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 scipy from 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 inclusive x = 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 line y = 3.45 * np.sin(1.334 * x) + np.random.normal(size = 40) # Test function with coefficients as parameters def 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_cov param, 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() function ans = (param[0]*(np.sin(param[1]*x))) '''Below 4 lines can be un-commented for plotting results using 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() Output: 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 np from 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() Output: 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.