Returns the mean value of a central approximation of the second derivative

    .. math::

        \\frac{1}{n} \\sum_{i=1,\ldots, n-1}  \\frac{1}{2} (x_{i+2} - 2 \\cdot x_{i+1} + x_i)

    :param x: the time series to calculate the feature of
    :type x: numpy.ndarray
    :return: the value of this feature
    :return type: float

def mean_second_derivative_central(x):
    """
    Returns the mean value of a central approximation of the second derivative

    .. math::

        \\frac{1}{n} \\sum_{i=1,\ldots, n-1}  \\frac{1}{2} (x_{i+2} - 2 \\cdot x_{i+1} + x_i)

    :param x: the time series to calculate the feature of
    :type x: numpy.ndarray
    :return: the value of this feature
    :return type: float
    """

    diff = (_roll(x, 1) - 2 * np.array(x) + _roll(x, -1)) / 2.0
    return np.mean(diff[1:-1])