Setting an exponential function through two data points with scipy curve_fit

Question

I want to set an exponential function y=x ** pwwith a constant pwso that it matches two data points. Function scipy curve_fitshould optimize adj1and adj2. I tried with the code below, but could not get it to work. A curve does not pass through data points. How can i fix this?

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

Here is the solution. I think lmfit is a good alternative to scipy to fit the curve.

from lmfit import minimize, Parameters, Parameter, report_fit
import numpy as np

# create data to be fitted
xf = [0.5,0.85] # two given datapoints to which the exponential function with power pw should fit
yf = [0.02,4]

# define objective function: returns the array to be minimized
def fcn2min(params, x, data):
    pw = params['pw'].value
    adj1 = params['adj1'].value
    adj2 = params['adj2'].value

    model = adj1 * np.power(x + adj2, pw)
    return model - data

pw=2

# create a set of Parameters
params = Parameters()
params.add('pw',   value= pw, vary=False)
params.add('adj1', value= 1)
params.add('adj2', value= 1)


# do fit, here with leastsq model
result = minimize(fcn2min, params, args=(xf, yf))

# calculate final result
final = yf + result.residual

# write error report
report_fit(result.params)
adj1=result.params['adj1']
adj2=result.params['adj2']

# try to plot results
x = np.linspace(0, 1, 100)
y = adj1 * np.power(x + adj2, pw)

import pylab
pylab.plot(xf, yf, 'ko')
pylab.plot(x, y, 'r')
pylab.show()

Answer 2

, : y1 = b (x1 + a) ^ p y2 = b (x2 + a) ^ p a 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()

, , .

Answer 3

, curve_fit . p0 , , ( p0 = [1., 1.]). , np.round .

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()