% The score distribution is considered as a mixture of two Gaussian % distributions. % % GMM.m The parameters of two Gaussian distributions are estimated by the % EM algorithm. % % Synopsis: [F, P, pi_1, pi_2, mu_1, sigma_1, mu_2, sigma_2] = GMM(score, X); % % Input: score = the score to be classified % X = the scores of the whole class % % Output: F = the probability of belonging to the secondary group % P = the probability of belonging to the leading group % pi_1 = prior of the leading group % mu_1 = mean of the leading group % sigma_1 = standard deviation of the leading group % pi_2 = prior of the secondary group % mu_2 = mean of the secondary group % sigma_2 = standard deviation of the secondary group % % Plot_GMM.m Plot the original score distribution and their approximation % by using two Gaussian distributions % Synopsis: Plot_GMM(X, pi_1, pi_2, mu_1, sigma_1, mu_2, sigma_2); % % Input: X = the scores of the whole class % pi_1 = prior of the leading group % mu_1 = mean of the leading group % sigma_1 = standard deviation of the leading group % pi_2 = prior of the secondary group % mu_2 = mean of the secondary group % sigma_2 = standard deviation of the secondary group clear all; close all; clc; theta = [.5 80 10 .5 40 10]; score_0 = 100; % the score of your homework 1 score_1 = 55; % the score of your quiz 1 score_2 = 55; % the score of your quiz 2 score_3 = 55; % the score of your Midterm score_4 = 55; % the score of your quiz 3 score_5 = 55; % the score of your quiz 4 score_6 = 55; % the score of your quiz 5 score_7 = 55; % the score of final exam %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Homework 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X0 = 100*ones(78,1)'; % the scores of the whole class X0(11) = 0; X0(17) = 0; X0(28) = 0; X0(41) = 60; X0(53) = 60; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % QUIZ 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X1 = [0,50,20,50,40,50,45,55,95,90,50,70,100,90,65,100,0,50,55,95,80,90,95,85,80,100,40,80,35,50,... 40,20,75,85,50,45,40,40,90,30,50,50,0,40,60,50,45,80,90,80,40,40,50,50,60,50,40,65,40,65,... 70,50,60,55,75,30,30,10,40,70,80,20,20,60,10,40,20,10]; [P, F, pri, mu, sigma] = GMM(score_1, X1, theta); sigma disp([' The score of your quiz 1 is ' num2str(score_1) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X1, pri, mu, sigma); title('Quiz 1'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % QUIZ 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X2 = [100,100,75,90,65,100,100,100,100,100,90,100,100,100,100,100,90,20,80,65,90,100,100,95,100,100 ... ,40,100,60,50,100,100,100,100,100,100,90,65,65,100,100,100,0,100,90,90,75,100,80,100,100,100 ... ,90,60,75,65,100,60,90,50,80,20,100,75,100,100,100,100,100,100,100,30,35,90,75,75,90,75]; [P, F, pri, mu, sigma] = GMM(score_2, X2, theta); disp([' The score of your quiz 2 is ' num2str(score_2) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X2, pri, mu, sigma); title('Quiz 2'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Midterm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X3=[69,73,77,68,80,77,60,64,73,90,40,68,81,93,73,92,77,80,66,68,65,75,73,75,80,64,77 ... ,72,25,57,68,57,58,75,56,57,69,70,79,55,72,77,78,42,92,38,54,67,25,58,12,78,68,65 ... ,51,78,79,76,63,68,74,33,77,73,98,68,63,50,43,75,94,56,35,30,56,62,25,41]; [P, F, pri, mu, sigma] = GMM(score_3, X3, theta); disp([' The score of your midterm is ' num2str(score_3) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X3, pri, mu, sigma); title('Midterm'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % QUIZ 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X4 = [100,100,100,70,95,100,100,100,100,90,60,85,95,100,100,95,65,95,100,95,75 ... ,95,100,100,100,95,95,100,95,100,100,100,95,100,100,100,100,100,75,100,0 ... ,95,100,100,90,95,100,70,100,100,100,100,100,100,100,100,95,100,80,70,70 ... ,50,100,90,100,100,100,100,95,100,100,95,0,100,95,100,50,100]; [P, F, pri, mu, sigma] = GMM(score_4, X4, theta); disp([' The score of your quiz 3 is ' num2str(score_4) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X4, pri, mu, sigma); title('Quiz 3'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % QUIZ 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X5 = [75,80,0,75,100,40,75,95,90,80,5,40,60,100,55,85,90,60,0,45,40,100,25,60,70 ... ,90,0,40,25,0,0,0,0,0,0,30,30,55,40,30,30,95,0,0,75,50,40,60,75,100,0,0,45,45 ... ,50,80,80,85,40,80,55,0,70,95,55,35,70,30,80,75,85,45,40,0,0,0,15,0]; [P, F, pri, mu, sigma] = GMM(score_5, X5, theta); disp([' The score of your quiz 4 is ' num2str(score_5) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X5, pri, mu, sigma); title('Quiz 4'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % QUIZ 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X6 = [80,100,70,100,50,70,50,50,70,80,80,50,0,50,0,50,100,0,0,20,0,0,50,0,80,80 ... ,80,0,0,0,20,0,0,30,50,0,0,30,70,0,0,60,0,0,100,50,70,80,100,50,0,0,50,50 ... ,80,80,100,50,100,30,80,0,50,50,100,30,50,70,100,100,100,30,0,100,40,80,0,50]; [P, F, pri, mu, sigma] = GMM(score_6, X6, theta); disp([' The score of your quiz 5 is ' num2str(score_6) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X6, pri, mu, sigma); title('Quiz 5'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Final exam %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % the scores of the whole class X7 = [68 59 87 76 51 64 69 79 88 78 85 56 68 95 62 90 85 30 79 46 58 64 74 46 93 93 64 ... 64 56 48 49 32 66 56 76 70 63 55 48 46 65 83 0 58 73 72 56 77 78 100 0 39 83 70 ... 75 83 90 78 92 75 89 0 88 78 83 61 62 42 96 90 87 73 45 49 55 77 39 25]; [P, F, pri, mu, sigma] = GMM(score_7, X7, theta); disp([' The score of your final exam is ' num2str(score_7) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(X7, pri, mu, sigma); title('Final Examination'), xlabel('Score'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Cumulated score %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% score = 0.05*score_0 + 0.07*(score_1 + score_2 + score_4 + score_5 + score_6)... + 0.3*(score_3 + score_7); % original score n_score = score*100/100; % normalized score % the cumulated scores of the whole class X = 0.05*X0 + 0.07*(X1 + X2 + X4 + X5 + X6) + 0.3*(X3 + X7) ; n_X = X*100/100; % normalized cumulated score [P, F, pri, mu, sigma] = GMM(n_score, n_X, theta); disp([' Your cumulated score is ' num2str(n_score) ]); disp([' The probability of belonging to the 1st group = ' num2str(P) ]); disp([' The probability of belonging to the 2nd group = ' num2str(F) ]); disp(' '); Plot_GMM(n_X, pri, mu, sigma); title('Cumulated score'), xlabel('Score'); % number of people whose final score are below 60 n = size(find(n_X < 60),2); disp([' There are ' num2str(n) ' people whose final scores are below 60.']); % Perform classification on the normalized cumulated scores p(1,:) = (sqrt(2*pi)*sigma(1))^(-1)*exp(-1*0.5*(n_X - mu(1)).^2/(sigma(1)^2)); p(2,:) = (sqrt(2*pi)*sigma(2))^(-1)*exp(-1*0.5*(n_X - mu(2)).^2/(sigma(2)^2)); tmp = p(1,:)*pri(1) + p(2,:)*pri(2); F = p(2,:)*pri(2)./tmp; n = size(find(F > .5),2); % number of people who are classified as the 2nd group disp([' There are ' num2str(n) ' people who are classified as the 2nd group.']); % mean of the normalized cumulated scores disp([' The mean score is ' num2str(mean(n_X)) '.']);