SavitzkyGolay fi的Scipy实现

Question

我在看scipy cookbook implementation of the Savitzky-Golay algorithm：

#!python
def savitzky_golay(y, window_size, order, deriv=0, rate=1):
    r"""Smooth (and optionally differentiate) data with a Savitzky-Golay filter.
    The Savitzky-Golay filter removes high frequency noise from data.
    It has the advantage of preserving the original shape and
    features of the signal better than other types of filtering
    approaches, such as moving averages techniques.
    Parameters
    ----------
    y : array_like, shape (N,)
        the values of the time history of the signal.
    window_size : int
        the length of the window. Must be an odd integer number.
    order : int
        the order of the polynomial used in the filtering.
        Must be less then `window_size` - 1.
    deriv: int
        the order of the derivative to compute (default = 0 means only smoothing)
    Returns
    -------
    ys : ndarray, shape (N)
        the smoothed signal (or it's n-th derivative).
    Notes
    -----
    The Savitzky-Golay is a type of low-pass filter, particularly
    suited for smoothing noisy data. The main idea behind this
    approach is to make for each point a least-square fit with a
    polynomial of high order over a odd-sized window centered at
    the point.
    Examples
    --------
    t = np.linspace(-4, 4, 500)
    y = np.exp( -t**2 ) + np.random.normal(0, 0.05, t.shape)
    ysg = savitzky_golay(y, window_size=31, order=4)
    import matplotlib.pyplot as plt
    plt.plot(t, y, label='Noisy signal')
    plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal')
    plt.plot(t, ysg, 'r', label='Filtered signal')
    plt.legend()
    plt.show()
    References
    ----------
    .. [1] A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of
       Data by Simplified Least Squares Procedures. Analytical
       Chemistry, 1964, 36 (8), pp 1627-1639.
    .. [2] Numerical Recipes 3rd Edition: The Art of Scientific Computing
       W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery
       Cambridge University Press ISBN-13: 9780521880688
    """
    import numpy as np
    from math import factorial

    try:
        window_size = np.abs(np.int(window_size))
        order = np.abs(np.int(order))
    except ValueError, msg:
        raise ValueError("window_size and order have to be of type int")
    if window_size % 2 != 1 or window_size < 1:
        raise TypeError("window_size size must be a positive odd number")
    if window_size < order + 2:
        raise TypeError("window_size is too small for the polynomials order")
    order_range = range(order+1)
    half_window = (window_size -1) // 2
    # precompute coefficients
    b = np.mat([[k**i for i in order_range] for k in range(-half_window, half_window+1)])
    m = np.linalg.pinv(b).A[deriv] * rate**deriv * factorial(deriv)
    # pad the signal at the extremes with
    # values taken from the signal itself
    firstvals = y[0] - np.abs( y[1:half_window+1][::-1] - y[0] )
    lastvals = y[-1] + np.abs(y[-half_window-1:-1][::-1] - y[-1])
    y = np.concatenate((firstvals, y, lastvals))
    return np.convolve( m[::-1], y, mode='valid')

这是让我困惑的部分：

^{pr2}$

我知道我们需要“pad”y，因为否则第一个window_size/2点将被排除，但是我看不到从y[0]中减去某个特定值的绝对差y[0]。在

我不认为绝对值应该在那里，否则，趋势会在水平方向上反映出来，如果开始上升，趋势会垂直反映出来。在

正如@ImportanceOfBeingErnest指出的，这可能是代码中的一个错误，从我链接到的页面的左侧可以看出。在

Answer 1

实际上，这种逻辑是不对的，这可以通过考虑y[0]和y[-1]为0的情况来最好地理解。我相信这是为了实现奇数反射，这样一阶导数在反射点处是连续的。正确的形式是

firstvals = 2*y[0] - y[1:half_window+1][::-1]
lastvals = 2*y[-1] - y[-half_window-1:-1][::-1]

或者，将反转和切片结合在一起

^{pr2}$

只是一个用户应该强调的是这个代码。实际的Scipy implementation of Savitzky-Golay filter完全不同。在