How to fit the following function using curve_fit

Question

I have the following function I need to solve:

np.exp((1-Y)/Y) = np.exp(c) -b*x

I defined the function as:

def function(x, b, c):

    np.exp((1-Y)/Y) = np.exp(c) -b*x
    return y

def function_solve(y, b, c):

    x = (np.exp(c)-np.exp((1-Y)/Y))/b
    return x

then I used:

x_data = [4, 6, 8, 10]

y_data = [0.86, 0.73, 0.53, 0.3] 

popt, pcov = curve_fit(function, x_data, y_data,(28.14,-0.25))
answer = function_solve(0.5, popt[0], popt[1])

I tried running the code and the error was:

can't assign to function call

The function I'm trying to solve is y = 1/ c*exp(-b*x) in the linear form. I have bunch of y_data and x_data, I want to get optimal values for c and b.

Answer 1

Some problems with your code have already been pointed out. Here is a solution:

First, you need to get the correct logarithmic expression of your original function:

y = 1 / (c * exp(-b * x))
y = exp(b * x) / c
ln(y) = b * x + ln(1/c)
ln(y) = b * x - ln(c)

If you want to use that in curve_fit, you need to define your function as follows:

def f_log(x, b, c_ln):
    return b * x - c_ln

I now show you the outcome for some randomly generated data (using b = 0.08 and c = 100.5) using the original function and then also the output for the data you provided:

[  8.17260899e-02   1.17566291e+02]

As you can see the fitted values are close to the original ones and the fit describes the data very well.

For your data it looks as follows:

[-0.094 -1.263]

Here is the code:

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


def f(x, b, c):
    return 1. / (c * np.exp(-b * x))


def f_log(x, b, c_ln):
    return b * x - c_ln

# some random data
b_org = 0.08
c_org = 100.5
x_data = np.linspace(0.01, 100., 50)
y_data = f(x_data, b_org, c_org) + np.random.normal(0, 0.5, len(x_data))

# fit the data
popt, pcov = curve_fit(f, x_data, y_data, p0=(0.1, 50))
print popt

# plot the data
xnew = np.linspace(0.01, 100., 5000)
plt.plot(x_data, y_data, 'bo')
plt.plot(xnew, f(xnew, *popt), 'r')
plt.show()

# your data
x_data = np.array([4, 6, 8, 10])
y_data = np.array([0.86, 0.73, 0.53, 0.3])

# fit the data
popt_log, pcov_log = curve_fit(f_log, x_data, y_data)
print popt_log

# plot the data
xnew = np.linspace(4, 10., 500)
plt.plot(x_data, y_data, 'bo')
plt.plot(xnew, f_log(xnew, *popt_log), 'r')
plt.show()