#!/usr/bin/env python3 import numpy as np import matplotlib.pyplot as plt def rotate(data, degree): # data: M x 2 theta = np.pi/180 * degree R = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]]) # rotation matrix return np.dot(data, R.T) def leastSquares(X, Y): # In this function, X is always the input, Y is always the output # X: M x (d+1), Y: M x 1, where d=1 here # return weights w # TODO: YOUR CODE HERE # closed form solution by matrix-vector representations only return w def model(X, w): # X: M x (d+1) # w: d+1 # return y_hat: M x 1 # TODO: YOUR CODE HERE return y_hat def generate_data(M, var1, var2, degree): # data generate involves two steps: # Step I: generating 2-D data, where two axis are independent # M (scalar): The number of data samples # var1 (scalar): variance of a # var2 (scalar): variance of b mu = [0, 0] Cov = [[var1, 0], [0, var2]] data = np.random.multivariate_normal(mu, Cov, M) # shape: M x 2 plt.figure() plt.scatter(data[:, 0], data[:, 1], color="blue") plt.xlim(-4, 4) plt.ylim(-4, 4) plt.xlabel('a') plt.ylabel('b') plt.tight_layout() plt.savefig('data_ab_'+str(var2)+'.jpg') # Step II: rotate data by 45 degree counter-clockwise, # so that the two dimensions are in fact correlated data = rotate(data, degree) plt.tight_layout() plt.figure() # plot the data points plt.scatter(data[:, 0], data[:, 1], color="blue") plt.xlim(-4, 4) plt.ylim(-4, 4) plt.xlabel('x') plt.ylabel('y') # plot the line where data are mostly generated around X_new = np.linspace(-5, 5, 100, endpoint=True).reshape([100, 1]) Y_new = np.tan(np.pi/180*degree)*X_new plt.plot(X_new, Y_new, color="blue", linestyle='dashed') plt.tight_layout() plt.savefig('data_xy_'+str(var2) + '_' + str(degree) + '.jpg') return data ########################### # Main code starts here ########################### # Settings M = 5000 var1 = 1 var2 = 0.8 degree = 45 data = generate_data(M, var1, var2, degree) ########## # Training the linear regression model predicting y from x (x2y) Input = data[:, 0].reshape((-1, 1)) # M x d, where d=1 # M x (d+1) augmented feature Input_aug = np.concatenate([Input, np.ones([M, 1])], axis=1) Output = data[:, 1].reshape((-1, 1)) # M x 1 w_x2y = leastSquares(Input_aug, Output) # (d+1) x 1, where d=1 print('Predicting y from x (x2y): weight=' + str(w_x2y[0, 0]), 'bias = ', str(w_x2y[1, 0])) # # Training the linear regression model predicting x from y (y2x) Input = data[:, 1].reshape((-1, 1)) # M x d, where d=1 # M x (d+1) augmented feature Input_aug = np.concatenate([Input, np.ones([M, 1])], axis=1) Output = data[:, 0].reshape((-1, 1)) # M x 1 w_y2x = leastSquares(Input_aug, Output) # (d+1) x 1, where d=1 print('Predicting x from y (y2x): weight=' + str(w_y2x[0, 0]), 'bias = ', str(w_y2x[1, 0])) # plot the data points plt.figure() X = data[:, 0].reshape((-1, 1)) # M x d, where d=1 Y = data[:, 1].reshape((-1, 1)) # M x d, where d=1 plt.scatter(X, Y, color="blue", marker='x') plt.xlim(-4, 4) plt.ylim(-4, 4) plt.xlabel('x') plt.ylabel('y') # plot the line where data are mostly generated around X_new = np.linspace(-4, 4, 100, endpoint=True).reshape([100, 1]) Y_new = np.tan(np.pi/180*degree)*X_new plt.plot(X_new, Y_new, color="blue", linestyle='dashed') # plot the prediction of y from x (x2y) # M x d, where d=1 X_new = np.linspace(-4, 4, 100, endpoint=True).reshape([100, 1]) # M x (d+1) augmented feature X_new_aug = np.concatenate([X_new, np.ones([X_new.shape[0], 1])], axis=1) plt.plot(X_new, model(X_new_aug, w_x2y), color="red", label="x2y") # plot the prediction of x from y (y2x) # M x d, where d=1 Y_new = np.linspace(-4, 4, 100, endpoint=True).reshape([100, 1]) # M x (d+1) augmented feature Y_new_aug = np.concatenate([X_new, np.ones([X_new.shape[0], 1])], axis=1) plt.plot(model(Y_new_aug, w_y2x), Y_new, color="green", label="y2x") plt.legend() plt.tight_layout() plt.savefig('Regression_model_' + str(var2) + '_' + str(degree) + '.jpg') 

PROBLEMS TO BE SOLVED:

With settings M = 5000, var1 = 1, var2 = 0.3, degree = 45, report the weight and bias for x2y and y2x regressions.

Three plots of regression models in a row, each with var2 = 0.1, 0.3, 0.8, respectively. (Other settings remain intact: M = 5000, var1 = 1, degree = 45).

We now set var2 = 0.1, but experiment with different rotation degrees. The student should design a controlled experimental protocol, plot three figures in a row with different settings, and describe the findings.