python SciPy curve_fit with np.exp返回pcov=inf

Question

我正在尝试使用scipy.optimize.curve\u fit优化指数拟合。但结果并不好。我的代码是：

def func(x, a, b, c):
  return a * np.exp(-b * x) + c

# xdata and data is obtain from another dataframe and their type is nparray

xdata =[36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70 ,71,72]
ydata = [4,4,4,6,6,13,22,22,26,28,38,48,55,65,65,92,112,134,171,210,267,307,353,436,669,669,818,1029,1219,1405,1617,1791,2032,2032,2182,2298,2389]

popt, pcov = curve_fit(func, xdata, ydata)
plt.plot(xdata, func(xdata, *popt), 'r-', label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))

plt.scatter(xdata, ydata, s=1)
plt.show()

然后我得到了这样的结果：

enter image description here

结果表明：

pcov = [[inf inf inf] [inf inf inf] [inf inf inf]]
popt = [1  1  611.83784]

我不知道如何使我的曲线拟合得很好。你能帮我吗？谢谢大家!

Answer 1

对指数函数进行拟合非常困难，因为指数的微小变化会导致结果的巨大差异。优化器在多个数量级上进行优化，与曲线上方的误差相比，原点附近的误差加权不相等

处理此问题的最简单方法是使用转换将指数数据转换为直线：

y' = np.log(y)

然后，您可以简单地使用numpy的polyfit函数拟合一条直线，而不需要使用更高级（更慢）的曲线拟合。如果愿意，可以将数据转换回线性空间进行分析。在这里，我编辑了您的代码，使用np.polyfit进行拟合，您可以看到拟合是合理的

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

# def func(x, a, b, c):
#   return a * np.exp(-b * x) + c

# xdata and data is obtain from another dataframe and their type is nparray

xdata = np.array([36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70 ,71,72])
ydata = np.array([4,4,4,6,6,13,22,22,26,28,38,48,55,65,65,92,112,134,171,210,267,307,353,436,669,669,818,1029,1219,1405,1617,1791,2032,2032,2182,2298,2389])

# popt, pcov = curve_fit(func, xdata, ydata)
# plt.plot(xdata, func(xdata, *popt), 'r-', label='fit: a=%5.3f, b=%5.3f, c=%5.3f' % tuple(popt))

# Fit a line (deg=1)
P, pcov = np.polyfit(xdata, np.log(ydata), deg=1, cov=True)
print(pcov)
plt.scatter(xdata, ydata, s=1)
plt.plot(xdata, np.exp(P[0]*xdata + P[1]), 'r-')

plt.legend()
plt.show()