aleatory.stats.vg#

aleatory.stats.vg = <aleatory.stats.variance_gamma.vg_gen object>#

A variance-gamma continuous random variable.

As an instance of the rv_continuous class, vg 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.

Methods#

rvs(r, theta, sigma, loc=0, scale=1, size=1, random_state=None): Random variates.
pdf(x, r, theta, sigma, loc=0, scale=1): Probability density function.
logpdf(x, r, theta, sigma, loc=0, scale=1): Log of the probability density function.
cdf(x, r, theta, sigma, loc=0, scale=1): Cumulative distribution function.
logcdf(x, r, theta, sigma, loc=0, scale=1): Log of the cumulative distribution function.
sf(x, r, theta, sigma, loc=0, scale=1): Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, r, theta, sigma, loc=0, scale=1): Log of the survival function.
ppf(q, r, theta, sigma, loc=0, scale=1): Percent point function (inverse of cdf — percentiles).
isf(q, r, theta, sigma, loc=0, scale=1): Inverse survival function (inverse of sf).
moment(order, r, theta, sigma, loc=0, scale=1): Non-central moment of the specified order.
stats(r, theta, sigma, loc=0, scale=1, moments=’mv’): Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(r, theta, sigma, 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=(r, theta, sigma), 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(r, theta, sigma, loc=0, scale=1): Median of the distribution.
mean(r, theta, sigma, loc=0, scale=1): Mean of the distribution.
var(r, theta, sigma, loc=0, scale=1): Variance of the distribution.
std(r, theta, sigma, loc=0, scale=1): Standard deviation of the distribution.
interval(confidence, r, theta, sigma, loc=0, scale=1): Confidence interval with equal areas around the median.

Notes#

The probability density function for vg is:

\[f(x, k, \lambda) = \exp(-(x^2 + \lambda^2)/2) (x/\lambda)^{k/2} \lambda I_{(k/2)-1}(\lambda x)\]

for \(x >= 0\), \(k > 0\) and \(\lambda \ge 0\). \(k\) specifies the degrees of freedom (denoted df in the implementation) and \(\lambda\) is the non-centrality parameter (denoted nc in the implementation). \(I_\nu\) denotes the modified Bessel function of first order of degree \(\nu\) (scipy.special.iv).

ncx takes df and nc as shape parameters.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, vg.pdf(x, r, theta, sigma, loc, scale) is identically equivalent to vg.pdf(y, r, theta, sigma) / scale with y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

Examples#

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

Calculate the first four moments:

>>> r, theta, sigma = 
>>> mean, var, skew, kurt = vg.stats(r, theta, sigma, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(vg.ppf(0.01, r, theta, sigma),
...                 vg.ppf(0.99, r, theta, sigma), 100)
>>> ax.plot(x, vg.pdf(x, r, theta, sigma),
...        'r-', lw=5, alpha=0.6, label='vg 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 = vg(r, theta, sigma)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = vg.ppf([0.001, 0.5, 0.999], r, theta, sigma)
>>> np.allclose([0.001, 0.5, 0.999], vg.cdf(vals, r, theta, sigma))
True

Generate random numbers:

>>> r = vg.rvs(r, theta, sigma, 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()