"""
======
Curves
======
Curve shape functions
"""
import numpy as np
[docs]def Gaussian(x, amplitude, center, width, baselevel=0):
"""
Gaussian curve
.. math::
f(x) = A * exp \left( - \\frac{(x - b)^{2}}{2c^2} \\right) + d
where:
* :math:`A` = amplitude
* :math:`b` = center
* :math:`c` = width (standard deviation)
* :math:`d` = baselevel
Parameters
----------
x : float or array-like
x-axis
amplitude : float
maximum peak height
center : float
peak center
width : float
peak width as standard deviation
baselevel : float, default 0
peak baselevel
Returns
-------
float, array-like
Gaussian curve evaluated at x
"""
return amplitude * np.exp(-((x - center) ** 2) / (2 * width**2)) + baselevel
[docs]def Lorentzian(x, amplitude, center, width, baselevel=0):
"""
Three-parameter Lorentzian curve
.. math::
f(x) = A\left(\\frac{c^2}{(x-b)^2 + c^2}\\right) + d
where:
* :math:`A` = amplitude
* :math:`b` = center
* :math:`c` = width (standard deviation)
* :math:`d` = baselevel
Parameters
----------
x : float or array-like
x-axis
amplitude : float
maximum peak height
center : float
peak center
width : float
peak width as half-width at full maximum
baselevel : float, default 0
peak baselevel
Returns
-------
float, array-like
Lorentzian curve evaluated at x
"""
return amplitude * width**2 / ((x - center) ** 2 + width**2) + baselevel
[docs]def GaussLorentz(x, amplitude, center, width, baselevel, shape):
"""
Mixed Gaussian-Lorentzian (Pseudo-Voigt) curve
.. math::
f(x) = Gaussian(x, A * \\alpha, b, c, d) + Lorentzian(x, A * (1 - \\alpha), b, c, d)
where:
* :math:`A` = amplitude
* :math:`b` = center
* :math:`c` = width (standard deviation)
* :math:`d` = baselevel
* :math:`\\alpha` = shape
with parameterisations from :py:func:`~ramCOH.signal_processing.curves.Gaussian` and :py:func:`~ramCOH.signal_processing.curves.Lorentzian`
Parameters
----------
x : float or array-like
x-axis
amplitude : float
maximum peak height
center : float
peak center
width : float
peak width as half-width at full maximum
baselevel : float, default 0
peak baselevel
shape : float
mixing parameter between 0 and 1
Returns
-------
float, array-like
Pseudo-Voigt curve evaluated at x
"""
return (
Gaussian(x, amplitude * (1 - shape), center, width, 0)
+ Lorentzian(x, amplitude * shape, center, width, 0)
+ baselevel
)
def sum_GaussLorentz(x, centers, amplitudes, widths, shapes, baselevels=0):
"""
add mixed Gauss-Lorentzian curves together
:meta private:
"""
peakAmount = len(centers)
if isinstance(baselevels, (int, float)):
baselevels = [baselevels] * peakAmount
if isinstance(shapes, (int, float)):
shapes = [shapes] * peakAmount
params = [
{"center": i, "amplitude": j, "width": k, "baselevel": l, "shape": m}
for i, j, k, l, m in zip(centers, amplitudes, widths, baselevels, shapes)
]
curves = GaussLorentz(x, **params[0])
for peak in params[1:]:
curves = curves + GaussLorentz(x, **peak)
return curves