%%% Section 1.1
%%% Line-fitting with LLSE 

clear
close all
m = 1/3;
b = -2;

x = [-2;4];

y = [x,ones(2,1)]*[m;b];

xl = [-10:1:10]';
Xl = [xl,ones(21,1)];
yl = Xl*[m;b];

figure(1)
plot(xl,yl)
hold on
scatter(x,y)
hold off
xlim([min(xl)-3,max(xl)+3])
ylim([min(yl)-3,max(yl)+3])
grid on
text(-5,0,['Y = ',num2str(m),'x - ',num2str(b)])


%%% LLSE Line Fitting

x = 20*(rand(24,1)-.5)
X = [x,ones(24,1)];
y = X*[m;b]+.5*randn(24,1);

z = inv(X'*X)*X'*y
yle = Xl*z;

figure(2)
plot(xl,yl,'-')
hold on
plot(xl,yle,'k--','Linewidth',2)
scatter(x,y)
hold off
xlim([min(xl)-3,max(xl)+3])
ylim([min(yl)-3,max(yl)+3])
grid on
text(-5,0,['Y = ',num2str(m),'x - ',num2str(b)])
text(0,-5,['Y = ',num2str(z(1)),'x - ',num2str(z(2))])
legend('true line','fitted line')

%%% m,b estimates.

for n1 = 1:10000
    x = 25*(rand(24,1)-.5)
X = [x,ones(24,1)];
y = X*[m;b]+.5*randn(24,1);

z = inv(X'*X)*X'*y;

z_llse(:,n1) = z;
end
figure(3)
% scatter(z_llse(1,:),z_llse(2,:),'Markeredgealpha',.3)
% 
% xlim([0 0.5]), ylim([-2.5 -1.5])
% xlabel('m')
% ylabel('b')
% hold on
% errorbar(mean(z_llse(1,:)'),mean(z_llse(2,:)'),std(z_llse(2,:)'),'-s','MarkerSize',10,...
%      'MarkerEdgeColor','red','MarkerFaceColor','red')
%  
histogram2(z_llse(1,:),z_llse(2,:),50,'DisplayStyle','tile')

xlabel('m')
ylabel('b')
xlim([0.2 0.5]), ylim([-2.5 -1.5])