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# Built-in Fitting Models in the `models` module

Lmfit provides several builtin fitting models in the `models` module. These pre-defined models each subclass from the `model.Model` class of the previous chapter and wrap relatively well-known functional forms, such as Gaussians, Lorentzian, and Exponentials that are used in a wide range of scientific domains. In fact, all the models are all based on simple, plain python functions defined in the `lineshapes` module. In addition to wrapping a function into a `model.Model`, these models also provide a `guess()` method that is intended to give a reasonable set of starting values from a data array that closely approximates the data to be fit.

As shown in the previous chapter, a key feature of the `mode.Model` class is that models can easily be combined to give a composite `model.Model`. Thus while some of the models listed here may seem pretty trivial (notably, `ConstantModel` and `LinearModel`), the main point of having these is to be able to used in composite models. For example, a Lorentzian plus a linear background might be represented as:

    >>> from lmfit.models import LinearModel, LorentzianModel
    >>> peak = LorentzianModel()
    >>> background  = LinearModel()
    >>> model = peak + background

All the models listed below are one dimensional, with an independent variable named `x`. Many of these models represent a function with a distinct peak, and so share common features. To maintain uniformity, common parameter names are used whenever possible. Thus, most models have a parameter called `amplitude` that represents the overall height (or area of) a peak or function, a `center` parameter that represents a peak centroid position, and a `sigma` parameter that gives a characteristic width. Some peak shapes also have a parameter `fwhm`, typically constrained by `sigma` to give the full width at half maximum.

After a list of builtin models, a few examples of their use is given.

## Peak-like models

There are many peak-like models available. These include `GaussianModel`, `LorentzianModel`, `VoigtModel` and some less commonly used variations. The `guess()` methods for all of these make a fairly crude guess for the value of `amplitude`, but also set a lower bound of 0 on the value of `sigma`.

### `GaussianModel`

class `GaussianModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Gaussian or normal distribution lineshape. Parameter names: `amplitude`, `center`, and `sigma`. In addition, a constrained parameter `fwhm` is included.

\[f(x; A, \mu, \sigma) = \frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{{2\sigma}^2}}]}\]

where the parameter `amplitude` corresponds to \(A\), `center` to \(\mu\), and `sigma` to \(\sigma\). The full width at half maximum is \(2\sigma\sqrt{2\ln{2}}\), approximately \(2.3548\sigma\)

### `LorentzianModel`

class `LorentzianModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Lorentzian or Cauchy-Lorentz distribution function. Parameter names: `amplitude`, `center`, and `sigma`. In addition, a constrained parameter `fwhm` is included.

\[f(x; A, \mu, \sigma) = \frac{A}{\pi} \big[\frac{\sigma}{(x - \mu)^2 + \sigma^2}\big]\]

where the parameter `amplitude` corresponds to \(A\), `center` to \(\mu\), and `sigma` to \(\sigma\). The full width at half maximum is \(2\sigma\).

### `VoigtModel`

class `VoigtModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Voigt distribution function. Parameter names: `amplitude`, `center`, and `sigma`. A `gamma` parameter is also available. By default, it is constrained to have value equal to `sigma`, though this can be varied independently. In addition, a constrained parameter `fwhm` is included. The definition for the Voigt function used here is

\[f(x; A, \mu, \sigma, \gamma) = \frac{A \textrm{Re}[w(z)]}{\sigma\sqrt{2 \pi}}\]

where

\[\begin{eqnarray*} z &=& \frac{x-\mu +i\gamma}{\sigma\sqrt{2}} \\ w(z) &=& e^{-z^2}{\operatorname{erfc}}(-iz) \end{eqnarray*}\]

and `erfc()` is the complimentary error function. As above, `amplitude` corresponds to \(A\), `center` to \(\mu\), and `sigma` to \(\sigma\). The parameter `gamma` corresponds to \(\gamma\). If `gamma` is kept at the default value (constrained to `sigma`), the full width at half maximum is approximately \(3.6013\sigma\).

### `PseudoVoigtModel`

class `PseudoVoigtModel`(missing=None[, prefix=''[, name=None[, **kws]]])

a model based on a pseudo-Voigt distribution function, which is a weighted sum of a Gaussian and Lorentzian distribution functions with that share values for `amplitude` (\(A\)), `center` (\(\mu\)) and full width at half maximum (and so have constrained values of `sigma` (\(\sigma\)). A parameter `fraction` (\(\alpha\)) controls the relative weight of the Gaussian and Lorentzian components, giving the full definition of

\[ f(x; A, \mu, \sigma, \alpha) = \frac{(1-\alpha)A}{\sigma_g\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{{2\sigma_g}^2}}]} + \frac{\alpha A}{\pi} \big[\frac{\sigma}{(x - \mu)^2 + \sigma^2}\big]\]

where \(\sigma_g = {\sigma}/{\sqrt{2\ln{2}}}\) so that the full width at half maximum of each component and of the sum is \(2\sigma\). The `guess()` function always sets the starting value for `fraction` at 0.5.

### `MoffatModel`

class `MoffatModel`(missing=None[, prefix=''[, name=None[, **kws]]])

a model based on a Moffat distribution function, the parameters are `amplitude` (\(A\)), `center` (\(\mu\)), a width parameter `sigma` (\(\sigma\)) and an exponent `beta` (\(\beta\)). For (\(\beta=1\)) the Moffat has a Lorentzian shape.

\[f(x; A, \mu, \sigma, \beta) = A \big[(\frac{x-\mu}{\sigma})^2+1\big]^{-\beta}\]

the full width have maximum is \(2\sigma\sqrt{2^{1/\beta}-1}\). `guess()` function always sets the starting value for `beta` to 1.

### `Pearson7Model`

class `Pearson7Model`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Pearson VII distribution. This is a Lorenztian-like distribution function. It has the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)), and also an `exponent` (\(m\)) in

\[f(x; A, \mu, \sigma, m) = \frac{A}{\sigma{\beta(m-\frac{1}{2}, \frac{1}{2})}} \bigl[1 + \frac{(x-\mu)^2}{\sigma^2} \bigr]^{-m}\]

where \(\beta\) is the beta function (see `scipy.special.beta()` in `scipy.special`). The `guess()` function always gives a starting value for `exponent` of 1.5.

### `StudentsTModel`

class `StudentsTModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Student’s t distribution function, with the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)) in

\[f(x; A, \mu, \sigma) = \frac{A \Gamma(\frac{\sigma+1}{2})} {\sqrt{\sigma\pi}\,\Gamma(\frac{\sigma}{2})} \Bigl[1+\frac{(x-\mu)^2}{\sigma}\Bigr]^{-\frac{\sigma+1}{2}}\]

where \(\Gamma(x)\) is the gamma function.

### `BreitWignerModel`

class `BreitWignerModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Breit-Wigner-Fano function. It has the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)), plus `q` (\(q\)) in

\[f(x; A, \mu, \sigma, q) = \frac{A (q\sigma/2 + x - \mu)^2}{(\sigma/2)^2 + (x - \mu)^2}\]

### `LognormalModel`

class `LognormalModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on the Log-normal distribution function. It has the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)) in

\[f(x; A, \mu, \sigma) = \frac{A e^{-(\ln(x) - \mu)/ 2\sigma^2}}{x}\]

### `DampedOcsillatorModel`

class `DampedOcsillatorModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on the Damped Harmonic Oscillator Amplitude. It has the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)) in

\[f(x; A, \mu, \sigma) = \frac{A}{\sqrt{ [1 - (x/\mu)^2]^2 + (2\sigma x/\mu)^2}}\]

### `ExponentialGaussianModel`

class `ExponentialGaussianModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model of an Exponentially modified Gaussian distribution. It has the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)), and also `gamma` (\(\gamma\)) in

\[f(x; A, \mu, \sigma, \gamma) = \frac{A\gamma}{2} \exp\bigl[\gamma({\mu - x + \gamma\sigma^2/2})\bigr] {\operatorname{erfc}}\Bigl(\frac{\mu + \gamma\sigma^2 - x}{\sqrt{2}\sigma}\Bigr)\]

where `erfc()` is the complimentary error function.

### `SkewedGaussianModel`

class `SkewedGaussianModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A variation of the above model, this is a Skewed normal distribution. It has the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)), and also `gamma` (\(\gamma\)) in

\[ f(x; A, \mu, \sigma, \gamma) = \frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{{2\sigma}^2}}]} \Bigl\{ 1 + {\operatorname{erf}}\bigl[ \frac{\gamma(x-\mu)}{\sigma\sqrt{2}} \bigr] \Bigr\}\]

where `erf()` is the error function.

### `DonaichModel`

class `DonaichModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model of an Doniach Sunjic asymmetric lineshape, used in photo-emission. With the usual parameters `amplitude` (\(A\)), `center` (\(\mu\)) and `sigma` (\(\sigma\)), and also `gamma` (\(\gamma\)) in

\[f(x; A, \mu, \sigma, \gamma) = A\frac{\cos\bigl[\pi\gamma/2 + (1-\gamma) \arctan{(x - \mu)}/\sigma\bigr]} {\bigr[1 + (x-\mu)/\sigma\bigl]^{(1-\gamma)/2}}\]

## Linear and Polynomial Models

These models correspond to polynomials of some degree. Of course, lmfit is a very inefficient way to do linear regression (see `numpy.polyfit()` or `scipy.stats.linregress()`), but these models may be useful as one of many components of composite model.

### `ConstantModel`

class `ConstantModel`(missing=None[, prefix=''[, name=None[, **kws]]])

  a class that consists of a single value, `c`. This is constant in the sense of having no dependence on the independent variable `x`, not in the sense of being non-varying. To be clear, `c` will be a variable Parameter.

### `LinearModel`

class `LinearModel`(missing=None[, prefix=''[, name=None[, **kws]]])

  a class that gives a linear model:

\[f(x; m, b) = m x + b\]

with parameters `slope` for \(m\) and `intercept` for \(b\).

### `QuadraticModel`

class `QuadraticModel`(missing=None[, prefix=''[, name=None[, **kws]]])

  a class that gives a quadratic model:

\[f(x; a, b, c) = a x^2 + b x + c\]

with parameters `a`, `b`, and `c`.

### `ParabolicModel`

class `ParabolicModel`(missing=None[, prefix=''[, name=None[, **kws]]])

  same as `QuadraticModel`.

### `PolynomialModel`

class `PolynomialModel`(degree, missing=None[, prefix=''[, name=None[, **kws]]])

  a class that gives a polynomial model up to `degree` (with maximum value of 7).

\[f(x; c_0, c_1, \ldots, c_7) = \sum_{i=0, 7} c_i x^i\]

with parameters `c0`, `c1`, ..., `c7`. The supplied `degree` will specify how many of these are actual variable parameters. This uses `numpy.polyval()` for its calculation of the polynomial.

## Step-like models

Two models represent step-like functions, and share many characteristics.

### `StepModel`

class `StepModel`(form='linear'[, missing=None[, prefix=''[, name=None[, **kws]]]])

A model based on a Step function, with four choices for functional form. The step function starts with a value 0, and ends with a value of \(A\) (`amplitude`), rising to \(A/2\) at \(\mu\) (`center`), with \(\sigma\) (`sigma`) setting the characteristic width. The supported functional forms are `linear` (the default), `atan` or `arctan` for an arc-tangent function, `erf` for an error function, or `logistic` for a logistic function. The forms are

\[\begin{eqnarray*} & f(x; A, \mu, \sigma, {\mathrm{form={}'linear{}'}}) & = A \min{[1, \max{(0, \alpha)}]} \\ & f(x; A, \mu, \sigma, {\mathrm{form={}'arctan{}'}}) & = A [1/2 + \arctan{(\alpha)}/{\pi}] \\ & f(x; A, \mu, \sigma, {\mathrm{form={}'erf{}'}}) & = A [1 + {\operatorname{erf}}(\alpha)]/2 \\ & f(x; A, \mu, \sigma, {\mathrm{form={}'logistic{}'}})& = A [1 - \frac{1}{1 + e^{\alpha}} ] \end{eqnarray*}\]

where \(\alpha = (x - \mu)/{\sigma}\).

### `RectangleModel`

class `RectangleModel`(form='linear'[, missing=None[, prefix=''[, name=None[, **kws]]]])

A model based on a Step-up and Step-down function of the same form. The same choices for functional form as for `StepModel` are supported, with `linear` as the default. The function starts with a value 0, and ends with a value of \(A\) (`amplitude`), rising to \(A/2\) at \(\mu_1\) (`center1`), with \(\sigma_1\) (`sigma1`) setting the characteristic width. It drops to rising to \(A/2\) at \(\mu_2\) (`center2`), with characteristic width \(\sigma_2\) (`sigma2`).

\[\begin{eqnarray*} &f(x; A, \mu, \sigma, {\mathrm{form={}'linear{}'}}) &= A \{ \min{[1, \max{(0, \alpha_1)}]} + \min{[-1, \max{(0, \alpha_2)}]} \} \\ &f(x; A, \mu, \sigma, {\mathrm{form={}'arctan{}'}}) &= A [\arctan{(\alpha_1)} + \arctan{(\alpha_2)}]/{\pi} \\ &f(x; A, \mu, \sigma, {\mathrm{form={}'erf{}'}}) &= A [{\operatorname{erf}}(\alpha_1) + {\operatorname{erf}}(\alpha_2)]/2 \\ &f(x; A, \mu, \sigma, {\mathrm{form={}'logistic{}'}}) &= A [1 - \frac{1}{1 + e^{\alpha_1}} - \frac{1}{1 + e^{\alpha_2}} ] \end{eqnarray*}\]

where \(\alpha_1 = (x - \mu_1)/{\sigma_1}\) and \(\alpha_2 = -(x - \mu_2)/{\sigma_2}\).

## Exponential and Power law models

### `ExponentialModel`

class `ExponentialModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on an exponential decay function. With parameters named `amplitude` (\(A\)), and `decay` (\(\tau\)), this has the form:

\[f(x; A, \tau) = A e^{-x/\tau}\]

### `PowerLawModel`

class `PowerLawModel`(missing=None[, prefix=''[, name=None[, **kws]]])

A model based on a Power Law. With parameters named `amplitude` (\(A\)), and `exponent` (\(k\)), this has the form:

\[f(x; A, k) = A x^k\]

## User-defined Models

As shown in the previous chapter (Modeling Data and Curve Fitting), it is fairly straightforward to build fitting models from parametrized python functions. The number of model classes listed so far in the present chapter should make it clear that this process is not too difficult. Still, it is sometimes desirable to build models from a user-supplied function. This may be especially true if model-building is built-in to some larger library or application for fitting in which the user may not be able to easily build and use a new model from python code.

The `ExpressionModel` allows a model to be built from a user-supplied expression. This uses the asteval module also used for mathematical constraints as discussed in Using Mathematical Constraints.

### `ExpressionModel`

class `ExpressionModel`(expr, independent_vars=None, init_script=None, **kws)

  A model using the user-supplied mathematical expression, which can be nearly any valid Python expresion.

  Parameters:

  - expr (string) – expression use to build model
  - independent_vars (`None` (default) or list of strings for independent variables.) – list of argument names in expression that are independent variables.
  - init_script (`None` (default) or string) – python script to run before parsing and evaluating expression.

with other parameters passed to `model.Model`, with the notable exception that `ExpressionModel` does not support the prefix argument.

Since the point of this model is that an arbitrary expression will be supplied, the determination of what are the parameter names for the model happens when the model is created. To do this, the expression is parsed, and all symbol names are found. Names that are already known (there are over 500 function and value names in the asteval namespace, including most python builtins, more than 200 functions inherited from numpy, and more than 20 common lineshapes defined in the `lineshapes` module) are not converted to parameters. Unrecognized name are expected to be names either of parameters or independent variables. If independent_vars is the default value of `None`, and if the expression contains a variable named x, that will be used as the independent variable. Otherwise, independent_vars must be given.

For example, if one creates an `ExpressionModel` as:

    >>> mod = ExpressionModel('off + amp * exp(-x/x0) * sin(x*phase)')

The name exp will be recognized as the exponent function, so the model will be interpreted to have parameters named off, amp, x0 and phase. In addition, x will be assumed to be the sole independent variable. In general, there is no obvious way to set default parameter values or parameter hints for bounds, so this will have to be handled explicitly.

To evaluate this model, you might do the following:

    >>> x = numpy.linspace(0, 10, 501)
    >>> params = mod.make_params(off=0.25, amp=1.0, x0=2.0, phase=0.04)
    >>> y = mod.eval(params, x=x)

While many custom models can be built with a single line expression (especially since the names of the lineshapes like gaussian, lorentzian and so on, as well as many numpy functions, are available), more complex models will inevitably require multiple line functions. You can include such Python code with the init_script argument. The text of this script is evaluated when the model is initialized (and before the actual expression is parsed), so that you can define functions to be used in your expression.

As a probably unphysical example, to make a model that is the derivative of a Gaussian function times the logarithm of a Lorentzian function you may could to define this in a script:

    >>> script = """
    def mycurve(x, amp, cen, sig):
        loren = lorentzian(x, amplitude=amp, center=cen, sigma=sig)
        gauss = gaussian(x, amplitude=amp, center=cen, sigma=sig)
        return log(loren)*gradient(gauss)/gradient(x)
    """

and then use this with `ExpressionModel` as:

    >>> mod = ExpressionModel('mycurve(x, height, mid, wid)',
                              init_script=script,
                              independent_vars=['x'])

As above, this will interpret the parameter names to be height, mid, and wid, and build a model that can be used to fit data.

## Example 1: Fit Peaked data to Gaussian, Lorentzian, and Voigt profiles

Here, we will fit data to three similar line shapes, in order to decide which might be the better model. We will start with a Gaussian profile, as in the previous chapter, but use the built-in `GaussianModel` instead of one we write ourselves. This is a slightly different version from the one in previous example in that the parameter names are different, and have built-in default values. So, we’ll simply use:

    from numpy import loadtxt
    from lmfit.models import GaussianModel

    data = loadtxt('test_peak.dat')
    x = data[:, 0]
    y = data[:, 1]

    mod = GaussianModel()
    pars = mod.guess(y, x=x)
    out  = mod.fit(y, pars, x=x)
    print(out.fit_report(min_correl=0.25))

which prints out the results:

    [[Model]]
        gaussian
    [[Fit Statistics]]
        # function evals   = 21
        # data points      = 401
        # variables        = 3
        chi-square         = 29.994
        reduced chi-square = 0.075
    [[Variables]]
        amplitude:   30.3135571 +/- 0.157126 (0.52%) (init= 29.08159)
        center:      9.24277049 +/- 0.007374 (0.08%) (init= 9.25)
        fwhm:        2.90156963 +/- 0.017366 (0.60%)  == '2.3548200*sigma'
        sigma:       1.23218319 +/- 0.007374 (0.60%) (init= 1.35)
    [[Correlations]] (unreported correlations are <  0.250)
        C(amplitude, sigma)          =  0.577

[We see a few interesting differences from the results of the previous
  chapter. First, the parameter names are longer. Second, there is a `fwhm` parameter, defined as \(\sim 2.355\sigma\). And third, the automated initial guesses are pretty good. A plot of the fit shows not such a great fit:

> Fit to peak with Gaussian (left) and Lorentzian (right) models.

suggesting that a different peak shape, with longer tails, should be used. Perhaps a Lorentzian would be better? To do this, we simply replace `GaussianModel` with `LorentzianModel` to get a `LorentzianModel`:

    from lmfit.models import LorentzianModel
    mod = LorentzianModel()
    pars = mod.guess(y, x=x)
    out  = mod.fit(y, pars, x=x)
    print(out.fit_report(min_correl=0.25))

Predictably, the first thing we try gives results that are worse:

    [[Model]]
        lorentzian
    [[Fit Statistics]]
        # function evals   = 25
        # data points      = 401
        # variables        = 3
        chi-square         = 53.754
        reduced chi-square = 0.135
    [[Variables]]
        amplitude:   38.9728645 +/- 0.313857 (0.81%) (init= 36.35199)
        center:      9.24438944 +/- 0.009275 (0.10%) (init= 9.25)
        fwhm:        2.30969034 +/- 0.026312 (1.14%)  == '2.0000000*sigma'
        sigma:       1.15484517 +/- 0.013156 (1.14%) (init= 1.35)
    [[Correlations]] (unreported correlations are <  0.250)
        C(amplitude, sigma)          =  0.709

with the plot shown on the right in the figure above.

A Voigt model does a better job. Using `VoigtModel`, this is as simple as:

    from lmfit.models import VoigtModel
    mod = VoigtModel()
    pars = mod.guess(y, x=x)
    out  = mod.fit(y, pars, x=x)
    print(out.fit_report(min_correl=0.25))

which gives:

    [[Model]]
        voigt
    [[Fit Statistics]]
        # function evals   = 17
        # data points      = 401
        # variables        = 3
        chi-square         = 14.545
        reduced chi-square = 0.037
    [[Variables]]
        amplitude:   35.7554017 +/- 0.138614 (0.39%) (init= 43.62238)
        center:      9.24411142 +/- 0.005054 (0.05%) (init= 9.25)
        fwhm:        2.62951718 +/- 0.013269 (0.50%)  == '3.6013100*sigma'
        gamma:       0.73015574 +/- 0.003684 (0.50%)  == 'sigma'
        sigma:       0.73015574 +/- 0.003684 (0.50%) (init= 0.8775)
    [[Correlations]] (unreported correlations are <  0.250)
        C(amplitude, sigma)          =  0.651

with the much better value for \(\chi^2\) and the obviously better match to the data as seen in the figure below (left).

> Fit to peak with Voigt model (left) and Voigt model with `gamma` varying independently of `sigma` (right).

The Voigt function has a \(\gamma\) parameter (`gamma`) that can be distinct from `sigma`. The default behavior used above constrains `gamma` to have exactly the same value as `sigma`. If we allow these to vary separately, does the fit improve? To do this, we have to change the `gamma` parameter from a constrained expression and give it a starting value:

    mod = VoigtModel()
    pars = mod.guess(y, x=x)
    pars['gamma'].set(value=0.7, vary=True, expr='')

    out  = mod.fit(y, pars, x=x)
    print(out.fit_report(min_correl=0.25))

which gives:

    [[Model]]
        voigt
    [[Fit Statistics]]
        # function evals   = 21
        # data points      = 401
        # variables        = 4
        chi-square         = 10.930
        reduced chi-square = 0.028
    [[Variables]]
        amplitude:   34.1914716 +/- 0.179468 (0.52%) (init= 43.62238)
        center:      9.24374845 +/- 0.004419 (0.05%) (init= 9.25)
        fwhm:        3.22385491 +/- 0.050974 (1.58%)  == '3.6013100*sigma'
        gamma:       0.52540157 +/- 0.018579 (3.54%) (init= 0.7)
        sigma:       0.89518950 +/- 0.014154 (1.58%) (init= 0.8775)
    [[Correlations]] (unreported correlations are <  0.250)
        C(amplitude, gamma)          =  0.821

and the fit shown on the right above.

Comparing the two fits with the Voigt function, we see that \(\chi^2\) is definitely improved with a separately varying `gamma` parameter. In addition, the two values for `gamma` and `sigma` differ significantly – well outside the estimated uncertainties. Even more compelling, reduced \(\chi^2\) is improved even though a fourth variable has been added to the fit. In the simplest statistical sense, this suggests that `gamma` is a significant variable in the model.

This example shows how easy it can be to alter and compare fitting models for simple problems. The example is included in the `doc_peakmodels.py` file in the examples directory.

## Example 2: Fit data to a Composite Model with pre-defined models

Here, we repeat the point made at the end of the last chapter that instances of `model.Model` class can be added together to make a composite model. By using the large number of built-in models available, it is therefore very simple to build models that contain multiple peaks and various backgrounds. An example of a simple fit to a noisy step function plus a constant:

After constructing step-like data, we first create a `StepModel` telling it to use the `erf` form (see details above), and a `ConstantModel`. We set initial values, in one case using the data and `guess()` method for the initial step function paramaters, and `make_params()` arguments for the linear component. After making a composite model, we run `fit()` and report the results, which give:

    [[Model]]
     Composite Model:
        step(prefix='step_',form='erf')
        linear(prefix='line_')
    [[Fit Statistics]]
        # function evals   = 49
        # data points      = 201
        # variables        = 5
        chi-square         = 633.465
        reduced chi-square = 3.232
    [[Variables]]
        line_intercept:   11.5685248 +/- 0.285611 (2.47%) (init= 10.72406)
        line_slope:       2.03270159 +/- 0.096041 (4.72%) (init= 0)
        step_amplitude:   112.270535 +/- 0.674790 (0.60%) (init= 136.3006)
        step_center:      3.12343845 +/- 0.005370 (0.17%) (init= 2.5)
        step_sigma:       0.67468813 +/- 0.011336 (1.68%) (init= 1.428571)
    [[Correlations]] (unreported correlations are <  0.100)
        C(step_amplitude, step_sigma)  =  0.564
        C(line_intercept, step_center)  =  0.428
        C(step_amplitude, step_center)  =  0.109

with a plot of

## Example 3: Fitting Multiple Peaks – and using Prefixes

As shown above, many of the models have similar parameter names. For composite models, this could lead to a problem of having parameters for different parts of the model having the same name. To overcome this, each `model.Model` can have a `prefix` attribute (normally set to a blank string) that will be put at the beginning of each parameter name. To illustrate, we fit one of the classic datasets from the NIST StRD suite involving a decaying exponential and two gaussians.

where we give a separate prefix to each model (they all have an `amplitude` parameter). The `prefix` values are attached transparently to the models.

MN—-: Note that the calls to `make_param()` used the bare name, without the prefix. We could have used them, but because we used the individual model `gauss1` and `gauss2`, there was no need.

Note also in the example here that we explicitly set bounds on many of the parameter values.

The fit results printed out are:

    [[Model]]
     Composite Model:
        gaussian(prefix='g1_')
        gaussian(prefix='g2_')
        exponential(prefix='exp_')
    [[Fit Statistics]]
        # function evals   = 55
        # data points      = 250
        # variables        = 8
        chi-square         = 1247.528
        reduced chi-square = 5.155
    [[Variables]]
        exp_amplitude:   99.0183291 +/- 0.537487 (0.54%) (init= 162.2102)
        exp_decay:       90.9508788 +/- 1.103104 (1.21%) (init= 93.24905)
        g1_amplitude:    4257.77384 +/- 42.38354 (1.00%) (init= 2000)
        g1_center:       107.030955 +/- 0.150068 (0.14%) (init= 105)
        g1_fwhm:         39.2609205 +/- 0.377907 (0.96%)  == '2.3548200*g1_sigma'
        g1_sigma:        16.6725781 +/- 0.160482 (0.96%) (init= 15)
        g2_amplitude:    2493.41747 +/- 36.16907 (1.45%) (init= 2000)
        g2_center:       153.270103 +/- 0.194665 (0.13%) (init= 155)
        g2_fwhm:         32.5128760 +/- 0.439860 (1.35%)  == '2.3548200*g2_sigma'
        g2_sigma:        13.8069474 +/- 0.186791 (1.35%) (init= 15)
    [[Correlations]] (unreported correlations are <  0.500)
        C(g1_amplitude, g1_sigma)    =  0.824
        C(g2_amplitude, g2_sigma)    =  0.815
        C(g1_sigma, g2_center)       =  0.684
        C(g1_amplitude, g2_center)   =  0.648
        C(g1_center, g2_center)      =  0.621
        C(g1_center, g1_sigma)       =  0.507

We get a very good fit to this challenging problem (described at the NIST site as of average difficulty, but the tests there are generally hard) by applying reasonable initial guesses and putting modest but explicit bounds on the parameter values. This fit is shown on the left:

One final point on setting initial values. From looking at the data itself, we can see the two Gaussian peaks are reasonably well separated but do overlap. Furthermore, we can tell that the initial guess for the decaying exponential component was poorly estimated because we used the full data range. We can simplify the initial parameter values by using this, and by defining an `index_of()` function to limit the data range. That is, with:

    def index_of(arrval, value):
        "return index of array *at or below* value "
        if value < min(arrval):  return 0
        return max(np.where(arrval<=value)[0])

    ix1 = index_of(x,  75)
    ix2 = index_of(x, 135)
    ix3 = index_of(x, 175)

    exp_mod.guess(y[:ix1], x=x[:ix1])
    gauss1.guess(y[ix1:ix2], x=x[ix1:ix2])
    gauss2.guess(y[ix2:ix3], x=x[ix2:ix3])

we can get a better initial estimate, and the fit converges in fewer steps, getting to identical values (to the precision printed out in the report), and without any bounds on parameters at all:

    [[Model]]
     Composite Model:
        gaussian(prefix='g1_')
        gaussian(prefix='g2_')
        exponential(prefix='exp_')
    [[Fit Statistics]]
        # function evals   = 46
        # data points      = 250
        # variables        = 8
        chi-square         = 1247.528
        reduced chi-square = 5.155
    [[Variables]]
        exp_amplitude:   99.0183281 +/- 0.537487 (0.54%) (init= 94.53724)
        exp_decay:       90.9508863 +/- 1.103105 (1.21%) (init= 111.1985)
        g1_amplitude:    4257.77321 +/- 42.38338 (1.00%) (init= 2126.432)
        g1_center:       107.030954 +/- 0.150067 (0.14%) (init= 106.5)
        g1_fwhm:         39.2609141 +/- 0.377905 (0.96%)  == '2.3548200*g1_sigma'
        g1_sigma:        16.6725754 +/- 0.160481 (0.96%) (init= 14.5)
        g2_amplitude:    2493.41766 +/- 36.16948 (1.45%) (init= 1878.892)
        g2_center:       153.270100 +/- 0.194667 (0.13%) (init= 150)
        g2_fwhm:         32.5128777 +/- 0.439866 (1.35%)  == '2.3548200*g2_sigma'
        g2_sigma:        13.8069481 +/- 0.186794 (1.35%) (init= 15)
    [[Correlations]] (unreported correlations are <  0.500)
        C(g1_amplitude, g1_sigma)    =  0.824
        C(g2_amplitude, g2_sigma)    =  0.815
        C(g1_sigma, g2_center)       =  0.684
        C(g1_amplitude, g2_center)   =  0.648
        C(g1_center, g2_center)      =  0.621
        C(g1_center, g1_sigma)       =  0.507

This example is in the file `doc_nistgauss2.py` in the examples folder, and the fit result shown on the right above shows an improved initial estimate of the data.