Module « scipy.stats »

Fonction pearsonr - module scipy.stats

Signature de la fonction pearsonr

def pearsonr(x, y) 

Description

pearsonr.__doc__

    Pearson correlation coefficient and p-value for testing non-correlation.

    The Pearson correlation coefficient [1]_ measures the linear relationship
    between two datasets.  The calculation of the p-value relies on the
    assumption that each dataset is normally distributed.  (See Kowalski [3]_
    for a discussion of the effects of non-normality of the input on the
    distribution of the correlation coefficient.)  Like other correlation
    coefficients, this one varies between -1 and +1 with 0 implying no
    correlation. Correlations of -1 or +1 imply an exact linear relationship.
    Positive correlations imply that as x increases, so does y. Negative
    correlations imply that as x increases, y decreases.

    The p-value roughly indicates the probability of an uncorrelated system
    producing datasets that have a Pearson correlation at least as extreme
    as the one computed from these datasets.

    Parameters
    ----------
    x : (N,) array_like
        Input array.
    y : (N,) array_like
        Input array.

    Returns
    -------
    r : float
        Pearson's correlation coefficient.
    p-value : float
        Two-tailed p-value.

    Warns
    -----
    PearsonRConstantInputWarning
        Raised if an input is a constant array.  The correlation coefficient
        is not defined in this case, so ``np.nan`` is returned.

    PearsonRNearConstantInputWarning
        Raised if an input is "nearly" constant.  The array ``x`` is considered
        nearly constant if ``norm(x - mean(x)) < 1e-13 * abs(mean(x))``.
        Numerical errors in the calculation ``x - mean(x)`` in this case might
        result in an inaccurate calculation of r.

    See Also
    --------
    spearmanr : Spearman rank-order correlation coefficient.
    kendalltau : Kendall's tau, a correlation measure for ordinal data.

    Notes
    -----
    The correlation coefficient is calculated as follows:

    .. math::

        r = \frac{\sum (x - m_x) (y - m_y)}
                 {\sqrt{\sum (x - m_x)^2 \sum (y - m_y)^2}}

    where :math:`m_x` is the mean of the vector :math:`x` and :math:`m_y` is
    the mean of the vector :math:`y`.

    Under the assumption that :math:`x` and :math:`m_y` are drawn from
    independent normal distributions (so the population correlation coefficient
    is 0), the probability density function of the sample correlation
    coefficient :math:`r` is ([1]_, [2]_):

    .. math::

        f(r) = \frac{{(1-r^2)}^{n/2-2}}{\mathrm{B}(\frac{1}{2},\frac{n}{2}-1)}

    where n is the number of samples, and B is the beta function.  This
    is sometimes referred to as the exact distribution of r.  This is
    the distribution that is used in `pearsonr` to compute the p-value.
    The distribution is a beta distribution on the interval [-1, 1],
    with equal shape parameters a = b = n/2 - 1.  In terms of SciPy's
    implementation of the beta distribution, the distribution of r is::

        dist = scipy.stats.beta(n/2 - 1, n/2 - 1, loc=-1, scale=2)

    The p-value returned by `pearsonr` is a two-sided p-value.  For a
    given sample with correlation coefficient r, the p-value is
    the probability that abs(r') of a random sample x' and y' drawn from
    the population with zero correlation would be greater than or equal
    to abs(r).  In terms of the object ``dist`` shown above, the p-value
    for a given r and length n can be computed as::

        p = 2*dist.cdf(-abs(r))

    When n is 2, the above continuous distribution is not well-defined.
    One can interpret the limit of the beta distribution as the shape
    parameters a and b approach a = b = 0 as a discrete distribution with
    equal probability masses at r = 1 and r = -1.  More directly, one
    can observe that, given the data x = [x1, x2] and y = [y1, y2], and
    assuming x1 != x2 and y1 != y2, the only possible values for r are 1
    and -1.  Because abs(r') for any sample x' and y' with length 2 will
    be 1, the two-sided p-value for a sample of length 2 is always 1.

    References
    ----------
    .. [1] "Pearson correlation coefficient", Wikipedia,
           https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
    .. [2] Student, "Probable error of a correlation coefficient",
           Biometrika, Volume 6, Issue 2-3, 1 September 1908, pp. 302-310.
    .. [3] C. J. Kowalski, "On the Effects of Non-Normality on the Distribution
           of the Sample Product-Moment Correlation Coefficient"
           Journal of the Royal Statistical Society. Series C (Applied
           Statistics), Vol. 21, No. 1 (1972), pp. 1-12.

    Examples
    --------
    >>> from scipy import stats
    >>> a = np.array([0, 0, 0, 1, 1, 1, 1])
    >>> b = np.arange(7)
    >>> stats.pearsonr(a, b)
    (0.8660254037844386, 0.011724811003954649)

    >>> stats.pearsonr([1, 2, 3, 4, 5], [10, 9, 2.5, 6, 4])
    (-0.7426106572325057, 0.1505558088534455)