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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 to most closely match some data. With `scipy`, such problems are commonly 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, but requires more effort than using `scipy.optimize.curve_fit()`.

Here we discuss lmfit’s `Model` class. This takes a model function – a function that calculates a model for some data – and provides methods to create parameters for that model and to fit data using that model function. This is closer in spirit to `scipy.optimize.curve_fit()`, but with the advantages of using `Parameters` and lmfit.

In addition to allowing you 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 function into high-level fitting models with the `Model` class, and using these to fit data.

## 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 provides a model function for a Gaussian profile, but here we’ll build our own. We start with a simple definition of the model function:

    >>> from numpy import sqrt, pi, exp, linspace
    >>>
    >>> def gaussian(x, amp, cen, wid):
    ...    return amp * exp(-(x-cen)**2 /wid)
    ...

We want to fit this objective function to data \(y(x)\) represented by the arrays `y` and `x`. This can be done easily wit `scipy.optimize.curve_fit()`:

    >>> from scipy.optimize import curve_fit
    >>>
    >>> x = linspace(-10,10)
    >>> y = y = gaussian(x, 2.33, 0.21, 1.51) + np.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

We sample random data point, 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 very useful. But it misses the benefits of lmfit.

To solve this with lmfit we would have to write an objective function. But such a function would be fairly simple (essentially, `data - model`, possibly with some weighting), and we would need to define and use appropriately named parameters. Though convenient, it is somewhat of a burden to keep the named parameter straight (on the other hand, with `scipy.optimize.curve_fit()` you are required to remember the parameter order). After doing this a few times it appears as a recurring pattern, and we can imagine automating this process. That’s where the `Model` class comes in.

`Model` allows us to easily wrap a model function such as the `gaussian` function. This automatically generate the appropriate residual function, and determines the corresponding parameter names from the function signature itself:

    >>> from lmfit import Model
    >>> gmod = Model(gaussian)
    >>> gmod.param_names
    set(['amp', 'wid', 'cen'])
    >>> gmod.independent_vars)
    ['x']

The Model `gmod` knows 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 parameters are named `amp`, `cen`, and `wid`, and `x` is the independent variable – all taken directly from the signature of the model function. As we will see below, you can specify what the independent variable is, and you can add or alter parameters, too.

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 doesn’t necessarily give them initial values – again, the model 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` 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(x=x, amp=10, cen=6.2, wid=0.75)

Admittedly, this a slightly long-winded way to calculate a Gaussian function. But now that the model is set up, we can also use its `fit()` method to fit this model to data, as with:

    >>> result = gmod.fit(y, x=x, amp=5, cen=5, wid=1)

Putting everything together, the script to do such a fit (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]]
        gaussian
    [[Fit Statistics]]
        # function evals   = 33
        # data points      = 101
        # variables        = 3
        chi-square         = 3.409
        reduced chi-square = 0.035
    [[Variables]]
        amp:   8.88021829 +/- 0.113594 (1.28%) (init= 5)
        cen:   5.65866102 +/- 0.010304 (0.18%) (init= 5)
        wid:   0.69765468 +/- 0.010304 (1.48%) (init= 1)
    [[Correlations]] (unreported correlations are <  0.100)
        C(amp, wid)                  =  0.577

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 2 lines of code:

    gmod = Model(gaussian)
    result = gmod.fit(y, 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`. This is much more expressive than `scipy.optimize.curve_fit()`:

    best_vals, covar = curve_fit(gaussian, x, y, p0=[5, 5, 1])

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 based on the user-supplied function. This uses introspection to automatically converting argument names of the function to Parameter names.

  Parameters:

  - func (callable) – model function to be wrapped
  - independent_vars (`None` (default) or list of strings.) – list of argument names to `func` that are independent variables.
  - param_names (`None` (default) or list of strings) – list of argument names to `func` that should be made into Parameters.
  - missing (one of `None` (default), ‘none’, ‘drop’, or ‘raise’.) – how to handle missing values.
  - prefix (string) – prefix to add to all parameter names to distinguish components in a `CompositeModel`.
  - name (`None` or string.) – name for the model. When `None` (default) the name is the same as the model function (`func`).
  - kws – additional keyword arguments to pass to model function.

Of course, the model function will have to return an array that will be the same size as the data being modeled. Generally this is handled by also specifying one or more independent variables.

### `Model` class Methods

`Model.``eval`(params=None[, **kws])

  evaluate the model function for a set of parameters and inputs.

  Parameters:

  - params (`None` (default) or Parameters) – parameters to use for fit.
  - kws – additional keyword arguments to pass to model function.

  Returns: | ndarray for model given the parameters and other arguments.

  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.

  Note that 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'[, scale_covar=True[, iter_cb=None[, **kws]]]]]])

  perform a fit of the model to the `data` array with a set of parameters.

  Parameters:

  - data (ndarray-like) – array of data to be fitted.
  - params (`None` (default) or Parameters) – parameters to use for fit.
  - weights (`None` (default) or ndarray-like.) – weights to use for residual calculation in fit.
  - method (string (default `leastsq`)) – name of fitting method to use. See Choosing Different Fitting Methods for details
  - scale_covar (bool (default `True`)) – whether to automatically scale covariance matrix (`leastsq` only)
  - iter_cb (callable or `None`) – function to be called at each fit iteration. See Using a Iteration Callback Function for details.
  - verbose (bool (default `True`)) – print a message when a new parameter is created due to a hint
  - kws – additional keyword arguments to pass to model function.

  Returns: | `ModelResult` object.

  If `params` is `None`, the internal `params` will be used. If it is supplied, these will replace the internal ones. If supplied, `weights` will be used to weight the calculated residual so that the quantitiy minimized in the least-squares sense is `weights*(data - fit)`. `weights` must be an ndarray-like object of same size and shape as `data`.

  Note that other arguments for the model function (including all the independent variables!) will need to be passed in using keyword arguments.

`Model.``guess`(data, **kws)

  Guess starting values for model parameters.

  > param data: | data array used to guess parameter values
  > type func: | ndarray
  > param kws: | additional options to pass to model function.
  > return: | `Parameters` with guessed initial values for each parameter.

  by default this is left to raise a `NotImplementedError`, but may be overwritten by subclasses. Generally, this method should take some values for `data` and use it to construct reasonable starting values for the parameters.

`Model.``make_params`(**kws)

  Create a set of parameters for model.

  > param kws: | optional keyword/value pairs to set initial values for parameters.
  > return: | `Parameters`.

  > The parameters may or may not have decent initial values for each parameter.

`Model.``set_param_hint`(name, value=None[, min=None[, max=None[, vary=True[, expr=None]]]])

  set hints to use when creating parameters with `Model.make_param()` for the named parameter. This is especially convenient for setting initial values. The `name` can include the models `prefix` or not.

  Parameters:

  - name (string) – parameter name.
  - value (float) – value for parameter
  - min (`None` or float) – lower bound for parameter value
  - max (`None` or float) – upper bound for parameter value
  - vary (boolean) – whether to vary parameter in fit.
  - expr (string) – mathematical expression for constraint

  See Using parameter hints.

### `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 explicitly set this, of course, and will need to 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’ll have to do this because none of the parameters have valid initial values. You 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:

    >>> 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, includind chisqr, redchi, aic, and bic.

class `ModelResult`

  Model fit is intended to be created and returned by `Model.fit()`.

### `ModelResult` methods

These methods are all inherited from `Minimize` or from `Model`.

`ModelResult.``eval`(**kwargs)

  evaluate the model using the best-fit parameters and supplied independent variables. The `**kwargs` arguments can be used to update parameter values and/or independent variables.

`ModelResult.``eval_components`(**kwargs)

  evaluate each component of a `CompositeModel`, returning an ordered dictionary of with the values for each component model. The returned dictionary will have keys of the model prefix or (if no prefix is given), the model name. The `**kwargs` arguments can be used to update parameter values and/or independent variables.

`ModelResult.``fit`(data=None[, params=None[, weights=None[, method=None[, **kwargs]]]])

  fit (or re-fit), optionally changing `data`, `params`, `weights`, or `method`, or changing the independent variable(s) with the `**kwargs` argument. See `Model.fit()` for argument descriptions, and note that any value of `None` defaults to the last used value.

`ModelResult.``fit_report`(modelpars=None[, show_correl=True[, `< min_correl=0.1]])

  return a printable fit report for the fit with fit statistics, best-fit values with uncertainties and correlations. As with `fit_report()`.

  Parameters:

  - modelpars – Parameters with “Known Values” (optional, default None)
  - show_correl – whether to show list of sorted correlations [`True`]
  - min_correl – smallest correlation absolute value to show [0.1]

`ModelResult.``conf_interval`(**kwargs)

  calculate the confidence intervals for the variable parameters using `confidence.conf_interval()`. All keyword arguments are passed to that function. The result is stored in `ci_out`, and so can be accessed without recalculating them.

`ModelResult.``ci_report`(with_offset=True)

  return a nicely formatted text report of the confidence intervals, as from `ci_report()`.

`ModelResult.``plot`(datafmt='o', fitfmt='-', initfmt='--', 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)

  Plot the fit results and residuals using matplotlib, if available. The plot will include two panels, one showing the fit residual, and the other with the data points, the initial fit curve, and the best-fit curve. If the fit model included weights or if `yerr` is specified, errorbars will also be plotted.

  Parameters:

  - datafmt (`None` or string.) – matplotlib format string for data curve.
  - fitfmt (`None` or string.) – matplotlib format string for best-fit curve.
  - initfmt – matplotlib format string for initial curve.
  - yerr (`None` or ndarray.) – array of uncertainties for data array.
  - numpoints (`None` or integer) – number of points to display
  - fig (`None` or matplotlib.figure.Figure) – matplotlib Figure to plot on.
  - data_kws (`None` or dictionary) – keyword arguments passed to plot for data curve.
  - fit_kws (`None` or dictionary) – keyword arguments passed to plot for best-fit curve.
  - init_kws (`None` or dictionary) – keyword arguments passed to plot for initial curve.
  - ax_res_kws (`None` or dictionary) – keyword arguments passed to creation of matplotlib axes for the residual plot.
  - ax_fit_kws (`None` or dictionary) – keyword arguments passed to creation of matplotlib axes for the fit plot.
  - fig_kws (`None` or dictionary) – keyword arguments passed to creation of matplotlib figure.

  Returns: | matplotlib.figure.Figure

  This combines `ModelResult.plot_fit()` and `ModelResult.plot_residual()`.

  If `yerr` is specified or if the fit model included weights, then matplotlib.axes.Axes.errorbar is used to plot the data. If `yerr` is not specified and the fit includes weights, `yerr` set to `1/self.weights`

  If `fig` is None then `matplotlib.pyplot.figure(**fig_kws)` is called.

`ModelResult.``plot_fit`(ax=None, datafmt='o', fitfmt='-', initfmt='--', yerr=None, numpoints=None, data_kws=None, fit_kws=None, init_kws=None, ax_kws=None)

  Plot the fit results using matplotlib, if available. The plot will include the data points, the initial fit curve, and the best-fit curve. If the fit model included weights or if `yerr` is specified, errorbars will also be plotted.

  Parameters:

  - ax (`None` or matplotlib.axes.Axes.) – matplotlib axes to plot on.
  - datafmt (`None` or string.) – matplotlib format string for data curve.
  - fitfmt (`None` or string.) – matplotlib format string for best-fit curve.
  - initfmt – matplotlib format string for initial curve.
  - yerr (`None` or ndarray.) – array of uncertainties for data array.
  - numpoints (`None` or integer) – number of points to display
  - data_kws (`None` or dictionary) – keyword arguments passed to plot for data curve.
  - fit_kws (`None` or dictionary) – keyword arguments passed to plot for best-fit curve.
  - init_kws (`None` or dictionary) – keyword arguments passed to plot for initial curve.
  - ax_kws (`None` or dictionary) – keyword arguments passed to creation of matplotlib axes.

  Returns: | matplotlib.axes.Axes

  For details about plot format strings and keyword arguments see documentation of `matplotlib.axes.Axes.plot()`.

  If `yerr` is specified or if the fit model included weights, then matplotlib.axes.Axes.errorbar is used to plot the data. If `yerr` is not specified and the fit includes weights, `yerr` set to `1/self.weights`

  If `ax` is None then `matplotlib.pyplot.gca(**ax_kws)` is called.

`ModelResult.``plot_residuals`(ax=None, datafmt='o', yerr=None, data_kws=None, fit_kws=None, ax_kws=None)

  Plot the fit residuals (data - fit) using matplotlib. If `yerr` is supplied or if the model included weights, errorbars will also be plotted.

  > param ax: | matplotlib axes to plot on.
  > type ax: | `None` or matplotlib.axes.Axes.
  > param datafmt: | matplotlib format string for data curve.
  > type datafmt: | `None` or string.
  > param yerr: | array of uncertainties for data array.
  > type yerr: | `None` or ndarray.
  > param numpoints:
  > | number of points to display
  > type numpoints: | `None` or integer
  > param data_kws: | keyword arguments passed to plot for data curve.
  > type data_kws: | `None` or dictionary
  > param fit_kws: | keyword arguments passed to plot for best-fit curve.
  > type fit_kws: | `None` or dictionary
  > param ax_kws: | keyword arguments passed to creation of matplotlib axes.
  > type ax_kws: | `None` or dictionary
  > returns: | matplotlib.axes.Axes

  > For details about plot format strings and keyword arguments see documentation of `matplotlib.axes.Axes.plot()`.

  > If `yerr` is specified or if the fit model included weights, then matplotlib.axes.Axes.errorbar is used to plot the data. If `yerr` is not specified and the fit includes weights, `yerr` set to `1/self.weights`

  > If `ax` is None then `matplotlib.pyplot.gca(**ax_kws)` is called.

### `ModelResult` attributes

`aic`

  floating point best-fit Akaike Information Criterion statistic (see MinimizerResult – the optimization result).

`best_fit`

  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`

  ndarray (square) covariance matrix returned from fit.

`data`

  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`

  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`

  ndarray for residual.

`scale_covar`

  boolean flag for whether to automatically scale covariance matrix.

`success`

  boolean value of whether fit succeeded.

`weights`

  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.

## 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. As an alternative 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
    [[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])

  Create a composite model from two models (left and right and an binary operator (op). Additional keywords are passed to `Model`.

  Parameters:

  - left (`Model`) – left-hand side Model
  - right (`Model`) – right-hand side Model
  - op (callable, and taking 2 arguments (left and right).) – binary operator

Normally, one does not have to explicitly create a `CompositeModel`, as doing:

    mod = Model(fcn1) + Model(fcn2) * Model(fcn3)

will automatically create a `CompositeModel`. In this example, mod.left will be Model(fcn1), mod.op will be `operator.add()`, and mod.right will be another CompositeModel that has a left attribute of Model(fcn2), an op of `operator.mul()`, and a right of Model(fcn3).

If you want to use a binary operator other than add, subtract, multiply, or divide that are supported through normal Python syntax, you’ll need to 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   = 25
        # data points      = 201
        # variables        = 3
        chi-square         = 21.692
        reduced chi-square = 0.110
    [[Variables]]
        amplitude:   0.62106099 +/- 0.001783 (0.29%) (init= 1)
        center:      4.49913218 +/- 0.009373 (0.21%) (init= 3.5)
        mid:         5 (fixed)
        sigma:       0.61936067 +/- 0.012977 (2.10%) (init= 1)
    [[Correlations]] (unreported correlations are <  0.100)
        C(amplitude, center)         =  0.336
        C(amplitude, sigma)          =  0.274

and shows the plots:

Using composite models with built-in or custom operators allows you to build complex models from testable sub-components.