I am looking to understand the procedure behind the determination of delta which is produced from the polyval function. I am not a stats buff so I am looking for some type of introduction or tutorial, I was unable to find a reference anywhere online (I have spent quite some time on google trying to find a explanation/tutorial/introduction). I can examine the calculation by typing 'edit polyval' but that only goes so far. It appears that delta is determined as follows:
E = V/R;% V is Vandermonde matrix for new X
e = sqrt(1+sum(E.*E,2));
delta = normr/sqrt(df)*e;
It is this procedure that I don't understand. I did find a vague reference to Numerical Recipes in C section 15.6 but no equations in that document follow the methodology in the polyval function.
Apart from the lack of theoretical understanding I have tried 1 piece of code which produce curious results. In the code snippet I create some data with large amount of noise and then extended the prediction into the future. I don't understand why delta increases as x increases. Delta at first decreases then increases in an apparent linear fashion. Why would delta increase linearly, or at all? I understand that delta represents prediction error estimates of the standard deviation at the future observation but the mechanism of how delta is computed is lost.
randn('seed',1);
%create initial data
x=[0,1,2,3,4,5,6]';
y = 2.5 * x + 80 + 50 * randn(length(x),1);
[p s] = polyfit(x, y, 1);
%create extended range
newX = 0:100;
[y_fit delta] = polyval(p, newX, s);
figure;
%plot data in blue
plot(x,y,'b*')
hold on
%add 2 sigma bounds
plot(newX, y_fit + 2 * delta, '--', newX, y_fit - 2 * delta, '--')
legend('Data','2 Sigma Bound')
%plot delta
figure;
plot(newX,delta)
