ChiSquare Distribution

class ChiSquareDistribution(amplitude: float = 1.0, degree_of_freedom: int | float = 1, loc: float = 0.0, scale: float = 1.0, normalize: bool = False)

Bases: BaseDistribution

Class for ChiSquareDistribution distribution.

Note

The ChiSquareDistribution is a special case of the GammaDistributionSR,

• \(\alpha_\text{gammaSR} = \text{dof} / 2\),

• \(\lambda_\text{gammaSR} = 0.5\).

Parameters:

• amplitude (float, optional) – The amplitude of the PDF. Defaults to 1.0. Ignored if normalize is True.

• degree_of_freedom (int or float, optional) – The degree of freedom for the chi-square distribution. Default is 1.0.

• loc (float, optional) – The location parameter, for shifting. Defaults to 0.0.

• normalize (bool, optional) – If True, the distribution is normalized so that the total area under the PDF equals 1. Defaults to False.

Raises:

• NegativeAmplitudeError – If the provided value of amplitude is negative.

• DegreeOfFreedomError – If the provided value of degree of freedom is either less than or equal to 0 or not an integer.

Examples:

Importing libraries

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import chi2

from pymultifit.distributions import ChiSquareDistribution

Generating a standard \(\chi^2(1)\) distribution with pyMultiFit and scipy.

x_values = np.linspace(start=0, stop=5, num=500)

y_multifit = ChiSquareDistribution(degree_of_freedom=1, normalize=True)
y_scipy = chi2

Plotting PDF and CDF

f, ax = plt.subplots(1, 2, figsize=(12, 5))

ax[0].plot(x_values, y_scipy.pdf(x=x_values, df=1), label='Scipy Chi2')
ax[0].plot(x_values, y_multifit.pdf(x_values), 'k:', label='pyMultiFit Chi2')
ax[0].set_ylabel('f(x)')

ax[1].plot(x_values, y_scipy.cdf(x=x_values, df=1), label='Scipy Chi2')
ax[1].plot(x_values, y_multifit.cdf(x_values), 'k:', label='pyMultiFit Chi2')
ax[1].set_ylabel('F(x)')

f.suptitle(r'$\chi^2$(1)')

for i in ax:
    i.set_xlabel('X')
    i.legend()
plt.tight_layout()

Generating a translated \(\chi^2(1)\) distribution with \(\text{loc} = 3\).

y_multifit = ChiSquareDistribution(degree_of_freedom=1, loc=3, normalize=True)

Plotting PDF and CDF

f, ax = plt.subplots(1, 2, figsize=(12, 5))

ax[0].plot(x_values, y_scipy.pdf(x=x_values, df=1, loc=3), label='Scipy translated Chi2')
ax[0].plot(x_values, y_multifit.pdf(x_values), 'k:', label='pyMultiFit translated Chi2')
ax[0].set_ylabel('f(x)')

ax[1].plot(x_values, y_scipy.cdf(x=x_values, df=1, loc=3), label='Scipy translated Chi2')
ax[1].plot(x_values, y_multifit.cdf(x_values), 'k:', label='pyMultiFit translated Chi2')
ax[1].set_ylabel('F(x)')

f.suptitle(r'$\chi^2$(1, loc=3)')

for i in ax:
    i.set_xlabel('X')
    i.legend()
plt.tight_layout()

Attributes:

mean

The mean of the distribution.

median

The median of the distribution.

mode

The mode of the distribution.

stddev

The standard deviation of the distribution.

variance

The variance of the distribution.

Methods

cdf(x)	Compute the cumulative density function (CDF) for the distribution.
logcdf(x)	Compute the log cumulative density function (logCDF) for the distribution.
logpdf(x)	Compute the log probability density function (logPDF) for the distribution.
pdf(x)	Compute the probability density function (PDF) for the distribution.
scipy_like(df[, loc, scale])	Instantiate ChiSquareDistribution with scipy parameterization.
stats()	Computes and returns the statistical properties of the distribution, including,

cdf(x)

Compute the cumulative density function (CDF) for the distribution.

Parameters:

x – Input array at which to evaluate the CDF.

logcdf(x)

Compute the log cumulative density function (logCDF) for the distribution.

Parameters:

x – Input array at which to evaluate the logCDF.

logpdf(x)

Compute the log probability density function (logPDF) for the distribution.

Parameters:

x – Input array at which to evaluate the logPDF.

pdf(x)

Compute the probability density function (PDF) for the distribution.

Parameters:

x – Input array at which to evaluate the PDF.

classmethod scipy_like(df: int | float, loc: float = 0.0, scale: float = 1.0)

Instantiate ChiSquareDistribution with scipy parameterization.

Parameters:

df: int or float

The degree of freedom for the ChiSquare distribution.

loc: float, optional

The location parameter. Defaults to 0.0.

scale: float, optional

The scale parameter. Defaults to 1.0

Returns:

ChiSquareDistribution

An instance of normalized ChiSquareDistribution.

stats()

Computes and returns the statistical properties of the distribution, including,

1. mean,

2. median,

3. variance, and

4. standard deviation.

Returns:

A dictionary containing statistical properties such as mean, variance, etc.

Return type:

Dict[str, float]

Notes

If any of the parameter is not computable for a distribution, this method returns None.

property mean

The mean of the distribution.

property median

The median of the distribution.

property mode

The mode of the distribution.

property stddev

The standard deviation of the distribution.

property variance

The variance of the distribution.

This class internally utilizes the following functions from utilities_d module:

• gamma_sr_pdf_()

• gamma_sr_cdf_()

Recommended Import

from pymultifit.distributions import ChiSquareDistribution

Full Import

from pymultifit.distributions.chiSquare_d import ChiSquareDistribution