SciPy | Curve Fitting

19 Sep, 2021
Given a Dataset comprising of a group of points, find the best fit representing the Data.We often have a dataset comprising of data following a general path, but each data has a standard deviation which makes them scattered across the line of best fit. We can get a single line using curve-fit() function. Using SciPy : Scipy is the scientific computing module of Python providing in-built functions on a lot of well-known Mathematical functions. The scipy.optimize package equips us with multiple optimization procedures. A detailed list of all functionalities of Optimize can be found on typing the following in the iPython console:  
help(scipy.optimize)
Among the most used are Least-Square minimization, curve-fitting, minimization of multivariate scalar functions etc.Curve Fitting Examples – Input :  
Output :  
Input :  
Output :  
As seen in the input, the Dataset seems to be scattered across a sine function in the first case and an exponential function in the second case, Curve-Fit gives legitimacy to the functions and determines the coefficients to provide the line of best fit.  Code showing the generation of the first example –  
Python3
import numpy as np # curve-fit() function imported from scipyfrom scipy.optimize import curve_fit from matplotlib import pyplot as plt # numpy.linspace with the given arguments# produce an array of 40 numbers between 0# and 10, both inclusivex = np.linspace(0, 10, num = 40)  # y is another array which stores 3.45 times# the sine of (values in x) * 1.334.# The random.normal() draws random sample# from normal (Gaussian) distribution to make# them scatter across the base liney = 3.45 * np.sin(1.334 * x) + np.random.normal(size = 40) # Test function with coefficients as parametersdef test(x, a, b):    return a * np.sin(b * x) # curve_fit() function takes the test-function# x-data and y-data as argument and returns# the coefficients a and b in param and# the estimated covariance of param in param_covparam, param_cov = curve_fit(test, x, y)  print("Sine function coefficients:")print(param)print("Covariance of coefficients:")print(param_cov) # ans stores the new y-data according to# the coefficients given by curve-fit() functionans = (param[0]*(np.sin(param[1]*x))) '''Below 4 lines can be un-commented for plotting resultsusing matplotlib as shown in the first example. ''' # plt.plot(x, y, 'o', color ='red', label ="data")# plt.plot(x, ans, '--', color ='blue', label ="optimized data")# plt.legend()# plt.show()
Sine function coefficients:
[ 3.66474998  1.32876756]
Covariance of coefficients:
[[  5.43766857e-02  -3.69114170e-05]
 [ -3.69114170e-05   1.02824503e-04]]
 
  Second example can be achieved by using the numpy exponential function shown as follows:  
Python3
x = np.linspace(0, 1, num = 40) y = 3.45 * np.exp(1.334 * x) + np.random.normal(size = 40) def test(x, a, b):    return a*np.exp(b*x) param, param_cov = curve_fit(test, x, y)
However, if the coefficients are too large, the curve flattens and fails to provide the best fit. The following code explains this fact: 
Python3
import numpy as npfrom scipy.optimize import curve_fit from matplotlib import pyplot as plt x = np.linspace(0, 10, num = 40) # The coefficients are much bigger.y = 10.45 * np.sin(5.334 * x) + np.random.normal(size = 40) def test(x, a, b):    return a * np.sin(b * x) param, param_cov = curve_fit(test, x, y) print("Sine function coefficients:")print(param)print("Covariance of coefficients:")print(param_cov) ans = (param[0]*(np.sin(param[1]*x))) plt.plot(x, y, 'o', color ='red', label ="data")plt.plot(x, ans, '--', color ='blue', label ="optimized data")plt.legend()plt.show()
Sine funcion coefficients:
[ 0.70867169  0.7346216 ]
Covariance of coefficients:
[[ 2.87320136 -0.05245869]
 [-0.05245869  0.14094361]]
 
 
The blue dotted line is undoubtedly the line with best-optimized distances from all points of the dataset, but it fails to provide a sine function with the best fit.Curve Fitting should not be confused with Regression. They both involve approximating data with functions. But the goal of Curve-fitting is to get the values for a Dataset through which a given set of explanatory variables can actually depict another variable. Regression is a special case of curve fitting but here you just don’t need a curve that fits the training data in the best possible way(which may lead to overfitting) but a model which is able to generalize the learning and thus predict new points efficiently. 
kk9826225
simmytarika5
Python-scipy
Machine Learning
Python
Machine Learning
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