% covariance in 2D:
x = 2.0*randn(1000, 1);
y = 0.5*x + 0.6*randn(1000, 1);
scatter(x, y)
cv = cov( [x, y] );
[v d] = eig(cv);
hold on
quiver([0 0], [0 0], v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 2);
axis( 'equal' );
hold off
pause

% covariance of independent coordinates in 2D:
x = 1.5*randn(1000, 1);
y = 0.8*randn(1000, 1);
scatter(x, y)
cv = cov( [x, y] );
[v d] = eig(cv);
hold on
quiver([0 0], [0 0], v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', 'r', 'LineWidth', 2);
axis( 'equal' );
hold off
pause

% covariance in 3D:
x = 2.0*randn(1000, 1);
y = -0.5*x + 0.2*randn(1000, 1);
z = 2.0*x + 0.6*y + 2.0*randn(1000, 1);
scatter3(x, y, z)
cv = cov( [x, y, z] );
[v d] = eig(cv);
hold on
quiver3([0 0 0], [0 0 0], [0 0 0], v(1,:).*sqrt(diag(d))', v(2,:).*sqrt(diag(d))', v(3,:).*sqrt(diag(d))', 'r', 'LineWidth', 2);
axis( 'equal' );
hold off
pause()