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_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, *pars)

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

  • pars – (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. Default True

  • st_term – If True, enable gaussian term. Default True

  • lt_term – If True, enable gaussian term. Default True

  • step_term – If True, enable gaussian term. Default True

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 curve G(x) and a Lorentzian curve L(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 Lorentz factor: 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 large x 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. Default True

  • st_term – If True, enable gaussian term. Default True

  • lt_term – If True, enable gaussian term. Default True

  • step_term – If True, enable gaussian term. Default True

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 curve G(x) and a Lorentzian curve L(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 Lorentz factor: 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 curve G(x) and a Lorentzian curve L(x) instead of their convolution.

  • height is the peak amplitudefor 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 Lorentz factor: 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_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