In practice, PXY is usually unknown and we use the empirical risk minimizer (ERM). We will reformulate the problem as a d-dimensional linear regression problem. First note that functions in Hd are parametrized by a vector b=[b0,b1,bd], we will use the notation fb. Similarly we will note aR3 the vector parametrizing g(x)=fa(x). We will also gather data points from the training sample in the following matrix and vector: X=111x1x2xNx1dx2dxN,y=[y0,y1,yN] These notations allow us to take advantage of the very effective linear algebra formalism. X is called the design matrix. Open the source code file hw1_code_source.py from the .zip folder. Using the function get_a get a value for a, and draw a sample x_train, y_train of size N=10 and a sample x_test, __test of size Ntest=1000 using the function draw_sample. 7. (2 Points) Write a function called least_square_estimator taking as input a design matrix XRN(d+1) and the corresponding vector yRN returning b^R(d+1). Your function should handle any value of N and d, and in particular return an error if Nd. (Drawing x at random from the uniform distribution makes it almost certain that any design matrix X with d1 we generate is full rank). 8. (1 Points) Recall the definition of the empical risk R^(f^) on a sample {xi,yi}i=1N for a prediction function f^. Write a function empirical_risk to compute the empirical risk of fb taking as input a design matrix XRN(d+1), a vector yRN and the vector bR(d+1) parametrizing the predictor. 9. (3 Points) Use your code to estimate b^ from x_train, y_train using d=5. Compare b^ and a. Make a single plot (Plot 1) of the plan (x,y) displaying the points in the training set, values of the true underlying function g(x) in [0,1] and values of the estimated function fb^(x) in [0,1]. Make sure to include a legend to your plot . 10. (1 Points) Now you can adjust d. What is the minimum value for which we get a "perfect fit"? How does this result relates with your conclusions on the approximation error above