def to_categorical(y, num_classes=None):
    """Converts a class vector (integers) to binary class matrix.
    E.g. for use with categorical_crossentropy.
    # Arguments
        y: class vector to be converted into a matrix
            (integers from 0 to num_classes).
        num_classes: total number of classes.
    # Returns
        A binary matrix representation of the input.
    """
    y = np.array(y, dtype='int').ravel()
    if not num_classes:
        num_classes = np.max(y) + 1
    n = y.shape[0]
    categorical = np.zeros((n, num_classes))
    categorical[np.arange(n), y] = 1
    return categorical


def normalize(x, axis=-1, order=2):
    """Normalizes a Numpy array.
    # Arguments
        x: Numpy array to normalize.
        axis: axis along which to normalize.
        order: Normalization order (e.g. 2 for L2 norm).
    # Returns
        A normalized copy of the array.
    """
    l2 = np.atleast_1d(np.linalg.norm(x, order, axis))
    l2[l2 == 0] = 1
    return x / np.expand_dims(l2, axis)