Built-in Fitting Models in the models module

Lmfit provides several built-in fitting models in the models module. These pre-defined models each subclass from the Model class of the previous chapter and wrap relatively well-known functional forms, such as Gaussian, Lorentzian, and Exponential that are used in a wide range of scientific domains. In fact, all the models are based on simple, plain Python functions defined in the lineshapes module. In addition to wrapping a function into a 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 Model class is that models can easily be combined to give a composite CompositeModel. 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 use them 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

Almost 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 intensity (or area of) a peak or function and a sigma parameter that gives a characteristic width.

After a list of built-in models, a few examples of their use are given.

Peak-like models

There are many peak-like models available. These include GaussianModel, LorentzianModel, VoigtModel, PseudoVoigtModel, and some less commonly used variations. Most of these models are unit-normalized and share the same parameter names so that you can easily switch between models and interpret the results. The amplitude parameter is the multiplicative factor for the unit-normalized peak lineshape, and so will represent the strength of that peak or the area under that curve. The center parameter will be the centroid x value. The sigma parameter is the characteristic width of the peak, with many functions using \((x-\mu)/\sigma\) where \(\mu\) is the centroid value. Most of these peak functions will have two additional parameters derived from and constrained by the other parameters. The first of these is fwhm which will hold the estimated “Full Width at Half Max” for the peak, which is often easier to compare between different models than sigma. The second of these is height which will contain the maximum value of the peak, typically the value at \(x = \mu\). Finally, each of these models has a guess() method that uses data to make a fairly crude but usually sufficient guess for the value of amplitude, center, and sigma, and sets a lower bound of 0 on the value of sigma.

GaussianModel

class GaussianModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Gaussian or normal distribution lineshape.

The model has three Parameters: amplitude, center, and sigma. In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively.

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

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 \(2.3548\sigma\).

For more information, see: https://en.wikipedia.org/wiki/Normal_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

LorentzianModel

class LorentzianModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Lorentzian or Cauchy-Lorentz distribution function.

The model has three Parameters: amplitude, center, and sigma. In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively.

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

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

For more information, see: https://en.wikipedia.org/wiki/Cauchy_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

SplitLorentzianModel

class SplitLorentzianModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Lorentzian or Cauchy-Lorentz distribution function.

The model has four parameters: amplitude, center, sigma, and sigma_r. In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively.

‘Split’ means that the width of the distribution is different between left and right slopes.

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

where the parameter amplitude corresponds to \(A\), center to \(\mu\), sigma to \(\sigma\), sigma_l to \(\sigma_l\), and \(H(x)\) is a Heaviside step function:

\[H(x) = 0 | x < 0, 1 | x \geq 0\]

The full width at half maximum is \(\sigma_l+\sigma_r\). Just as with the Lorentzian model, integral of this function from -.inf to +.inf equals to amplitude.

For more information, see: https://en.wikipedia.org/wiki/Cauchy_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

VoigtModel

class VoigtModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Voigt distribution function.

The model has four Parameters: amplitude, center, sigma, and gamma. By default, gamma is constrained to have a value equal to sigma, though it can be varied independently. In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively. 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 complementary 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 \(3.6013\sigma\).

For more information, see: https://en.wikipedia.org/wiki/Voigt_profile

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

PseudoVoigtModel

class PseudoVoigtModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a pseudo-Voigt distribution function.

This is a weighted sum of a Gaussian and Lorentzian distribution function that share values for amplitude (\(A\)), center (\(\mu\)), and full width at half maximum fwhm (and so has constrained values of sigma (\(\sigma\)) and height (maximum peak height). The 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.

For more information, see: https://en.wikipedia.org/wiki/Voigt_profile#Pseudo-Voigt_Approximation

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

MoffatModel

class MoffatModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on the Moffat distribution function.

The model has four Parameters: amplitude (\(A\)), center (\(\mu\)), a width parameter sigma (\(\sigma\)), and an exponent beta (\(\beta\)). In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively.

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

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

Note that for (\(\beta=1\)) the Moffat has a Lorentzian shape. For more information, see: https://en.wikipedia.org/wiki/Moffat_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

Pearson7Model

class Pearson7Model(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Pearson VII distribution.

The model has four parameters: amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and exponent (\(m\)). In addition, parameters fwhm and height are included as constraints to report estimates for the full width at half maximum and maximum peak height, respectively.

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

where \(\beta\) is the beta function (see scipy.special.beta). The guess() function always gives a starting value for exponent of 1.5. In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively.

For more information, see: https://en.wikipedia.org/wiki/Pearson_distribution#The_Pearson_type_VII_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

StudentsTModel

class StudentsTModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Student’s t-distribution function.

The model has three Parameters: amplitude (\(A\)), center (\(\mu\)), and sigma (\(\sigma\)). In addition, parameters fwhm and height are included as constraints to report full width at half maximum and maximum peak height, respectively.

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

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

For more information, see: https://en.wikipedia.org/wiki/Student%27s_t-distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

BreitWignerModel

class BreitWignerModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Breit-Wigner-Fano function.

The model has four Parameters: amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and q (\(q\)).

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

For more information, see: https://en.wikipedia.org/wiki/Fano_resonance

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

LognormalModel

class LognormalModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on the Log-normal distribution function.

The modal has three Parameters amplitude (\(A\)), center (\(\mu\)), and sigma (\(\sigma\)). In addition, parameters fwhm and height are included as constraints to report estimates of full width at half maximum and maximum peak height, respectively.

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

For more information, see: https://en.wikipedia.org/wiki/Lognormal

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

DampedOscillatorModel

class DampedOscillatorModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on the Damped Harmonic Oscillator Amplitude.

The model has three Parameters: amplitude (\(A\)), center (\(\mu\)), and sigma (\(\sigma\)). In addition, the parameter height is included as a constraint to report the maximum peak height.

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

For more information, see: https://en.wikipedia.org/wiki/Harmonic_oscillator#Amplitude_part

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

DampedHarmonicOscillatorModel

class DampedHarmonicOscillatorModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a variation of the Damped Harmonic Oscillator.

The model follows the definition given in DAVE/PAN (see: https://www.ncnr.nist.gov/dave) and has four Parameters: amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and gamma (\(\gamma\)). In addition, parameters fwhm and height are included as constraints to report estimates for full width at half maximum and maximum peak height, respectively.

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

where \(\gamma=kT\), k is the Boltzmann constant in \(evK^-1\), and T is the temperature in \(K\).

For more information, see: https://en.wikipedia.org/wiki/Harmonic_oscillator

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

ExponentialGaussianModel

class ExponentialGaussianModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model of an Exponentially modified Gaussian distribution.

The model has four Parameters: amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and gamma (\(\gamma\)).

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

where erfc() is the complementary error function.

For more information, see: https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

SkewedGaussianModel

class SkewedGaussianModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A skewed Gaussian model, using a skewed normal distribution.

The model has four Parameters: amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and gamma (\(\gamma\)).

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

where erf() is the error function.

For more information, see: https://en.wikipedia.org/wiki/Skew_normal_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

SkewedVoigtModel

class SkewedVoigtModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A skewed Voigt model, modified using a skewed normal distribution.

The model has five Parameters amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and gamma (\(\gamma\)), as usual for a Voigt distribution, and adds a new Parameter skew.

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

where erf() is the error function.

For more information, see: https://en.wikipedia.org/wiki/Skew_normal_distribution

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

ThermalDistributionModel

class ThermalDistributionModel(independent_vars=['x'], prefix='', nan_policy='raise', form='bose', **kwargs)

Return a thermal distribution function.

Variable form defines the kind of distribution as below with three Parameters: amplitude (\(A\)), center (\(x_0\)), and kt (\(kt\)). The following distributions are available:

  • ‘bose’ : Bose-Einstein distribution (default)

  • ‘maxwell’ : Maxwell-Boltzmann distribution

  • ‘fermi’ : Fermi-Dirac distribution

The functional forms are defined as:

\begin{eqnarray*} & f(x; A, x_0, kt, {\mathrm{form={}'bose{}'}}) & = \frac{1}{A \exp(\frac{x - x_0}{kt}) - 1} \\ & f(x; A, x_0, kt, {\mathrm{form={}'maxwell{}'}}) & = \frac{1}{A \exp(\frac{x - x_0}{kt})} \\ & f(x; A, x_0, kt, {\mathrm{form={}'fermi{}'}}) & = \frac{1}{A \exp(\frac{x - x_0}{kt}) + 1} ] \end{eqnarray*}

Notes

  • kt should be defined in the same units as x (\(k_B = 8.617\times10^{-5}\) eV/K).

  • set \(kt<0\) to implement the energy loss convention common in scattering research.

For more information, see: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disfcn.html

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

DoniachModel

class DoniachModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model of an Doniach Sunjic asymmetric lineshape.

This model is used in photo-emission and has four Parameters: amplitude (\(A\)), center (\(\mu\)), sigma (\(\sigma\)), and gamma (\(\gamma\)). In addition, parameter height is included as a constraint to report maximum peak height.

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

For more information, see: https://www.casaxps.com/help_manual/line_shapes.htm

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

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 a composite model.

ConstantModel

class ConstantModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

Constant model, with a single Parameter: c.

Note that 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 Parameter that will be varied in the fit (by default, of course).

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

LinearModel

class LinearModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

Linear model, with two Parameters: intercept and slope.

Defined as:

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

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

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

QuadraticModel

class QuadraticModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A quadratic model, with three Parameters: a, b, and c.

Defined as:

\[f(x; a, b, c) = a x^2 + b x + c\]
Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

PolynomialModel

class PolynomialModel(degree=7, independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A polynomial model with up to 7 Parameters, specified by degree.

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

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.

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

Periodic Models

These models correspond to periodic functions.

SineModel

class SineModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a sinusoidal lineshape.

The model has three Parameters: amplitude, frequency, and shift.

\[f(x; A, \phi, f) = A \sin (f x + \phi)\]

where the parameter amplitude corresponds to \(A\), frequency to \(f\), and shift to \(\phi\). All are constrained to be non-negative, and shift additionally to be smaller than \(2\pi\).

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

Step-like models

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

StepModel

class StepModel(independent_vars=['x'], prefix='', nan_policy='raise', form='linear', **kwargs)

A model based on a Step function.

The model has three Parameters: amplitude (\(A\)), center (\(\mu\)), and sigma (\(\sigma\)).

There are four choices for form:

The step function starts with a value 0 and ends with a value of \(A\) rising to \(A/2\) at \(\mu\), with \(\sigma\) setting the characteristic width. The functional forms are defined as:

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

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

RectangleModel

class RectangleModel(independent_vars=['x'], prefix='', nan_policy='raise', form='linear', **kwargs)

A model based on a Step-up and Step-down function.

The model has five Parameters: amplitude (\(A\)), center1 (\(\mu_1\)), center2 (\(\mu_2\)), sigma1 (\(\sigma_1\)), and sigma2 (\(\sigma_2\)).

There are four choices for form, which is used for both the Step up and the Step down:

The function starts with a value 0 and transitions to a value of \(A\), taking the value \(A/2\) at \(\mu_1\), with \(\sigma_1\) setting the characteristic width. The function then transitions again to the value \(A/2\) at \(\mu_2\), with \(\sigma_2\) setting the characteristic width. The functional forms are defined as:

\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 \(\alpha_2 = -(x - \mu_2)/{\sigma_2}\).

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

Exponential and Power law models

ExponentialModel

class ExponentialModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on an exponential decay function.

The model has two Parameters: amplitude (\(A\)) and decay (\(\tau\)) and is defined as:

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

For more information, see: https://en.wikipedia.org/wiki/Exponential_decay

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

PowerLawModel

class PowerLawModel(independent_vars=['x'], prefix='', nan_policy='raise', **kwargs)

A model based on a Power Law.

The model has two Parameters: amplitude (\(A\)) and exponent (\(k\)) and is defined as:

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

For more information, see: https://en.wikipedia.org/wiki/Power_law

Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

Two dimensional Peak-like models

The one example of a two-dimensional peak is a two-dimensional Gaussian.

Gaussian2dModel

class Gaussian2dModel(independent_vars=['x', 'y'], prefix='', nan_policy='raise', **kwargs)

A model based on a two-dimensional Gaussian function.

The model has two independent variables x and y and five Parameters: amplitude, centerx, sigmax, centery, and sigmay. In addition, parameters fwhmx, fwhmy, and height are included as constraints to report the maximum peak height and the two full width at half maxima, respectively.

\[f(x, y; A, \mu_x, \sigma_x, \mu_y, \sigma_y) = A g(x; A=1, \mu_x, \sigma_x) g(y; A=1, \mu_y, \sigma_y)\]

where subfunction \(g(x; A, \mu, \sigma)\) is a Gaussian lineshape:

\[g(x; A, \mu, \sigma) = \frac{A}{\sigma\sqrt{2\pi}} e^{[{-{(x-\mu)^2}/{{2\sigma}^2}}]}.\]
Parameters
  • independent_vars (list of str, optional) – Arguments to the model function that are independent variables default is [‘x’, ‘y’]).

  • prefix (str, optional) – String to prepend to parameter names, needed to add two Models that have parameter names in common.

  • nan_policy ({'raise', 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kwargs (optional) – Keyword arguments to pass to Model.

Notes

1. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

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, nan_policy='raise', **kws)

ExpressionModel class.

Generate a model from user-supplied expression.

Parameters
  • expr (str) – Mathematical expression for model.

  • independent_vars (list of str or None, optional) – Variable names to use as independent variables.

  • init_script (str or None, optional) – Initial script to run in asteval interpreter.

  • nan_policy ({'raise, 'propagate', 'omit'}, optional) – How to handle NaN and missing values in data. See Notes below.

  • **kws (optional) – Keyword arguments to pass to Model.

Notes

  1. each instance of ExpressionModel will create and use its own version of an asteval interpreter.

  2. prefix is not supported for ExpressionModel.

3. nan_policy sets what to do when a NaN or missing value is seen in the data. Should be one of:

  • ‘raise’ : raise a ValueError (default)

  • ‘propagate’ : do nothing

  • ‘omit’ : drop missing data

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 built-ins, more than 200 functions inherited from NumPy, and more than 20 common lineshapes defined in the lineshapes module) are not converted to parameters. Unrecognized names are expected to be names of either 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:

from lmfit.models import ExpressionModel

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:

from numpy import exp, linspace, sin

x = 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 Peak 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 writing one 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. We will 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]]
    Model(gaussian)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 25
    # data points      = 401
    # variables        = 3
    chi-square         = 29.9943157
    reduced chi-square = 0.07536260
    Akaike info crit   = -1033.77437
    Bayesian info crit = -1021.79248
[[Variables]]
    amplitude:  30.3135620 +/- 0.15712686 (0.52%) (init = 43.62238)
    center:     9.24277047 +/- 0.00737496 (0.08%) (init = 9.25)
    sigma:      1.23218359 +/- 0.00737496 (0.60%) (init = 1.35)
    fwhm:       2.90157056 +/- 0.01736670 (0.60%) == '2.3548200*sigma'
    height:     9.81457817 +/- 0.05087283 (0.52%) == '0.3989423*amplitude/max(1e-15, sigma)'
[[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 are fwhm and height parameters, to give the full-width-at-half-maximum and maximum peak height, respectively. And third, the automated initial guesses are pretty good. A plot of the fit:

_images/builtin_models_7_0.svg

shows a decent match to the data – the fit worked with no explicit setting of initial parameter values. Looking more closely, the fit is not perfect, especially in the tails of the peak, 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()

with the rest of the script as above. Perhaps predictably, the first thing we try gives results that are worse by comparing the fit statistics:

[[Model]]
    Model(lorentzian)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 21
    # data points      = 401
    # variables        = 3
    chi-square         = 53.7535387
    reduced chi-square = 0.13505914
    Akaike info crit   = -799.830322
    Bayesian info crit = -787.848438
[[Variables]]
    amplitude:  38.9727645 +/- 0.31386183 (0.81%) (init = 54.52798)
    center:     9.24438944 +/- 0.00927619 (0.10%) (init = 9.25)
    sigma:      1.15483925 +/- 0.01315659 (1.14%) (init = 1.35)
    fwhm:       2.30967850 +/- 0.02631318 (1.14%) == '2.0000000*sigma'
    height:     10.7421156 +/- 0.08633945 (0.80%) == '0.3183099*amplitude/max(1e-15, sigma)'
[[Correlations]] (unreported correlations are < 0.250)
    C(amplitude, sigma) =  0.709

and also by visual inspection of the fit to the data (figure below).

_images/builtin_models_10_0.svg

The tails are now too big, and the value for \(\chi^2\) almost doubled. A Voigt model does a better job. Using VoigtModel, this is as simple as using:

from lmfit.models import VoigtModel
mod = VoigtModel()

with all the rest of the script as above. This gives:

[[Model]]
    Model(voigt)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 21
    # data points      = 401
    # variables        = 3
    chi-square         = 14.5448627
    reduced chi-square = 0.03654488
    Akaike info crit   = -1324.00615
    Bayesian info crit = -1312.02427
[[Variables]]
    amplitude:  35.7554146 +/- 0.13861321 (0.39%) (init = 65.43358)
    center:     9.24411150 +/- 0.00505482 (0.05%) (init = 9.25)
    sigma:      0.73015627 +/- 0.00368460 (0.50%) (init = 0.8775)
    gamma:      0.73015627 +/- 0.00368460 (0.50%) == 'sigma'
    fwhm:       2.62950494 +/- 0.00806900 (0.31%) == '1.0692*gamma+sqrt(0.8664*gamma**2+5.545083*sigma**2)'
    height:     10.2203969 +/- 0.03009415 (0.29%) == '(amplitude/(max(1e-15, sigma*sqrt(2*pi))))*wofz((1j*gamma)/(max(1e-15, sigma*sqrt(2)))).real'
[[Correlations]] (unreported correlations are < 0.250)
    C(amplitude, sigma) =  0.651

which has a much better value for \(\chi^2\) and the other goodness-of-fit measures, and an obviously better match to the data as seen in the figure below (left).

_images/builtin_models_13_0.svg

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

Can we do better? 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 using something like:

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

which gives:

[[Model]]
    Model(voigt)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 21
    # data points      = 401
    # variables        = 4
    chi-square         = 10.9301767
    reduced chi-square = 0.02753193
    Akaike info crit   = -1436.57602
    Bayesian info crit = -1420.60017
[[Variables]]
    amplitude:  34.1914737 +/- 0.17946860 (0.52%) (init = 65.43358)
    center:     9.24374847 +/- 0.00441903 (0.05%) (init = 9.25)
    sigma:      0.89518909 +/- 0.01415450 (1.58%) (init = 0.8775)
    gamma:      0.52540199 +/- 0.01857955 (3.54%) (init = 0.7)
    fwhm:       2.72573635 +/- 0.01363980 (0.50%) == '1.0692*gamma+sqrt(0.8664*gamma**2+5.545083*sigma**2)'
    height:     10.0872204 +/- 0.03482129 (0.35%) == '(amplitude/(max(1e-15, sigma*sqrt(2*pi))))*wofz((1j*gamma)/(max(1e-15, sigma*sqrt(2)))).real'
[[Correlations]] (unreported correlations are < 0.250)
    C(sigma, gamma)     = -0.928
    C(amplitude, gamma) =  0.821
    C(amplitude, sigma) = -0.651

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. 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. In addition, we can use both the Akaike or Bayesian Information Criteria (see Akaike and Bayesian Information Criteria) to assess how likely the model with variable gamma is to explain the data than the model with gamma fixed to the value of sigma. According to theory, \(\exp(-(\rm{AIC1}-\rm{AIC0})/2)\) gives the probability that a model with AIC1 is more likely than a model with AIC0. For the two models here, with AIC values of -1436 and -1324 (Note: if we had more carefully set the value for weights based on the noise in the data, these values might be positive, but there difference would be roughly the same), this says that the model with gamma fixed to sigma has a probability less than 5.e-25 of being the better model.

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 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:

# <examples/doc_builtinmodels_stepmodel.py>
import matplotlib.pyplot as plt
import numpy as np

from lmfit.models import LinearModel, StepModel

x = np.linspace(0, 10, 201)
y = np.ones_like(x)
y[:48] = 0.0
y[48:77] = np.arange(77-48)/(77.0-48)
np.random.seed(0)
y = 110.2 * (y + 9e-3*np.random.randn(x.size)) + 12.0 + 2.22*x

step_mod = StepModel(form='erf', prefix='step_')
line_mod = LinearModel(prefix='line_')

pars = line_mod.make_params(intercept=y.min(), slope=0)
pars += step_mod.guess(y, x=x, center=2.5)

mod = step_mod + line_mod
out = mod.fit(y, pars, x=x)

print(out.fit_report())

plt.plot(x, y, 'b')
plt.plot(x, out.init_fit, 'k--', label='initial fit')
plt.plot(x, out.best_fit, 'r-', label='best fit')
plt.legend(loc='best')
plt.show()
# <end examples/doc_builtinmodels_stepmodel.py>

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 parameters, and make_params() arguments for the linear component. After making a composite model, we run fit() and report the results, which gives:

[[Model]]
    (Model(step, prefix='step_', form='erf') + Model(linear, prefix='line_'))
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 49
    # data points      = 201
    # variables        = 5
    chi-square         = 593.709622
    reduced chi-square = 3.02913072
    Akaike info crit   = 227.700173
    Bayesian info crit = 244.216698
[[Variables]]
    line_slope:      1.87164656 +/- 0.09318713 (4.98%) (init = 0)
    line_intercept:  12.0964833 +/- 0.27606235 (2.28%) (init = 11.58574)
    step_amplitude:  112.858376 +/- 0.65392947 (0.58%) (init = 134.7378)
    step_center:     3.13494792 +/- 0.00516615 (0.16%) (init = 2.5)
    step_sigma:      0.67392841 +/- 0.01091168 (1.62%) (init = 1.428571)
[[Correlations]] (unreported correlations are < 0.100)
    C(line_slope, step_amplitude)     = -0.879
    C(step_amplitude, step_sigma)     =  0.564
    C(line_slope, step_sigma)         = -0.457
    C(line_intercept, step_center)    =  0.427
    C(line_slope, line_intercept)     = -0.309
    C(line_slope, step_center)        = -0.234
    C(line_intercept, step_sigma)     = -0.137
    C(line_intercept, step_amplitude) = -0.117
    C(step_amplitude, step_center)    =  0.109

with a plot of

_images/builtin_models_17_0.svg

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 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.

# <examples/doc_builtinmodels_nistgauss.py>
import matplotlib.pyplot as plt
import numpy as np

from lmfit.models import ExponentialModel, GaussianModel

dat = np.loadtxt('NIST_Gauss2.dat')
x = dat[:, 1]
y = dat[:, 0]

exp_mod = ExponentialModel(prefix='exp_')
pars = exp_mod.guess(y, x=x)

gauss1 = GaussianModel(prefix='g1_')
pars.update(gauss1.make_params())

pars['g1_center'].set(value=105, min=75, max=125)
pars['g1_sigma'].set(value=15, min=3)
pars['g1_amplitude'].set(value=2000, min=10)

gauss2 = GaussianModel(prefix='g2_')
pars.update(gauss2.make_params())

pars['g2_center'].set(value=155, min=125, max=175)
pars['g2_sigma'].set(value=15, min=3)
pars['g2_amplitude'].set(value=2000, min=10)

mod = gauss1 + gauss2 + exp_mod

init = mod.eval(pars, x=x)
out = mod.fit(y, pars, x=x)

print(out.fit_report(min_correl=0.5))

fig, axes = plt.subplots(1, 2, figsize=(12.8, 4.8))
axes[0].plot(x, y, 'b')
axes[0].plot(x, init, 'k--', label='initial fit')
axes[0].plot(x, out.best_fit, 'r-', label='best fit')
axes[0].legend(loc='best')

comps = out.eval_components(x=x)
axes[1].plot(x, y, 'b')
axes[1].plot(x, comps['g1_'], 'g--', label='Gaussian component 1')
axes[1].plot(x, comps['g2_'], 'm--', label='Gaussian component 2')
axes[1].plot(x, comps['exp_'], 'k--', label='Exponential component')
axes[1].legend(loc='best')

plt.show()
# <end examples/doc_builtinmodels_nistgauss.py>

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

Note that the calls to make_param() used the bare name, without the prefix. We could have used the prefixes, 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]]
    ((Model(gaussian, prefix='g1_') + Model(gaussian, prefix='g2_')) + Model(exponential, prefix='exp_'))
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 46
    # data points      = 250
    # variables        = 8
    chi-square         = 1247.52821
    reduced chi-square = 5.15507524
    Akaike info crit   = 417.864631
    Bayesian info crit = 446.036318
[[Variables]]
    exp_amplitude:  99.0183283 +/- 0.53748735 (0.54%) (init = 162.2102)
    exp_decay:      90.9508860 +/- 1.10310509 (1.21%) (init = 93.24905)
    g1_amplitude:   4257.77319 +/- 42.3833645 (1.00%) (init = 2000)
    g1_center:      107.030954 +/- 0.15006786 (0.14%) (init = 105)
    g1_sigma:       16.6725753 +/- 0.16048161 (0.96%) (init = 15)
    g1_fwhm:        39.2609138 +/- 0.37790530 (0.96%) == '2.3548200*g1_sigma'
    g1_height:      101.880231 +/- 0.59217099 (0.58%) == '0.3989423*g1_amplitude/max(1e-15, g1_sigma)'
    g2_amplitude:   2493.41771 +/- 36.1694731 (1.45%) (init = 2000)
    g2_center:      153.270101 +/- 0.19466743 (0.13%) (init = 155)
    g2_sigma:       13.8069484 +/- 0.18679415 (1.35%) (init = 15)
    g2_fwhm:        32.5128783 +/- 0.43986659 (1.35%) == '2.3548200*g2_sigma'
    g2_height:      72.0455934 +/- 0.61722094 (0.86%) == '0.3989423*g2_amplitude/max(1e-15, g2_sigma)'
[[Correlations]] (unreported correlations are < 0.100)
    C(g1_amplitude, g1_sigma)      =  0.824
    C(g2_amplitude, g2_sigma)      =  0.815
    C(exp_amplitude, exp_decay)    = -0.695
    C(g1_sigma, g2_center)         =  0.684
    C(g1_center, g2_amplitude)     = -0.669
    C(g1_center, g2_sigma)         = -0.652
    C(g1_amplitude, g2_center)     =  0.648
    C(g1_center, g2_center)        =  0.621
    C(g1_center, g1_sigma)         =  0.507
    C(exp_decay, g1_amplitude)     = -0.507
    C(g1_sigma, g2_amplitude)      = -0.491
    C(g2_center, g2_sigma)         = -0.489
    C(g1_sigma, g2_sigma)          = -0.483
    C(g2_amplitude, g2_center)     = -0.476
    C(exp_decay, g2_amplitude)     = -0.427
    C(g1_amplitude, g1_center)     =  0.418
    C(g1_amplitude, g2_sigma)      = -0.401
    C(g1_amplitude, g2_amplitude)  = -0.307
    C(exp_amplitude, g2_amplitude) =  0.282
    C(exp_decay, g1_sigma)         = -0.252
    C(exp_decay, g2_sigma)         = -0.233
    C(exp_amplitude, g2_sigma)     =  0.171
    C(exp_decay, g2_center)        = -0.151
    C(exp_amplitude, g1_amplitude) =  0.148
    C(exp_decay, g1_center)        =  0.105

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

_images/builtin_models_20_0.svg

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 (see below).

_images/builtin_models_22_0.svg

The fit converges to the same answer, giving to identical values (to the precision printed out in the report), but in fewer steps, and without any bounds on parameters at all:

[[Model]]
    ((Model(gaussian, prefix='g1_') + Model(gaussian, prefix='g2_')) + Model(exponential, prefix='exp_'))
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 37
    # data points      = 250
    # variables        = 8
    chi-square         = 1247.52821
    reduced chi-square = 5.15507524
    Akaike info crit   = 417.864631
    Bayesian info crit = 446.036318
[[Variables]]
    exp_amplitude:  99.0183270 +/- 0.53748905 (0.54%) (init = 94.53724)
    exp_decay:      90.9508890 +/- 1.10310483 (1.21%) (init = 111.1985)
    g1_amplitude:   4257.77343 +/- 42.3836433 (1.00%) (init = 3189.648)
    g1_center:      107.030956 +/- 0.15006873 (0.14%) (init = 106.5)
    g1_sigma:       16.6725765 +/- 0.16048227 (0.96%) (init = 14.5)
    g1_fwhm:        39.2609166 +/- 0.37790686 (0.96%) == '2.3548200*g1_sigma'
    g1_height:      101.880230 +/- 0.59217233 (0.58%) == '0.3989423*g1_amplitude/max(1e-15, g1_sigma)'
    g2_amplitude:   2493.41733 +/- 36.1696906 (1.45%) (init = 2818.337)
    g2_center:      153.270101 +/- 0.19466905 (0.13%) (init = 150)
    g2_sigma:       13.8069461 +/- 0.18679534 (1.35%) (init = 15)
    g2_fwhm:        32.5128728 +/- 0.43986939 (1.35%) == '2.3548200*g2_sigma'
    g2_height:      72.0455948 +/- 0.61722328 (0.86%) == '0.3989423*g2_amplitude/max(1e-15, g2_sigma)'
[[Correlations]] (unreported correlations are < 0.100)
    C(g1_amplitude, g1_sigma)      =  0.824
    C(g2_amplitude, g2_sigma)      =  0.815
    C(exp_amplitude, exp_decay)    = -0.695
    C(g1_sigma, g2_center)         =  0.684
    C(g1_center, g2_amplitude)     = -0.669
    C(g1_center, g2_sigma)         = -0.652
    C(g1_amplitude, g2_center)     =  0.648
    C(g1_center, g2_center)        =  0.621
    C(g1_center, g1_sigma)         =  0.507
    C(exp_decay, g1_amplitude)     = -0.507
    C(g1_sigma, g2_amplitude)      = -0.491
    C(g2_center, g2_sigma)         = -0.489
    C(g1_sigma, g2_sigma)          = -0.483
    C(g2_amplitude, g2_center)     = -0.476
    C(exp_decay, g2_amplitude)     = -0.427
    C(g1_amplitude, g1_center)     =  0.418
    C(g1_amplitude, g2_sigma)      = -0.401
    C(g1_amplitude, g2_amplitude)  = -0.307
    C(exp_amplitude, g2_amplitude) =  0.282
    C(exp_decay, g1_sigma)         = -0.252
    C(exp_decay, g2_sigma)         = -0.233
    C(exp_amplitude, g2_sigma)     =  0.171
    C(exp_decay, g2_center)        = -0.151
    C(exp_amplitude, g1_amplitude) =  0.148
    C(exp_decay, g1_center)        =  0.105

This script is in the file doc_builtinmodels_nistgauss2.py in the examples folder, and the figure above shows an improved initial estimate of the data.