Savitzky-Golay滤波器的Scipy实现 [英] Scipy implementation of Savitzky-Golay filter
问题描述
我正在查看 Savitzky-Golay算法的科学食谱实现:
#!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')
这是令我困惑的部分:
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))
我知道我们需要'填充'y，因为否则将排除前一个window_size/2点，但是我看不到从
I get that we need to 'pad' y, since otherwise the first window_size/2 points would be excluded, but I don't see the point of subtracting a particular value's absolute difference with y[0] from y[0].
我认为绝对值不应该存在，否则，如果趋势从增加开始，则水平反映，如果从减少开始，则垂直趋势.
I don't think the absolute value should be there, as otherwise, the trend gets mirrored horizontally if it starts by increasing, and vertically if it starts by decreasing.
如@ImportanceOfBeingErnest所指出的那样，这可能是代码中的错别字，可以通过查看我链接到的页面中图的左侧来看到.
As pointed it out by @ImportanceOfBeingErnest, this may be a typo in the code, as can be seen by looking at the left hand side of the plot in the page I linked to.
推荐答案
实际上，这种逻辑是不正确的，可以通过考虑y [0]和y [-1]为0的情况来最好地看出这一点.相信这样做的目的是实现奇数反射，以便一阶导数在反射点处是连续的.正确的格式是
Indeed, this logic isn't right, which can be best seen by considering the case of y[0] and y[-1] being 0. I believe the intent was to achieve odd reflection, so that the first derivative would be continuous at the reflection point. The correct form for that is
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，
or, combining reversing and slicing in one step,
firstvals = 2*y[0] - y[half_window:0:-1]
lastvals = 2*y[-1] - y[-2:-half_window-2:-1]
我应该强调，这只是用户提供的一些代码.实际的 Savitzky-Golay过滤器的Scipy实现是完全不同.
I should emphasize this is just some code contributed by a user. The actual Scipy implementation of Savitzky-Golay filter is entirely different.
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