Write a MATLAB program that implements the least-square regression for the singl

Question

Write a MATLAB program that implements the least-square regression for the single-variable polynomial model f=a0+a1x+a2x2 +...+anxn? given a set of data [x] and [y] of some lengths. Your MATLAB code submission should have the following three M files: me2016_PolyReRression.m This file is a general function so that a functional call of [A err] = me2016_PolyRegression(X, Y, d) should return the polynomial coefficients A=[a0, a1. a2,..., ad] T and the square error err; where the input parameters include X as input column vector of [x], Y as input column vector of [y], and d is the degree of polynomial. The implementation must be based on the matrix form as discussed in the class, instead of nested loops with element-by-element operations. To solve the linear equation CA=B, you should use the single matlab command of "A=CB".

Explanation / Answer

function [MSE_train_mean,MSE_train_std,MSE_test_mean,MSE_test_std,w_star] = perform_lsr(X,Y,Z) % 604 days for the training set (29/05/2009 - 18/10/2011) % 145 days for the evaluation set (19/10/2011 - 16/05/2012) Xtran = X(1:end-145,:); Ytran = Y(1:end-145,:); Ztran = Z(1:end-145,:); Xeval = X(end-144:end,:); Yeval = Y(end-144:end,:); Zeval = Z(end-144:end,:); % Set number of runs. runs = 20; % Set number of folds. folds = 5; % Initialize auxiliary matrices. MSE_train = zeros(runs,folds); MSE_test = zeros(runs,folds); w_star = zeros(runs,folds,size(Xtran,2)); % Perform runs. for r=1:runs, % Split original dataset into training and test set. split_assignments = cross_val_split(folds,size(Xtran,1)); % Perform folds. for f=1:folds, % Assign explanatory variables (x). x_train = Xtran((split_assignments(:,f)==0),:); x_test = Xtran((split_assignments(:,f)==1),:); % Assign respond variable (y). y_train = Ytran((split_assignments(:,f)==0),:); y_test = Ytran((split_assignments(:,f)==1),:); % Retrieve size of matrices. [l_train_n,l_train_m] = size(x_train); [l_test_n,l_test_m] = size(x_test); % Estimate parameters (w0,w1,...) of the 1st order model from training set. w_star(r,f,:) = (x_train' * x_train) (x_train' * y_train); w_star_temp = w_star(r,f,:); w_star_temp = w_star_temp(:); % Apply the learned weights on both training and test sets and compute the % corresponding MSEs. MSE_train(r,f) = (1 / l_train_n) * (w_star_temp' * (x_train') * x_train * w_star_temp - 2 * y_train' * x_train * w_star_temp + y_train' * y_train); MSE_test(r,f) = (1 / l_test_n) * (w_star_temp' * (x_test') * x_test * w_star_temp - 2 * y_test' * x_test * w_star_temp + y_test' * y_test); end end % % Plot both training and test sets' MSEs as a function of runs and folds. % figure, mesh(MSE_train), title('Training Set`s MSEs vs Runs and Folds'); % figure, mesh(MSE_test), title('Testing Set`s MSEs vs Runs and Folds'); % % % Plot both training and test sets' mean of MSEs accompanied by their % % corresponding std of MSEs. % figure, errorbar(1:runs,mean(MSE_train,2),std(MSE_train,0,2)); % figure, errorbar(1:runs,mean(MSE_test,2),std(MSE_test,0,2)); % Average over folds and then over runs. MSE_train_mean = mean(mean(MSE_train,2),1); MSE_train_std = std(mean(MSE_train,2),0,1); MSE_test_mean = mean(mean(MSE_test,2),1); MSE_test_std = std(mean(MSE_test,2),0,1); w_star_temp = mean(mean(w_star,2),1); w_star = w_star_temp(:); % % Calculate in-sample residuals. % y_hat_tran = Xtran * w_star; % e_tran = Ytran - y_hat_tran; % % Check that they sum up to zero. % fprintf('In-sample residuals sum up to %i. ',sum(e_tran)); % % Finally, plot them. % figure, scatter(1:size(e_tran,1),e_tran); % % % Calculate out-of-sample residuals. % y_hat_eval = Xeval * w_star; % e_eval = Yeval - y_hat_eval; % % Check that they sum up to zero. % fprintf('Out-of-sample residuals sum up to %i. ',sum(e_eval)); % % Finally, plot them. % figure, scatter(1:size(e_eval,1),e_eval); % Print summary statistics. fprintf('Over %i runs, %i folds, and regarding the training set, the mean and standard deviation is %.10f and %.10f respectively. ',runs,folds,MSE_train_mean,MSE_train_std); fprintf('Over %i runs, %i folds, and regarding the testing set, the mean and standard deviation is %.10f and %.10f respectively. ',runs,folds,MSE_test_mean,MSE_test_std); end

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