Modeling Data and Curve Fitting¶
A common use of leastsquares minimization is curve fitting, where one
has a parametrized model function meant to explain some phenomena and wants
to adjust the numerical values for the model so that it most closely
matches some data. With scipy
, such problems are typically solved
with scipy.optimize.curve_fit, which is a wrapper around
scipy.optimize.leastsq. Since lmfit’s
minimize()
is also a highlevel wrapper around
scipy.optimize.leastsq it can be used for curvefitting problems.
While it offers many benefits over scipy.optimize.leastsq, using
minimize()
for many curvefitting problems still
requires more effort than using scipy.optimize.curve_fit.
The Model
class in lmfit provides a simple and flexible approach
to curvefitting problems. Like scipy.optimize.curve_fit, a
Model
uses a model function – a function that is meant to
calculate a model for some phenomenon – and then uses that to best match
an array of supplied data. Beyond that similarity, its interface is rather
different from scipy.optimize.curve_fit, for example in that it uses
Parameters
, but also offers several other
important advantages.
In addition to allowing you to turn any model function into a curvefitting
method, lmfit also provides canonical definitions for many known line shapes
such as Gaussian or Lorentzian peaks and Exponential decays that are widely
used in many scientific domains. These are available in the models
module that will be discussed in more detail in the next chapter
(Builtin Fitting Models in the models module). We mention it here as you may want to
consult that list before writing your own model. For now, we focus on
turning Python functions into highlevel fitting models with the
Model
class, and using these to fit data.
Motivation and simple example: Fit data to Gaussian profile¶
Let’s start with a simple and common example of fitting data to a Gaussian
peak. As we will see, there is a buitin GaussianModel
class that
can help do this, but here we’ll build our own. We start with a simple
definition of the model function:
>>> from numpy import sqrt, pi, exp, linspace, random
>>>
>>> def gaussian(x, amp, cen, wid):
... return amp * exp((xcen)**2 /wid)
We want to use this function to fit to data \(y(x)\) represented by the arrays y and x. With scipy.optimize.curve_fit, this would be:
>>> from scipy.optimize import curve_fit
>>>
>>> x = linspace(10,10, 101)
>>> y = gaussian(x, 2.33, 0.21, 1.51) + random.normal(0, 0.2, len(x))
>>>
>>> init_vals = [1, 0, 1] # for [amp, cen, wid]
>>> best_vals, covar = curve_fit(gaussian, x, y, p0=init_vals)
>>> print best_vals
That is, we create data, make an initial guess of the model values, and run scipy.optimize.curve_fit with the model function, data arrays, and initial guesses. The results returned are the optimal values for the parameters and the covariance matrix. It’s simple and useful, but it misses the benefits of lmfit.
With lmfit, we create a Model
that wraps the gaussian model
function, which automatically generates the appropriate residual function,
and determines the corresponding parameter names from the function
signature itself:
>>> from lmfit import Model
>>> gmodel = Model(gaussian)
>>> gmodel.param_names
set(['amp', 'wid', 'cen'])
>>> gmodel.independent_vars
['x']
As you can see, the Model gmodel determined the names of the parameters
and the independent variables. By default, the first argument of the
function is taken as the independent variable, held in
independent_vars
, and the rest of the functions positional
arguments (and, in certain cases, keyword arguments – see below) are used
for Parameter names. Thus, for the gaussian function above, the
independent variable is x, and the parameters are named amp,
cen, and wid, and – all taken directly from the signature of the
model function. As we will see below, you can modify the default
assignment of independent variable / arguments and specify yourself what
the independent variable is and which function arguments should be identified
as parameter names.
The Parameters are not created when the model is created. The model knows
what the parameters should be named, but not anything about the scale and
range of your data. You will normally have to make these parameters and
assign initial values and other attributes. To help you do this, each
model has a make_params()
method that will generate parameters with
the expected names:
>>> params = gmod.make_params()
This creates the Parameters
but does not
automaticaly give them initial values since it has no idea what the scale
should be. You can set initial values for parameters with keyword
arguments to make_params()
:
>>> params = gmod.make_params(cen=5, amp=200, wid=1)
or assign them (and other parameter properties) after the
Parameters
class has been created.
A Model
has several methods associated with it. For example, one
can use the eval()
method to evaluate the model or the fit()
method to fit data to this model with a Parameter
object. Both of
these methods can take explicit keyword arguments for the parameter values.
For example, one could use eval()
to calculate the predicted
function:
>>> x = linspace(0, 10, 201)
>>> y = gmod.eval(params, x=x)
or with:
>>> y = gmod.eval(x=x, cen=6.5, amp=100, wid=2.0)
Admittedly, this a slightly longwinded way to calculate a Gaussian
function, given that you could have called your gaussian function
directly. But now that the model is set up, we can use its
fit()
method to fit this model to data, as with:
>>> result = gmod.fit(y, params)
or with:
>>> result = gmod.fit(y, cen=6.5, amp=100, wid=2.0)
Putting everything together, (included in the
examples
folder with the source code) is:
#!/usr/bin/env python
#<examples/doc_model1.py>
from numpy import sqrt, pi, exp, linspace, loadtxt
from lmfit import Model
import matplotlib.pyplot as plt
data = loadtxt('model1d_gauss.dat')
x = data[:, 0]
y = data[:, 1]
def gaussian(x, amp, cen, wid):
"1d gaussian: gaussian(x, amp, cen, wid)"
return (amp/(sqrt(2*pi)*wid)) * exp((xcen)**2 /(2*wid**2))
gmodel = Model(gaussian)
result = gmodel.fit(y, x=x, amp=5, cen=5, wid=1)
print(result.fit_report())
plt.plot(x, y, 'bo')
plt.plot(x, result.init_fit, 'k')
plt.plot(x, result.best_fit, 'r')
plt.show()
#<end examples/doc_model1.py>
which is pretty compact and to the point. The returned result will be
a ModelResult
object. As we will see below, this has many
components, including a fit_report()
method, which will show:
[[Model]]
Model(gaussian)
[[Fit Statistics]]
# function evals = 31
# data points = 101
# variables = 3
chisquare = 3.409
reduced chisquare = 0.035
Akaike info crit = 336.264
Bayesian info crit = 328.418
[[Variables]]
amp: 5.07800631 +/ 0.064957 (1.28%) (init= 5)
cen: 5.65866112 +/ 0.010304 (0.18%) (init= 5)
wid: 0.97344373 +/ 0.028756 (2.95%) (init= 1)
[[Correlations]] (unreported correlations are < 0.100)
C(amp, wid) = 0.577
As the script shows, the result will also have init_fit
for the fit
with the initial parameter values and a best_fit
for the fit with
the best fit parameter values. These can be used to generate the following
plot:
which shows the data in blue dots, the best fit as a solid red line, and the initial fit as a dashed black line.
Note that the model fitting was really performed with:
gmodel = Model(gaussian)
result = gmodel.fit(y, params, x=x, amp=5, cen=5, wid=1)
These lines clearly express that we want to turn the gaussian function
into a fitting model, and then fit the \(y(x)\) data to this model,
starting with values of 5 for amp, 5 for cen and 1 for wid. In
addition, all the other features of lmfit are included:
Parameters
can have bounds and constraints and
the result is a rich object that can be reused to explore the model fit in
detail.
The Model
class¶
The Model
class provides a general way to wrap a predefined
function as a fitting model.

class
Model
(func, independent_vars=None, param_names=None, missing='none', prefix='', name=None, **kws)¶ Create a model from a usersupplied model function.
The model function will normally take an independent variable (generally, the first argument) and a series of arguments that are meant to be parameters for the model. It will return an array of data to model some data as for a curvefitting problem.
Parameters:  func (callable) – Function to be wrapped.
 independent_vars (list of str, optional) – Arguments to func that are independent variables (default is None).
 param_names (list of str, optional) – Names of arguments to func that are to be made into parameters (default is None).
 missing (str, optional) –
How to handle NaN and missing values in data. One of:
 ‘none’ or None : Do not check for null or missing values (default).
 ‘drop’ : Drop null or missing observations in data. If pandas is installed, pandas.isnull is used, otherwise numpy.isnan is used.
 ‘raise’ : Raise a (more helpful) exception when data contains null or missing values.
 prefix (str, optional) – Prefix used for the model.
 name (str, optional) – Name for the model. When None (default) the name is the same as the model function (func).
 **kws (dict, optional) – Additional keyword arguments to pass to model function.
Notes
1. Parameter names are inferred from the function arguments, and a residual function is automatically constructed.
2. The model function must return an array that will be the same size as the data being modeled.
Examples
The model function will normally take an independent variable (generally, the first argument) and a series of arguments that are meant to be parameters for the model. Thus, a simple peak using a Gaussian defined as:
>>> import numpy as np >>> def gaussian(x, amp, cen, wid): ... return amp * np.exp((xcen)**2 / wid)
can be turned into a Model with:
>>> gmodel = Model(gaussian)
this will automatically discover the names of the independent variables and parameters:
>>> print(gmodel.param_names, gmodel.independent_vars) ['amp', 'cen', 'wid'], ['x']
Model
class Methods¶

Model.
eval
(params=None, **kwargs)¶ Evaluate the model with supplied parameters and keyword arguments.
Parameters:  params (Parameters, optional) – Parameters to use in Model.
 **kwargs (optional) – Additional keyword arguments to pass to model function.
Returns: Value of model given the parameters and other arguments.
Return type: Notes
1. if params is None, the values for all parameters are expected to be provided as keyword arguments. If params is given, and a keyword argument for a parameter value is also given, the keyword argument will be used.
2. all nonparameter arguments for the model function, including all the independent variables will need to be passed in using keyword arguments.

Model.
fit
(data, params=None, weights=None, method='leastsq', iter_cb=None, scale_covar=True, verbose=False, fit_kws=None, **kwargs)¶ Fit the model to the data using the supplied Parameters.
Parameters:  data (array_like) – Array of data to be fit.
 params (Parameters, optional) – Parameters to use in fit (default is None).
 weights (array_like of same size as data, optional) – Weights to use for the calculation of the fit residual (default is None).
 method (str, optional) – Name of fitting method to use (default is ‘leastsq’).
 iter_cb (callable, optional) – Callback function to call at each iteration (default is None).
 scale_covar (bool, optional) – Whether to automatically scale the covariance matrix when calculating uncertainties (default is True, leastsq method only).
 verbose (bool, optional) – Whether to print a message when a new parameter is added because of a hint (default is True).
 fit_kws (dict, optional) – Options to pass to the minimizer being used.
 **kwargs (optional) – Arguments to pass to the model function, possibly overriding params.
Returns: Return type: Examples
Take t to be the independent variable and data to be the curve we will fit. Use keyword arguments to set initial guesses:
>>> result = my_model.fit(data, tau=5, N=3, t=t)
Or, for more control, pass a Parameters object.
>>> result = my_model.fit(data, params, t=t)
Keyword arguments override Parameters.
>>> result = my_model.fit(data, params, tau=5, t=t)
Notes
1. if params is None, the values for all parameters are expected to be provided as keyword arguments. If params is given, and a keyword argument for a parameter value is also given, the keyword argument will be used.
2. all nonparameter arguments for the model function, including all the independent variables will need to be passed in using keyword arguments.
3. Parameters (however passed in), are copied on input, so the original Parameter objects are unchanged, and the updated values are in the returned ModelResult.

Model.
guess
(data, **kws)¶ Guess starting values for the parameters of a model.
This is not implemented for all models, but is available for many of the builtin models.
Parameters:  data (array_like) – Array of data to use to guess parameter values.
 **kws (optional) – Additional keyword arguments, passed to model function.
Returns: params
Return type: Notes
Should be implemented for each model subclass to run self.make_params(), update starting values and return a Parameters object.
Raises: NotImplementedError

Model.
make_params
(verbose=False, **kwargs)¶ Create a Parameters object for a Model.
Parameters:  verbose (bool, optional) – Whether to print out messages (default is False).
 **kwargs (optional) – Parameter names and initial values.
Returns: params
Return type: Notes
1. The parameters may or may not have decent initial values for each parameter.
2. This applies any default values or parameter hints that may have been set.

Model.
set_param_hint
(name, **kwargs)¶ Set hints to use when creating parameters with make_params() for the named parameter.
This is especially convenient for setting initial values. The name can include the models prefix or not. The hint given can also include optional bounds and constraints
(value, vary, min, max, expr)
, which will be used by make_params() when building default parameters.Parameters:  name (string) – Parameter name.
 **kwargs (optional) –
Arbitrary keyword arguments, needs to be a Parameter attribute. Can be any of the following:
 value : float, optional
 Numerical Parameter value.
 vary : bool, optional
 Whether the Parameter is varied during a fit (default is True).
 min : float, optional
 Lower bound for value (default is numpy.inf, no lower bound).
 max : float, optional
 Upper bound for value (default is numpy.inf, no upper bound).
 expr : str, optional
 Mathematical expression used to constrain the value during the fit.
Example
>>> model = GaussianModel() >>> model.set_param_hint('sigma', min=0)
Model
class Attributes¶

func
¶ The model function used to calculate the model.

independent_vars
¶ List of strings for names of the independent variables.

missing
¶ Describes what to do for missing values. The choices are:
 None: Do not check for null or missing values (default).
 ‘none’: Do not check for null or missing values.
 ‘drop’: Drop null or missing observations in data. If pandas is installed,
pandas.isnull()
is used, otherwisenumpy.isnan()
is used.  ‘raise’: Raise a (more helpful) exception when data contains null or missing values.

name
¶ Name of the model, used only in the string representation of the model. By default this will be taken from the model function.

opts
¶ Extra keyword arguments to pass to model function. Normally this will be determined internally and should not be changed.

param_hints
¶ Dictionary of parameter hints. See Using parameter hints.

param_names
¶ List of strings of parameter names.

prefix
¶ Prefix used for namemangling of parameter names. The default is ‘’. If a particular
Model
has arguments amplitude, center, and sigma, these would become the parameter names. Using a prefix of ‘g1_’ would convert these parameter names to g1_amplitude, g1_center, and g1_sigma. This can be essential to avoid name collision in composite models.
Determining parameter names and independent variables for a function¶
The Model
created from the supplied function func will create
a Parameters
object, and names are inferred from the function
arguments, and a residual function is automatically constructed.
By default, the independent variable is take as the first argument to the function. You can, of course, explicitly set this, and will need to do so if the independent variable is not first in the list, or if there are actually more than one independent variables.
If not specified, Parameters are constructed from all positional arguments and all keyword arguments that have a default value that is numerical, except the independent variable, of course. Importantly, the Parameters can be modified after creation. In fact, you will have to do this because none of the parameters have valid initial values. In addition, one can place bounds and constraints on Parameters, or fix their values.
Explicitly specifying independent_vars
¶
As we saw for the Gaussian example above, creating a Model
from a
function is fairly easy. Let’s try another one:
>>> from lmfit import Model
>>> import numpy as np
>>> def decay(t, tau, N):
... return N*np.exp(t/tau)
...
>>> decay_model = Model(decay)
>>> print decay_model.independent_vars
['t']
>>> for pname, par in decay_model.params.items():
... print pname, par
...
tau <Parameter 'tau', None, bounds=[None:None]>
N <Parameter 'N', None, bounds=[None:None]>
Here, t is assumed to be the independent variable because it is the first argument to the function. The other function arguments are used to create parameters for the model.
If you want tau to be the independent variable in the above example, you can say so:
>>> decay_model = Model(decay, independent_vars=['tau'])
>>> print decay_model.independent_vars
['tau']
>>> for pname, par in decay_model.params.items():
... print pname, par
...
t <Parameter 't', None, bounds=[None:None]>
N <Parameter 'N', None, bounds=[None:None]>
You can also supply multiple values for multidimensional functions with multiple independent variables. In fact, the meaning of independent variable here is simple, and based on how it treats arguments of the function you are modeling:
 independent variable
 A function argument that is not a parameter or otherwise part of the
model, and that will be required to be explicitly provided as a
keyword argument for each fit with
Model.fit()
or evaluation withModel.eval()
.
Note that independent variables are not required to be arrays, or even floating point numbers.
Functions with keyword arguments¶
If the model function had keyword parameters, these would be turned into Parameters if the supplied default value was a valid number (but not None, True, or False).
>>> def decay2(t, tau, N=10, check_positive=False):
... if check_small:
... arg = abs(t)/max(1.e9, abs(tau))
... else:
... arg = t/tau
... return N*np.exp(arg)
...
>>> mod = Model(decay2)
>>> for pname, par in mod.params.items():
... print pname, par
...
t <Parameter 't', None, bounds=[None:None]>
N <Parameter 'N', 10, bounds=[None:None]>
Here, even though N is a keyword argument to the function, it is turned into a parameter, with the default numerical value as its initial value. By default, it is permitted to be varied in the fit – the 10 is taken as an initial value, not a fixed value. On the other hand, the check_positive keyword argument, was not converted to a parameter because it has a boolean default value. In some sense, check_positive becomes like an independent variable to the model. However, because it has a default value it is not required to be given for each model evaluation or fit, as independent variables are.
Defining a prefix for the Parameters¶
As we will see in the next chapter when combining models, it is sometimes
necessary to decorate the parameter names in the model, but still have them
be correctly used in the underlying model function. This would be
necessary, for example, if two parameters in a composite model (see
Composite Models : adding (or multiplying) Models or examples in the next chapter) would have
the same name. To avoid this, we can add a prefix to the
Model
which will automatically do this mapping for us.
>>> def myfunc(x, amplitude=1, center=0, sigma=1):
...
>>> mod = Model(myfunc, prefix='f1_')
>>> for pname, par in mod.params.items():
... print pname, par
...
f1_amplitude <Parameter 'f1_amplitude', None, bounds=[None:None]>
f1_center <Parameter 'f1_center', None, bounds=[None:None]>
f1_sigma <Parameter 'f1_sigma', None, bounds=[None:None]>
You would refer to these parameters as f1_amplitude and so forth, and the model will know to map these to the amplitude argument of myfunc.
Initializing model parameters¶
As mentioned above, the parameters created by Model.make_params()
are
generally created with invalid initial values of None. These values
must be initialized in order for the model to be evaluated or used in a
fit. There are four different ways to do this initialization that can be
used in any combination:
 You can supply initial values in the definition of the model function.
 You can initialize the parameters when creating parameters with
Model.make_params()
. You can give parameter hints with
Model.set_param_hint()
. You can supply initial values for the parameters when you use the
Model.eval()
orModel.fit()
methods.
Of course these methods can be mixed, allowing you to overwrite initial values at any point in the process of defining and using the model.
Initializing values in the function definition¶
To supply initial values for parameters in the definition of the model function, you can simply supply a default value:
>>> def myfunc(x, a=1, b=0):
>>> ...
instead of using:
>>> def myfunc(x, a, b):
>>> ...
This has the advantage of working at the function level – all parameters with keywords can be treated as options. It also means that some default initial value will always be available for the parameter.
Initializing values with Model.make_params()
¶
When creating parameters with Model.make_params()
you can specify initial
values. To do this, use keyword arguments for the parameter names and
initial values:
>>> mod = Model(myfunc)
>>> pars = mod.make_params(a=3, b=0.5)
Initializing values by setting parameter hints¶
After a model has been created, but prior to creating parameters with
Model.make_params()
, you can set parameter hints. These allows you to set
not only a default initial value but also to set other parameter attributes
controlling bounds, whether it is varied in the fit, or a constraint
expression. To set a parameter hint, you can use Model.set_param_hint()
,
as with:
>>> mod = Model(myfunc)
>>> mod.set_param_hint('a', value = 1.0)
>>> mod.set_param_hint('b', value = 0.3, min=0, max=1.0)
>>> pars = mod.make_params()
Parameter hints are discussed in more detail in section Using parameter hints.
Initializing values when using a model¶
Finally, you can explicitly supply initial values when using a model. That
is, as with Model.make_params()
, you can include values
as keyword arguments to either the Model.eval()
or Model.fit()
methods:
>>> y1 = mod.eval(x=x, a=7.0, b=2.0)
>>> out = mod.fit(x=x, pars, a=3.0, b=0.0)
These approaches to initialization provide many opportunities for setting
initial values for parameters. The methods can be combined, so that you
can set parameter hints but then change the initial value explicitly with
Model.fit()
.
Using parameter hints¶
After a model has been created, you can give it hints for how to create
parameters with Model.make_params()
. This allows you to set not only a
default initial value but also to set other parameter attributes
controlling bounds, whether it is varied in the fit, or a constraint
expression. To set a parameter hint, you can use Model.set_param_hint()
,
as with:
>>> mod = Model(myfunc)
>>> mod.set_param_hint('a', value = 1.0)
>>> mod.set_param_hint('b', value = 0.3, min=0, max=1.0)
Parameter hints are stored in a model’s param_hints
attribute,
which is simply a nested dictionary:
>>> print mod.param_hints
{'a': {'value': 1}, 'b': {'max': 1.0, 'value': 0.3, 'min': 0}}
You can change this dictionary directly, or with the Model.set_param_hint()
method. Either way, these parameter hints are used by Model.make_params()
when making parameters.
An important feature of parameter hints is that you can force the creation of new parameters with parameter hints. This can be useful to make derived parameters with constraint expressions. For example to get the fullwidth at half maximum of a Gaussian model, one could use a parameter hint of:
>>> mod = Model(gaussian)
>>> mod.set_param_hint('fwhm', expr='2.3548*sigma')
The ModelResult
class¶
A ModelResult
(which had been called ModelFit prior to version
0.9) is the object returned by Model.fit()
. It is a subclass of
Minimizer
, and so contains many of the fit results.
Of course, it knows the Model
and the set of
Parameters
used in the fit, and it has methods to
evaluate the model, to fit the data (or refit the data with changes to
the parameters, or fit with different or modified data) and to print out a
report for that fit.
While a Model
encapsulates your model function, it is fairly
abstract and does not contain the parameters or data used in a particular
fit. A ModelResult
does contain parameters and data as well as
methods to alter and redo fits. Thus the Model
is the idealized
model while the ModelResult
is the messier, more complex (but perhaps
more useful) object that represents a fit with a set of parameters to data
with a model.
A ModelResult
has several attributes holding values for fit
results, and several methods for working with fits. These include
statistics inherited from Minimizer
useful for
comparing different models, including chisqr, redchi, aic, and bic.

class
ModelResult
(model, params, data=None, weights=None, method='leastsq', fcn_args=None, fcn_kws=None, iter_cb=None, scale_covar=True, **fit_kws)¶ Result from the Model fit.
This has many attributes and methods for viewing and working with the results of a fit using Model. It inherits from Minimizer, so that it can be used to modify and rerun the fit for the Model.
Parameters:  model (Model) – Model to use.
 params (Parameters) – Parameters with initial values for model.
 data (array_like, optional) – Data to be modeled.
 weights (array_like, optional) – Weights to multiply (datamodel) for fit residual.
 method (str, optional) – Name of minimization method to use (default is ‘leastsq’).
 fcn_args (sequence, optional) – Positional arguments to send to model function.
 fcn_dict (dict, optional) – Keyword arguments to send to model function.
 iter_cb (callable, optional) – Function to call on each iteration of fit.
 scale_covar (bool, optional) – Whether to scale covariance matrix for uncertainty evaluation.
 **fit_kws (optional) – Keyword arguments to send to minimization routine.
ModelResult
methods¶

ModelResult.
eval
(params=None, **kwargs)¶ Evaluate model function.
Parameters:  params (Parameters, optional) – Parameters to use.
 **kwargs (optional) – Options to send to Model.eval()
Returns: out – Array for evaluated model.
Return type:

ModelResult.
eval_components
(params=None, **kwargs)¶ Evaluate each component of a composite model function.
Parameters:  params (Parameters, optional) – Parameters, defaults to ModelResult.params
 **kwargs (optional) – Leyword arguments to pass to model function.
Returns: Keys are prefixes of component models, and values are the estimated model value for each component of the model.
Return type: OrderedDict

ModelResult.
fit
(data=None, params=None, weights=None, method=None, **kwargs)¶ Reperform fit for a Model, given data and params.
Parameters:  data (array_like, optional) – Data to be modeled.
 params (Parameters, optional) – Parameters with initial values for model.
 weights (array_like, optional) – Weights to multiply (datamodel) for fit residual.
 method (str, optional) – Name of minimization method to use (default is ‘leastsq’).
 **kwargs (optional) – Keyword arguments to send to minimization routine.

ModelResult.
fit_report
(modelpars=None, show_correl=True, min_correl=0.1, sort_pars=False)¶ Return a printable fit report.
The report contains fit statistics and bestfit values with uncertainties and correlations.
Parameters:  modelpars (Parameters, optional) – Known Model Parameters.
 show_correl (bool, optional) – Whether to show list of sorted correlations (default is True).
 min_correl (float, optional) – Smallest correlation in absolute value to show (default is 0.1).
 sort_pars (callable, optional) – Whether to show parameter names sorted in alphanumerical order (default is False). If False, then the parameters will be listed in the order as they were added to the Parameters dictionary. If callable, then this (one argument) function is used to extract a comparison key from each list element.
Returns: text – Multiline text of fit report.
Return type: See also

ModelResult.
conf_interval
(**kwargs)¶ Calculate the confidence intervals for the variable parameters.
Confidence intervals are calculated using the
confidence.conf_interval()
function and keyword arguments (**kwargs) are passed to that function. The result is stored in theci_out
attribute so that it can be accessed without recalculating them.

ModelResult.
ci_report
(with_offset=True, ndigits=5, **kwargs)¶ Return a nicely formatted text report of the confidence intervals.
Parameters: Returns: Text of formatted report on confidence intervals.
Return type:

ModelResult.
eval_uncertainty
(params=None, sigma=1, **kwargs)¶ Evaluate the uncertainty of the model function from the uncertainties for the bestfit parameters. This can be used to give confidence bands for the model.
Parameters:  params (Parameters, optional) – Parameters, defaults to ModelResult.params.
 sigma (float, optional) – Confidence level, i.e. how many sigma (default is 1).
 **kwargs (optional) – Values of options, independent variables, etcetera.
Returns: Uncertainty at each value of the model.
Return type: Example
>>> out = model.fit(data, params, x=x) >>> dely = out.eval_confidence_band(x=x) >>> plt.plot(x, data) >>> plt.plot(x, out.best_fit) >>> plt.fill_between(x, out.best_fitdely, ... out.best_fit+dely, color='#888888')
Notes
 This is based on the excellent and clear example from https://www.astro.rug.nl/software/kapteyn/kmpfittutorial.html#confidenceandpredictionintervals, which references the original work of: J. Wolberg,Data Analysis Using the Method of Least Squares, 2006, Springer
 The value of sigma is number of sigma values, and is converted to a probability. Values or 1, 2, or 3 give probalities of 0.6827, 0.9545, and 0.9973, respectively. If the sigma value is < 1, it is interpreted as the probability itself. That is, sigma=1 and sigma=0.6827 will give the same results, within precision errors.

ModelResult.
plot
(*args, **kws)¶ Plot the fit results and residuals using matplotlib, if available.
The method will produce a matplotlib figure with both results of the fit and the residuals plotted. If the fit model included weights, errorbars will also be plotted.
Parameters:  datafmt (str, optional) – Matplotlib format string for data points.
 fitfmt (str, optional) – Matplotlib format string for fitted curve.
 initfmt (str, optional) – Matplotlib format string for initial conditions for the fit.
 xlabel (str, optional) – Matplotlib format string for labeling the xaxis.
 ylabel (str, optional) – Matplotlib format string for labeling the yaxis.
 yerr (numpy.ndarray, optional) – Array of uncertainties for data array.
 numpoints (int, optional) – If provided, the final and initial fit curves are evaluated not only at data points, but refined to contain numpoints points in total.
 fig (matplotlib.figure.Figure, optional) – The figure to plot on. The default is None, which means use the current pyplot figure or create one if there is none.
 data_kws (dict, optional) – Keyword arguments passed on to the plot function for data points.
 fit_kws (dict, optional) – Keyword arguments passed on to the plot function for fitted curve.
 init_kws (dict, optional) – Keyword arguments passed on to the plot function for the initial conditions of the fit.
 ax_res_kws (dict, optional) – Keyword arguments for the axes for the residuals plot.
 ax_fit_kws (dict, optional) – Keyword arguments for the axes for the fit plot.
 fig_kws (dict, optional) – Keyword arguments for a new figure, if there is one being created.
Returns: Return type: A tuple with matplotlib’s Figure and GridSpec objects.
Notes
The method combines ModelResult.plot_fit and ModelResult.plot_residuals.
If yerr is specified or if the fit model included weights, then matplotlib.axes.Axes.errorbar is used to plot the data. If yerr is not specified and the fit includes weights, yerr set to 1/self.weights
If fig is None then matplotlib.pyplot.figure(**fig_kws) is called, otherwise fig_kws is ignored.
See also
ModelResult.plot_fit()
 Plot the fit results using matplotlib.
ModelResult.plot_residuals()
 Plot the fit residuals using matplotlib.

ModelResult.
plot_fit
(*args, **kws)¶ Plot the fit results using matplotlib, if available.
The plot will include the data points, the initial fit curve, and the bestfit curve. If the fit model included weights or if yerr is specified, errorbars will also be plotted.
Parameters:  ax (matplotlib.axes.Axes, optional) – The axes to plot on. The default in None, which means use the current pyplot axis or create one if there is none.
 datafmt (str, optional) – Matplotlib format string for data points.
 fitfmt (str, optional) – Matplotlib format string for fitted curve.
 initfmt (str, optional) – Matplotlib format string for initial conditions for the fit.
 xlabel (str, optional) – Matplotlib format string for labeling the xaxis.
 ylabel (str, optional) – Matplotlib format string for labeling the yaxis.
 yerr (numpy.ndarray, optional) – Array of uncertainties for data array.
 numpoints (int, optional) – If provided, the final and initial fit curves are evaluated not only at data points, but refined to contain numpoints points in total.
 data_kws (dict, optional) – Keyword arguments passed on to the plot function for data points.
 fit_kws (dict, optional) – Keyword arguments passed on to the plot function for fitted curve.
 init_kws (dict, optional) – Keyword arguments passed on to the plot function for the initial conditions of the fit.
 ax_kws (dict, optional) – Keyword arguments for a new axis, if there is one being created.
Returns: Return type: matplotlib.axes.Axes
Notes
For details about plot format strings and keyword arguments see documentation of matplotlib.axes.Axes.plot.
If yerr is specified or if the fit model included weights, then matplotlib.axes.Axes.errorbar is used to plot the data. If yerr is not specified and the fit includes weights, yerr set to 1/self.weights
If ax is None then matplotlib.pyplot.gca(**ax_kws) is called.
See also
ModelResult.plot_residuals()
 Plot the fit residuals using matplotlib.
ModelResult.plot()
 Plot the fit results and residuals using matplotlib.

ModelResult.
plot_residuals
(*args, **kws)¶ Plot the fit residuals using matplotlib, if available.
If yerr is supplied or if the model included weights, errorbars will also be plotted.
Parameters:  ax (matplotlib.axes.Axes, optional) – The axes to plot on. The default in None, which means use the current pyplot axis or create one if there is none.
 datafmt (str, optional) – Matplotlib format string for data points.
 yerr (numpy.ndarray, optional) – Array of uncertainties for data array.
 data_kws (dict, optional) – Keyword arguments passed on to the plot function for data points.
 fit_kws (dict, optional) – Keyword arguments passed on to the plot function for fitted curve.
 ax_kws (dict, optional) – Keyword arguments for a new axis, if there is one being created.
Returns: Return type: matplotlib.axes.Axes
Notes
For details about plot format strings and keyword arguments see documentation of matplotlib.axes.Axes.plot.
If yerr is specified or if the fit model included weights, then matplotlib.axes.Axes.errorbar is used to plot the data. If yerr is not specified and the fit includes weights, yerr set to 1/self.weights
If ax is None then matplotlib.pyplot.gca(**ax_kws) is called.
See also
ModelResult.plot_fit()
 Plot the fit results using matplotlib.
ModelResult.plot()
 Plot the fit results and residuals using matplotlib.
ModelResult
attributes¶

aic
¶ Floating point bestfit Akaike Information Criterion statistic (see MinimizerResult – the optimization result).

best_fit
¶ numpy.ndarray result of model function, evaluated at provided independent variables and with bestfit parameters.

best_values
¶ Dictionary with parameter names as keys, and bestfit values as values.

bic
¶ Floating point bestfit Bayesian Information Criterion statistic (see MinimizerResult – the optimization result).

chisqr
¶ Floating point bestfit chisquare statistic (see MinimizerResult – the optimization result).

ci_out
¶ Confidence interval data (see Calculation of confidence intervals) or None if the confidence intervals have not been calculated.

covar
¶ numpy.ndarray (square) covariance matrix returned from fit.

data
¶ numpy.ndarray of data to compare to model.

errorbars
¶ Boolean for whether error bars were estimated by fit.

ier
¶ Integer returned code from scipy.optimize.leastsq.

init_fit
¶ numpy.ndarray result of model function, evaluated at provided independent variables and with initial parameters.

init_params
¶ Initial parameters.

init_values
¶ Dictionary with parameter names as keys, and initial values as values.

iter_cb
¶ Optional callable function, to be called at each fit iteration. This must take take arguments of
(params, iter, resid, *args, **kws)
, where params will have the current parameter values, iter the iteration, resid the current residual array, and *args and **kws as passed to the objective function. See Using a Iteration Callback Function.

jacfcn
¶ Optional callable function, to be called to calculate Jacobian array.

lmdif_message
¶ String message returned from scipy.optimize.leastsq.

message
¶ String message returned from
minimize()
.

method
¶ String naming fitting method for
minimize()
.

ndata
¶ Integer number of data points.

nfev
¶ Integer number of function evaluations used for fit.

nfree
¶ Integer number of free parameters in fit.

nvarys
¶ Integer number of independent, freely varying variables in fit.

params
¶ Parameters used in fit. Will have bestfit values.

redchi
¶ Floating point reduced chisquare statistic (see MinimizerResult – the optimization result).

residual
¶ numpy.ndarray for residual.

scale_covar
¶ Boolean flag for whether to automatically scale covariance matrix.

success
¶ Boolean value of whether fit succeeded.

weights
¶ numpy.ndarray (or None) of weighting values to be used in fit. If not None, it will be used as a multiplicative factor of the residual array, so that
weights*(data  fit)
is minimized in the leastsquares sense.
Calculating uncertainties in the model function¶
We return to the first example above and ask not only for the
uncertainties in the fitted parameters but for the range of values that
those uncertainties mean for the model function itself. We can use the
ModelResult.eval_uncertainty()
method of the model result object to
evaluate the uncertainty in the model with a specified level for
\(sigma\).
That is, adding:
dely = result.eval_uncertainty(sigma=3)
plt.fill_between(x, result.best_fitdely, result.best_fit+dely, color="#ABABAB")
to the example fit to the Gaussian at the beginning of this chapter will give \(3sigma\) bands for the bestfit Gaussian, and produce the figure below.
Composite Models : adding (or multiplying) Models¶
One of the more interesting features of the Model
class is that
Models can be added together or combined with basic algebraic operations
(add, subtract, multiply, and divide) to give a composite model. The
composite model will have parameters from each of the component models,
with all parameters being available to influence the whole model. This
ability to combine models will become even more useful in the next chapter,
when prebuilt subclasses of Model
are discussed. For now, we’ll
consider a simple example, and build a model of a Gaussian plus a line, as
to model a peak with a background. For such a simple problem, we could just
build a model that included both components:
def gaussian_plus_line(x, amp, cen, wid, slope, intercept):
"line + 1d gaussian"
gauss = (amp/(sqrt(2*pi)*wid)) * exp((xcen)**2 /(2*wid**2))
line = slope * x + intercept
return gauss + line
and use that with:
mod = Model(gaussian_plus_line)
But we already had a function for a gaussian function, and maybe we’ll discover that a linear background isn’t sufficient which would mean the model function would have to be changed.
Instead, lmfit allows models to be combined into a CompositeModel
.
As an alternative to including a linear background in our model function,
we could define a linear function:
def line(x, slope, intercept):
"a line"
return slope * x + intercept
and build a composite model with just:
mod = Model(gaussian) + Model(line)
This model has parameters for both component models, and can be used as:
#!/usr/bin/env python
#<examples/model_doc2.py>
from numpy import sqrt, pi, exp, loadtxt
from lmfit import Model
import matplotlib.pyplot as plt
data = loadtxt('model1d_gauss.dat')
x = data[:, 0]
y = data[:, 1] + 0.25*x  1.0
def gaussian(x, amp, cen, wid):
"1d gaussian: gaussian(x, amp, cen, wid)"
return (amp/(sqrt(2*pi)*wid)) * exp((xcen)**2 /(2*wid**2))
def line(x, slope, intercept):
"line"
return slope * x + intercept
mod = Model(gaussian) + Model(line)
pars = mod.make_params( amp=5, cen=5, wid=1, slope=0, intercept=1)
result = mod.fit(y, pars, x=x)
print(result.fit_report())
plt.plot(x, y, 'bo')
plt.plot(x, result.init_fit, 'k')
plt.plot(x, result.best_fit, 'r')
plt.show()
#<end examples/model_doc2.py>
which prints out the results:
[[Model]]
(Model(gaussian) + Model(line))
[[Fit Statistics]]
# function evals = 44
# data points = 101
# variables = 5
chisquare = 2.579
reduced chisquare = 0.027
Akaike info crit = 360.457
Bayesian info crit = 347.381
[[Variables]]
amp: 8.45931061 +/ 0.124145 (1.47%) (init= 5)
cen: 5.65547872 +/ 0.009176 (0.16%) (init= 5)
intercept: 0.96860201 +/ 0.033522 (3.46%) (init= 1)
slope: 0.26484403 +/ 0.005748 (2.17%) (init= 0)
wid: 0.67545523 +/ 0.009916 (1.47%) (init= 1)
[[Correlations]] (unreported correlations are < 0.100)
C(amp, wid) = 0.666
C(cen, intercept) = 0.129
and shows the plot on the left.
On the left, data is shown in blue dots, the total fit is shown in solid
red line, and the initial fit is shown as a black dashed line. In the
figure on the right, the data is again shown in blue dots, and the Gaussian
component shown as a black dashed line, and the linear component shown as a
red dashed line. These components were generated after the fit using the
Models ModelResult.eval_components()
method of the result:
comps = result.eval_components()
which returns a dictionary of the components, using keys of the model name
(or prefix if that is set). This will use the parameter values in
result.params and the independent variables (x) used during the
fit. Note that while the ModelResult
held in result does store the
best parameters and the best estimate of the model in result.best_fit,
the original model and parameters in pars are left unaltered.
You can apply this composite model to other data sets, or evaluate the model at other values of x. You may want to do this to give a finer or coarser spacing of data point, or to extrapolate the model outside the fitting range. This can be done with:
xwide = np.linspace(5, 25, 3001)
predicted = mod.eval(x=xwide)
In this example, the argument names for the model functions do not overlap.
If they had, the prefix argument to Model
would have allowed
us to identify which parameter went with which component model. As we will
see in the next chapter, using composite models with the builtin models
provides a simple way to build up complex models.

class
CompositeModel
(left, right, op[, **kws])¶ Combine two models (left and right) with a binary operator (op) into a CompositeModel.
Normally, one does not have to explicitly create a CompositeModel, but can use normal Python operators +, ‘‘, *, and / to combine components as in:
>>> mod = Model(fcn1) + Model(fcn2) * Model(fcn3)
Parameters: Notes
 The two models must use the same independent variable.
Note that when using builtin Python binary operators, a
CompositeModel
will automatically be constructed for you. That is,
doing:
mod = Model(fcn1) + Model(fcn2) * Model(fcn3)
will create a CompositeModel
. Here, left will be Model(fcn1),
op will be operator.add()
, and right will be another
CompositeModel that has a left attribute of Model(fcn2), an op of
operator.mul()
, and a right of Model(fcn3).
To use a binary operator other than ‘+’, ‘‘, ‘*’, or ‘/’ you can
explicitly create a CompositeModel
with the appropriate binary
operator. For example, to convolve two models, you could define a simple
convolution function, perhaps as:
import numpy as np
def convolve(dat, kernel):
# simple convolution
npts = min(len(dat), len(kernel))
pad = np.ones(npts)
tmp = np.concatenate((pad*dat[0], dat, pad*dat[1]))
out = np.convolve(tmp, kernel, mode='valid')
noff = int((len(out)  npts)/2)
return (out[noff:])[:npts]
which extends the data in both directions so that the convolving kernel function gives a valid result over the data range. Because this function takes two array arguments and returns an array, it can be used as the binary operator. A full script using this technique is here:
#!/usr/bin/env python
#<examples/model_doc3.py>
import numpy as np
from lmfit import Model, CompositeModel
from lmfit.lineshapes import step, gaussian
import matplotlib.pyplot as plt
# create data from broadened step
npts = 201
x = np.linspace(0, 10, npts)
y = step(x, amplitude=12.5, center=4.5, sigma=0.88, form='erf')
y = y + np.random.normal(size=npts, scale=0.35)
def jump(x, mid):
"heaviside step function"
o = np.zeros(len(x))
imid = max(np.where(x<=mid)[0])
o[imid:] = 1.0
return o
def convolve(arr, kernel):
# simple convolution of two arrays
npts = min(len(arr), len(kernel))
pad = np.ones(npts)
tmp = np.concatenate((pad*arr[0], arr, pad*arr[1]))
out = np.convolve(tmp, kernel, mode='valid')
noff = int((len(out)  npts)/2)
return out[noff:noff+npts]
#
# create Composite Model using the custom convolution operator
mod = CompositeModel(Model(jump), Model(gaussian), convolve)
pars = mod.make_params(amplitude=1, center=3.5, sigma=1.5, mid=5.0)
# 'mid' and 'center' should be completely correlated, and 'mid' is
# used as an integer index, so a very poor fit variable:
pars['mid'].vary = False
# fit this model to data array y
result = mod.fit(y, params=pars, x=x)
print(result.fit_report())
plot_components = False
# plot results
plt.plot(x, y, 'bo')
if plot_components:
# generate components
comps = result.eval_components(x=x)
plt.plot(x, 10*comps['jump'], 'k')
plt.plot(x, 10*comps['gaussian'], 'r')
else:
plt.plot(x, result.init_fit, 'k')
plt.plot(x, result.best_fit, 'r')
plt.show()
# #<end examples/model_doc3.py>
which prints out the results:
[[Model]]
(Model(jump) <function convolve at 0x109ee4488> Model(gaussian))
[[Fit Statistics]]
# function evals = 27
# data points = 201
# variables = 3
chisquare = 22.091
reduced chisquare = 0.112
Akaike info crit = 437.837
Bayesian info crit = 427.927
[[Variables]]
mid: 5 (fixed)
sigma: 0.64118585 +/ 0.013233 (2.06%) (init= 1.5)
center: 4.51633608 +/ 0.009567 (0.21%) (init= 3.5)
amplitude: 0.62654849 +/ 0.001813 (0.29%) (init= 1)
[[Correlations]] (unreported correlations are < 0.100)
C(center, amplitude) = 0.344
C(sigma, amplitude) = 0.280
and shows the plots:
Using composite models with builtin or custom operators allows you to build complex models from testable subcomponents.