Fit functions#
This module provides fit functions.
List of fit functions:#
sum_gauss()
sum_agauss()
sum_splitgauss()
sum_fastagauss()
sum_apvoigt()
sum_pvoigt()
sum_splitpvoigt()
sum_splitpvoigt2()
sum_lorentz()
sum_alorentz()
sum_splitlorentz()
sum_stepdown()
sum_stepup()
sum_slit()
sum_ahypermet()
sum_fastahypermet()
Full documentation:#
- atan_stepup(x, a, b, c)#
Step up function using an inverse tangent.
- Parameters:
x (numpy array) – Independent variable where the function is calculated
a – Height of the step up
b – Center of the step up
c – Parameter related to the slope of the step. A lower
c
value yields a sharper step.
- Returns:
a * (0.5 + (arctan((x - b) / c) / pi))
- Return type:
numpy array
- periodic_gauss(x, *params)#
Return a sum of gaussian functions defined by (npeaks, delta, height, centroid, fwhm), where:
npeaks is the number of gaussians peaks
delta is the constant distance between 2 peaks
height is the peak amplitude of all the gaussians
centroid is the peak x-coordinate of the first gaussian
fwhm is the full-width at half maximum for all the gaussians
- Parameters:
x – Independent variable where the function is calculated
params – (npeaks, delta, height, centroid, fwhm)
- Returns:
Sum of
npeaks
gaussians
- sum_agauss(x, *params)#
Return a sum of gaussian functions defined by (area, centroid, fwhm), where:
area is the area underneath the peak
centroid is the peak x-coordinate
fwhm is the full-width at half maximum
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of gaussian parameters (length must be a multiple of 3): (area1, centroid1, fwhm1, area2, centroid2, fwhm2,…)
- Returns:
Array of sum of gaussian functions at each
x
coordinate.
- sum_ahypermet(x, *params, gaussian_term=True, st_term=True, lt_term=True, step_term=True)#
Return a sum of ahypermet functions. defined by (area, position, fwhm, st_area_r, st_slope_r, lt_area_r, lt_slope_r, step_height_r).
area is the area underneath the gaussian peak
position is the center of the various peaks and the position of the step down
fwhm is the full-width at half maximum of the terms
st_area_r is factor between the gaussian area and the area of the short tail term
st_slope_r is a ratio related to the slope of the short tail in the low
x
values (the lower, the steeper)lt_area_r is ratio between the gaussian area and the area of the long tail term
lt_slope_r is a ratio related to the slope of the long tail in the low
x
values (the lower, the steeper)step_height_r is the ratio between the height of the step down and the gaussian height
A hypermet function is a sum of four functions (terms):
a gaussian term
a long tail term
a short tail term
a step down term
- Parameters:
x (numpy.ndarray) – Independent variable where the hypermets are calculated
params – Array of hypermet parameters (length must be a multiple of 8): (area1, position1, fwhm1, st_area_r1, st_slope_r1, lt_area_r1, lt_slope_r1, step_height_r1…)
gaussian_term – If
True
, enable gaussian term. DefaultTrue
st_term – If
True
, enable short tail term. DefaultTrue
lt_term – If
True
, enable long tail term. DefaultTrue
step_term – If
True
, enable step term. DefaultTrue
- Returns:
Array of sum of hypermet functions at each
x
coordinate
- sum_alorentz(x, *params)#
Return a sum of Lorentz distributions, also known as Cauchy distribution, defined by (area, centroid, fwhm).
area is the area underneath the peak
centroid is the peak x-coordinate for both functions
fwhm is the full-width at half maximum
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of Lorentz parameters (length must be a multiple of 3): (area1, centroid1, fwhm1,…)
- Returns:
Array of sum of Lorentz functions at each
x
coordinate
- sum_apvoigt(x, *params)#
Return a sum of pseudo-Voigt functions, defined by (area, centroid, fwhm, eta).
The pseudo-Voigt profile
PV(x)
is an approximation of the Voigt profile using a linear combination of a Gaussian curveG(x)
and a Lorentzian curveL(x)
instead of their convolution.area is the area underneath both G(x) and L(x)
centroid is the peak x-coordinate for both functions
fwhm is the full-width at half maximum of both functions
eta is the Lorentzian fraction: PV(x) = eta * L(x) + (1 - eta) * G(x)
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of pseudo-Voigt parameters (length must be a multiple of 4): (area1, centroid1, fwhm1, eta1, area2, centroid2, fwhm2, eta2,…)
- Returns:
Array of sum of pseudo-Voigt functions at each
x
coordinate
- sum_fastagauss(x, *params)#
Return a sum of gaussian functions defined by (area, centroid, fwhm), where:
area is the area underneath the peak
centroid is the peak x-coordinate
fwhm is the full-width at half maximum
This implementation differs from
sum_agauss()
by the usage of a lookup table with precalculated exponential values. This might speed up the computation for large numbers of individual gaussian functions.- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of gaussian parameters (length must be a multiple of 3): (area1, centroid1, fwhm1, area2, centroid2, fwhm2,…)
- Returns:
Array of sum of gaussian functions at each
x
coordinate.
- sum_fastahypermet(x, *params, gaussian_term=True, st_term=True, lt_term=True, step_term=True)#
Return a sum of hypermet functions defined by (area, position, fwhm, st_area_r, st_slope_r, lt_area_r, lt_slope_r, step_height_r).
area is the area underneath the gaussian peak
position is the center of the various peaks and the position of the step down
fwhm is the full-width at half maximum of the terms
st_area_r is factor between the gaussian area and the area of the short tail term
st_slope_r is a parameter related to the slope of the short tail in the low
x
values (the lower, the steeper)lt_area_r is factor between the gaussian area and the area of the long tail term
lt_slope_r is a parameter related to the slope of the long tail in the low
x
values (the lower, the steeper)step_height_r is the factor between the height of the step down and the gaussian height
A hypermet function is a sum of four functions (terms):
a gaussian term
a long tail term
a short tail term
a step down term
This function differs from
sum_ahypermet()
by the use of a lookup table for calculating exponentials. This offers better performance when calculating many functions for largex
arrays.- Parameters:
x (numpy.ndarray) – Independent variable where the hypermets are calculated
params – Array of hypermet parameters (length must be a multiple of 8): (area1, position1, fwhm1, st_area_r1, st_slope_r1, lt_area_r1, lt_slope_r1, step_height_r1…)
gaussian_term – If
True
, enable gaussian term. DefaultTrue
st_term – If
True
, enable short tail term. DefaultTrue
lt_term – If
True
, enable long tail term. DefaultTrue
step_term – If
True
, enable step term. DefaultTrue
- Returns:
Array of sum of hypermet functions at each
x
coordinate
- sum_gauss(x, *params)#
Return a sum of gaussian functions defined by (height, centroid, fwhm), where:
height is the peak amplitude
centroid is the peak x-coordinate
fwhm is the full-width at half maximum
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of gaussian parameters (length must be a multiple of 3): (height1, centroid1, fwhm1, height2, centroid2, fwhm2,…)
- Returns:
Array of sum of gaussian functions at each
x
coordinate.
- sum_lorentz(x, *params)#
Return a sum of Lorentz distributions, also known as Cauchy distribution, defined by (height, centroid, fwhm).
height is the peak amplitude
centroid is the peak x-coordinate
fwhm is the full-width at half maximum
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of Lorentz parameters (length must be a multiple of 3): (height1, centroid1, fwhm1,…)
- Returns:
Array of sum Lorentz functions at each
x
coordinate
- sum_pvoigt(x, *params)#
Return a sum of pseudo-Voigt functions, defined by (height, centroid, fwhm, eta).
The pseudo-Voigt profile
PV(x)
is an approximation of the Voigt profile using a linear combination of a Gaussian curveG(x)
and a Lorentzian curveL(x)
instead of their convolution.height is the peak amplitude of G(x) and L(x)
centroid is the peak x-coordinate for both functions
fwhm is the full-width at half maximum of both functions
eta is the Lorentzian fraction: PV(x) = eta * L(x) + (1 - eta) * G(x)
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of pseudo-Voigt parameters (length must be a multiple of 4): (height1, centroid1, fwhm1, eta1, height2, centroid2, fwhm2, eta2,…)
- Returns:
Array of sum of pseudo-Voigt functions at each
x
coordinate
- sum_slit(x, *params)#
Return a sum of slit functions. defined by (height, position, fwhm, beamfwhm).
height is the slit’s amplitude
position is the center of the slit’s x-coordinate
fwhm is the full-width at half maximum of the slit
beamfwhm is the full-width at half maximum of the derivative, which is a measure of the sharpness of the edges of the slit
- Parameters:
x (numpy.ndarray) – Independent variable where the slits are calculated
params – Array of slit parameters (length must be a multiple of 4): (height1, centroid1, fwhm1, beamfwhm1,…)
- Returns:
Array of sum of slit functions at each
x
coordinate
- sum_splitgauss(x, *params)#
Return a sum of gaussian functions defined by (area, centroid, fwhm1, fwhm2), where:
height is the peak amplitude
centroid is the peak x-coordinate
fwhm1 is the full-width at half maximum for the distribution when
x < centroid
fwhm2 is the full-width at half maximum for the distribution when
x > centroid
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of gaussian parameters (length must be a multiple of 4): (height1, centroid1, fwhm11, fwhm21, height2, centroid2, fwhm12, fwhm22,…)
- Returns:
Array of sum of split gaussian functions at each
x
coordinate
- sum_splitlorentz(x, *params)#
Return a sum of split Lorentz distributions, defined by (height, centroid, fwhm1, fwhm2).
height is the peak amplitude
centroid is the peak x-coordinate for both functions
fwhm1 is the full-width at half maximum for
x < centroid
fwhm2 is the full-width at half maximum for
x > centroid
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of Lorentz parameters (length must be a multiple of 4): (height1, centroid1, fwhm11, fwhm21…)
- Returns:
Array of sum of Lorentz functions at each
x
coordinate
- sum_splitpvoigt(x, *params)#
Return a sum of split pseudo-Voigt functions, defined by (height, centroid, fwhm1, fwhm2, eta).
The pseudo-Voigt profile
PV(x)
is an approximation of the Voigt profile using a linear combination of a Gaussian curveG(x)
and a Lorentzian curveL(x)
instead of their convolution.height is the peak amplitude for G(x) and L(x)
centroid is the peak x-coordinate for both functions
fwhm1 is the full-width at half maximum of both functions when
x < centroid
fwhm2 is the full-width at half maximum of both functions when
x > centroid
eta is the Lorentzian fraction: PV(x) = eta * L(x) + (1 - eta) * G(x)
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of pseudo-Voigt parameters (length must be a multiple of 5): (height1, centroid1, fwhm11, fwhm21, eta1,…)
- Returns:
Array of sum of split pseudo-Voigt functions at each
x
coordinate
- sum_splitpvoigt2(x, *params)#
Return a sum of split pseudo-Voigt functions, defined by (height, centroid, fwhm1, fwhm2, eta1, eta2).
The pseudo-Voigt profile
PV(x)
is an approximation of the Voigt profile using a linear combination of a Gaussian curveG(x)
and a Lorentzian curveL(x)
instead of their convolution.height is the peak amplitude for G(x) and L(x)
centroid is the peak x-coordinate for both functions
fwhm1 is the full-width at half maximum of both functions when
x < centroid
fwhm2 is the full-width at half maximum of both functions when
x > centroid
eta1 is the Lorentzian fraction when
x < centroid
eta2 is the Lorentzian fraction when
x > centroid
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of pseudo-Voigt parameters (length must be a multiple of 6): (height1, centroid1, fwhm11, fwhm21, eta11, eta21,…)
- Returns:
Array of sum of split pseudo-Voigt functions at each
x
coordinate
- sum_stepdown(x, *params)#
Return a sum of stepdown functions. defined by (height, centroid, fwhm).
height is the step’s amplitude
centroid is the step’s x-coordinate
fwhm is the full-width at half maximum for the derivative, which is a measure of the sharpness of the step-down’s edge
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of stepdown parameters (length must be a multiple of 3): (height1, centroid1, fwhm1,…)
- Returns:
Array of sum of stepdown functions at each
x
coordinate
- sum_stepup(x, *params)#
Return a sum of stepup functions. defined by (height, centroid, fwhm).
height is the step’s amplitude
centroid is the step’s x-coordinate
fwhm is the full-width at half maximum for the derivative, which is a measure of the sharpness of the step-up’s edge
- Parameters:
x (numpy.ndarray) – Independent variable where the gaussians are calculated
params – Array of stepup parameters (length must be a multiple of 3): (height1, centroid1, fwhm1,…)
- Returns:
Array of sum of stepup functions at each
x
coordinate