Generating Sample Data:
Let us now generate some sample data using our previously developed functions. First, build a polynomial in the form of:
Next, create gaussian noise to simulate a gaussian process:
The resultant equation is a combination of the noise and the polynomial:
In python we will utilize numpy’s np.random.normal(mean,std,n_samples), and simply add it to the evaluated polynomial. It is important to note that there is nothing special about the choice of the polynomial function; it is simply arbitrary and for the sake of presentation.
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| """ | |
| Polynomials | |
| @author: Abdullah Alnuaimi | |
| """ | |
| import numpy as np | |
| import matplotlib.pyplot as plt | |
| def fit_poly(a,k): | |
| '''returns a function of the dot product (A=V.a) ''' | |
| A=lambda x,a=a,k=k:[[a*n**k for a,k in zip(a,k)] for n in x] | |
| return A | |
| def evaluate_poly(x,A): | |
| ''' evaluates A=V.a,stores it in matrix form, and | |
| returns a list y(x)=[A0,..An]''' | |
| y=[sum(i) for i in A(x)] | |
| return y,A(x) | |
| ############################## Main ######################################### | |
| # y(x)=x+x^2-0.2x^3 | |
| coefficients=[0,1,1,–.2] # polynomial | |
| degree=[0,1,2,3] | |
| A=fit_poly(coefficients,degree) # returns A(x0)…A(Xn) | |
| # Evaluate the functions and returns p(x) | |
| x=np.linspace(0,5,100) | |
| p,_=evaluate_poly(x,A) | |
| # Fix the random seed and add gauss. noise to the data. | |
| np.random.seed(seed=1337) | |
| e=np.random.normal(0,.7,len(x)) | |
| y=p+e | |
| #Plotting | |
| plt.scatter(x,y,label='y(x)=x+x^2-0.2x^3+e(x)',marker='.') | |
| plt.plot(x,p,label='p(x)=x+x^2-0.2x^3',color='red') | |
| plt.xlabel('x') | |
| plt.ylabel('y(x)') | |
| plt.legend() | |
| plt.grid() | |
| #np.savetxt('data.csv', (x,y), delimiter=',') |
Now we are ready to take on the task of linear regression. (TL;DR) If you don’t care about generating the data for this tutorial you can just download it as a csv and follow along in the next section.
Github repository :
https://github.com/b00033811/ml-uae
The file that contains the output is Linear_Regression/data.csv
ML-UAE 