Here is a start on analyzing your problem. It doesn't get all the way
to your goal, but perhaps others on the list can help... -- DFB.
------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
------------------------------------------------------------
The problem rephrased:
Given a A + b B ---> c X
To find {p, q} such that p A + q B ---> X
When {a, b, c} are fixed numbers, p = a/c and q = b/c
When {a, b, c} contain known uncertainties
{a+-s(a), b+-s(b), c+-s(c)}: one desires {p+-s(p), q+-s(q)}
where {s(p), s(q)} are defined consistently with {s(a) s(b) s(c)}.
Q1. Are the uncertainties {s(a) s(b) s(c)} standard errors of the
estimates of the coefficients, or 95% confidence intervals, or what?
Q2. Are the coefficients estimated assumed to be distributed
normally, or to follow some other distribution?
In the absence of better strategy, one can always use interval
arithmetic to obtain numerical uncertainties about {p, q}:
since {a, b, c} are all positive, AND their uncertainties are all
smaller than their estimates,
p_max = [a + s(a)] / [c - s(c)], p_min = [a - s(a)] / [c + s(c)];
q_max = [b + s(b)] / [c - s(c)], q_min = [b - s(b)] / [c + s(c)].
and the interval [p_min, p_max] can be expressed as p*+-d(p);
but this is not a statistical interval, and d(p) cannot be expected to
be the same sort of thing as {s(a) s(b) s(c)}: it is probably larger,
perhaps considerably larger, than s(p) would be. However, it may be
better than nothing. (Similarly for q .)
For this approach, it is a serious problem that the uncertainty in b
is almost as large as the estimate of b : q will have very wide
boundaries, q_min being near 0 and q_max being more than twice the value
(0.18) calculated as though {b c} were known with no uncertainty.
(But that will be a serious problem in any case.)
For a proper confidence interval on p one would need to know the
presumed distributions of a and c (e.g., normal (= Gaussian) with
mean m(a) and standard deviation s(a), etc.), AND what that implies
about the distribution of p = a/c (e.g., normal with mean m(p) and
standard deviation s(p); or chi with parameters {m(p), s(p), df};
or whatever; where m(p) and s(p) are functions of the parameters of
a and c. (This last bit I've never had occasion to work out, but
probably someone else on the list would know.)
======= Original message =================
Date: 18 Apr 2004 14:35:58 -0700
From: Juergen Fritz <[EMAIL PROTECTED]>
To: [EMAIL PROTECTED]
Subject: Re: Division of confidence interval limits
A bit more information in detail:
0.2813+-0.0984 A + 0.0192+-0.0184 B ----> 0.1052+-0.0509 X
is a biochemical reaction I obtain from a parameter identification
(using several Matlab optimizers). I use the Fisher Information matrix
to "a posteriori" estimate the confidence limits for all parameters as
shown in the reaction.
As of now, I have never been interested in the confidence limits, so I
could easily normalize a reaction such as
0.2813 A + 0.0192 B ----> 0.1052 X
with respect to X to obtain
2.67 A + 0.18 B --> X .
Now I am very much interested in the confidence limits (or intervals) of
all parameters.
The identification procedure yields:
0.2813+-0.0984 A + 0.0192+-0.0184 B ----> 0.1052+-0.0509 X
Is there a possibility to calculate the coefficients of A and B such
that the biochemical reaction can be rewritten as?
xx A + yy B ----> X
without having to change the parameter estimation procedure just by
using the output as I obtain it so far. I realize that it is
problematic, that the coefficient of X is now deterministic equal to
one, where the uncertainties expressed by the confidence limits should
now be included in the coefficients of the other reaction components A
and B.
The normalization is necessary to compare the identification results
with a reference reaction scheme.
Can you think of a proper solution to this problem?
Any help on that greatly appreciated. I hope these explanations better
explain my problem.
Thanks in advance
Juergen U. Fritz ( [EMAIL PROTECTED] )
.
.
=================================================================
Instructions for joining and leaving this list, remarks about the
problem of INAPPROPRIATE MESSAGES, and archives are available at:
. http://jse.stat.ncsu.edu/ .
=================================================================
