Hello,
my math courses are long away ... could someone tell me which of these two
approaches is the right one? An experiment produced n pairs of points (xi,
yi) for which a first order model is postulated
yi = [1 xi] [b; a]
The parameters are found by letting
A = [ ones(size(x)) x]; B = y; th = A\B; [b, a] = deal(th)
A covariance matrix can be found by
S2y= sumsq(B-A*th)/(length(y)-2); %# a posteriori variance
Cth = inv(A'*A)*S2y;
From here, I want to get, for a given, errorless x0, what is an 90%
confidence interval for y0.
First way: from error propagation technique, I can compute the variance of
y0 as
Sy0 = [1 x0]Cth[1; x0]; %# notice that this depends from x0
but then I start from S2y, which has n-2 degrees of freedom, to [db da]
which is bivariate normal, to Sy0 which is univariate normal. Can I assume
that (y-y0)^2/Sy0 is chi-square with one d.f. ?
Second way: [db da] is a bivariate normal, so, applying the concept of
Mahalanobis distance,
[db da] inv(Cth) [db; da] is chi-square with 2 d.f, confidence intervals
are ellipse. From there, I compute dy = db + x0 da; and I choose the pair
[db, da] belonging to the ellipse which gives rise to an extrema in dy. I
then conclude that the 90% interval on (db, da), the ellipse, is mapped to
[y0-ymin, y0+ymax].
What is the correct approach ? Any pointer to litterature ?
Regards
Pascal