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scipy.stats.

# weightedtau

scipy.stats.weightedtau(x, y, rank=True, weigher=None, additive=True, *, axis=None, nan_policy='propagate', keepdims=False)

  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 [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 \(\tau_\mathrm h\): 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, 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 \(r\) is mapped to weight \(1/(r+1)\).

    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.

    axis int or None, default: None

      If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If `None`, the input will be raveled before computing the statistic.

    nan_policy{‘propagate’, ‘omit’, ‘raise’}

      Defines how to handle input NaNs.

      - `propagate`: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

      - `omit`: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

      - `raise`: if a NaN is present, a `ValueError` will be raised.

    keepdims bool, default: False

      If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

  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).

  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 \(O(n \log n)\), mergesort-based algorithm [1] that is a weighted extension of Knight’s algorithm for Kendall’s \(\tau\) [2]. It can compute Shieh’s weighted \(\tau\) [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.

  Added in version 0.19.0.

  Beginning in SciPy 1.9, `np.matrix` inputs (not recommended for new code) are converted to `np.ndarray` before the calculation is performed. In this case, the output will be a scalar or `np.ndarray` of appropriate shape rather than a 2D `np.matrix`. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or `np.ndarray` rather than a masked array with `mask=False`.

  Array API Standard Support

  `weightedtau` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ⛔
  PyTorch | ⛔ | ⛔
  JAX | ⛔ | ⛔
  Dask | ⛔ | n/a

  > See Support for the array API standard for more information.

  References

  [1] (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.

  [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)