Gamma Distribution (SR)#

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

Bases: BaseDistribution

Class for Gamma distribution with shape and rate 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.
rate (float, optional) – The rate parameter, \(\lambda\). 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.
NegativeRateError – If the provided value of rate 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.
`pdf`(x)	Compute the probability density function (PDF) for the distribution.
`scipy_like`(a[, loc, scale])	Instantiate GammaDistributionSR 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 GammaDistributionSR

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

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

y_multifit = GammaDistributionSR(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 SR')
ax[0].plot(x_values, y_multifit.pdf(x_values), 'k:', label='pyMultiFit Gamma SR')
ax[0].set_ylabel('f(x)')

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

f.suptitle('GammaSR(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 = GammaDistributionSR(shape=1.5, rate=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=1 / 0.2, loc=3), label='Scipy translated Gamma SR')
ax[0].plot(x_values, y_multifit.pdf(x_values), 'k:', label='pyMultiFit translated Gamma SR')
ax[0].set_ylabel('f(x)')

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

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

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

cdf(x: array) → array[source]#

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

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

pdf(x: array) → array[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)[source]#

Instantiate GammaDistributionSR with scipy parametrization.

Parameters:

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

Returns:

“GammaDistributionSR”: An instance of normalized GammaDistributionSR.

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#: 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.

Note

The GammaDistributionSR encompasses the following specific cases:

ExponentialDistribution:
- \(\alpha = 1\), and
- \(\lambda_\text{gammaSR} = \lambda_\text{expon}\).
GammaDistributionSS:
- \(\alpha_\text{gammaSR} = \alpha_\text{gammaSS}\)
- \(\lambda = \theta^{-1}\).
UniformDistribution:
- \(\alpha = 1\), and
- \(\lambda = 1\).

This class internally utilizes the following functions from utilities_d module:

Recommended Import#

from pymultifit.distributions import GammaDistributionSR

Full Import#

from pymultifit.distributions.gamma_d import GammaDistributionSR