Fitting exponential function through two data points with scipy curve_fit

Question

I want to fit an exponential function y=x ** pw with a constant pw to fit through two datapoints. The scipy curve_fit function should optimise adj1 and adj2. I have tried with the code below but couldn't get it to work. The curve does not go through the datapoints. How can I fix it?

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

def func(x, adj1,adj2):
    return np.round(((x+adj1) ** pw) * adj2, 2)

x = [0.5,0.85] # two given datapoints to which the exponential function with power pw should fit
y = [0.02,4]

pw=15
popt, pcov = curve_fit(func, x, y)

xf=np.linspace(0,1,50)

plt.figure()
plt.plot(x, y, 'ko', label="Original Data")
plt.plot(xf, func(xf, *popt), 'r-', label="Fitted Curve")
plt.show()

Answer 1

If you want to find the two parameters in your objective function from only two data points, this isn't necessarily a problem for least-squares fitting: just solve the simultaneous equations y1 = b(x1+a)^p and y2 = b(x2+a)^p for the parameters a and b:

import numpy as np
import matplotlib.pyplot as plt

def func(x, adj1,adj2):
    return ((x+adj1) ** pw) * adj2

x = [0.5,0.85] # two given datapoints to which the exponential function with power pw should fit
y = [0.02,4]

pw = 15
A = np.exp(np.log(y[0]/y[1])/pw)
a = (x[0] - x[1]*A)/(A-1)
b = y[0]/(x[0]+a)**pw

xf=np.linspace(0,1,50)
plt.figure()
plt.plot(x, y, 'ko', label="Original Data")
plt.plot(xf, func(xf, a, b), 'r-', label="Fitted Curve")
plt.show()

The resulting function will pass through both points exactly, of course.

Answer 2

It's just because the round method is killing curve_fit's ability to search the space. Small perturbations of p0 will always give an equal result, so it stops searching immediatly and will always return whatever you give it as a starting point (by default p0 = [1.,1.]). The solution is to simply remove np.round from your function.

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

def func(x, adj1,adj2):
    return ((x+adj1) ** pw) * adj2

x = [0.5,0.85] # two given datapoints to which the exponential function with power pw should fit
y = [0.02,4]

pw=15
popt, pcov = curve_fit(func, x, y)

xf=np.linspace(0,1,50)

plt.figure()
plt.plot(x, y, 'ko', label="Original Data")
plt.plot(xf, func(xf, *popt), 'r-', label="Fitted Curve")
plt.show()