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_fitfrom 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_fitfrom matplotlib import pyplot as pltx = 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 function 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.
