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Curve fit in Python

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In Python, we can perform curve fit by using scipy.optimize library.

Syntax:

scipy.optimize.curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, check_finite=True, bounds=(- inf, inf), method=None, jac=None, *, full_output=False, **kwargs)

Parameters:

  • f (callable function): The model function, f(X, . . .). It must take the independent variable as the first argument and the parameters to fit as separate remaining arguments.
  • xdata (array_like or object): The independent variable where the data is measured. Should usually be an M-length sequence or an (k,M)-shaped array in case of functions with k predictors (multiple independent variables).
  • ydata (array_like): The dependent data, a length M array – nominally f(xdata, . . .).

  The curve_fit uses the non-linear least squares method by default to fit a function, f, to the data points.

Defining Model function

We define the function (curve) to which we want to fit our data. Here, a and b are parameters that define the curve. In this example, we choose  y=(a(x_2)^2+b(x_2)^2) as our model function.

Python3




def f(X, a, b):
    x_1, x_2 = X
    return a*x_1**2 + b*x_2**2
 






                        









													
													
												

												
												

													
													
Initializing the  independent(y) and dependent(X) data

													
													
												

												
												

													
													
In this step, we initialize the independent data x_1 and x_2 using np.linspace(0, 4, 50) which creates an evenly spaced array over a specified interval. In the case of multiple independent variables X = (x_1, x_2, ). Then we initialize dependent data y using the model function and adding noise (np.random.random(50)*4) to it.

													
													
												

												
												

													
													




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import numpy as np


import matplotlib.pyplot as plt


from scipy.optimize import curve_fit


x_1 = np.linspace(0, 4, 50)


x_2 = np.linspace(0, 4, 50)


X = (x_1, x_2)


y = f(X, 2, 4)


 


# Adding Noise


y = y + np.random.random(50)*4


 


# plotting the data points


fig = plt.figure()


fig.set_figwidth(40)


fig.set_figheight(10)


ax = plt.axes(projection='3d')


ax.set_xlabel('x_1', fontsize=12, color='green')


ax.set_ylabel('x_2', fontsize=12, color='green')


ax.set_zlabel('y', fontsize=12, color='green')


ax.scatter3D(x_1, x_2, y, color='green')


plt.title("Plot of data points")


plt.show()



















                        







                        









													
													
												

												
												

													
													
Output:

													
													
												

												
												

													
													
 

													
													
												

												
												

													
													
The curve_fit() method returns the following output:

													
													
												

												
												

													
													
    

  • popt (array): Optimal values for the parameters so that the sum of the squared residuals of f(xdata, *popt) – ydata is minimized.
    • 

  • pcov2-D (array): The estimated covariance of popt. The diagonals provide the variance of the parameter estimate. 
    • 


													
													
												

												
												

													
													




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popt, pcov = curve_fit(f, X, y)


popt



















                        







                        









													
													
												

												
												

													
													
Output:

													
													
												

												
												

													
													
array([-96.89634526, 103.10365474])

													
													
												

												
											

											
											

												
Python – Scipy curve_fit with multiple independent variables

												
												

													
													
Curve fitting examines the relationship between one or more predictors (independent variables) and a response variable (dependent variable), with the goal of defining a “best fit” model of the relationship. It is the process of constructing a mathematical function, that has the best fit to a series of data points possibly subject to constraints.

													
													
												

												
											

										


										
										

											
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												Visualizing Results