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def genhyperbolic(*args, **kwds)
A generalized hyperbolic continuous random variable.
As an instance of the `rv_continuous` class, `genhyperbolic` 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, a, b, loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, p, a, b, loc=0, scale=1)
Probability density function.
logpdf(x, p, a, b, loc=0, scale=1)
Log of the probability density function.
cdf(x, p, a, b, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, p, a, b, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, p, a, b, loc=0, scale=1)
Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, p, a, b, loc=0, scale=1)
Log of the survival function.
ppf(q, p, a, b, loc=0, scale=1)
Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, p, a, b, loc=0, scale=1)
Inverse survival function (inverse of ``sf``).
moment(order, p, a, b, loc=0, scale=1)
Non-central moment of the specified order.
stats(p, a, b, loc=0, scale=1, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(p, a, b, loc=0, scale=1)
(Differential) entropy of the RV.
fit(data)
Parameter estimates for generic data.
See `scipy.stats.rv_continuous.fit <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#scipy.stats.rv_continuous.fit>`__ for detailed documentation of the
keyword arguments.
expect(func, args=(p, 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(p, a, b, loc=0, scale=1)
Median of the distribution.
mean(p, a, b, loc=0, scale=1)
Mean of the distribution.
var(p, a, b, loc=0, scale=1)
Variance of the distribution.
std(p, a, b, loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, p, a, b, loc=0, scale=1)
Confidence interval with equal areas around the median.
See Also
--------
t, norminvgauss, geninvgauss, laplace, cauchy
Notes
-----
The probability density function for `genhyperbolic` is:
.. math::
f(x, p, a, b) =
\frac{(a^2 - b^2)^{p/2}}
{\sqrt{2\pi}a^{p-1/2}
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}}
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.kv`)
`genhyperbolic` takes ``p`` as a tail parameter,
``a`` as a shape parameter,
``b`` as a skewness parameter.
The probability density above is defined in the "standardized" form. To shift
and/or scale the distribution use the ``loc`` and ``scale`` parameters.
Specifically, ``genhyperbolic.pdf(x, p, a, b, loc, scale)`` is identically
equivalent to ``genhyperbolic.pdf(y, p, 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.
The original parameterization of the Generalized Hyperbolic Distribution
is found in [1]_ as follows
.. math::
f(x, \lambda, \alpha, \beta, \delta, \mu) =
\frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)}
e^{\beta (x - \mu)} \times \frac{K_{\lambda - 1/2}
(\alpha \sqrt{\delta^2 + (x - \mu)^2})}
{(\sqrt{\delta^2 + (x - \mu)^2} / \alpha)^{1/2 - \lambda}}
for :math:`x \in ( - \infty; \infty)`,
:math:`\gamma := \sqrt{\alpha^2 - \beta^2}`,
:math:`\lambda, \mu \in ( - \infty; \infty)`,
:math:`\delta \ge 0, |\beta| < \alpha` if :math:`\lambda \ge 0`,
:math:`\delta > 0, |\beta| \le \alpha` if :math:`\lambda < 0`.
The location-scale-based parameterization implemented in
SciPy is based on [2]_, where :math:`a = \alpha\delta`,
:math:`b = \beta\delta`, :math:`p = \lambda`,
:math:`scale=\delta` and :math:`loc=\mu`
Moments are implemented based on [3]_ and [4]_.
For the distributions that are a special case such as Student's t,
it is not recommended to rely on the implementation of genhyperbolic.
To avoid potential numerical problems and for performance reasons,
the methods of the specific distributions should be used.
References
----------
.. [1] O. Barndorff-Nielsen, "Hyperbolic Distributions and Distributions
on Hyperbolae", Scandinavian Journal of Statistics, Vol. 5(3),
pp. 151-157, 1978. https://www.jstor.org/stable/4615705
.. [2] Eberlein E., Prause K. (2002) The Generalized Hyperbolic Model:
Financial Derivatives and Risk Measures. In: Geman H., Madan D.,
Pliska S.R., Vorst T. (eds) Mathematical Finance - Bachelier
Congress 2000. Springer Finance. Springer, Berlin, Heidelberg.
:doi:`10.1007/978-3-662-12429-1_12`
.. [3] Scott, David J, Würtz, Diethelm, Dong, Christine and Tran,
Thanh Tam, (2009), Moments of the generalized hyperbolic
distribution, MPRA Paper, University Library of Munich, Germany,
https://EconPapers.repec.org/RePEc:pra:mprapa:19081.
.. [4] E. Eberlein and E. A. von Hammerstein. Generalized hyperbolic
and inverse Gaussian distributions: Limiting cases and approximation
of processes. FDM Preprint 80, April 2003. University of Freiburg.
https://freidok.uni-freiburg.de/fedora/objects/freidok:7974/datastreams/FILE1/content
Examples
--------
>>> import numpy as np
>>> from scipy.stats import genhyperbolic
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> p, a, b = 0.5, 1.5, -0.5
>>> mean, var, skew, kurt = genhyperbolic.stats(p, a, b, moments='mvsk')
Display the probability density function (``pdf``):
>>> x = np.linspace(genhyperbolic.ppf(0.01, p, a, b),
... genhyperbolic.ppf(0.99, p, a, b), 100)
>>> ax.plot(x, genhyperbolic.pdf(x, p, a, b),
... 'r-', lw=5, alpha=0.6, label='genhyperbolic 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 = genhyperbolic(p, a, b)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = genhyperbolic.ppf([0.001, 0.5, 0.999], p, a, b)
>>> np.allclose([0.001, 0.5, 0.999], genhyperbolic.cdf(vals, p, a, b))
True
Generate random numbers:
>>> r = genhyperbolic.rvs(p, 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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