Vous êtes un professionnel et vous avez besoin d'une formation ? Calcul scientifique
avec Python Voir le programme détaillé

Module « scipy.special »

Fonction huber - module scipy.special

Signature de la fonction huber

def huber(*args, **kwargs) 

Description

help(scipy.special.huber)

huber(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

huber(delta, r, out=None)

Huber loss function.

.. math:: \text{huber}(\delta, r) = \begin{cases} \infty & \delta < 0  \\
          \frac{1}{2}r^2 & 0 \le \delta, | r | \le \delta \\
          \delta ( |r| - \frac{1}{2}\delta ) & \text{otherwise} \end{cases}

Parameters
----------
delta : ndarray
    Input array, indicating the quadratic vs. linear loss changepoint.
r : ndarray
    Input array, possibly representing residuals.
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    The computed Huber loss function values.

See Also
--------
pseudo_huber : smooth approximation of this function

Notes
-----
`huber` is useful as a loss function in robust statistics or machine
learning to reduce the influence of outliers as compared to the common
squared error loss, residuals with a magnitude higher than `delta` are
not squared [1]_.

Typically, `r` represents residuals, the difference
between a model prediction and data. Then, for :math:`|r|\leq\delta`,
`huber` resembles the squared error and for :math:`|r|>\delta` the
absolute error. This way, the Huber loss often achieves
a fast convergence in model fitting for small residuals like the squared
error loss function and still reduces the influence of outliers
(:math:`|r|>\delta`) like the absolute error loss. As :math:`\delta` is
the cutoff between squared and absolute error regimes, it has
to be tuned carefully for each problem. `huber` is also
convex, making it suitable for gradient based optimization.

.. versionadded:: 0.15.0

References
----------
.. [1] Peter Huber. "Robust Estimation of a Location Parameter",
       1964. Annals of Statistics. 53 (1): 73 - 101.

Examples
--------
Import all necessary modules.

>>> import numpy as np
>>> from scipy.special import huber
>>> import matplotlib.pyplot as plt

Compute the function for ``delta=1`` at ``r=2``

>>> huber(1., 2.)
1.5

Compute the function for different `delta` by providing a NumPy array or
list for `delta`.

>>> huber([1., 3., 5.], 4.)
array([3.5, 7.5, 8. ])

Compute the function at different points by providing a NumPy array or
list for `r`.

>>> huber(2., np.array([1., 1.5, 3.]))
array([0.5  , 1.125, 4.   ])

The function can be calculated for different `delta` and `r` by
providing arrays for both with compatible shapes for broadcasting.

>>> r = np.array([1., 2.5, 8., 10.])
>>> deltas = np.array([[1.], [5.], [9.]])
>>> print(r.shape, deltas.shape)
(4,) (3, 1)

>>> huber(deltas, r)
array([[ 0.5  ,  2.   ,  7.5  ,  9.5  ],
       [ 0.5  ,  3.125, 27.5  , 37.5  ],
       [ 0.5  ,  3.125, 32.   , 49.5  ]])

Plot the function for different `delta`.

>>> x = np.linspace(-4, 4, 500)
>>> deltas = [1, 2, 3]
>>> linestyles = ["dashed", "dotted", "dashdot"]
>>> fig, ax = plt.subplots()
>>> combined_plot_parameters = list(zip(deltas, linestyles))
>>> for delta, style in combined_plot_parameters:
...     ax.plot(x, huber(delta, x), label=fr"$\delta={delta}$", ls=style)
>>> ax.legend(loc="upper center")
>>> ax.set_xlabel("$x$")
>>> ax.set_title(r"Huber loss function $h_{\delta}(x)$")
>>> ax.set_xlim(-4, 4)
>>> ax.set_ylim(0, 8)
>>> plt.show()

Vous êtes un professionnel et vous avez besoin d'une formation ? Programmation Python
Les fondamentaux Voir le programme détaillé