# scipy.stats.weightedtau#

Compute a weighted version of Kendall’s $$\tau$$.

The weighted $$\tau$$ is a weighted version of Kendall’s $$\tau$$ 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, $$\tau_\mathrm h$$, which has been shown to provide the best balance between important and unimportant elements .

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 $$\tau_\mathrm h$$: an exchange between elements with rank $$r$$ and $$s$$ (starting from zero) has weight $$1/(r+1) + 1/(s+1)$$.

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 $$\tau$$ 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, yarray_like

Arrays of scores, of the same shape. If arrays are not 1-D, they will be flattened to 1-D.

rankarray_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).

weighercallable, 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 $$r$$ is mapped to weight $$1/(r+1)$$.

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:

statisticfloat

The weighted $$\tau$$ correlation index.

pvaluefloat

Presently np.nan, as the null distribution of the statistic is unknown (even in the additive hyperbolic case).

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 $$O(n \log n)$$, mergesort-based algorithm  that is a weighted extension of Knight’s algorithm for Kendall’s $$\tau$$ . It can compute Shieh’s weighted $$\tau$$  between rankings without ties (i.e., permutations) by setting additive and rank to False, as the definition given in  is a generalization of Shieh’s.

NaNs are considered the smallest possible score.

New in version 0.19.0.

References

 (1,2,3)

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.



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.



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)