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def levy_stable(*args, **params)
A Levy-stable continuous random variable.
As an instance of the `rv_continuous` class, `levy_stable` 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(alpha, beta, loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, alpha, beta, loc=0, scale=1)
Probability density function.
logpdf(x, alpha, beta, loc=0, scale=1)
Log of the probability density function.
cdf(x, alpha, beta, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, alpha, beta, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, alpha, beta, loc=0, scale=1)
Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(x, alpha, beta, loc=0, scale=1)
Log of the survival function.
ppf(q, alpha, beta, loc=0, scale=1)
Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, alpha, beta, loc=0, scale=1)
Inverse survival function (inverse of ``sf``).
moment(order, alpha, beta, loc=0, scale=1)
Non-central moment of the specified order.
stats(alpha, beta, loc=0, scale=1, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(alpha, beta, 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=(alpha, beta), 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(alpha, beta, loc=0, scale=1)
Median of the distribution.
mean(alpha, beta, loc=0, scale=1)
Mean of the distribution.
var(alpha, beta, loc=0, scale=1)
Variance of the distribution.
std(alpha, beta, loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, alpha, beta, loc=0, scale=1)
Confidence interval with equal areas around the median.
See Also
--------
levy, levy_l, cauchy, norm
Notes
-----
The distribution for `levy_stable` has characteristic function:
.. math::
\varphi(t, \alpha, \beta, c, \mu) =
e^{it\mu -|ct|^{\alpha}(1-i\beta\operatorname{sign}(t)\Phi(\alpha, t))}
where two different parameterizations are supported. The first :math:`S_1`:
.. math::
\Phi = \begin{cases}
\tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\
-{\frac {2}{\pi }}\log |t|&\alpha =1
\end{cases}
The second :math:`S_0`:
.. math::
\Phi = \begin{cases}
-\tan \left({\frac {\pi \alpha }{2}}\right)(|ct|^{1-\alpha}-1)
&\alpha \neq 1\\
-{\frac {2}{\pi }}\log |ct|&\alpha =1
\end{cases}
The probability density function for `levy_stable` is:
.. math::
f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \varphi(t)e^{-ixt}\,dt
where :math:`-\infty < t < \infty`. This integral does not have a known
closed form.
`levy_stable` generalizes several distributions. Where possible, they
should be used instead. Specifically, when the shape parameters
assume the values in the table below, the corresponding equivalent
distribution should be used.
========= ======== ===========
``alpha`` ``beta`` Equivalent
========= ======== ===========
1/2 -1 `levy_l`
1/2 1 `levy`
1 0 `cauchy`
2 any `norm` (with ``scale=sqrt(2)``)
========= ======== ===========
Evaluation of the pdf uses Nolan's piecewise integration approach with the
Zolotarev :math:`M` parameterization by default. There is also the option
to use direct numerical integration of the standard parameterization of the
characteristic function or to evaluate by taking the FFT of the
characteristic function.
The default method can changed by setting the class variable
``levy_stable.pdf_default_method`` to one of 'piecewise' for Nolan's
approach, 'dni' for direct numerical integration, or 'fft-simpson' for the
FFT based approach. For the sake of backwards compatibility, the methods
'best' and 'zolotarev' are equivalent to 'piecewise' and the method
'quadrature' is equivalent to 'dni'.
The parameterization can be changed by setting the class variable
``levy_stable.parameterization`` to either 'S0' or 'S1'.
The default is 'S1'.
To improve performance of piecewise and direct numerical integration one
can specify ``levy_stable.quad_eps`` (defaults to 1.2e-14). This is used
as both the absolute and relative quadrature tolerance for direct numerical
integration and as the relative quadrature tolerance for the piecewise
method. One can also specify ``levy_stable.piecewise_x_tol_near_zeta``
(defaults to 0.005) for how close x is to zeta before it is considered the
same as x [NO]. The exact check is
``abs(x0 - zeta) < piecewise_x_tol_near_zeta*alpha**(1/alpha)``. One can
also specify ``levy_stable.piecewise_alpha_tol_near_one`` (defaults to
0.005) for how close alpha is to 1 before being considered equal to 1.
To increase accuracy of FFT calculation one can specify
``levy_stable.pdf_fft_grid_spacing`` (defaults to 0.001) and
``pdf_fft_n_points_two_power`` (defaults to None which means a value is
calculated that sufficiently covers the input range).
Further control over FFT calculation is available by setting
``pdf_fft_interpolation_degree`` (defaults to 3) for spline order and
``pdf_fft_interpolation_level`` for determining the number of points to use
in the Newton-Cotes formula when approximating the characteristic function
(considered experimental).
Evaluation of the cdf uses Nolan's piecewise integration approach with the
Zolatarev :math:`S_0` parameterization by default. There is also the option
to evaluate through integration of an interpolated spline of the pdf
calculated by means of the FFT method. The settings affecting FFT
calculation are the same as for pdf calculation. The default cdf method can
be changed by setting ``levy_stable.cdf_default_method`` to either
'piecewise' or 'fft-simpson'. For cdf calculations the Zolatarev method is
superior in accuracy, so FFT is disabled by default.
Fitting estimate uses quantile estimation method in [MC]. MLE estimation of
parameters in fit method uses this quantile estimate initially. Note that
MLE doesn't always converge if using FFT for pdf calculations; this will be
the case if alpha <= 1 where the FFT approach doesn't give good
approximations.
Any non-missing value for the attribute
``levy_stable.pdf_fft_min_points_threshold`` will set
``levy_stable.pdf_default_method`` to 'fft-simpson' if a valid
default method is not otherwise set.
.. warning::
For pdf calculations FFT calculation is considered experimental.
For cdf calculations FFT calculation is considered experimental. Use
Zolatarev's method instead (default).
The probability density above is defined in the "standardized" form. To
shift and/or scale the distribution use the ``loc`` and ``scale``
parameters.
Generally ``levy_stable.pdf(x, alpha, beta, loc, scale)`` is identically
equivalent to ``levy_stable.pdf(y, alpha, beta) / scale`` with
``y = (x - loc) / scale``, except in the ``S1`` parameterization if
``alpha == 1``. In that case ``levy_stable.pdf(x, alpha, beta, loc, scale)``
is identically equivalent to ``levy_stable.pdf(y, alpha, beta) / scale`` with
``y = (x - loc - 2 * beta * scale * np.log(scale) / np.pi) / scale``.
See [NO2]_ Definition 1.8 for more information.
Note that shifting the location of a distribution
does not make it a "noncentral" distribution.
References
----------
.. [MC] McCulloch, J., 1986. Simple consistent estimators of stable
distribution parameters. Communications in Statistics - Simulation and
Computation 15, 11091136.
.. [WZ] Wang, Li and Zhang, Ji-Hong, 2008. Simpson's rule based FFT method
to compute densities of stable distribution.
.. [NO] Nolan, J., 1997. Numerical Calculation of Stable Densities and
distributions Functions.
.. [NO2] Nolan, J., 2018. Stable Distributions: Models for Heavy Tailed
Data.
.. [HO] Hopcraft, K. I., Jakeman, E., Tanner, R. M. J., 1999. Lévy random
walks with fluctuating step number and multiscale behavior.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import levy_stable
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> alpha, beta = 1.8, -0.5
>>> mean, var, skew, kurt = levy_stable.stats(alpha, beta, moments='mvsk')
Display the probability density function (``pdf``):
>>> x = np.linspace(levy_stable.ppf(0.01, alpha, beta),
... levy_stable.ppf(0.99, alpha, beta), 100)
>>> ax.plot(x, levy_stable.pdf(x, alpha, beta),
... 'r-', lw=5, alpha=0.6, label='levy_stable 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 = levy_stable(alpha, beta)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of ``cdf`` and ``ppf``:
>>> vals = levy_stable.ppf([0.001, 0.5, 0.999], alpha, beta)
>>> np.allclose([0.001, 0.5, 0.999], levy_stable.cdf(vals, alpha, beta))
True
Generate random numbers:
>>> r = levy_stable.rvs(alpha, beta, 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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