fit a curve

scipy.interpolate.splprep

# python - Interpolating a closed curve using scipy - Stack Overflow
# https://stackoverflow.com/questions/33962717/interpolating-a-closed-curve-using-scipy
import numpy as np
from scipy import interpolate
from matplotlib import pyplot as plt
 
x = np.array([23, 24, 24, 25, 25])
y = np.array([13, 12, 13, 12, 13])
 
# append the starting x,y coordinates
x = np.r_[x, x[0]]
y = np.r_[y, y[0]]
 
# fit splines to x=f(u) and y=g(u), treating both as periodic. also note that s=0
# is needed in order to force the spline fit to pass through all the input points.
tck, u = interpolate.splprep([x, y], s=0, per=True)
 
# evaluate the spline fits for 1000 evenly spaced distance values
xi, yi = interpolate.splev(np.linspace(0, 1, 1000), tck)
 
# plot the result
fig, ax = plt.subplots(1, 1)
ax.plot(x, y, 'or')
ax.plot(xi, yi, '-b')

numpy.polyfit

ref:
numpy.polyfit — NumPy v1.14 Manual
https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.polyfit.html

minimizing the sum of the squared residuals

#!/usr/bin/env python3
# -*- coding: UTF-8 -*-
# @Time    : 8/20/2018 11:36 PM
# @Author  : yusisc (yusisc@gmail.com)
# scipy.optimize.curve_fit — SciPy v0.19.0 Reference Guide
# https://docs.scipy.org/doc/scipy-0.19.0/reference/generated/scipy.optimize.curve_fit.html
 
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
 
 
# define the data to be fit with some noise
np.random.seed(1)
xdata = np.linspace(0, 4, 50)
y = func(xdata, 2.5, 1.3, 0.5)
y_noise = 0.2 * np.random.normal(size=xdata.size)
ydata = y + y_noise
plt.plot(xdata, ydata, 'b-', label='data')
# Fit for the parameters a, b, c of the function func
 
popt, pcov = curve_fit(func, xdata, ydata)
plt.plot(xdata, func(xdata, *popt), 'r-', label='fit')
 
# Constrain the optimization to the region of 0 < a < 3, 0 < b < 2 and 0 < c < 1:
popt, pcov = curve_fit(func, xdata, ydata, bounds=(0, [3., 2., 1.]))
plt.plot(xdata, func(xdata, *popt), 'g--', label='fit-with-bounds')
 
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()

More

Fitting data — SciPy Cookbook documentation
https://scipy-cookbook.readthedocs.io/items/FittingData.html