Lorentz1D

class astropy.modeling.functional_models.Lorentz1D(amplitude=1, x_0=0, fwhm=1, **kwargs)

Bases: Fittable1DModel

One dimensional Lorentzian model.

Parameters:

amplitudepython:float or Quantity.

Peak value - for a normalized profile (integrating to 1), set amplitude = 2 / (np.pi * fwhm)

x_0python:float or Quantity.

Position of the peak

fwhmpython:float or Quantity.

Full width at half maximum (FWHM)

Other Parameters:

fixeda python:dict, optional

A dictionary {parameter_name: boolean} of parameters to not be varied during fitting. True means the parameter is held fixed. Alternatively the fixed property of a parameter may be used.

tiedpython:dict, optional

A dictionary {parameter_name: callable} of parameters which are linked to some other parameter. The dictionary values are callables providing the linking relationship. Alternatively the tied property of a parameter may be used.

boundspython:dict, optional

A dictionary {parameter_name: value} of lower and upper bounds of parameters. Keys are parameter names. Values are a list or a tuple of length 2 giving the desired range for the parameter. Alternatively, the min and max properties of a parameter may be used.

eqconspython:list, optional

A list of functions of length n such that eqcons[j](x0,*args) == 0.0 in a successfully optimized problem.

ineqconspython:list, optional

A list of functions of length n such that ieqcons[j](x0,*args) >= 0.0 is a successfully optimized problem.

See also

Gaussian1D, Box1D, RickerWavelet1D

Notes

Either all or none of input x, position x_0 and fwhm must be provided consistently with compatible units or as unitless numbers.

Model formula:

\[f(x) = \frac{A \gamma^{2}}{\gamma^{2} + \left(x - x_{0}\right)^{2}}\]

where \(\gamma\) is half of given FWHM.

Examples

import numpy as np
import matplotlib.pyplot as plt

from astropy.modeling.models import Lorentz1D

plt.figure()
s1 = Lorentz1D()
r = np.arange(-5, 5, .01)

for factor in range(1, 4):
    s1.amplitude = factor
    plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

plt.axis([-5, 5, -1, 4])
plt.show()

(png, svg, pdf)

Attributes Summary

amplitude
fwhm
input_units	This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted).
param_names	Names of the parameters that describe models of this type.
x_0

Methods Summary

evaluate(x, amplitude, x_0, fwhm)	One dimensional Lorentzian model function
fit_deriv(x, amplitude, x_0, fwhm)	One dimensional Lorentzian model derivative with respect to parameters

Attributes Documentation

amplitude = Parameter('amplitude', value=1.0)

fwhm = Parameter('fwhm', value=1.0)

input_units

param_names = ('amplitude', 'x_0', 'fwhm')

Names of the parameters that describe models of this type.

The parameters in this tuple are in the same order they should be passed in when initializing a model of a specific type. Some types of models, such as polynomial models, have a different number of parameters depending on some other property of the model, such as the degree.

When defining a custom model class the value of this attribute is automatically set by the Parameter attributes defined in the class body.

x_0 = Parameter('x_0', value=0.0)

Methods Documentation

static evaluate(x, amplitude, x_0, fwhm)

One dimensional Lorentzian model function

static fit_deriv(x, amplitude, x_0, fwhm)

One dimensional Lorentzian model derivative with respect to parameters