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Generalized Hyperbolic Distribution

doc/source/tutorial/stats/continuous_genhyperbolic.rst

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.. _continuous-genhyperbolic:

Generalized Hyperbolic Distribution

The Generalized Hyperbolic Distribution is defined as the normal variance-mean mixture with Generalized Inverse Gaussian distribution as the mixing distribution. The "hyperbolic" characterization refers to the fact that the shape of the log-probability distribution can be described as a hyperbola. Hyperbolic distributions are sometime referred to as semi-fat tailed because their probability density decrease slower than "sub-hyperbolic" distributions (e.g. normal distribution, whose log-probability decreases quadratically), but faster than other "extreme value" distributions (e.g. pareto distribution, whose log-probability decreases logarithmically).

Functions

Different parameterizations exist in the literature; SciPy implements the "4th parametrization" in Prause (1999).

.. math:: :nowrap:

\begin{eqnarray*}
    f(x, p, a, b) & = &
    \frac{(a^2 - b^2)^{p/2}}
    {\sqrt{2\pi}a^{p-0.5}
    K_p\Big(\sqrt{a^2 - b^2}\Big)}
    e^{bx} \times \frac{K_{p - 1/2}
    (a \sqrt{1 + x^2})}
    {(\sqrt{1 + x^2})^{1/2 - p}}
\end{eqnarray*}

for:

  • :math:x, p \in ( - \infty; \infty)
  • :math:|b| < a if :math:p \ge 0
  • :math:|b| \le a if :math:p < 0
  • :math:K_{p}(.) denotes the modified Bessel function of the second kind and order :math:p (scipy.special.kn)

The probability density above is defined in the "standardized" form. To shift and/or scale the distribution use the :math:\text{loc} and :math:\text{scale} parameters. Specifically, :math:f(x, p, a, b, \text{loc}, \text{scale}) is identically equivalent to :math:\frac{1}{\text{scale}}f(y, p, a, b) with :math:y = \frac{1}{\text{scale}}(x - \text{loc}).

This parameterization derives from the original :math:(\lambda, \alpha, \beta, \delta, \mu) parameterization in Barndorff (1978) by setting:

  • :math:\lambda = p
  • :math:\alpha = \frac{a}{\delta} = \frac{\hat{\alpha}}{\delta}
  • :math:\beta = \frac{b}{\delta} = \frac{\hat{\beta}}{\delta}
  • :math:\delta = \text{scale}
  • :math:\mu = \text{location}

Random variates for the scipy.stats.genhyperbolic can be efficiently sampled from the above-mentioned normal variance-mean mixture where scipy.stats.geninvgauss is parametrized as :math:GIG\Big(p = p, b = \sqrt{\hat{\alpha}^2 - \hat{\beta}^2}, \text{loc} = \text{location}, \text{scale} = \frac{1}{\sqrt{\hat{\alpha}^2 - \hat{\beta}^2}}\Big) so that: :math:GH(p, \hat{\alpha}, \hat{\beta}) = \hat{\beta} \cdot GIG + \sqrt{GIG} \cdot N(0,1)

The "generalized" characterization suggests the fact that this distribution is a superclass of several other probability distribution, for instance:

  • :math:f(p = -\nu/2, a = 0, b = 0, \text{loc} = 0, \text{scale} = \sqrt{\nu}) has a Student's t-distribution (scipy.stats.t) with :math:\nu degrees of freedom.
  • :math:f(p = 1, a = \hat{\alpha}, b = \hat{\beta}, \text{loc} = \mu, \text{scale} = \delta) has a Hyperbolic Distribution.
  • :math:f(p = - 1/2, a = \hat{\alpha}, b = \hat{\beta}, \text{loc} = \mu, \text{scale} = \delta) has a Normal Inverse Gaussian Distribution (scipy.stats.norminvgauss).
  • :math:f(p = 1, a = \delta, b = 0, loc = \mu, \text{scale} = \delta) has a Laplace Distribution (scipy.stats.laplace) for :math:\delta \rightarrow 0

Examples

It is useful to understand how the parameters affect the shape of the distribution. While it is fairly straightforward to interpret the meaning of :math:b as the skewness, understanding the difference between :math:a and :math:p is not as obvious, as both affect the kurtosis of the distribution. :math:a can be interpreted as the speed of the decay of the probability density (where :math:a > 1 the asymptotic decay is faster than :math:log_e and vice versa) or - equivalently - as the slope of the log-probability hyperbola asymptote (where :math:a > 1 decay is faster than :math:|1| and vice versa). :math:p can be seen as the width of the shoulders of the probability density distribution (where :math:p < 1 results in narrow shoulders and vice versa) or - equivalently - as the shape of the log-probability hyperbola, which is convex for :math:p < 1 and concave otherwise.

.. code-block:: python

import numpy as np
from matplotlib import pyplot as plt
from scipy import stats

p, a, b, loc, scale = 1, 1, 0, 0, 1
x = np.linspace(-10, 10, 100)

# plot GH for different values of p
plt.figure(0)
plt.title("Generalized Hyperbolic | -10 < p < 10")
plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        label = 'GH(p=1, a=1, b=0, loc=0, scale=1)')
plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'red', alpha = 0.5, label='GH(p>1, a=1, b=0, loc=0, scale=1)')
[plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'red', alpha = 0.1) for p in np.linspace(1, 10, 10)]
plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'blue', alpha = 0.5, label='GH(p<1, a=1, b=0, loc=0, scale=1)')
[plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'blue', alpha = 0.1) for p in np.linspace(-10, 1, 10)]
plt.plot(x, stats.norm.pdf(x, loc, scale), label = 'N(loc=0, scale=1)')
plt.plot(x, stats.laplace.pdf(x, loc, scale), label = 'Laplace(loc=0, scale=1)')
plt.plot(x, stats.pareto.pdf(x+1, 1, loc, scale), label = 'Pareto(a=1, loc=0, scale=1)')
plt.ylim(1e-15, 1e2)
plt.yscale('log')
plt.legend(bbox_to_anchor=(1.1, 1))
plt.subplots_adjust(right=0.5)

# plot GH for different values of a
plt.figure(1)
plt.title("Generalized Hyperbolic | 0 < a < 10")
plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        label = 'GH(p=1, a=1, b=0, loc=0, scale=1)')
plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'blue', alpha = 0.5, label='GH(p=1, a>1, b=0, loc=0, scale=1)')
[plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'blue', alpha = 0.1) for a in np.linspace(1, 10, 10)]
plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'red', alpha = 0.5, label='GH(p=1, 0<a<1, b=0, loc=0, scale=1)')
[plt.plot(x, stats.genhyperbolic.pdf(x, p, a, b, loc, scale),
        color = 'red', alpha = 0.1) for a in np.linspace(0, 1, 10)]
plt.plot(x, stats.norm.pdf(x, loc, scale),  label = 'N(loc=0, scale=1)')
plt.plot(x, stats.laplace.pdf(x, loc, scale), label = 'Laplace(loc=0, scale=1)')
plt.plot(x, stats.pareto.pdf(x+1, 1, loc, scale), label = 'Pareto(a=1, loc=0, scale=1)')
plt.ylim(1e-15, 1e2)
plt.yscale('log')
plt.legend(bbox_to_anchor=(1.1, 1))
plt.subplots_adjust(right=0.5)

plt.show()

References

Implementation: scipy.stats.genhyperbolic