Module « scipy.stats »

Fonction pearson3 - module scipy.stats

Signature de la fonction pearson3

def pearson3(*args, **kwds) 

Description

pearson3.__doc__

A pearson type III continuous random variable.

    As an instance of the `rv_continuous` class, `pearson3` 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(skew, loc=0, scale=1, size=1, random_state=None)
        Random variates.
    pdf(x, skew, loc=0, scale=1)
        Probability density function.
    logpdf(x, skew, loc=0, scale=1)
        Log of the probability density function.
    cdf(x, skew, loc=0, scale=1)
        Cumulative distribution function.
    logcdf(x, skew, loc=0, scale=1)
        Log of the cumulative distribution function.
    sf(x, skew, loc=0, scale=1)
        Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
    logsf(x, skew, loc=0, scale=1)
        Log of the survival function.
    ppf(q, skew, loc=0, scale=1)
        Percent point function (inverse of ``cdf`` --- percentiles).
    isf(q, skew, loc=0, scale=1)
        Inverse survival function (inverse of ``sf``).
    moment(n, skew, loc=0, scale=1)
        Non-central moment of order n
    stats(skew, loc=0, scale=1, moments='mv')
        Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
    entropy(skew, 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=(skew,), 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(skew, loc=0, scale=1)
        Median of the distribution.
    mean(skew, loc=0, scale=1)
        Mean of the distribution.
    var(skew, loc=0, scale=1)
        Variance of the distribution.
    std(skew, loc=0, scale=1)
        Standard deviation of the distribution.
    interval(alpha, skew, loc=0, scale=1)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    Notes
    -----
    The probability density function for `pearson3` is:

    .. math::

        f(x, \kappa) = \frac{|\beta|}{\Gamma(\alpha)}
                       (\beta (x - \zeta))^{\alpha - 1}
                       \exp(-\beta (x - \zeta))

    where:

    .. math::

            \beta = \frac{2}{\kappa}

            \alpha = \beta^2 = \frac{4}{\kappa^2}

            \zeta = -\frac{\alpha}{\beta} = -\beta

    :math:`\Gamma` is the gamma function (`scipy.special.gamma`).
    Pass the skew :math:`\kappa` into `pearson3` as the shape parameter
    ``skew``.

    The probability density above is defined in the "standardized" form. To shift
    and/or scale the distribution use the ``loc`` and ``scale`` parameters.
    Specifically, ``pearson3.pdf(x, skew, loc, scale)`` is identically
    equivalent to ``pearson3.pdf(y, skew) / 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 pearson3
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four moments:
    
    >>> skew = 0.1
    >>> mean, var, skew, kurt = pearson3.stats(skew, moments='mvsk')
    
    Display the probability density function (``pdf``):
    
    >>> x = np.linspace(pearson3.ppf(0.01, skew),
    ...                 pearson3.ppf(0.99, skew), 100)
    >>> ax.plot(x, pearson3.pdf(x, skew),
    ...        'r-', lw=5, alpha=0.6, label='pearson3 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 = pearson3(skew)
    >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
    
    Check accuracy of ``cdf`` and ``ppf``:
    
    >>> vals = pearson3.ppf([0.001, 0.5, 0.999], skew)
    >>> np.allclose([0.001, 0.5, 0.999], pearson3.cdf(vals, skew))
    True
    
    Generate random numbers:
    
    >>> r = pearson3.rvs(skew, size=1000)
    
    And compare the histogram:
    
    >>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()
    

    References
    ----------
    R.W. Vogel and D.E. McMartin, "Probability Plot Goodness-of-Fit and
    Skewness Estimation Procedures for the Pearson Type 3 Distribution", Water
    Resources Research, Vol.27, 3149-3158 (1991).

    L.R. Salvosa, "Tables of Pearson's Type III Function", Ann. Math. Statist.,
    Vol.1, 191-198 (1930).

    "Using Modern Computing Tools to Fit the Pearson Type III Distribution to
    Aviation Loads Data", Office of Aviation Research (2003).