scipy.stats.geninvgauss#

scipy.stats.geninvgauss=<scipy.stats._continuous_distns.geninvgauss_gen object>[source] #

A Generalized Inverse Gaussian continuous random variable.

As an instance of the rv_continuous class, geninvgauss 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(p, b, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, p, b, loc=0, scale=1)

Probability density function.

logpdf(x, p, b, loc=0, scale=1)

Log of the probability density function.

cdf(x, p, b, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, p, b, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, p, b, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, p, b, loc=0, scale=1)

Log of the survival function.

ppf(q, p, b, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, p, b, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, p, b, loc=0, scale=1)

Non-central moment of the specified order.

stats(p, b, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(p, b, 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=(p, b), 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(p, b, loc=0, scale=1)

Median of the distribution.

mean(p, b, loc=0, scale=1)

Mean of the distribution.

var(p, b, loc=0, scale=1)

Variance of the distribution.

std(p, b, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, p, b, loc=0, scale=1)

Confidence interval with equal areas around the median.

Notes

The probability density function for geninvgauss is:

\[f(x, p, b) = x^{p-1} \exp(-b (x + 1/x) / 2) / (2 K_p(b))\]

where x > 0, p is a real number and b > 0([1]). \(K_p\) is the modified Bessel function of second kind of order p (scipy.special.kv).

The probability density above is defined in the "standardized" form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, geninvgauss.pdf(x, p, b, loc, scale) is identically equivalent to geninvgauss.pdf(y, p, b) / 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.

The inverse Gaussian distribution stats.invgauss(mu) is a special case of geninvgauss with p = -1/2, b = 1 / mu and scale = mu.

Generating random variates is challenging for this distribution. The implementation is based on [2].

References

[1 ]

O. Barndorff-Nielsen, P. Blaesild, C. Halgreen, "First hitting time models for the generalized inverse gaussian distribution", Stochastic Processes and their Applications 7, pp. 49–54, 1978.

[2 ]

W. Hoermann and J. Leydold, "Generating generalized inverse Gaussian random variates", Statistics and Computing, 24(4), p. 547–557, 2014.

Examples

>>> importnumpyasnp
>>> fromscipy.statsimport geninvgauss
>>> importmatplotlib.pyplotasplt
>>> fig, ax = plt.subplots(1, 1)

Get the support:

>>> p, b = 2.3, 1.5
>>> lb, ub = geninvgauss.support(p, b)

Calculate the first four moments:

>>> mean, var, skew, kurt = geninvgauss.stats(p, b, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

>>> vals = geninvgauss.ppf([0.001, 0.5, 0.999], p, b)
>>> np.allclose([0.001, 0.5, 0.999], geninvgauss.cdf(vals, p, b))
True

Generate random numbers:

>>> r = geninvgauss.rvs(p, b, 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()

../../_images/scipy-stats-geninvgauss-1.png