def gradient_descent(X, y, theta, alpha, num_iters): m = y.size J_history = np.zeros(shape=(num_iters, 1)) for i in range(num_iters): predictions = X.dot(theta).flatten() errors_x1 = (predictions - y) * X[:, 0] errors_x2 = (predictions - y) * X[:, 1] theta[0][0] = theta[0][0] - alpha * (1.0 / m) * errors_x1.sum() theta[1][0] = theta[1][0] - alpha * (1.0 / m) * errors_x2.sum() J_history[i, 0] = compute_cost(X, y, theta) return theta, J_history
#and replace all ones in second column by X values #first column stays one : remember x subscript zero is all ones. just to make it mathematically convinient x = np.ones(shape=(m, 2)) x[:, 1] = X #we need to intialize theta with some dummy values, lets assume its all zeros theta = np.zeros(shape=(2, 1)) #set number of iterations for updating gradient discent iterations = 50000 #set learning rate alpha = 0.01 #display initial cost print compute_cost(x, y, theta) theta, J_history = gradient_descent(x, y, theta, alpha, iterations) print theta #Predict values for population sizes of 35,000 and 70,000 predict1 = np.array([1, 3.5]).dot(theta).flatten() print 'For population = 35,000, we predict a profit of %f' % (predict1 * 10000) predict2 = np.array([1, 5.0]).dot(theta).flatten() print 'For population = 50,000, we predict a profit of %f' % (predict2 * 10000) #Plot the results result = x.dot(theta).flatten() pl.plot(data[:, 0], result) pl.show()
