使用curve_fit在python中拟合数据

Question

I'm trying to fit data in python to obtain the coefficients for the best fit.

The equation I need to fit is:

Vs = a*(qt^b) (fs^c) (ov^d)

Whereby I have the data for qt, fs and ov and need to obtain the values for a,b,c,d.

The code I'm using is:

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

qt = [10867.073,    8074.986,   2208.366,   3066.566,   2945.326,   4795.766,   2249.813,   2018.3]
fs = [229.6,    17.4,   5.3,    0.1,    0.1,    0.1,    0.1,    0.1]     
ov = [19.159,   29.054, 37.620, 44.854, 51.721, 58.755, 65.622, 72.492]  
Vs = [149.787,  125.3962,   133.927,    110.047,    149.787,    137.809,    201.506,    154.925] 
d = [1.018, 1.518,  2.0179, 2.517,  3.017,  3.517,  4.018,  4.52]
 
def func(a, b, c, d):   
return a*qt**b*fs**c*ov**d
 
popt, pcov = curve_fit(func, Vs, d) 
print(popt[0], popt[1], popt[2])
plt.plot(Vs, d, 'ro',label="Original Data") 
plt.plot(Vs, func(Vs,*popt), label="Fitted Curve") 
plt.gca().invert_yaxis() 
plt.show()

Which produces the following output (Significant figures cut by me):

-0.333528 -0.1413381 -0.3553966

I was hoping to get something more like below where the data it fitted but it hasn't been done perfectly (note the one below is just an example and is not correct).

Answer 1

The main hitch is the graphical representation of the result. What is drawn has no signifiance: Even with perfect data and with perfect fitting the points will appear verry scattered. This is misleading.

Better draw (Vs from computation) divided by (Vs from data) and compare to 1 which should be the exact value if the fitting was perfect.

Note that your problem is a simple linear regression in logarithm scale.

ln(Vs) = A + b * ln(qt) + c * ln(fs) + d * ln(ov)

Linear regression straight away gives A, b, c, d and a = exp(A).