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scipy.stats.

# expectile

scipy.stats.expectile(a, alpha=0.5, *, weights=None, axis=None, nan_policy='propagate', keepdims=False)

  Compute the expectile at the specified level.

  Expectiles are a generalization of the expectation in the same way as quantiles are a generalization of the median. The expectile at level alpha = 0.5 is the mean (average). See Notes for more details.

  Parameters:
    a array_like

      Array containing numbers whose expectile is desired.

    alpha float, default: 0.5

      The level of the expectile; e.g., `alpha=0.5` gives the mean.

    weights array_like, optional

      An array of weights associated with the values in a. The weights must be broadcastable to the same shape as a. Default is None, which gives each value a weight of 1.0. An integer valued weight element acts like repeating the corresponding observation in a that many times. See Notes for more details.

    axis int or None, default: 0

      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:
    expectile ndarray

      The empirical expectile at level `alpha`.

  See also

  `numpy.mean`

    Arithmetic average

  `numpy.quantile`

    Quantile

  Notes

  In general, the expectile at level \(\alpha\) of a random variable \(X\) with cumulative distribution function (CDF) \(F\) is given by the unique solution \(t\) of:

  \[\alpha E((X - t)_+) = (1 - \alpha) E((t - X)_+) \,.\]

  Here, \((x)_+ = \max(0, x)\) is the positive part of \(x\). This equation can be equivalently written as:

  \[\alpha \int_t^\infty (x - t)\mathrm{d}F(x) = (1 - \alpha) \int_{-\infty}^t (t - x)\mathrm{d}F(x) \,.\]

  The empirical expectile at level \(\alpha\) (`alpha`) of a sample \(a_i\) (the array a) is defined by plugging in the empirical CDF of a. Given sample or case weights \(w\) (the array weights), it reads \(F_a(x) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{a_i \leq x}\) with indicator function \(1_{A}\). This leads to the definition of the empirical expectile at level `alpha` as the unique solution \(t\) of:

  \[\alpha \sum_{i=1}^n w_i (a_i - t)_+ = (1 - \alpha) \sum_{i=1}^n w_i (t - a_i)_+ \,.\]

  For \(\alpha=0.5\), this simplifies to the weighted average. Furthermore, the larger \(\alpha\), the larger the value of the expectile.

  As a final remark, the expectile at level \(\alpha\) can also be written as a minimization problem. One often used choice is

  \[\operatorname{argmin}_t E(\lvert 1_{t\geq X} - \alpha\rvert(t - X)^2) \,.\]

  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

  `expectile` 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

  `expectile` also accepts MArrays backed by the backends indicated above; masked values will be treated as though they were not present.

  See Support for the array API standard for more information.

  References

  [1]

  W. K. Newey and J. L. Powell (1987), “Asymmetric Least Squares Estimation and Testing,” Econometrica, 55, 819-847.

  [2]

  T. Gneiting (2009). “Making and Evaluating Point Forecasts,” Journal of the American Statistical Association, 106, 746 - 762. DOI:10.48550/arXiv.0912.0902

  Examples

      >>> import numpy as np
      >>> from scipy.stats import expectile
      >>> a = [1, 4, 2, -1]
      >>> expectile(a, alpha=0.5) == np.mean(a)
      True
      >>> expectile(a, alpha=0.2)
      0.42857142857142855
      >>> expectile(a, alpha=0.8)
      2.5714285714285716
      >>> weights = [1, 3, 1, 1]
      >>> expectile(a, alpha=0.8, weights=weights)
      3.3333333333333335