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Module « scipy.stats »

Fonction expectile - module scipy.stats

Signature de la fonction expectile

def expectile(a, alpha=0.5, *, weights=None) 

Description

help(scipy.stats.expectile)

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.

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 :math:`\alpha` of a random variable
:math:`X` with cumulative distribution function (CDF) :math:`F` is given
by the unique solution :math:`t` of:

.. math::

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

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

.. math::

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

.. math::

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

For :math:`\alpha=0.5`, this simplifies to the weighted average.
Furthermore, the larger :math:`\alpha`, the larger the value of the
expectile.

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

.. math::

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

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]

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