# scipy.special.huber#

scipy.special.huber(delta, r, out=None) = <ufunc 'huber'>#

Huber loss function.

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

Input array, indicating the quadratic vs. linear loss changepoint.

rndarray

Input array, possibly representing residuals.

outndarray, optional

Optional output array for the function values

Returns:
scalar or ndarray

The computed Huber loss function values.

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 $$|r|\leq\delta$$, huber resembles the squared error and for $$|r|>\delta$$ 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 ($$|r|>\delta$$) like the absolute error loss. As $$\delta$$ 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.

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()