scipy.stats.norminvgauss — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.norminvgauss = <scipy.stats._continuous_distns.norminvgauss_gen object>[source]#

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.

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 withy = (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 isinvgauss(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

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()

Methods