%
% Simulerer data fra ulike regresjonsmodeller, estimerer 
% residualer i en enkel lineær regresjonsmodell og 
% lager residualplott
%
 
n = 100;
figure(1);
figure(2);
pause;
 
% betrakter først modellen Y = 0.5 + 1.5x + epsilon, epsilon ~ n(.;0,1)
x = unifrnd(-1,1,n,1);
y = 0.5 + 1.5 * x + normrnd(0,1,n,1);
figure(1);
plot(x,y,'.');
title('kryssplott');
 
betahat = sum((x-mean(x)) .* y) / sum((x-mean(x)).^2);
alphahat = mean(y) - betahat * mean(x);
residualhat = y - alphahat - betahat * x;
hold on;
plot([min(x) max(x)],alphahat + betahat * [min(x) max(x)],'r-');
hold off;
figure(2);
plot(x,residualhat,'.');
title('residualplott');
hold on;
plot([min(x) max(x)],[0 0],'r-');
hold off;
pause;
 
 
% betrakter så en modell med ikke-lineær forventning
x = unifrnd(-1,1,n,1);
y = 0.5 + 1.5 * x + 4 * x.^2 + normrnd(0,1,n,1);
figure(1);
plot(x,y,'.');
title('kryssplott');
 
betahat = sum((x-mean(x)) .* y) / sum((x-mean(x)).^2);
alphahat = mean(y) - betahat * mean(x);
residualhat = y - alphahat - betahat * x;
hold on;
plot([min(x) max(x)],alphahat + betahat * [min(x) max(x)],'r-');
hold off;
figure(2);
plot(x,residualhat,'.');
title('residualplott');
hold on;
plot([min(x) max(x)],[0 0],'r-');
hold off;
pause;
 
 
% en annen modell med ikke-lineær forventning
x = unifrnd(-1,1,n,1);
y = exp(2.5 * x) + normrnd(0,1,n,1);
figure(1);
plot(x,y,'.');
title('kryssplott');
 
 
betahat = sum((x-mean(x)) .* y) / sum((x-mean(x)).^2);
alphahat = mean(y) - betahat * mean(x);
residualhat = y - alphahat - betahat * x;
hold on;
plot([min(x) max(x)],alphahat + betahat * [min(x) max(x)],'r-');
hold off;
figure(2);
plot(x,residualhat,'.');
title('residualplott');
hold on;
plot([min(x) max(x)],[0 0],'r-');
hold off;
pause;
 
 
% så en modell med ikke-konstant varians
x = unifrnd(-1,1,n,1);
y = 0.5 + 1.5 * x + 0.5 * (x.^2 + 0.1) .* normrnd(0,1,n,1);
figure(1);
plot(x,y,'.');
title('kryssplott');
 
 
betahat = sum((x-mean(x)) .* y) / sum((x-mean(x)).^2);
alphahat = mean(y) - betahat * mean(x);
residualhat = y - alphahat - betahat * x;
hold on;
plot([min(x) max(x)],alphahat + betahat * [min(x) max(x)],'r-');
hold off;
figure(2);
plot(x,residualhat,'.');
title('residualplott');
hold on;
plot([min(x) max(x)],[0 0],'r-');
hold off;
pause;
 
 
 
 
 
% så en modell med skjev fordeling
x = unifrnd(-1,1,n,1);
y = 0.5 + 1.5 * x + 0.5 * (exprnd(1,n,1) - 1);
figure(1);
plot(x,y,'.');
title('kryssplott');
 
 
betahat = sum((x-mean(x)) .* y) / sum((x-mean(x)).^2);
alphahat = mean(y) - betahat * mean(x);
residualhat = y - alphahat - betahat * x;
hold on;
plot([min(x) max(x)],alphahat + betahat * [min(x) max(x)],'r-');
hold off;
figure(2);
plot(x,residualhat,'.');
title('residualplott');
hold on;
plot([min(x) max(x)],[0 0],'r-');
hold off;
pause