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Module « scipy.stats »

Fonction weightedtau - module scipy.stats

Signature de la fonction weightedtau

def weightedtau(x, y, rank=True, weigher=None, additive=True) 

Description

help(scipy.stats.weightedtau)

Compute a weighted version of Kendall's :math:`\tau`.

The weighted :math:`\tau` is a weighted version of Kendall's
:math:`\tau` in which exchanges of high weight are more influential than
exchanges of low weight. The default parameters compute the additive
hyperbolic version of the index, :math:`\tau_\mathrm h`, which has
been shown to provide the best balance between important and
unimportant elements [1]_.

The weighting is defined by means of a rank array, which assigns a
nonnegative rank to each element (higher importance ranks being
associated with smaller values, e.g., 0 is the highest possible rank),
and a weigher function, which assigns a weight based on the rank to
each element. The weight of an exchange is then the sum or the product
of the weights of the ranks of the exchanged elements. The default
parameters compute :math:`\tau_\mathrm h`: an exchange between
elements with rank :math:`r` and :math:`s` (starting from zero) has
weight :math:`1/(r+1) + 1/(s+1)`.

Specifying a rank array is meaningful only if you have in mind an
external criterion of importance. If, as it usually happens, you do
not have in mind a specific rank, the weighted :math:`\tau` is
defined by averaging the values obtained using the decreasing
lexicographical rank by (`x`, `y`) and by (`y`, `x`). This is the
behavior with default parameters. Note that the convention used
here for ranking (lower values imply higher importance) is opposite
to that used by other SciPy statistical functions.

Parameters
----------
x, y : array_like
    Arrays of scores, of the same shape. If arrays are not 1-D, they will
    be flattened to 1-D.
rank : array_like of ints or bool, optional
    A nonnegative rank assigned to each element. If it is None, the
    decreasing lexicographical rank by (`x`, `y`) will be used: elements of
    higher rank will be those with larger `x`-values, using `y`-values to
    break ties (in particular, swapping `x` and `y` will give a different
    result). If it is False, the element indices will be used
    directly as ranks. The default is True, in which case this
    function returns the average of the values obtained using the
    decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`).
weigher : callable, optional
    The weigher function. Must map nonnegative integers (zero
    representing the most important element) to a nonnegative weight.
    The default, None, provides hyperbolic weighing, that is,
    rank :math:`r` is mapped to weight :math:`1/(r+1)`.
additive : bool, optional
    If True, the weight of an exchange is computed by adding the
    weights of the ranks of the exchanged elements; otherwise, the weights
    are multiplied. The default is True.

Returns
-------
res: SignificanceResult
    An object containing attributes:

    statistic : float
       The weighted :math:`\tau` correlation index.
    pvalue : float
       Presently ``np.nan``, as the null distribution of the statistic is
       unknown (even in the additive hyperbolic case).

See Also
--------
kendalltau : Calculates Kendall's tau.
spearmanr : Calculates a Spearman rank-order correlation coefficient.
theilslopes : Computes the Theil-Sen estimator for a set of points (x, y).

Notes
-----
This function uses an :math:`O(n \log n)`, mergesort-based algorithm
[1]_ that is a weighted extension of Knight's algorithm for Kendall's
:math:`\tau` [2]_. It can compute Shieh's weighted :math:`\tau` [3]_
between rankings without ties (i.e., permutations) by setting
`additive` and `rank` to False, as the definition given in [1]_ is a
generalization of Shieh's.

NaNs are considered the smallest possible score.

.. versionadded:: 0.19.0

References
----------
.. [1] Sebastiano Vigna, "A weighted correlation index for rankings with
       ties", Proceedings of the 24th international conference on World
       Wide Web, pp. 1166-1176, ACM, 2015.
.. [2] W.R. Knight, "A Computer Method for Calculating Kendall's Tau with
       Ungrouped Data", Journal of the American Statistical Association,
       Vol. 61, No. 314, Part 1, pp. 436-439, 1966.
.. [3] Grace S. Shieh. "A weighted Kendall's tau statistic", Statistics &
       Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998.

Examples
--------
>>> import numpy as np
>>> from scipy import stats
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> res = stats.weightedtau(x, y)
>>> res.statistic
-0.56694968153682723
>>> res.pvalue
nan
>>> res = stats.weightedtau(x, y, additive=False)
>>> res.statistic
-0.62205716951801038

NaNs are considered the smallest possible score:

>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, np.nan]
>>> res = stats.weightedtau(x, y)
>>> res.statistic
-0.56694968153682723

This is exactly Kendall's tau:

>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> res = stats.weightedtau(x, y, weigher=lambda x: 1)
>>> res.statistic
-0.47140452079103173

>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> stats.weightedtau(x, y, rank=None)
SignificanceResult(statistic=-0.4157652301037516, pvalue=nan)
>>> stats.weightedtau(y, x, rank=None)
SignificanceResult(statistic=-0.7181341329699028, pvalue=nan)

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