% EE3025 S2001: Recitation 8
% N. Sidiropoulos, March 8, 2001.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First part of recitation: Linear MMSE estimation of one RV from two RVs.
% This part uses the weather data of Recitation 7 - these should be in 
% your working directory.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all;

load AMES94.weather;
load MASONCITY94.weather;
load NEVADA94.weather;
load ANAMOSA94.weather;

% AMES94 is a 184 (days) x 5 matrix.
% Column 1 is date; column 3 Tmax, column 4 Tmin, column 5 precipitation
% and similarly for the other three matrices

% Drop first few and last several days (season transients!):

AMES94 = AMES94(20:150,:);
MASONCITY94 = MASONCITY94(20:150,:);
NEVADA94 = NEVADA94(20:150,:);

figure(1);
clf;
subplot(2,2,1);
plot(AMES94(:,4));
legend('AMES94,Tmin');
subplot(2,2,2);
plot(NEVADA94(:,4));
legend('NEVADA94,Tmin');
subplot(2,2,4);
plot(MASONCITY94(:,4));
legend('MASONCITY94,Tmin');

% Linear MMSE estimation of one RV from two RVs.
% Cf. Extra Problem in HW5. 

x = AMES94(:,4);
y = MASONCITY94(:,4);
z = NEVADA94(:,4);

% Either: 
% A = [mean(y.*y) mean(y.*z) mean(y)
%     mean(y.*z) mean(z.*z) mean(z)
%     mean(y) mean(z) 1];
% d = [mean(x.*y); mean(x.*z); mean(x)];
% 
% or, to see dependence on corco:

a11 = (std(y))^2+(mean(y))^2;
a12 = std(y)*std(z)*corco(y,z)+mean(y)*mean(z);
a13 = mean(y);
a21 = a12;
a22 = (std(z))^2+(mean(z))^2;
a23 = mean(z);
a31 = a13;
a32 = a23;
a33 = 1; 

A = [a11 a12 a13
     a21 a22 a23
     a31 a32 a33];

d1 = std(x)*std(y)*corco(x,y)+mean(x)*mean(y);
d2 = std(x)*std(z)*corco(x,z)+mean(x)*mean(z);
d3 = mean(x);

d = [d1
     d2 
     d3];

w = inv(A)*d;

a = w(1);
b = w(2);
c = w(3);

xhat = a*y+b*z+c;

figure(2);
clf;
subplot(2,1,1);
plot(x,'o');
hold on;
plot(xhat,'r*-');
legend('x (Tmin, AMES)','xhat (LMMSE estimate, based on Tmin, MASONCITY, NECADA');
subplot(2,1,2);
plot(x-xhat,'ko-');
legend('estimation error');

% resulting squared error:

esaccruedYZ = (1/length(x))*norm(x-xhat,'fro')^2

% Compare with estimation based on x only:

a = corco(x,y)*(std(x)/std(y));
b = mean(x) - a*mean(y);
xhat = a*y+b;

esaccruedY = (1/length(x))*norm(x-xhat,'fro')^2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Second part of recitation: Joint Gaussian PDF plots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all;
figure(3);

% We will employ the bivariate Gaussian distribution with zero means,
% variances 1, and correlation coefficient rho. The matlab syntax 
% for this density function is:

% 1/(2*pi*sqrt(1-rho^2))*exp(-(X^2-2*rho*X*Y+Y^2)/(2*(1-rho^2)))

% Follow the script:

x=-4:.2:4; % This creates your x-axis
y=x; % This creates your y-axis
[X,Y]=meshgrid(x,y); % X and Y are 41x41 matrices; mesh(X), mesh(Y)
                     % to understand what's happening here

rho=0;
Z=1/(2*pi*sqrt(1-rho^2))*exp(-(X.^2-2*rho*X.*Y+Y.^2)/(2*(1-rho^2)));
colormap(gray);
mesh(X,Y,Z); 

% more plots of the same thing:

surfl(X,Y,Z);

surfl(X,Y,Z), shading interp

contour(X,Y,Z); 

contour(X,Y,Z,40);

% Repeat the above with rho = 0.5:

rho = 0.5;

Z=1/(2*pi*sqrt(1-rho^2))*exp(-(X.^2-2*rho*X.*Y+Y.^2)/(2*(1-rho^2)));
colormap(gray);
mesh(X,Y,Z); 

% more plots of the same thing:

surfl(X,Y,Z);

surfl(X,Y,Z), shading interp

contour(X,Y,Z); 

contour(X,Y,Z,40);

% What do you observe?

% Repeat the above with rho = 0.9, and then rho = -0.9. What do you observe?

% Repeat the above with rho = 0.99. What do you observe?

% Generate the Z for rho = 0.5:

rho = 0.5;

Z=1/(2*pi*sqrt(1-rho^2))*exp(-(X.^2-2*rho*X.*Y+Y.^2)/(2*(1-rho^2)));

z1=Z(:,15);
z2=Z(:,20);
z3=Z(:,25);
z4=Z(:,30);
plot(y,z1,y,z2,y,z3,y,z4)

% What do these plots represent?

z1=Z(15,:)';
z2=Z(20,:)';
z3=Z(25,:)';
z4=Z(30,:)'; 
plot(x,z1,x,z2,x,z3,x,z4)

% What do these plots represent?

% Answer (scaled) conditional PDFs.