使用scipy.optimize curve_fit拟合指数曲线时遇协方差无法估计问题

问题现象 #

使用scipy.optimize.curve_fit拟合实验数据时触发警告：

optimizeWarning: Covariance of the parameters could not be estimated
  warnings.warn('Covariance of the parameters could not be estimated',

输出结果显示拟合参数popt与初始猜测值完全一致，协方差矩阵pcov全部为无穷大：

Popt
[ 1.00000000e+00  7.80761109e-04 -1.00000000e+02]
Pcov
[[inf inf inf]
 [inf inf inf]
 [inf inf inf]]

显然curve_fit未能完成有效拟合，无法计算参数协方差，得到的参数并非最优解。数据可视化见配图，对应原始数据存储在CSV文件中。

拟合代码 #

# using curve_fit
from scipy.optimize import curve_fit
import numpy as np

# exponential curve
def _1_func(x,  a0,b0,beta):
    """
    calculates the exponential curve shifted by bo and scaled by a0
    beta is exponential 
    """
    y = a0 * np.exp( beta * x ) + b0
    return y

# the code to fit
# initial guess for exp fitting params

numpoints = spectrum_one.shape[0]

x = F[1:numpoints] # zero element is not used
y = np.absolute(spectrum_one[1:numpoints])/signal_size

# making an initial guess
a0 = 1
b0 = y.mean()
beta = -100  

p0 = [a0, b0, beta]

popt, pcov = curve_fit(_1_func, x, y, p0=p0)
perr = np.sqrt(np.diag(pcov)) # errors

print('Popt')
print(popt)
print('Pcov')
print(pcov)

疑问 #

• 为何会出现协方差无法估计的错误？
• 是否需要对数据进行缩放处理？

内容的提问来源于stack exchange，提问作者twistfire