scipy.stats.gennorm#

scipy.stats.gennorm = <scipy.stats._continuous_distns.gennorm_gen object>[source]#

A generalized normal continuous random variable.

As an instance of the rv_continuous class, gennorm object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

See also

laplace: Laplace distribution
norm: normal distribution

Notes

The probability density function for gennorm is [1]:

\[f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),\]

where \(x\) is a real number, \(\beta > 0\) and \(\Gamma\) is the gamma function (scipy.special.gamma).

gennorm takes beta as a shape parameter for \(\beta\). For \(\beta = 1\), it is identical to a Laplace distribution. For \(\beta = 2\), it is identical to a normal distribution (with scale=1/sqrt(2)).

References

[1]

“Generalized normal distribution, Version 1”, https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1

[2]

Nardon, Martina, and Paolo Pianca. “Simulation techniques for generalized Gaussian densities.” Journal of Statistical Computation and Simulation 79.11 (2009): 1317-1329

[3]

Wicklin, Rick. “Simulate data from a generalized Gaussian distribution” in The DO Loop blog, September 21, 2016, https://blogs.sas.com/content/iml/2016/09/21/simulate-generalized-gaussian-sas.html

Examples

>>> import numpy as np
>>> from scipy.stats import gennorm
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> beta = 1.3
>>> mean, var, skew, kurt = gennorm.stats(beta, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(gennorm.ppf(0.01, beta),
...                 gennorm.ppf(0.99, beta), 100)
>>> ax.plot(x, gennorm.pdf(x, beta),
...        'r-', lw=5, alpha=0.6, label='gennorm pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = gennorm(beta)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = gennorm.ppf([0.001, 0.5, 0.999], beta)
>>> np.allclose([0.001, 0.5, 0.999], gennorm.cdf(vals, beta))
True

Generate random numbers:

>>> r = gennorm.rvs(beta, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Methods

rvs(beta, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, beta, loc=0, scale=1)	Probability density function.
logpdf(x, beta, loc=0, scale=1)	Log of the probability density function.
cdf(x, beta, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, beta, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, beta, loc=0, scale=1)	Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
logsf(x, beta, loc=0, scale=1)	Log of the survival function.
ppf(q, beta, loc=0, scale=1)	Percent point function (inverse of `cdf` — percentiles).
isf(q, beta, loc=0, scale=1)	Inverse survival function (inverse of `sf`).
moment(order, beta, loc=0, scale=1)	Non-central moment of the specified order.
stats(beta, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(beta, loc=0, scale=1)	(Differential) entropy of the RV.
fit(data)	Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
expect(func, args=(beta,), loc=0, scale=1, lb=None, ub=None, conditional=False, kwds)**	Expected value of a function (of one argument) with respect to the distribution.
median(beta, loc=0, scale=1)	Median of the distribution.
mean(beta, loc=0, scale=1)	Mean of the distribution.
var(beta, loc=0, scale=1)	Variance of the distribution.
std(beta, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, beta, loc=0, scale=1)	Confidence interval with equal areas around the median.