python curve_fit 的灾难性拟合

Question

I need to fit data (x axes: sigma, y axes: Mbh) with an exponential model. This is my code:

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

#define my data
Mbh = np.array([1.8e6,2.5e6,4.5e7,3.7e7,4.4e7,1.5e7,1.4e7,4.1e7, 1.0e9,2.1e8,1.0e8,1.0e8,1.6e7,1.9e8,3.9e7,5.2e8,3.1e8,3.0e8,7.0e7,1.1e8,3.0e9,5.6e7,7.8e7,2.0e9,1.7e8,1.4e7,2.4e8,5.3e8,3.3e8,3.5e6,2.5e9])
sigma = np.array([103,75,160,209,205,151,175,140,230,205,145,206,143,182,130,315,242,225,186,190,375,162,152,385,177,90,234,290,266,67,340])

#define my model to fit 
def Mbh02(alpha, sigma, beta):
    return alpha * np.exp(beta*sigma);

#calculate the fit parameter:
#for second model
popt02, pcov02 = curve_fit(Mbh02, sigma, Mbh, p0=[1, 0.058])
print(f'Parameter of the second function : {popt02}')


sigma_plot = [103,75,160,209,205,151,175,140,230,205,145,206,143,182,130,315,242,225,186,190,375,162,152,385,177,90,234,290,266,67,340]
sigma_plot.sort()
sigma_plot = np.array(sigma_plot)

#plot model with data with
plt.figure(figsize=(6,6))


plt.scatter(sigma, Mbh * 1e-9, marker = '+', color ='black', label = 'Data')
plt.plot(sigma_plot , Mbh02(alpha = popt02[0], sigma = sigma_plot,  beta = popt02[1]) * 1e-9, color='orange', ls ='-', label ='2. fit')

plt.ylabel(r'$M_{BH}$ in $M_\odot *10^9$ unit', fontsize=16)
plt.xlabel(r'$\sigma$', fontsize=16)
# plt.ylim(-1,10)
plt.title('Plot of the black hole mass $M_{BH}$ \nagainst the velocity dispersion $\sigma$ \nfor different elliptical galaxies', fontsize=18)

plt.grid(True)
plt.legend()
plt.show()

and I get the following parameter:

print(popt01) = [16.13278858  0.91788691]

which looks:

If I try to find the parameter manually, and plotting them with:

plt.plot(sigma_plot , (1 * np.exp(0.058 * sigma_plot)) * 1e-9, ls ='--', label ='2. fit manual')

I get the following plot which is much better:

What is the problem? Why is curve_fit not working and giving such parameter?

Answer 1

In the curve_fit documentation , it says

Assumes ydata = f(xdata, *params) + eps

So if you change your function definition so that the x data is first in your function, it will work:

def Mbh02(sigma, alpha, beta):
    return alpha * np.exp(beta*sigma);

# Rest of code

plt.plot(sigma_plot , Mbh02(sigma_plot, *popt02) * 1e-9, color='orange', ls ='-')

Answer 2

Have you tried fitting the log(Mbh) with a linear fit instead of fitting the exp. model directly? This usually gives a lot of stability.

import numpy as np
import matplotlib.pyplot as plt

Mbh = np.array([1.8e6,2.5e6,4.5e7,3.7e7,4.4e7,1.5e7,1.4e7,4.1e7, 1.0e9,2.1e8,1.0e8,1.0e8,1.6e7,1.9e8,3.9e7,5.2e8,3.1e8,3.0e8,7.0e7,1.1e8,3.0e9,5.6e7,7.8e7,2.0e9,1.7e8,1.4e7,2.4e8,5.3e8,3.3e8,3.5e6,2.5e9])
sigma = np.array([103,75,160,209,205,151,175,140,230,205,145,206,143,182,130,315,242,225,186,190,375,162,152,385,177,90,234,290,266,67,340])

plt.figure(2)
plt.plot(sigma,Mbh,'.')

lnMbh= np.log(Mbh)

p = np.polyfit(sigma,lnMbh,1)

plt.plot(sigma, np.exp(np.polyval(p,sigma)),'*')

alpha = np.log(p[0])
beta = p[1]