在python中平滑曲线，边界没有错误？

Question

考虑与两个numpy数组x和y相关联的以下曲线：

如何在python中正确地平滑它，xmax附近没有问题？ （如果我应用高斯滤波器，曲线最后会上升）

数据在这里（两列）：http://lite4.framapad.org/p/xqhpGJpV5R

Answer 1

如果您的所有数据在日志空间中都在缓慢修改，我会执行以下操作：

1. 在线性刻度上对对数数据进行严重下采样
2. caclulate a smoothing spline
3. 转换回线性比例

e.g：

import numpy as np
from scipy.interpolate import interp1d, splrep, splev
import pylab

x = np.log10(x)
y = np.log10(y)

ip = interp1d(x,y)
xi = np.linspace(x.min(),x.max(),10)
yi = ip(xi)

tcl = splrep(xi,yi,s=1)
xs = np.linspace(x.min(), x.max(), 100)
ys = splev(xs, tcl)

xi = np.power(10,xi)
yi = np.power(10,yi)
xs = np.power(10,xs)
ys = np.power(10,ys)

f = pylab.figure()
pl = f.add_subplot(111)
pl.loglog(aset.x,aset.y,alpha=0.4)
pl.loglog(xi,yi,'go--',linewidth=1, label='linear')
pl.loglog(xs,ys,'r-',linewidth=1, label='spline')
pl.legend(loc=0)
f.show()

这会产生：

Answer 2

最简单的方法是在过滤之前去掉信号。你所看到的边缘效应主要是由于信号不是静止的（即它有一个斜率）。

首先，我们来说明问题：

import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import gaussian_filter1d

x, y = np.loadtxt('spectrum.dat').T

# Smooth with a guassian filter
smooth = gaussian_filter1d(y, 10)

fig, ax = plt.subplots()
ax.loglog(x, y, color='black')
ax.loglog(x, smooth, color='red')
plt.show()

哎哟！边缘效应在数据的末端（右手大小）特别糟糕，因为这是斜率最陡的地方。如果你在开始时有一个更陡峭的坡度，那么你也会看到更强的边缘效应。

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好消息是，有很多方法可以解决这个问题。 @ChristianK。的答案显示了如何使用平滑样条来有效地执行低通滤波器。我将演示如何使用其他一些信号处理方法来完成同样的事情。哪个“最好”都取决于您的需求。平滑样条线是直截了当的。使用“发烧友”信号处理方法可以精确控制滤除的频率。

您的数据在对数日志空间中看起来像抛物线，所以让我们在对数日志空间中使用二阶多项式对其进行去趋势，然后应用滤波器。

作为一个简单的例子：

import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import gaussian_filter1d

x, y = np.loadtxt('spectrum.dat').T

# Let's detrend by fitting a second-order polynomial in log space
# (Note that your data looks like a parabola in log-log space.)
logx, logy = np.log(x), np.log(y)
model = np.polyfit(logx, logy, 2)
trend = np.polyval(model, logx)

# Smooth with a guassian filter
smooth = gaussian_filter1d(logy - trend, 10)

# Add the trend back in and convert back to linear space
smooth = np.exp(smooth + trend)

fig, ax = plt.subplots()
ax.loglog(x, y, color='black')
ax.loglog(x, smooth, color='red')
plt.show()

请注意，我们仍然有一些边缘效果。这是因为我使用的高斯滤波器引起相移。如果我们真的想要变幻想，我们可以解决问题，然后使用零相位滤波器来进一步减少边缘效应。

import numpy as np
import matplotlib.pyplot as plt
import scipy.signal as signal

def main():
    x, y = np.loadtxt('spectrum.dat').T

    logx, logy = np.log(x), np.log(y)
    smooth_log = detrend_zero_phase(logx, logy)
    smooth = np.exp(smooth_log)

    fig, ax = plt.subplots()
    ax.loglog(x, y, 'k-')
    ax.loglog(x, smooth, 'r-')
    plt.show()

def zero_phase(y):
    # Low-pass filter...
    b, a = signal.butter(3, 0.05)

    # Filtfilt applies the filter twice to avoid phase shifts.
    return signal.filtfilt(b, a, y)

def detrend_zero_phase(x, y):
    # Fit a second order polynomial (Can't just use scipy.signal.detrend here,
    # because we need to know what the trend is to add it back in.)
    model = np.polyfit(x, y, 2)
    trend = np.polyval(model, x)

    # Apply a zero-phase filter to the detrended curve.
    smooth = zero_phase(y - trend)

    # Add the trend back in
    return smooth + trend

main()