Module « scipy.stats »

Fonction levy_stable - module scipy.stats

Signature de la fonction levy_stable

def levy_stable(*args, **kwds) 

Description

levy_stable.__doc__

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(n, alpha, beta, loc=0, scale=1)
        Non-central moment of order n
    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(alpha, alpha, beta, loc=0, scale=1)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    See Also
    --------
    levy, levy_l

    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:

    .. math::

        \Phi = \begin{cases}
                \tan \left({\frac {\pi \alpha }{2}}\right)&\alpha \neq 1\\
                -{\frac {2}{\pi }}\log |t|&\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.

    For evaluation of pdf we use either Zolotarev :math:`S_0` parameterization with integration,
    direct integration of standard parameterization of characteristic function or FFT of
    characteristic function. If set to other than None and if number of points is greater than
    ``levy_stable.pdf_fft_min_points_threshold`` (defaults to None) we use FFT otherwise we use one
    of the other methods.

    The default method is 'best' which uses Zolotarev's method if alpha = 1 and integration of
    characteristic function otherwise. The default method can be changed by setting
    ``levy_stable.pdf_default_method`` to either 'zolotarev', 'quadrature' or 'best'.

    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 a value that covers the
    input range * 4). Setting ``pdf_fft_n_points_two_power`` to 16 should be sufficiently accurate
    in most cases at the expense of CPU time.

    For evaluation of cdf we use Zolatarev :math:`S_0` parameterization with integration or integral of
    the pdf FFT interpolated spline. The settings affecting FFT calculation are the same as
    for pdf calculation. Setting the threshold to ``None`` (default) will disable FFT. 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; so it's best that ``pdf_fft_min_points_threshold`` is left unset.

    .. warning::

        For pdf calculations implementation of Zolatarev is unstable for values where alpha = 1 and
        beta != 0. In this case the quadrature method is recommended. FFT calculation is also
        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.
    Specifically, ``levy_stable.pdf(x, alpha, beta, loc, scale)`` is identically
    equivalent to ``levy_stable.pdf(y, alpha, beta) / 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.

    References
    ----------
    .. [MC] McCulloch, J., 1986. Simple consistent estimators of stable distribution parameters.
       Communications in Statistics - Simulation and Computation 15, 11091136.
    .. [MS] Mittnik, S.T. Rachev, T. Doganoglu, D. Chenyao, 1999. Maximum likelihood estimation
       of stable Paretian models, Mathematical and Computer Modelling, Volume 29, Issue 10,
       1999, Pages 275-293.
    .. [BS] Borak, S., Hardle, W., Rafal, W. 2005. Stable distributions, Economic Risk.

    Examples
    --------
    >>> 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, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()