python - 即使提供了良好的猜测，scipy curve_fit 也根本无法正确拟合？

Question

我正在尝试将指数修改的高斯函数拟合到某些数据。数据在顶部提供。

我有以下代码:

import matplotlib.pyplot as plt
import numpy as np
from scipy.optimize import curve_fit
from scipy.special import erfc

bins = [-46.82455, -46.41738, -46.01021, -45.60304, -45.19587, -44.7887,  -44.38153,
 -43.97436, -43.56719, -43.16002, -42.75285, -42.34568, -41.93851, -41.53134,
 -41.12417, -40.717,   -40.30983, -39.90266, -39.49549, -39.08832, -38.68115,
 -38.27398, -37.86681, -37.45964, -37.05247, -36.6453,  -36.23813, -35.83096,
 -35.42379, -35.01662, -34.60945, -34.20228, -33.79511, -33.38794, -32.98077,
 -32.5736,  -32.16643, -31.75926, -31.35209, -30.94492, -30.53775, -30.13058,
 -29.72341, -29.31624, -28.90907, -28.5019,  -28.09473, -27.68756, -27.28039,
 -26.87322, -26.46605, -26.05888, -25.65171, -25.24454, -24.83737, -24.4302,
 -24.02303, -23.61586, -23.20869, -22.80152, -22.39435, -21.98718, -21.58001,
 -21.17284, -20.76567, -20.3585,  -19.95133, -19.54416, -19.13699, -18.72982,
 -18.32265, -17.91548, -17.50831, -17.10114, -16.69397, -16.2868,  -15.87963,
 -15.47246, -15.06529, -14.65812, -14.25095, -13.84378, -13.43661, -13.02944,
 -12.62227, -12.2151,  -11.80793, -11.40076, -10.99359, -10.58642, -10.17925,
  -9.77208,  -9.36491,  -8.95774,  -8.55057,  -8.1434,   -7.73623,  -7.32906,
  -6.92189,  -6.51472,  -6.10755,  -5.70038,  -5.29321,  -4.88604,  -4.47887,
  -4.0717,   -3.66453,  -3.25736,  -2.85019,  -2.44302,  -2.03585,  -1.62868,
  -1.22151,  -0.81434,  -0.40717,   0.0,   0.40717,   0.81434,   1.22151,
   1.62868,   2.03585,   2.44302,   2.85019,   3.25736,   3.66453,   4.0717,
   4.47887,   4.88604,   5.29321,   5.70038,   6.10755,   6.51472,   6.92189,
   7.32906,   7.73623,   8.1434,    8.55057,   8.95774,   9.36491,   9.77208,
  10.17925,  10.58642,  10.99359,  11.40076,  11.80793,  12.2151,   12.62227,
  13.02944,  13.43661,  13.84378,  14.25095,  14.65812,  15.06529,  15.47246,
  15.87963,  16.2868,   16.69397,  17.10114,  17.50831,  17.91548,  18.32265,
  18.72982,  19.13699,  19.54416,  19.95133,  20.3585,   20.76567,  21.17284,
  21.58001,  21.98718,  22.39435,  22.80152,  23.20869,  23.61586,  24.02303,
  24.4302,   24.83737,  25.24454,  25.65171,  26.05888,  26.46605,  26.87322,
  27.28039,  27.68756,  28.09473,  28.5019,   28.90907,  29.31624,  29.72341,
  30.13058,  30.53775,  30.94492,  31.35209,  31.75926,  32.16643,  32.5736,
  32.98077,  33.38794,  33.79511,  34.20228,  34.60945,  35.01662,  35.42379,
  35.83096,  36.23813,  36.6453,   37.05247,  37.45964,  37.86681,  38.27398,
  38.68115,  39.08832,  39.49549,  39.90266,  40.30983,  40.717,    41.12417,
  41.53134,  41.93851,  42.34568,  42.75285,  43.16002,  43.56719,  43.97436,
  44.38153,  44.7887,   45.19587,  45.60304,  46.01021,  46.41738]

counts = [0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 9.82318271e-04, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 9.82318271e-04, 0.00000000e+00, 9.82318271e-04, 0.00000000e+00,
 0.00000000e+00, 9.82318271e-04, 0.00000000e+00, 9.82318271e-04,
 9.82318271e-04, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 9.82318271e-04, 0.00000000e+00, 0.00000000e+00, 9.82318271e-04,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 9.82318271e-04, 0.00000000e+00,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 9.82318271e-04,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 9.82318271e-04,
 0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
 0.00000000e+00, 9.82318271e-04, 0.00000000e+00, 0.00000000e+00,
 1.96463654e-03, 9.82318271e-04, 9.82318271e-04, 0.00000000e+00,
 9.82318271e-04, 1.96463654e-03, 9.82318271e-04, 9.82318271e-04,
 7.85854617e-03, 9.82318271e-03, 1.27701375e-02, 1.47347741e-02,
 1.76817289e-02, 2.75049116e-02, 3.14341847e-02, 4.32220039e-02,
 5.79567780e-02, 6.77799607e-02, 9.43025540e-02, 1.29666012e-01,
 1.48330059e-01, 1.87622790e-01, 2.07269155e-01, 2.54420432e-01,
 3.00589391e-01, 3.33005894e-01, 4.03732809e-01, 4.72495088e-01,
 5.22593320e-01, 5.99214145e-01, 6.34577603e-01, 7.04322200e-01,
 8.18271120e-01, 8.58546169e-01, 9.26326130e-01, 9.65618861e-01,
 9.35166994e-01, 9.76424361e-01, 9.39096267e-01, 1.00000000e+00,
 9.67583497e-01, 9.36149312e-01, 9.13555992e-01, 9.38113949e-01,
 8.35952849e-01, 8.31041257e-01, 8.33988212e-01, 7.54420432e-01,
 7.17092338e-01, 6.12966601e-01, 6.22789784e-01, 5.37328094e-01,
 4.76424361e-01, 4.35166994e-01, 3.89980354e-01, 3.53634578e-01,
 3.47740668e-01, 3.51669941e-01, 2.87819253e-01, 2.67190570e-01,
 3.04518664e-01, 2.60314342e-01, 2.70137525e-01, 2.65225933e-01,
 2.65225933e-01, 2.67190570e-01, 3.06483301e-01, 2.72102161e-01,
 2.61296660e-01, 2.57367387e-01, 2.45579568e-01, 2.67190570e-01,
 2.25933202e-01, 2.28880157e-01, 2.21021611e-01, 2.23968566e-01,
 1.95481336e-01, 1.80746562e-01, 1.56188605e-01, 1.53241650e-01,
 1.23772102e-01, 1.47347741e-01, 1.26719057e-01, 8.93909627e-02,
 7.17092338e-02, 8.84086444e-02, 6.28683694e-02, 6.97445972e-02,
 6.58153242e-02, 4.61689587e-02, 4.51866405e-02, 4.61689587e-02,
 4.32220039e-02, 4.61689587e-02, 4.22396857e-02, 3.92927308e-02,
 3.43811395e-02, 3.14341847e-02, 2.45579568e-02, 3.53634578e-02,
 2.94695481e-02, 3.53634578e-02, 2.75049116e-02, 2.16110020e-02,
 3.14341847e-02, 3.63457760e-02, 1.96463654e-02, 2.94695481e-02,
 2.35756385e-02, 2.84872299e-02, 2.35756385e-02, 2.55402750e-02,
 2.06286837e-02, 1.86640472e-02, 3.33988212e-02, 1.96463654e-02,
 2.35756385e-02, 1.86640472e-02, 1.96463654e-02, 2.25933202e-02,
 2.45579568e-02, 2.84872299e-02, 1.96463654e-02, 1.96463654e-02,
 1.86640472e-02, 1.76817289e-02, 1.47347741e-02, 1.96463654e-02,
 2.65225933e-02, 2.06286837e-02, 2.45579568e-02, 2.06286837e-02,
 2.35756385e-02, 1.47347741e-02, 2.06286837e-02, 6.87622790e-03,
 1.27701375e-02, 1.86640472e-02, 1.66994106e-02, 2.35756385e-02,
 1.17878193e-02, 1.96463654e-02, 9.82318271e-03, 1.47347741e-02,
 1.08055010e-02, 1.17878193e-02, 1.27701375e-02, 1.27701375e-02,
 1.37524558e-02, 1.08055010e-02, 1.27701375e-02, 5.89390963e-03,
 1.08055010e-02, 9.82318271e-03]

def exp_mod_gauss(x, m, s, l):
    y = 0.5*l*np.exp(0.5*l*(2*m+l*s*s-2*x))*erfc((m+l*s*s-x)/(np.sqrt(2)*s))
    return y
    #l=Lambda, s=Sigma, m=Mu

bins=np.asarray(bins, dtype='float')
counts=np.asarray(counts, dtype='float')

popt, pcov = curve_fit(exp_mod_gauss, bins, counts, p0=[-3.5,2.8736,0.1548])
fitted_func = exp_mod_gauss(bins, popt[0], popt[1], popt[2])
#fitted_func = exp_mod_gauss(bins, -3.5, 2.8736, 0.1548) #used for manual example
plt.plot(bins, counts, 'o', markersize=1) #plot actual counts
plt.plot(bins, fitted_func/max(fitted_func), '-') #plot fitted func/scaled
plt.show()

如果我按照编写的 scipy 运行代码，我会得到以下结果:

这显然不是很好。

但是，如果我注释掉使用 curve_fit 参数的fitting_func 行并使用我在初始猜测中提供的参数(-3.5、2.876、0.1548)，那么我会得到以下结果:

因此，即使我为 curve_fit 提供了最初的猜测(这基本上是我正在寻找的答案)，它也会失败。我通过在 Matlab 中执行完全相同的过程获得了良好的拟合参数，但我不想使用 Matlab。我想使用Python。

有人知道这里发生了什么吗？

非常感谢。

Answer 1

事实证明，该拟合需要 EMG 功能具有额外的自由度才能发挥作用。这样它就可以扩展到数据。如果我将肌电图函数修改为:

def exp_mod_gauss(x, b, m, s, l):
    y = b*(0.5*l*np.exp(0.5*l*(2*m+l*s*s-2*x))*erfc((m+l*s*s-x)/(np.sqrt(2)*s)))
    return y
    #l=Lambda, s=Sigma, m=Mu, #b=scaling

因此添加 b 项来缩放峰值对其进行了排序。我提供了猜测 [1,-1,1,0] 现在符合我预期的各种数据。