# 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. a * (0.5 + (arctan((x - b) / c) / pi)) 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) 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,…) 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 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,…) 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,…) 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,…) 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 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,…) 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,…) 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,…) 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,…) 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,…) 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…) 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,…) 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,…) 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,…) Array of sum of stepup functions at each x coordinate