Polyfit with scaling - Q regarding polyval

[Ben Abbott]

I'm moving past polyfit, and looking into what is needed for polyval.

I'm finding that my grasp of statistics is short of what I need, and  
thus, I am in need of some statistics expertise.

Matlab's polyval.m and polyconf.m each are able to provide confidence/ 
prediction intervals. If I grasp the intended operation of these  
routines

	[y, delta] = polyval (p, x, s);

is equivalent to either

	[y, delta] = polyconf (p, x, s, 'alpha', 0.5, 'predopt', 'curve',  
'simopt', 'off');

I'm attempting to modify Paul Kienzle's routine, polyconf.m, support  
the Matlab syntax. Unfortunately, I have no idea what is intended by  
"simultandous" and "nonsimultaneous" intervals ... well except that  
Paul's appear to be "simultaneous" (I reached that conclusion after  
Googling the subject and browsing a few papers).

For example, http://www.uwgb.edu/zornm/1997-01%20Anal.%20Chem.%2069,%203069-3075.pdf

In the paper above, eqn (3) describes the "nonsimultaneous tolerance  
interval".

The routine polyconf needs to do calculate eqn (3). The routine  
polyval also needs something similar, but does not require alpha to  
vary (I think ... please correct me if I'm wrong).

So, I'm looking for two things.

First, I'd like a convenient way to calculate equation (3) in that  
above paper for this applicaiton? I'm guessing it might look something  
like this?

         t = t_inv (1-alpha/2, s.df) + norminv (??) * sqrt (s.df /  
chi2inv (alpha/2, s.df));

... probably not ... since it really is a wild guess on my part ;-)

Second, I'd like a confirmation that polyval is intended to return  
nonsimultaneous tolerances (my comparison of Paul's polyocnf and  
Matlab's polyval have led me to that guess/conclusion).

For the 2nd, if someone has a copy of Matlab's statistics toolbox  
please let me know that the code below returns. ;-)

r = 0:10:50;
p = poly (r);
p = p / max(abs(p));
x = linspace(0,50,11);
y = polyval(p,x) + 0.5*rand(size(x));
[pf, s] = polyfit (x, y, numel(r));
[y1, delta1] = polyval (pf, x, s);
[y2, delta2] = polyconf (pf, x, s, 'alpha', 0.5, 'predopt', 'curve',  
'simopt', 'off');
[y3, delta3] = polyconf (pf, x, s, 'alpha', 0.5, 'predopt', 'curve',  
'simopt', 'on');

disp ([delta1; delta2; delta3])

TiA
Ben


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