Gamma Distribution#

class GammaDistribution(amplitude: float = 1.0, shape: float = 1.0, scale: float = 1.0, loc: float = 0.0, normalize: bool = False)[source]#

Bases: BaseDistribution

Class for Gamma distribution with shape and scale parameters.

Parameters:

amplitude (float, optional) – The amplitude of the PDF. Default is 1.0. Ignored if normalize is True.
shape (float, optional) – The shape parameter, \(\alpha\). Defaults to 1.0.
scale (float, optional) – The rate parameter, \(\theta\). Defaults to 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.
NegativeShapeError – If the provided value of shape is negative.
NegativeScaleError – If the provided value of scale is negative.

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.
`from_scipy_params`(a[, loc, scale])	Instantiate GammaDistributionSS with scipy parametrization.
`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`(a[, loc, scale])	Instantiate GammaDistributionSS with scipy parametrization.
`stats`()	Computes and returns the statistical properties of the distribution, including,

Examples

Importing libraries:

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

from pymultifit.distributions import GammaDistribution

Generating a standard GammaSS(\(\alpha =1.5, \lambda = 1\)) distribution with pyMultiFit and scipy:

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

y_multifit = GammaDistribution(shape=1.5, normalize=True)
y_scipy = gamma

Plotting PDF and CDF:

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

ax[0].plot(x_values, y_scipy.pdf(x=x_values, a=1.5), label='Scipy Gamma SS')
ax[0].plot(x_values, y_multifit.pdf(x_values), 'k:', label='pyMultiFit Gamma SS')
ax[0].set_ylabel('f(x)')

ax[1].plot(x_values, y_scipy.cdf(x=x_values, a=1.5), label='Scipy Gamma SS')
ax[1].plot(x_values, y_multifit.cdf(x_values), 'k:', label='pyMultiFit Gamma SS')
ax[1].set_ylabel('F(x)')

f.suptitle('GammaSS(1.5)')

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

Generating a translated Gamma(\(\alpha=1.5, \lambda=0.2\)) distribution with \(\text{loc} = 3\):

y_multifit = GammaDistribution(shape=1.5, scale=0.2, 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, a=1.5, scale=0.2, loc=3), label='Scipy translated Gamma SS')
ax[0].plot(x_values, y_multifit.pdf(x_values), 'k:', label='pyMultiFit translated Gamma SS')
ax[0].set_ylabel('f(x)')

ax[1].plot(x_values, y_scipy.cdf(x=x_values, a=1.5, scale=0.2, loc=3), label='Scipy translated Gamma SS')
ax[1].plot(x_values, y_multifit.cdf(x_values), 'k:', label='pyMultiFit translated Gamma SS')
ax[1].set_ylabel('F(x)')

f.suptitle(r'Gamma SS(1.5, 0.2, 3)')

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

cdf(x: ndarray) → ndarray[source]#

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

Parameters:: x – Input array at which to evaluate the CDF.

classmethod from_scipy_params(a: float, loc: float = 0.0, scale: float = 1.0) → GammaDistribution[source]#

Instantiate GammaDistributionSS with scipy parametrization.

Parameters:

a: float: The shape parameter.
loc: float, optional: The location parameter. Defaults to 0.0.
scale: float, optional: The scaling parameter. Defaults to 1.0.

Returns:

GammaDistribution: An instance of normalized GammaDistributionSS.

logcdf(x: ndarray) → ndarray[source]#

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

Parameters:: x – Input array at which to evaluate the logCDF.

logpdf(x: ndarray) → ndarray[source]#

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

Parameters:: x – Input array at which to evaluate the logPDF.

pdf(x: ndarray) → ndarray[source]#

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

Parameters:: x – Input array at which to evaluate the PDF.

classmethod scipy_like(a: float, loc: float = 0.0, scale: float = 1.0) → GammaDistribution[source]#

Instantiate GammaDistributionSS with scipy parametrization.

Parameters:

a: float: The shape parameter.
loc: float, optional: The location parameter. Defaults to 0.0.
scale: float, optional: The scaling parameter. Defaults to 1.0.

Returns:

GammaDistribution: An instance of normalized GammaDistributionSS.

Deprecated since version 1.0.7: Use from_scipy_params instead of scipy_like. scipy_like will be removed in a future release.

stats() → Dict[str, float][source]#

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

mean,
median,
variance, and
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: float | None#: The mean of the distribution.

property median: float | None#: The median of the distribution.

property mode: float | None#: The mode of the distribution.

property stddev: float | None#: The standard deviation of the distribution.

property variance: float | None#: The variance of the distribution.

Note

The GammaDistribution encompasses the following specific cases:

ExponentialDistribution:
- \(\alpha = 1\), and
- \(\theta_\text{gamma} = \dfrac{1}{\lambda_\text{expon}}\).
UniformDistribution:
- \(\alpha = 1\), and
- \(\theta = 1\).

This class internally utilizes the following functions from utilities_d module:

Recommended Import#

from pymultifit.distributions import GammaDistribution

Full Import#

from pymultifit.distributions.gamma_d import GammaDistribution