scipy.stats.norminvgauss

scipy.stats.norminvgauss = <scipy.stats._continuous_distns.norminvgauss_gen object>

A Normal Inverse Gaussian continuous random variable.

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

Notes

The probability density function for norminvgauss is:

\[f(x, a, b) = \frac{a \, K_1(a \sqrt{1 + x^2})}{\pi \sqrt{1 + x^2}} \, \exp(\sqrt{a^2 - b^2} + b x)\]

where \(x\) is a real number, the parameter \(a\) is the tail heaviness and \(b\) is the asymmetry parameter satisfying \(a > 0\) and \(|b| <= a\). \(K_1\) is the modified Bessel function of second kind (scipy.special.k1).

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

A normal inverse Gaussian random variable Y with parameters a and b can be expressed as a normal mean-variance mixture: Y = b * V + sqrt(V) * X where X is norm(0,1) and V is invgauss(mu=1/sqrt(a**2 - b**2)). This representation is used to generate random variates.

Another common parametrization of the distribution (see Equation 2.1 in [2]) is given by the following expression of the pdf:

\[g(x, \alpha, \beta, \delta, \mu) = \frac{\alpha\delta K_1\left(\alpha\sqrt{\delta^2 + (x - \mu)^2}\right)} {\pi \sqrt{\delta^2 + (x - \mu)^2}} \, e^{\delta \sqrt{\alpha^2 - \beta^2} + \beta (x - \mu)}\]

In SciPy, this corresponds to a = alpha * delta, b = beta * delta, loc = mu, scale=delta.

References

[1]

O. Barndorff-Nielsen, “Hyperbolic Distributions and Distributions on Hyperbolae”, Scandinavian Journal of Statistics, Vol. 5(3), pp. 151-157, 1978.

[2]

O. Barndorff-Nielsen, “Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling”, Scandinavian Journal of Statistics, Vol. 24, pp. 1-13, 1997.

Examples

Try it in your browser!

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

Calculate the first four moments:

>>> a, b = 1.25, 0.5
>>> mean, var, skew, kurt = norminvgauss.stats(a, b, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = norminvgauss.rvs(a, 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()

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