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Fonction softmax - module scipy.special

Signature de la fonction softmax

def softmax(x, axis=None) 

Description

softmax.__doc__

    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`.

    .. versionadded:: 1.2.0

    Examples
    --------
    >>> 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()
    1.0000000000000002

    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.])