Modeling Data and Curve Fitting¶
A common use of least-squares 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 high-level wrapper around
scipy.optimize.leastsq it can be used for curve-fitting problems.
While it offers many benefits over scipy.optimize.leastsq, using
minimize()
for many curve-fitting problems still
requires more effort than using scipy.optimize.curve_fit.
The Model
class in lmfit provides a simple and flexible approach
to curve-fitting 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 curve-fitting
method, lmfit also provides canonical definitions for many known lineshapes
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
(Built-in 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 high-level 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 built-in 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 exp, linspace, random
def gaussian(x, amp, cen, wid):
return amp * exp(-(x-cen)**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, x.size)
init_vals = [1, 0, 1] # for [amp, cen, wid]
best_vals, covar = curve_fit(gaussian, x, y, p0=init_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)
print(f'parameter names: {gmodel.param_names}')
print(f'independent variables: {gmodel.independent_vars}')
parameter names: ['amp', 'cen', 'wid']
independent variables: ['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 nothing 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 = gmodel.make_params()
This creates the Parameters
but does not
automatically 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 = gmodel.make_params(cen=0.3, amp=3, wid=1.25)
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_eval = linspace(0, 10, 201)
y_eval = gmodel.eval(params, x=x_eval)
or with:
y_eval = gmodel.eval(x=x_eval, cen=6.5, amp=100, wid=2.0)
Admittedly, this a slightly long-winded 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 = gmodel.fit(y, params, x=x)
or with:
result = gmodel.fit(y, x=x, cen=0.5, amp=10, wid=2.0)
Putting everything together, included in the examples
folder with the
source code, is:
# <examples/doc_model_gaussian.py>
import matplotlib.pyplot as plt
from numpy import exp, loadtxt, pi, sqrt
from lmfit import Model
data = loadtxt('model1d_gauss.dat')
x = data[:, 0]
y = data[:, 1]
def gaussian(x, amp, cen, wid):
"""1-d gaussian: gaussian(x, amp, cen, wid)"""
return (amp / (sqrt(2*pi) * wid)) * exp(-(x-cen)**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, 'o')
plt.plot(x, result.init_fit, '--', label='initial fit')
plt.plot(x, result.best_fit, '-', label='best fit')
plt.legend()
plt.show()
# <end examples/doc_model_gaussian.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]]
# fitting method = leastsq
# function evals = 33
# data points = 101
# variables = 3
chi-square = 3.40883599
reduced chi-square = 0.03478404
Akaike info crit = -336.263713
Bayesian info crit = -328.418352
[[Variables]]
amp: 8.88021830 +/- 0.11359493 (1.28%) (init = 5)
cen: 5.65866102 +/- 0.01030495 (0.18%) (init = 5)
wid: 0.69765468 +/- 0.01030495 (1.48%) (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 green line, and the initial fit as a dashed orange 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 pre-defined
function as a fitting model.
- class Model(func, independent_vars=None, param_names=None, nan_policy='raise', prefix='', name=None, **kws)¶
Create a model from a user-supplied 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 curve-fitting problem.
- Parameters
func (callable) – Function to be wrapped.
independent_vars (
list
ofstr
, optional) – Arguments to func that are independent variables (default is None).param_names (
list
ofstr
, optional) – Names of arguments to func that are to be made into parameters (default is None).nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.
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.
3. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:
‘raise’ : raise a ValueError (default)
‘propagate’ : do nothing
‘omit’ : drop missing data
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(-(x-cen)**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
numpy.ndarray, float, int or complex
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 non-parameter arguments for the model function, including all the independent variables will need to be passed in using keyword arguments.
3. The return type depends on the model function. For many of the built-models it is a numpy.ndarray, with the exception of ConstantModel and ComplexConstantModel, which return a float/int or complex value.
- Model.fit(data, params=None, weights=None, method='leastsq', iter_cb=None, scale_covar=True, verbose=False, fit_kws=None, nan_policy=None, calc_covar=True, max_nfev=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, optional) – Weights to use for the calculation of the fit residual (default is None). Must have the same size as data.
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).
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.
nan_policy ({'raise', 'propagate', 'omit'}, optional) – What to do when encountering NaNs when fitting Model.
calc_covar (bool, optional) – Whether to calculate the covariance matrix (default is True) for solvers other than ‘leastsq’ and ‘least_squares’. Requires the
numdifftools
package to be installed.max_nfev (int or None, optional) – Maximum number of function evaluations (default is None). The default value depends on the fitting method.
**kwargs (optional) – Arguments to pass to the model function, possibly overriding parameters.
- Returns
- 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 non-parameter 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.
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)
- Model.guess(data, x, **kws)¶
Guess starting values for the parameters of a Model.
This is not implemented for all models, but is available for many of the built-in models.
- Parameters
data (array_like) – Array of data (i.e., y-values) to use to guess parameter values.
x (array_like) – Array of values for the independent variable (i.e., x-values).
**kws (optional) – Additional keyword arguments, passed to model function.
- Returns
Initial, guessed values for the parameters of a Model.
- Return type
- Raises
NotImplementedError – If the guess method is not implemented for a Model.
Notes
Should be implemented for each model subclass to run self.make_params(), update starting values and return a Parameters object.
Changed in version 1.0.3: Argument
x
is now explicitly required to estimate starting values.
- 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 – Parameters object for the Model.
- 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().
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 (str) – Parameter name.
**kwargs (optional) –
Arbitrary keyword arguments, needs to be a Parameter attribute. Can be any of the following:
- valuefloat, optional
Numerical Parameter value.
- varybool, optional
Whether the Parameter is varied during a fit (default is True).
- minfloat, optional
Lower bound for value (default is
-numpy.inf
, no lower bound).
- maxfloat, optional
Upper bound for value (default is
numpy.inf
, no upper bound).
- exprstr, 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.
- nan_policy¶
Describes what to do for NaNs that indicate missing values in the data. The choices are:
'raise'
: Raise aValueError
(default)'propagate'
: Do not check for NaNs or missing values. The fit will try to ignore them.'omit'
: Remove NaNs or missing observations in data. If pandas is installed,pandas.isnull()
is used, otherwisenumpy.isnan()
is used.
- 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 name-mangling of parameter names. The default is
''
. If a particularModel
has argumentsamplitude
,center
, andsigma
, these would become the parameter names. Using a prefix of'g1_'
would convert these parameter names tog1_amplitude
,g1_center
, andg1_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 taken 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 is actually more than one independent variable.
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:
import numpy as np
from lmfit import Model
def decay(t, tau, N):
return N*np.exp(-t/tau)
decay_model = Model(decay)
print(f'independent variables: {decay_model.independent_vars}')
params = decay_model.make_params()
print('\nParameters:')
for pname, par in params.items():
print(pname, par)
independent variables: ['t']
Parameters:
tau <Parameter 'tau', value=-inf, bounds=[-inf:inf]>
N <Parameter 'N', value=-inf, bounds=[-inf:inf]>
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(f'independent variables: {decay_model.independent_vars}')
params = decay_model.make_params()
print('\nParameters:')
for pname, par in params.items():
print(pname, par)
independent variables: ['tau']
Parameters:
t <Parameter 't', value=-inf, bounds=[-inf:inf]>
N <Parameter 'N', value=-inf, bounds=[-inf:inf]>
You can also supply multiple values for multi-dimensional 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.e-9, abs(tau))
else:
arg = t/tau
return N*np.exp(arg)
mod = Model(decay2)
params = mod.make_params()
print('Parameters:')
for pname, par in params.items():
print(pname, par)
Parameters:
tau <Parameter 'tau', value=-inf, bounds=[-inf:inf]>
N <Parameter 'N', value=10, bounds=[-inf:inf]>
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):
# function definition, for now just ``pass``
pass
mod = Model(myfunc, prefix='f1_')
params = mod.make_params()
print('Parameters:')
for pname, par in params.items():
print(pname, par)
Parameters:
f1_amplitude <Parameter 'f1_amplitude', value=1, bounds=[-inf:inf]>
f1_center <Parameter 'f1_center', value=0, bounds=[-inf:inf]>
f1_sigma <Parameter 'f1_sigma', value=1, bounds=[-inf:inf]>
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):
return a*x + 10*a - b
instead of using:
def myfunc(x, a, b):
return a*x + 10*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:
x = linspace(0, 10, 100)
y_eval = mod.eval(x=x, a=7.0, b=-2.0)
y_sim = y_eval + random.normal(0, 0.2, x.size)
out = mod.fit(y_sim, pars, x=x, 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('Parameter hints:')
for pname, par in mod.param_hints.items():
print(pname, par)
Parameter hints:
a {'value': 1.0}
b {'value': 0.3, 'min': 0, 'max': 1.0}
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 full-width 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')
Saving and Loading Models¶
New in version 0.9.8.
It is sometimes desirable to save a Model
for later use outside of
the code used to define the model. Lmfit provides a save_model()
function that will save a Model
to a file. There is also a
companion load_model()
function that can read this file and
reconstruct a Model
from it.
Saving a model turns out to be somewhat challenging. The main issue is that
Python is not normally able to serialize a function (such as the model
function making up the heart of the Model) in a way that can be
reconstructed into a callable Python object. The dill
package can
sometimes serialize functions, but with the limitation that it can be used
only in the same version of Python. In addition, class methods used as
model functions will not retain the rest of the class attributes and
methods, and so may not be usable. With all those warnings, it should be
emphasized that if you are willing to save or reuse the definition of the
model function as Python code, then saving the Parameters and rest of the
components that make up a model presents no problem.
If the dill
package is installed, the model function will be saved using
it. But because saving the model function is not always reliable, saving a
model will always save the name of the model function. The load_model()
takes an optional funcdefs
argument that can contain a dictionary of
function definitions with the function names as keys and function objects as
values. If one of the dictionary keys matches the saved name, the
corresponding function object will be used as the model function. With this
approach, if you save a model and can provide the code used for the model
function, the model can be saved and reliably reloaded and used.
- save_model(model, fname)¶
Save a Model to a file.
- load_model(fname, funcdefs=None)¶
Load a saved Model from a file.
As a simple example, one can save a model as:
# <examples/doc_model_savemodel.py>
import numpy as np
from lmfit.model import Model, save_model
def mysine(x, amp, freq, shift):
return amp * np.sin(x*freq + shift)
sinemodel = Model(mysine)
pars = sinemodel.make_params(amp=1, freq=0.25, shift=0)
save_model(sinemodel, 'sinemodel.sav')
# <end examples/doc_model_savemodel.py>
To load that later, one might do:
# <examples/doc_model_loadmodel.py>
import matplotlib.pyplot as plt
import numpy as np
from lmfit.model import load_model
def mysine(x, amp, freq, shift):
return amp * np.sin(x*freq + shift)
data = np.loadtxt('sinedata.dat')
x = data[:, 0]
y = data[:, 1]
model = load_model('sinemodel.sav', funcdefs={'mysine': mysine})
params = model.make_params(amp=3, freq=0.52, shift=0)
params['shift'].max = 1
params['shift'].min = -1
params['amp'].min = 0.0
result = model.fit(y, params, x=x)
print(result.fit_report())
plt.plot(x, y, 'o')
plt.plot(x, result.best_fit, '-')
plt.show()
# <end examples/doc_model_loadmodel.py>
See also Saving and Loading ModelResults.
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 re-fit 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 re-do 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, nan_policy='raise', calc_covar=True, max_nfev=None, **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 re-run 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
(data-model)
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.
nan_policy ({'raise', 'propagate', 'omit'}, optional) – What to do when encountering NaNs when fitting Model.
calc_covar (bool, optional) – Whether to calculate the covariance matrix (default is True) for solvers other than ‘leastsq’ and ‘least_squares’. Requires the
numdifftools
package to be installed.max_nfev (int or None, optional) – Maximum number of function evaluations (default is None). The default value depends on the fitting method.
**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
Array or value for the evaluated model.
- Return type
numpy.ndarray, float, int, or complex
- 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) – Keyword 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
- ModelResult.fit(data=None, params=None, weights=None, method=None, nan_policy=None, **kwargs)¶
Re-perform 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
(data-model)
for fit residual.method (str, optional) – Name of minimization method to use (default is ‘leastsq’).
nan_policy ({'raise', 'propagate', 'omit'}, optional) – What to do when encountering NaNs when fitting Model.
**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 best-fit 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
Multi-line text of fit report.
- Return type
- 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 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.
This can be used to give confidence bands for the model from the uncertainties in the best-fit parameters.
- 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
Notes
This is based on the excellent and clear example from https://www.astro.rug.nl/software/kapteyn/kmpfittutorial.html#confidence-and-prediction-intervals, 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 of 1, 2, or 3 give probabilities 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
andsigma=0.6827
will give the same results, within precision errors.
Examples
>>> out = model.fit(data, params, x=x) >>> dely = out.eval_uncertainty(x=x) >>> plt.plot(x, data) >>> plt.plot(x, out.best_fit) >>> plt.fill_between(x, out.best_fit-dely, ... out.best_fit+dely, color='#888888')
- ModelResult.plot(datafmt='o', fitfmt='-', initfmt='--', xlabel=None, ylabel=None, yerr=None, numpoints=None, fig=None, data_kws=None, fit_kws=None, init_kws=None, ax_res_kws=None, ax_fit_kws=None, fig_kws=None, show_init=False, parse_complex='abs', title=None)¶
Plot the fit results and residuals using matplotlib.
The method will produce a matplotlib figure (if package available) with both results of the fit and the residuals plotted. If the fit model included weights, errorbars will also be plotted. To show the initial conditions for the fit, pass the argument
show_init=True
.- 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 x-axis.
ylabel (str, optional) – Matplotlib format string for labeling the y-axis.
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 to the plot function for data points.
fit_kws (dict, optional) – Keyword arguments passed to the plot function for fitted curve.
init_kws (dict, optional) – Keyword arguments passed 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 a new one is created.
show_init (bool, optional) – Whether to show the initial conditions for the fit (default is False).
parse_complex ({'abs', 'real', 'imag', 'angle'}, optional) – How to reduce complex data for plotting. Options are one of: ‘abs’ (default), ‘real’, ‘imag’, or ‘angle’, which correspond to the NumPy functions with the same name.
title (str, optional) – Matplotlib format string for figure title.
- Returns
- Return type
See also
ModelResult.plot_fit
Plot the fit results using matplotlib.
ModelResult.plot_residuals
Plot the fit residuals using matplotlib.
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 model returns complex data, yerr is treated the same way that weights are in this case.
If fig is None then matplotlib.pyplot.figure(**fig_kws) is called, otherwise fig_kws is ignored.
- ModelResult.plot_fit(ax=None, datafmt='o', fitfmt='-', initfmt='--', xlabel=None, ylabel=None, yerr=None, numpoints=None, data_kws=None, fit_kws=None, init_kws=None, ax_kws=None, show_init=False, parse_complex='abs', title=None)¶
Plot the fit results using matplotlib, if available.
The plot will include the data points, the initial fit curve (optional, with
show_init=True
), and the best-fit 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 x-axis.
ylabel (str, optional) – Matplotlib format string for labeling the y-axis.
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 to the plot function for data points.
fit_kws (dict, optional) – Keyword arguments passed to the plot function for fitted curve.
init_kws (dict, optional) – Keyword arguments passed to the plot function for the initial conditions of the fit.
ax_kws (dict, optional) – Keyword arguments for a new axis, if a new one is created.
show_init (bool, optional) – Whether to show the initial conditions for the fit (default is False).
parse_complex ({'abs', 'real', 'imag', 'angle'}, optional) – How to reduce complex data for plotting. Options are one of: ‘abs’ (default), ‘real’, ‘imag’, or ‘angle’, which correspond to the NumPy functions with the same name.
title (str, optional) – Matplotlib format string for figure title.
- Returns
- Return type
See also
ModelResult.plot_residuals
Plot the fit residuals using matplotlib.
ModelResult.plot
Plot the fit results and residuals using matplotlib.
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 model returns complex data, yerr is treated the same way that weights are in this case.
If ax is None then matplotlib.pyplot.gca(**ax_kws) is called.
- ModelResult.plot_residuals(ax=None, datafmt='o', yerr=None, data_kws=None, fit_kws=None, ax_kws=None, parse_complex='abs', title=None)¶
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 to the plot function for data points.
fit_kws (dict, optional) – Keyword arguments passed to the plot function for fitted curve.
ax_kws (dict, optional) – Keyword arguments for a new axis, if a new one is created.
parse_complex ({'abs', 'real', 'imag', 'angle'}, optional) – How to reduce complex data for plotting. Options are one of: ‘abs’ (default), ‘real’, ‘imag’, or ‘angle’, which correspond to the NumPy functions with the same name.
title (str, optional) – Matplotlib format string for figure title.
- Returns
- Return type
See also
ModelResult.plot_fit
Plot the fit results using matplotlib.
ModelResult.plot
Plot the fit results and residuals using matplotlib.
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 model returns complex data, yerr is treated the same way that weights are in this case.
If ax is None then matplotlib.pyplot.gca(**ax_kws) is called.
ModelResult
attributes¶
- aic¶
Floating point best-fit Akaike Information Criterion statistic (see MinimizerResult – the optimization result).
- best_fit¶
numpy.ndarray result of model function, evaluated at provided independent variables and with best-fit parameters.
- best_values¶
Dictionary with parameter names as keys, and best-fit values as values.
- bic¶
Floating point best-fit Bayesian Information Criterion statistic (see MinimizerResult – the optimization result).
- chisqr¶
Floating point best-fit chi-square 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)
, whereparams
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()
.
- call_kws¶
Dict of keyword arguments actually send to underlying solver with
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 contain the best-fit values.
- redchi¶
Floating point reduced chi-square 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 notNone
, it will be used as a multiplicative factor of the residual array, so thatweights*(data - fit)
is minimized in the least-squares 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_fit-dely, result.best_fit+dely, color="#ABABAB",
label='3-$\sigma$ uncertainty band')
to the example fit to the Gaussian at the beginning of this chapter will give 3-\(\sigma\) bands for the best-fit Gaussian, and produce the figure below.
Saving and Loading ModelResults¶
New in version 0.9.8.
As with saving models (see section Saving and Loading Models), it is
sometimes desirable to save a ModelResult
, either for later use or
to organize and compare different fit results. Lmfit provides a
save_modelresult()
function that will save a ModelResult
to
a file. There is also a companion load_modelresult()
function that
can read this file and reconstruct a ModelResult
from it.
As discussed in section Saving and Loading Models, there are challenges to
saving model functions that may make it difficult to restore a saved a
ModelResult
in a way that can be used to perform a fit.
Use of the optional funcdefs
argument is generally the most
reliable way to ensure that a loaded ModelResult
can be used to
evaluate the model function or redo the fit.
- save_modelresult(modelresult, fname)¶
Save a ModelResult to a file.
- Parameters
modelresult (ModelResult) – ModelResult to be saved.
fname (str) – Name of file for saved ModelResult.
- load_modelresult(fname, funcdefs=None)¶
Load a saved ModelResult from a file.
- Parameters
- Returns
ModelResult object loaded from file.
- Return type
An example of saving a ModelResult
is:
# <examples/doc_model_savemodelresult.py>
import numpy as np
from lmfit.model import save_modelresult
from lmfit.models import GaussianModel
data = np.loadtxt('model1d_gauss.dat')
x = data[:, 0]
y = data[:, 1]
gmodel = GaussianModel()
result = gmodel.fit(y, x=x, amplitude=5, center=5, sigma=1)
save_modelresult(result, 'gauss_modelresult.sav')
print(result.fit_report())
# <end examples/doc_model_savemodelresult.py>
To load that later, one might do:
# <examples/doc_model_loadmodelresult.py>
import matplotlib.pyplot as plt
import numpy as np
from lmfit.model import load_modelresult
data = np.loadtxt('model1d_gauss.dat')
x = data[:, 0]
y = data[:, 1]
result = load_modelresult('gauss_modelresult.sav')
print(result.fit_report())
plt.plot(x, y, 'o')
plt.plot(x, result.best_fit, '-')
plt.show()
# <end examples/doc_model_loadmodelresult.py>
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 pre-built 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 + 1-d gaussian"""
gauss = (amp / (sqrt(2*pi) * wid)) * exp(-(x-cen)**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:
# <examples/doc_model_two_components.py>
import matplotlib.pyplot as plt
from numpy import exp, loadtxt, pi, sqrt
from lmfit import Model
data = loadtxt('model1d_gauss.dat')
x = data[:, 0]
y = data[:, 1] + 0.25*x - 1.0
def gaussian(x, amp, cen, wid):
"""1-d gaussian: gaussian(x, amp, cen, wid)"""
return (amp / (sqrt(2*pi) * wid)) * exp(-(x-cen)**2 / (2*wid**2))
def line(x, slope, intercept):
"""a 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, 'o')
plt.plot(x, result.init_fit, '--', label='initial fit')
plt.plot(x, result.best_fit, '-', label='best fit')
plt.legend()
plt.show()
# <end examples/doc_model_two_components.py>
which prints out the results:
[[Model]]
(Model(gaussian) + Model(line))
[[Fit Statistics]]
# fitting method = leastsq
# function evals = 44
# data points = 101
# variables = 5
chi-square = 2.57855517
reduced chi-square = 0.02685995
Akaike info crit = -360.457020
Bayesian info crit = -347.381417
[[Variables]]
amp: 8.45931062 +/- 0.12414515 (1.47%) (init = 5)
cen: 5.65547873 +/- 0.00917678 (0.16%) (init = 5)
wid: 0.67545524 +/- 0.00991686 (1.47%) (init = 1)
slope: 0.26484404 +/- 0.00574892 (2.17%) (init = 0)
intercept: -0.96860202 +/- 0.03352202 (3.46%) (init = 1)
[[Correlations]] (unreported correlations are < 0.100)
C(slope, intercept) = -0.795
C(amp, wid) = 0.666
C(amp, intercept) = -0.222
C(amp, slope) = -0.169
C(cen, slope) = -0.162
C(wid, intercept) = -0.148
C(cen, intercept) = 0.129
C(wid, slope) = -0.113
and shows the plot on the left.
On the left, data is shown in blue dots, the total fit is shown in solid green line, and the initial fit is shown as a orange dashed line. The figure on the right shows again the data in blue dots, the Gaussian component as a orange dashed line and the linear component as a green dashed line. It is created using the following code:
comps = result.eval_components()
plt.plot(x, y, 'o')
plt.plot(x, comps['gaussian'], '--', label='Gaussian component')
plt.plot(x, comps['line'], '--', label='Line component')
The components were generated after the fit using the
ModelResult.eval_components()
method of the result
, 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 = linspace(-5, 25, 3001)
predicted = mod.eval(result.params, 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 built-in models
provides a simple way to build up complex models.
- class CompositeModel(left, right, op[, **kws])¶
Combine two models (left and right) with binary operator (op).
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 built-in 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 of two arrays"""
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:
# <examples/doc_model_composite.py>
import matplotlib.pyplot as plt
import numpy as np
from lmfit import CompositeModel, Model
from lmfit.lineshapes import gaussian, step
# create data from broadened step
x = np.linspace(0, 10, 201)
y = step(x, amplitude=12.5, center=4.5, sigma=0.88, form='erf')
np.random.seed(0)
y = y + np.random.normal(scale=0.35, size=x.size)
def jump(x, mid):
"""Heaviside step function."""
o = np.zeros(x.size)
imid = max(np.where(x <= mid)[0])
o[imid:] = 1.0
return o
def convolve(arr, kernel):
"""Simple convolution of two arrays."""
npts = min(arr.size, kernel.size)
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())
# generate components
comps = result.eval_components(x=x)
# plot results
fig, axes = plt.subplots(1, 2, figsize=(12.8, 4.8))
axes[0].plot(x, y, 'bo')
axes[0].plot(x, result.init_fit, 'k--', label='initial fit')
axes[0].plot(x, result.best_fit, 'r-', label='best fit')
axes[0].legend()
axes[1].plot(x, y, 'bo')
axes[1].plot(x, 10*comps['jump'], 'k--', label='Jump component')
axes[1].plot(x, 10*comps['gaussian'], 'r-', label='Gaussian component')
axes[1].legend()
plt.show()
# <end examples/doc_model_composite.py>
which prints out the results:
[[Model]]
(Model(jump) <function convolve at 0x7f76614caf70> Model(gaussian))
[[Fit Statistics]]
# fitting method = leastsq
# function evals = 25
# data points = 201
# variables = 3
chi-square = 24.7562335
reduced chi-square = 0.12503148
Akaike info crit = -414.939746
Bayesian info crit = -405.029832
[[Variables]]
mid: 5 (fixed)
amplitude: 0.62508459 +/- 0.00189732 (0.30%) (init = 1)
center: 4.50853671 +/- 0.00973231 (0.22%) (init = 3.5)
sigma: 0.59576118 +/- 0.01348582 (2.26%) (init = 1.5)
[[Correlations]] (unreported correlations are < 0.100)
C(amplitude, center) = 0.329
C(amplitude, sigma) = 0.268
and shows the plots:
Using composite models with built-in or custom operators allows you to build complex models from testable sub-components.