Built-in Functions

ezfit provides several commonly used model functions optimized with numba.

Numba-optimized functions for fitting.

ezfit.functions.exponential(x, a, b)

Exponential function.

Computes an exponential relationship of the form y = a * exp(b * x), where a is the amplitude and b is the decay/growth rate.

Parameters:

• x (array-like) – Independent variable values.

• a (float) – Amplitude (prefactor) of the exponential.

• b (float) – Decay rate (if negative) or growth rate (if positive).

Returns:

Dependent variable values computed as y = a * exp(b * x).

Return type:

array-like

References

[exp_func_ref]

Exponential function. Wikipedia. https://en.wikipedia.org/wiki/Exponential_function

[exp_decay_ref]

Exponential decay. Wikipedia. https://en.wikipedia.org/wiki/Exponential_decay

Examples

>>> import numpy as np
>>> from ezfit import exponential
>>> x = np.linspace(0, 10, 100)
>>> y = exponential(x, a=1.0, b=-0.5)  # Exponential decay

ezfit.functions.gaussian(x, amplitude, center, fwhm)

Gaussian (normal) distribution function.

Computes a Gaussian peak with specified amplitude, center position, and full width at half maximum (FWHM).

The function is defined as:

G(x) = amplitude * exp[-4 * ln(2) * ((x - center) / fwhm)^2]

At x = center, the function reaches its maximum value of amplitude. The half maximum occurs at |x-center| = fwhm / 2.

Parameters:

• x (array-like) – Independent variable values.

• amplitude (float) – Peak amplitude (maximum value at x = center).

• center (float) – Center position (mean) of the Gaussian.

• fwhm (float) – Full width at half maximum. Must be positive.

Returns:

Dependent variable values of the Gaussian function.

Return type:

array-like

References

[gauss_norm_ref]

Normal distribution. Wikipedia. https://en.wikipedia.org/wiki/Normal_distribution

[gauss_func_ref]

Gaussian function. Wikipedia. https://en.wikipedia.org/wiki/Gaussian_function

Examples

>>> import numpy as np
>>> from ezfit import gaussian
>>> x = np.linspace(-5, 5, 100)
>>> y = gaussian(x, amplitude=1.0, center=0.0, fwhm=2.0)

ezfit.functions.linear(x, m, b)

Linear function.

The function is defined as:

y = m * x + b

where m is the slope and b is the y-intercept.

Parameters:

• x (array-like) – Independent variable values.

• m (float) – Slope of the line.

• b (float) – Y-intercept (value when x = 0).

Returns:

Dependent variable values computed as y = m * x + b.

Return type:

array-like

References

[linear_func_ref]

Linear function. Wikipedia. https://en.wikipedia.org/wiki/Linear_function

[linear_eq_ref]

Linear equation. Wikipedia. https://en.wikipedia.org/wiki/Linear_equation

Examples

>>> import numpy as np
>>> from ezfit import linear
>>> x = np.linspace(0, 10, 100)
>>> y = linear(x, m=2.0, b=1.0)

ezfit.functions.lorentzian(x, amplitude, center, fwhm)

Lorentzian (Cauchy) distribution function.

Computes a Lorentzian peak with specified amplitude, center position, and full width at half maximum (FWHM).

The function is defined as:

L(x) = amplitude * [ (fwhm/2)^2 / ((x - center)^2 + (fwhm/2)^2) ]

At x = center, the function reaches its maximum value of amplitude. The half maximum occurs at |x-center| = fwhm / 2.

Parameters:

• x (array-like) – Independent variable values.

• amplitude (float) – Peak amplitude (maximum value at x = center).

• center (float) – Center position (location parameter) of the Lorentzian.

• fwhm (float) – Full width at half maximum. Must be positive.

Returns:

Dependent variable values of the Lorentzian function.

Return type:

array-like

References

[lorentz_cauchy_ref]

Cauchy distribution. Wikipedia. https://en.wikipedia.org/wiki/Cauchy_distribution

[lorentz_line_ref]

Spectral line shape. Wikipedia. https://en.wikipedia.org/wiki/Spectral_line_shape#Lorentzian

Examples

>>> import numpy as np
>>> from ezfit import lorentzian
>>> x = np.linspace(-5, 5, 100)
>>> y = lorentzian(x, amplitude=1.0, center=0.0, fwhm=2.0)

ezfit.functions.power_law(x, a, b)

Power law function.

Computes a power law relationship of the form y = a * x^b, where a is the coefficient and b is the exponent.

Parameters:

• x (array-like) – Independent variable values.

• a (float) – Coefficient (prefactor) of the power law.

• b (float) – Exponent of the power law.

Returns:

Dependent variable values computed as y = a * x^b.

Return type:

array-like

References

[power_law_ref]

Power law. Wikipedia. https://en.wikipedia.org/wiki/Power_law

Examples

>>> import numpy as np
>>> from ezfit import power_law
>>> x = np.linspace(1, 10, 100)
>>> y = power_law(x, a=2.0, b=1.5)

ezfit.functions.pseudo_voigt(x, height, center, fwhm, eta)

Pseudo-Voigt profile function.

Computes a pseudo-Voigt profile, which is a weighted linear combination of a Gaussian and a Lorentzian profile. This function is commonly used to model spectral line shapes and diffraction peaks.

The function is defined as:

y = height * [(1 - eta) * G + eta * L]

where G is a normalized Gaussian and L is a normalized Lorentzian, both with the same FWHM and center position. The mixing parameter eta controls the relative contribution of each component (0 = pure Gaussian, 1 = pure Lorentzian).

Parameters:

• x (array-like) – Independent variable values.

• height (float) – Peak height (maximum value at x = center).

• center (float) – Center position of the profile.

• fwhm (float) – Full width at half maximum. Must be positive.

• eta (float) – Mixing parameter between 0 and 1. Controls the relative contribution of Gaussian (1-eta) and Lorentzian (eta) components. - eta = 0: Pure Gaussian - eta = 1: Pure Lorentzian - 0 < eta < 1: Mixed profile

Returns:

Dependent variable values of the pseudo-Voigt function.

Return type:

array-like

References

[voigt_profile_ref]

Voigt profile. Wikipedia. https://en.wikipedia.org/wiki/Voigt_profile

[pseudo_voigt_ref]

Pseudo-Voigt profile. Wikipedia. https://en.wikipedia.org/wiki/Voigt_profile#Pseudo-Voigt_approximation

Examples

>>> import numpy as np
>>> from ezfit import pseudo_voigt
>>> x = np.linspace(-5, 5, 100)
>>> # 50% Gaussian, 50% Lorentzian
>>> y = pseudo_voigt(x, height=1.0, center=0.0, fwhm=2.0, eta=0.5)

Available Functions

Linear

ezfit.functions.linear(x, m, b)

Linear function.

The function is defined as:

y = m * x + b

where m is the slope and b is the y-intercept.

Parameters:

• x (array-like) – Independent variable values.

• m (float) – Slope of the line.

• b (float) – Y-intercept (value when x = 0).

Returns:

Dependent variable values computed as y = m * x + b.

Return type:

array-like

References

[linear_func_ref]

Linear function. Wikipedia. https://en.wikipedia.org/wiki/Linear_function

[linear_eq_ref]

Linear equation. Wikipedia. https://en.wikipedia.org/wiki/Linear_equation

Examples

>>> import numpy as np
>>> from ezfit import linear
>>> x = np.linspace(0, 10, 100)
>>> y = linear(x, m=2.0, b=1.0)

Exponential

ezfit.functions.exponential(x, a, b)

Exponential function.

Computes an exponential relationship of the form y = a * exp(b * x), where a is the amplitude and b is the decay/growth rate.

Parameters:

• x (array-like) – Independent variable values.

• a (float) – Amplitude (prefactor) of the exponential.

• b (float) – Decay rate (if negative) or growth rate (if positive).

Returns:

Dependent variable values computed as y = a * exp(b * x).

Return type:

array-like

References

[exp_func_ref]

Exponential function. Wikipedia. https://en.wikipedia.org/wiki/Exponential_function

[exp_decay_ref]

Exponential decay. Wikipedia. https://en.wikipedia.org/wiki/Exponential_decay

Examples

>>> import numpy as np
>>> from ezfit import exponential
>>> x = np.linspace(0, 10, 100)
>>> y = exponential(x, a=1.0, b=-0.5)  # Exponential decay

Power Law

ezfit.functions.power_law(x, a, b)

Power law function.

Computes a power law relationship of the form y = a * x^b, where a is the coefficient and b is the exponent.

Parameters:

• x (array-like) – Independent variable values.

• a (float) – Coefficient (prefactor) of the power law.

• b (float) – Exponent of the power law.

Returns:

Dependent variable values computed as y = a * x^b.

Return type:

array-like

References

[power_law_ref]

Power law. Wikipedia. https://en.wikipedia.org/wiki/Power_law

Examples

>>> import numpy as np
>>> from ezfit import power_law
>>> x = np.linspace(1, 10, 100)
>>> y = power_law(x, a=2.0, b=1.5)

Gaussian

ezfit.functions.gaussian(x, amplitude, center, fwhm)

Gaussian (normal) distribution function.

Computes a Gaussian peak with specified amplitude, center position, and full width at half maximum (FWHM).

The function is defined as:

G(x) = amplitude * exp[-4 * ln(2) * ((x - center) / fwhm)^2]

At x = center, the function reaches its maximum value of amplitude. The half maximum occurs at |x-center| = fwhm / 2.

Parameters:

• x (array-like) – Independent variable values.

• amplitude (float) – Peak amplitude (maximum value at x = center).

• center (float) – Center position (mean) of the Gaussian.

• fwhm (float) – Full width at half maximum. Must be positive.

Returns:

Dependent variable values of the Gaussian function.

Return type:

array-like

References

[gauss_norm_ref]

Normal distribution. Wikipedia. https://en.wikipedia.org/wiki/Normal_distribution

[gauss_func_ref]

Gaussian function. Wikipedia. https://en.wikipedia.org/wiki/Gaussian_function

Examples

>>> import numpy as np
>>> from ezfit import gaussian
>>> x = np.linspace(-5, 5, 100)
>>> y = gaussian(x, amplitude=1.0, center=0.0, fwhm=2.0)

Lorentzian

ezfit.functions.lorentzian(x, amplitude, center, fwhm)

Lorentzian (Cauchy) distribution function.

Computes a Lorentzian peak with specified amplitude, center position, and full width at half maximum (FWHM).

The function is defined as:

L(x) = amplitude * [ (fwhm/2)^2 / ((x - center)^2 + (fwhm/2)^2) ]

At x = center, the function reaches its maximum value of amplitude. The half maximum occurs at |x-center| = fwhm / 2.

Parameters:

• x (array-like) – Independent variable values.

• amplitude (float) – Peak amplitude (maximum value at x = center).

• center (float) – Center position (location parameter) of the Lorentzian.

• fwhm (float) – Full width at half maximum. Must be positive.

Returns:

Dependent variable values of the Lorentzian function.

Return type:

array-like

References

[lorentz_cauchy_ref]

Cauchy distribution. Wikipedia. https://en.wikipedia.org/wiki/Cauchy_distribution

[lorentz_line_ref]

Spectral line shape. Wikipedia. https://en.wikipedia.org/wiki/Spectral_line_shape#Lorentzian

Examples

>>> import numpy as np
>>> from ezfit import lorentzian
>>> x = np.linspace(-5, 5, 100)
>>> y = lorentzian(x, amplitude=1.0, center=0.0, fwhm=2.0)

Pseudo-Voigt

ezfit.functions.pseudo_voigt(x, height, center, fwhm, eta)

Pseudo-Voigt profile function.

Computes a pseudo-Voigt profile, which is a weighted linear combination of a Gaussian and a Lorentzian profile. This function is commonly used to model spectral line shapes and diffraction peaks.

The function is defined as:

y = height * [(1 - eta) * G + eta * L]

where G is a normalized Gaussian and L is a normalized Lorentzian, both with the same FWHM and center position. The mixing parameter eta controls the relative contribution of each component (0 = pure Gaussian, 1 = pure Lorentzian).

Parameters:

• x (array-like) – Independent variable values.

• height (float) – Peak height (maximum value at x = center).

• center (float) – Center position of the profile.

• fwhm (float) – Full width at half maximum. Must be positive.

• eta (float) – Mixing parameter between 0 and 1. Controls the relative contribution of Gaussian (1-eta) and Lorentzian (eta) components. - eta = 0: Pure Gaussian - eta = 1: Pure Lorentzian - 0 < eta < 1: Mixed profile

Returns:

Dependent variable values of the pseudo-Voigt function.

Return type:

array-like

References

[voigt_profile_ref]

Voigt profile. Wikipedia. https://en.wikipedia.org/wiki/Voigt_profile

[pseudo_voigt_ref]

Pseudo-Voigt profile. Wikipedia. https://en.wikipedia.org/wiki/Voigt_profile#Pseudo-Voigt_approximation

Examples

>>> import numpy as np
>>> from ezfit import pseudo_voigt
>>> x = np.linspace(-5, 5, 100)
>>> # 50% Gaussian, 50% Lorentzian
>>> y = pseudo_voigt(x, height=1.0, center=0.0, fwhm=2.0, eta=0.5)

Examples

Using Built-in Functions

from ezfit import gaussian
import pandas as pd

df = pd.read_csv("peak_data.csv")

# Fit a Gaussian peak
model, ax, _ = df.fit(
    gaussian, "x", "y", "yerr",
    amplitude={"value": 10.0, "min": 0},
    center={"value": 5.0},
    fwhm={"value": 2.0, "min": 0}
)

Combining Functions

from ezfit import gaussian
import numpy as np

def two_peaks(x, A1, c1, w1, A2, c2, w2, B):
    """Sum of two Gaussians plus baseline"""
    return gaussian(x, A1, c1, w1) + gaussian(x, A2, c2, w2) + B

model, ax, _ = df.fit(two_peaks, "x", "y", "yerr", ...)