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# 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 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 (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 buit-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 sqrt, pi, 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, 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 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 = 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:

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
        chi-square         = 3.409
        reduced chi-square = 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 pre-defined 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 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 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(-(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

  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.

`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: | 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)

  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.

`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 built-in 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: | Parameters

  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: | Parameters

  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)

  See Using parameter hints.

`Model.``print_param_hints`(colwidth=8)

  Print a nicely aligned text-table of parameter hints.

  Parameters: | colwidth (int, optional) – Width of each column, except for first and last columns.

### `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, otherwise `numpy.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 name-mangling 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 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 with `Model.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)
    >>> 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:

> 1. You can supply initial values in the definition of the model function.
> 2. You can initialize the parameters when creating parameters with `Model.make_params()`.
> 3. You can give parameter hints with `Model.set_param_hint()`.
> 4. You can supply initial values for the parameters when you use the `Model.eval()` or `Model.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 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')

## 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, **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.
  - **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: | numpy.ndarray

`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)

  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’).
  - **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: | text – Multi-line text of fit report.
  Return type: | str

  See also

  `fit_report()`

`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 the `ci_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:

  - with_offset (bool, optional) – Whether to subtract best value from all other values (default is True).
  - ndigits (int, optional) – Number of significant digits to show (default is 5).
  - **kwargs (optional) – Keyword arguments that are passed to the conf_interval function.

  Returns: | Text of formatted report on confidence intervals.
  Return type: | str

`ModelResult.``eval_uncertainty`(params=None, sigma=1, **kwargs)

  Evaluate the uncertainty of the model function from the uncertainties for the best-fit 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: | numpy.ndarray

  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_fit-dely,
      ...                 out.best_fit+dely, color='#888888')

  Notes

  1. 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
  2. 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, **kwargs)

`ModelResult.``plot_fit`(*args, **kwargs)

`ModelResult.``plot_residuals`(*args, **kwargs)

### `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)`, 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()`.

`model`

  Instance of `Model` used for model.

`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 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 not None, it will be used as a multiplicative factor of the residual array, so that `weights*(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")

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.

## 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:

which prints out the results:

    [[Model]]
        (Model(gaussian) + Model(line))
    [[Fit Statistics]]
        # function evals   = 44
        # data points      = 101
        # variables        = 5
        chi-square         = 2.579
        reduced chi-square = 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 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 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:

  - left (Model) – Left-hand model.
  - right (Model) – Right-hand model.
  - op (callable binary operator) – Operator to combine left and right models.
  - **kws (optional) – Additional keywords are passed to Model when creating this new model.

  Notes

  1. 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:

which prints out the results:

    [[Model]]
        (Model(jump) <function convolve at 0x109ee4488> Model(gaussian))
    [[Fit Statistics]]
        # function evals   = 27
        # data points      = 201
        # variables        = 3
        chi-square         = 22.091
        reduced chi-square = 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 built-in or custom operators allows you to build complex models from testable sub-components.