, the params will be unchanged. The bestfit 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 tominimize()
 kws – keyword arguments. Must match
kws
argument tominimize()
Returns: residual array (generally datamodel) to be minimized in the leastsquares sense.
Return type: numpy array. The length of this array cannot change between calls.
 params (
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
LevenbergMarquardt 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 leastsquares 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.e10:
period = sign(period)*1.e10
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.e10)
but putting this directly in the function with:
if abs(period) < 1.e10:
period = sign(period)*1.e10
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 LevenbergMarquardt 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 wellbehaved for most curvefitting 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 tominimize()
orMinimizer.minimize()
LevenbergMarquardt leastsq
NelderMead nelder
LBFGSB lbfgsb
Powell powell
Conjugate Gradient cg
NewtonCG newton
COBYLA cobyla
Truncated Newton tnc
Dogleg dogleg
Sequential Linear Squares Programming slsqp
Differential Evolution differential_evolution
Note
The objective function for the LevenbergMarquardt 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 LevenbergMarquardt 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 bestfit 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
andcorrel
.

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

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 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
¶ chisquare: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\)

redchi
¶ reduced chisquare: \(\chi^2_{\nu}= {\chi^2} / {(N  N_{\rm varys})}\)

aic
¶ Akaike Information Criterion statistic (see below)

bic
¶ Bayesian Information Criterion statistic (see below).
GoodnessofFit Statistics¶
Table of Fit Results: These values, including the standard GoodnessofFit statistics, are all attributes of theMinimizerResult
object returned byminimize()
orMinimizer.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  chisquare: \(\chi^2 = \sum_i^N [{\rm Resid}_i]^2\) 
redchi  reduced chisquare: \(\chi^2_{\nu}= {\chi^2} / {(N  N_{\rm varys})}\) 
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 chisquare and reduced chisquare 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 \(1\sigma\)
errorbar) 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 chisquare and reduced chisquare statistics:
where \(r\) is the residual array returned by the objective function
(likely to be (datamodel)/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
When comparing fits with different numbers of varying parameters, one typically selects the model with lowest reduced chisquare, 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 usersupplied code to be run at each iteration, and can be used to abort a fit.

iter_cb(params, iter, resid, *args, **kws):
usersupplied 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 tominimize()
 kws – keyword arguments. Must match
kws
argument tominimize()
Returns: residual array (generally datamodel) to be minimized in the leastsquares sense.
Return type: None
for normal behavior, any value likeTrue
to abort fit. params (
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 nonzero 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
[az_][az09_]*
and is not a python reserved word. Each value must beParameter
.  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 chisquare (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 unaltered ‘omit’  the nonfinite 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()
orscalar_minimize()
.Parameters:  method (str.) – name of fitting method. Must be one of the names in Table of Supported Fitting Methods
 params (
Parameters
or None) – aParameters
dictionary for starting values
Returns: 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 LevenbergMarquardt algorithm. Keywords will be passed directly to optimize.leastsq. By default, numerical derivatives are used, and the following arguments are set:
leastsq()
argDefault Value Description xtol 1.e7 Relative error in the approximate solution ftol 1.e7 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='NelderMead', params=None, hess=None, tol=None, **kws)¶ perform fit with any of the scalar minimization algorithms supported by optimize.minimize.
scalar_minimize()
argDefault Value Description method NelderMead
fitting method tol 1.e7 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 aMinimizerResult
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) – aParameters
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 MetropolisHastings 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 givenMinimizer
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: theParameters
used to create the sampler must not change inbetween calls toemcee()
. Alteration ofParameters
would include changedmin
,max
,vary
andexpr
attributes. This may happen, for example, if you use an alteredParameters
object and call theminimize()
method inbetween calls toemcee()
.  workers (int or Poollike) – For parallelization of sampling. It can be any Poollike object with a map method that follows the same calling sequence as the builtin map function. If int is given as the argument, then a multiprocessingbased 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 logposterior 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.
Returns: MinimizerResult
object containing updated params, statistics, etc. TheMinimizerResult
also contains thechain
,flatchain
andlnprob
attributes. Thechain
andflatchain
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. Theflatchain
attribute is apandas.DataFrame
of the flattened chain, chain.reshape(1, nvarys). To access flattened chain values for a particular parameter use result.flatchain[parname]. Thelnprob
attribute contains the log probability for each sample inchain
. 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 logposterior 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 ‘loglikelihood’ and \(\ln p(F_{true})\) is the ‘logprior’. The default logprior encodes prior information already known about the model. This method assumes that the logprior probability is np.inf (impossible) if the one of the parameters is outside its limits. The logprior probability term is zero if all the parameters are inside their bounds (known as a uniform prior). The loglikelihood function is given by [1]:
\[\ln p(DF_{true}) = \frac{1}{2}\sum_n \left[\frac{\left(g_n(F_{true})  D_n \right)^2}{s_n^2}+\ln (2\pi s_n^2)\right]\]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 logposterior probability, \(\ln p(F_{true}  D)\). However, since the inbuilt logprior term is zero, the objective function can also just return the loglikelihood, unless you wish to create a nonuniform prior.If a float value is returned by the objective function then this value is assumed by default to be the logposterior 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 loglikelihood (and logposterior 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 logposterior 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.
[1] (1, 2) http://dan.iel.fm/emcee/current/user/line/  params (
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 logposterior probability. The logposterior probability is a sum of the logprior probability and loglikelihood functions. The logprior 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
>>> mi.params.add('f', value=1, min=0.001, max=2)
>>> # This is the loglikelihood 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 2sigma 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]))
2 sigma spread 0.298878202908
Getting and Printing Fit Reports¶

fit_report
(result, modelpars=None, show_correl=True, min_correl=0.1)¶ generate and return text of report of bestfit values, uncertainties, and correlations from fit.
Parameters:  result –
MinimizerResult
object as returned byminimize()
.  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, goodnessoffit statistics will not be included. result –

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()
p_true.add('amp', value=14.0)
p_true.add('period', value=5.46)
p_true.add('shift', value=0.123)
p_true.add('decay', value=0.032)
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()
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})
print(fit_report(out))
#<end of examples/doc_withreport.py>
which would write out:
[[Fit Statistics]]
# function evals = 85
# data points = 1001
# variables = 4
chisquare = 498.812
reduced chisquare = 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