## 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*