Re: 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