Non-linear curve-fitting program in python

Question

I would like to find and plot a function f that represents a curve fitted on some number of set points that I already know, x and y.

After some research I started experimenting with scipy.optimize and curve_fit but on the reference guide I found that the program uses a function to fit the data instead and it assumes ydata = f(xdata, *params) + eps.

So my question is this: What do I have to change in my code to use the curve_fit or any other library to find the function of the curve using my set points? (note: I want to know the function as well so I can integrate later for my project and plot it). I know that its going to be a decaying exponencial function but don't know the exact parameters. This is what I tried in my program:

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

    def func(x, a, b, c):
        return a * np.exp(-b * x) + c

    xdata = np.array([0.2, 0.5, 0.8, 1])
    ydata = np.array([6, 1, 0.5, 0.2])
    plt.plot(xdata, ydata, 'b-', label='data')
    popt, pcov = curve_fit(func, xdata, ydata)
    plt.plot(xdata, func(xdata, *popt), 'r-', label='fit')

    plt.xlabel('x')
    plt.ylabel('y')
    plt.legend()
    plt.show()

Am currently developing this project on a Raspberry Pi, if it changes anything. And would like to use least squares method since is great and precise, but any other method that works well is welcome. Again, this is based on the reference guide of scipy library. Also, I get the following graph, which is not even a curve: Graph and curve based on set points

Answer 1

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

def func(x, a, b, c):
    return a * np.exp(-b * x) + c

#c is a constant so taking the derivative makes it go to zero
def deriv(x, a, b, c):
    return -a * b * np.exp(-b * x)

#Integrating gives you another c coefficient (offset) let's call it c1 and set it equal to zero by default
def integ(x, a, b, c, c1 = 0):
    return -a/b * np.exp(-b * x) + c*x + c1

#There are only 4 (x,y) points here
xdata = np.array([0.2, 0.5, 0.8, 1])
ydata = np.array([6, 1, 0.5, 0.2])

#curve_fit already uses "non-linear least squares to fit a function, f, to data"
popt, pcov = curve_fit(func, xdata, ydata)
a,b,c = popt #these are the optimal parameters for fitting your 4 data points

#Now get more x values to plot the curve along so it looks like a curve
step = 0.01
fit_xs = np.arange(min(xdata),max(xdata),step)

#Plot the results
plt.plot(xdata, ydata, 'bx', label='data')
plt.plot(fit_xs, func(fit_xs,a,b,c), 'r-', label='fit')
plt.plot(fit_xs, deriv(fit_xs,a,b,c), 'g-', label='deriv')
plt.plot(fit_xs, integ(fit_xs,a,b,c), 'm-', label='integ')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()