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Module « scipy.special »

Fonction softmax - module scipy.special

Signature de la fonction softmax

def softmax(x, axis=None) 

Description

help(scipy.special.softmax)

Compute the softmax function.

The softmax function transforms each element of a collection by
computing the exponential of each element divided by the sum of the
exponentials of all the elements. That is, if `x` is a one-dimensional
numpy array::

    softmax(x) = np.exp(x)/sum(np.exp(x))

Parameters
----------
x : array_like
    Input array.
axis : int or tuple of ints, optional
    Axis to compute values along. Default is None and softmax will be
    computed over the entire array `x`.

Returns
-------
s : ndarray
    An array the same shape as `x`. The result will sum to 1 along the
    specified axis.

Notes
-----
The formula for the softmax function :math:`\sigma(x)` for a vector
:math:`x = \{x_0, x_1, ..., x_{n-1}\}` is

.. math:: \sigma(x)_j = \frac{e^{x_j}}{\sum_k e^{x_k}}

The `softmax` function is the gradient of `logsumexp`.

The implementation uses shifting to avoid overflow. See [1]_ for more
details.

.. versionadded:: 1.2.0

References
----------
.. [1] P. Blanchard, D.J. Higham, N.J. Higham, "Accurately computing the
   log-sum-exp and softmax functions", IMA Journal of Numerical Analysis,
   Vol.41(4), :doi:`10.1093/imanum/draa038`.

Examples
--------
>>> import numpy as np
>>> from scipy.special import softmax
>>> np.set_printoptions(precision=5)

>>> x = np.array([[1, 0.5, 0.2, 3],
...               [1,  -1,   7, 3],
...               [2,  12,  13, 3]])
...

Compute the softmax transformation over the entire array.

>>> m = softmax(x)
>>> m
array([[  4.48309e-06,   2.71913e-06,   2.01438e-06,   3.31258e-05],
       [  4.48309e-06,   6.06720e-07,   1.80861e-03,   3.31258e-05],
       [  1.21863e-05,   2.68421e-01,   7.29644e-01,   3.31258e-05]])

>>> m.sum()
1.0

Compute the softmax transformation along the first axis (i.e., the
columns).

>>> m = softmax(x, axis=0)

>>> m
array([[  2.11942e-01,   1.01300e-05,   2.75394e-06,   3.33333e-01],
       [  2.11942e-01,   2.26030e-06,   2.47262e-03,   3.33333e-01],
       [  5.76117e-01,   9.99988e-01,   9.97525e-01,   3.33333e-01]])

>>> m.sum(axis=0)
array([ 1.,  1.,  1.,  1.])

Compute the softmax transformation along the second axis (i.e., the rows).

>>> m = softmax(x, axis=1)
>>> m
array([[  1.05877e-01,   6.42177e-02,   4.75736e-02,   7.82332e-01],
       [  2.42746e-03,   3.28521e-04,   9.79307e-01,   1.79366e-02],
       [  1.22094e-05,   2.68929e-01,   7.31025e-01,   3.31885e-05]])

>>> m.sum(axis=1)
array([ 1.,  1.,  1.])

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