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Fonction kappa4 - module scipy.stats

Signature de la fonction kappa4

def kappa4(*args, **kwds) 

Description

kappa4.__doc__

Kappa 4 parameter distribution.

    As an instance of the `rv_continuous` class, `kappa4` 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(h, k, loc=0, scale=1, size=1, random_state=None)
        Random variates.
    pdf(x, h, k, loc=0, scale=1)
        Probability density function.
    logpdf(x, h, k, loc=0, scale=1)
        Log of the probability density function.
    cdf(x, h, k, loc=0, scale=1)
        Cumulative distribution function.
    logcdf(x, h, k, loc=0, scale=1)
        Log of the cumulative distribution function.
    sf(x, h, k, loc=0, scale=1)
        Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
    logsf(x, h, k, loc=0, scale=1)
        Log of the survival function.
    ppf(q, h, k, loc=0, scale=1)
        Percent point function (inverse of ``cdf`` --- percentiles).
    isf(q, h, k, loc=0, scale=1)
        Inverse survival function (inverse of ``sf``).
    moment(n, h, k, loc=0, scale=1)
        Non-central moment of order n
    stats(h, k, loc=0, scale=1, moments='mv')
        Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
    entropy(h, k, 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=(h, k), 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(h, k, loc=0, scale=1)
        Median of the distribution.
    mean(h, k, loc=0, scale=1)
        Mean of the distribution.
    var(h, k, loc=0, scale=1)
        Variance of the distribution.
    std(h, k, loc=0, scale=1)
        Standard deviation of the distribution.
    interval(alpha, h, k, loc=0, scale=1)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    Notes
    -----
    The probability density function for kappa4 is:

    .. math::

        f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}

    if :math:`h` and :math:`k` are not equal to 0.

    If :math:`h` or :math:`k` are zero then the pdf can be simplified:

    h = 0 and k != 0::

        kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                              exp(-(1.0 - k*x)**(1.0/k))

    h != 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

    h = 0 and k = 0::

        kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

    kappa4 takes :math:`h` and :math:`k` as shape parameters.

    The kappa4 distribution returns other distributions when certain
    :math:`h` and :math:`k` values are used.

    +------+-------------+----------------+------------------+
    | h    | k=0.0       | k=1.0          | -inf<=k<=inf     |
    +======+=============+================+==================+
    | -1.0 | Logistic    |                | Generalized      |
    |      |             |                | Logistic(1)      |
    |      |             |                |                  |
    |      | logistic(x) |                |                  |
    +------+-------------+----------------+------------------+
    |  0.0 | Gumbel      | Reverse        | Generalized      |
    |      |             | Exponential(2) | Extreme Value    |
    |      |             |                |                  |
    |      | gumbel_r(x) |                | genextreme(x, k) |
    +------+-------------+----------------+------------------+
    |  1.0 | Exponential | Uniform        | Generalized      |
    |      |             |                | Pareto           |
    |      |             |                |                  |
    |      | expon(x)    | uniform(x)     | genpareto(x, -k) |
    +------+-------------+----------------+------------------+

    (1) There are at least five generalized logistic distributions.
        Four are described here:
        https://en.wikipedia.org/wiki/Generalized_logistic_distribution
        The "fifth" one is the one kappa4 should match which currently
        isn't implemented in scipy:
        https://en.wikipedia.org/wiki/Talk:Generalized_logistic_distribution
        https://www.mathwave.com/help/easyfit/html/analyses/distributions/gen_logistic.html
    (2) This distribution is currently not in scipy.

    References
    ----------
    J.C. Finney, "Optimization of a Skewed Logistic Distribution With Respect
    to the Kolmogorov-Smirnov Test", A Dissertation Submitted to the Graduate
    Faculty of the Louisiana State University and Agricultural and Mechanical
    College, (August, 2004),
    https://digitalcommons.lsu.edu/gradschool_dissertations/3672

    J.R.M. Hosking, "The four-parameter kappa distribution". IBM J. Res.
    Develop. 38 (3), 25 1-258 (1994).

    B. Kumphon, A. Kaew-Man, P. Seenoi, "A Rainfall Distribution for the Lampao
    Site in the Chi River Basin, Thailand", Journal of Water Resource and
    Protection, vol. 4, 866-869, (2012).
    :doi:`10.4236/jwarp.2012.410101`

    C. Winchester, "On Estimation of the Four-Parameter Kappa Distribution", A
    Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March
    2000).
    http://www.nlc-bnc.ca/obj/s4/f2/dsk2/ftp01/MQ57336.pdf

    The probability density above is defined in the "standardized" form. To shift
    and/or scale the distribution use the ``loc`` and ``scale`` parameters.
    Specifically, ``kappa4.pdf(x, h, k, loc, scale)`` is identically
    equivalent to ``kappa4.pdf(y, h, k) / 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.

    Examples
    --------
    >>> from scipy.stats import kappa4
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four moments:
    
    >>> h, k = 0.1, 0
    >>> mean, var, skew, kurt = kappa4.stats(h, k, moments='mvsk')
    
    Display the probability density function (``pdf``):
    
    >>> x = np.linspace(kappa4.ppf(0.01, h, k),
    ...                 kappa4.ppf(0.99, h, k), 100)
    >>> ax.plot(x, kappa4.pdf(x, h, k),
    ...        'r-', lw=5, alpha=0.6, label='kappa4 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 = kappa4(h, k)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    
    Check accuracy of ``cdf`` and ``ppf``:
    
    >>> vals = kappa4.ppf([0.001, 0.5, 0.999], h, k)
    >>> np.allclose([0.001, 0.5, 0.999], kappa4.cdf(vals, h, k))
    True
    
    Generate random numbers:
    
    >>> r = kappa4.rvs(h, k, size=1000)
    
    And compare the histogram:
    
    >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()