Thread View: sci.math.num-analysis
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Started by Rex Tayloe
Thu, 10 Sep 1992 01:06
HELP: A least-squares problem with nonlinear constraints
Author: Rex Tayloe
Date: Thu, 10 Sep 1992 01:06
Date: Thu, 10 Sep 1992 01:06
62 lines
3960 bytes
3960 bytes
I have been struggling with a problem that I have encountered in my thesis
analysis and would like to pose the problem to see if there might be a kind
soul out there with some helpful advice.
I am analyzing data from an experiment where we collide antiprotons with
protons and look at several hyperon channels. My particular goal is to find
the antiproton,proton->lambda,sigma (pbar,p->LS) events. And, because they
are very similar, I am also looking at antiproton,proton->lambda,lambdabar
(pbar,p->LL) events.
So, for each event I want to do a least-squares fit subject to
energy-momentum conservation at each vertex (3*4 equations for LL, 4*4 for LS).
The least-squares sum I want to minimize is:
(ymeas-yfit)V(ymeas-yfit)
transpose---^^^^^^^^^^
where ymeas = vector of measured parameters (direction cosines, beam momentum)
yfit = vector of *improved* values
V = covariance matrix of the measured parameters (not diagonal
because the measured parameters are correlated)
Then, after fitting each event to the LL and LS hypothesis, I decide whether
it was a LL, LS, or neither. OK, I think that I have the general idea. The
problems are the specifics. I will list some questions that have come up in my
failures in trying to implement this plan. Comments and/or advice is
appreciated.
1) Is anyone familiar with the method on p.308 of "Probability and
Statistics in Particle Physics" by Frodesen et. al? (I dont know the proper
name of the method). It works well for most events using the LL hypothesis but
fails for the LS hypothesis due to the fact that one of the matricies is very
close to singular and cannot be inverted. What am I doing wrong?
2) My next try was with the NAGlib routine, E04UCF. It didnt do as well on
some events actually. It misses some minima that the above algorithm finds.
I feel that it would work if I could limit the step size. I tried changing
the "STEP LIMIT" to a small number but it didn't do what I expected. Any ideas?
3) Are there any more good books out there for this stuff? I really like
the above-mentioned statistics book, but can't handle "Practical Optimization"
by Gill et al. for example. Something more in-line for a physics grad student
who needs to hurry up and solve this problem so that he can graduate?
4) Are there guidelines for a reasonable covariance matrix? I find myself
looking at the darn thing not being able to tell if it is right (other than the
diagonal terms are the error**2). Should it be positive definite or something
like that? Mine always have a fairly large condition number (~10000-100000) Is
this a signal of something wrong?
Well, if you made it this far through my note I am glad. Please post
comments or E-mail me if you prefer. I really would like to talk to someone
with experience about this.
Thanks
__ __ ____ __
/__/\ /__/| /____/ \ /__/| Rex Tayloe TAYLOE@uinpla.npl.uiuc.edu
| \\| | || __ \ /|| | | Nuclear Physics Lab TAYLOE@uiucnpl.bitnet
| |\ \| | || |__) |/ | | |__ U of Illinois UINPLA::(47165::)TAYLOE
| | \ | || ___/ | |/__ /| Champaign Il 61820 (317)333-3190
|_|/ \ _|/ |_|/ |_____|/
Re: HELP: A least-squares problem with nonlinear constraints
Author: Rex Tayloe
Date: Thu, 10 Sep 1992 02:18
Date: Thu, 10 Sep 1992 02:18
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In article <0096060C.3C725B20@uinpla.npl.uiuc.edu>, I wrote
>
> I have been struggling with a problem that I have encountered in my thesis
>analysis and would like to pose the problem to see if there might be a kind
>soul out there with some helpful advice.
>
> I am analyzing data from an experiment where we collide antiprotons with
>protons and look at several hyperon channels. My particular goal is to find
>the antiproton,proton->lambda,sigma (pbar,p->LS) events. And, because they
>are very similar, I am also looking at antiproton,proton->lambda,lambdabar
>(pbar,p->LL) events.
>
> So, for each event I want to do a least-squares fit subject to
>energy-momentum conservation at each vertex (3*4 equations for LL, 4*4 for LS).
>The least-squares sum I want to minimize is:
> (ymeas-yfit)V(ymeas-yfit)
> transpose---^^^^^^^^^^
I meant:
(ymeas-yfit)V(-1)(ymeas-yfit)
transpose---^^^^^^^^^^
__ __ ____ __
/__/\ /__/| /____/ \ /__/| Rex Tayloe TAYLOE@uinpla.npl.uiuc.edu
| \\| | || __ \ /|| | | Nuclear Physics Lab TAYLOE@uiucnpl.bitnet
| |\ \| | || |__) |/ | | |__ U of Illinois UINPLA::(47165::)TAYLOE
| | \ | || ___/ | |/__ /| Champaign Il 61820 (317)333-3190
|_|/ \ _|/ |_|/ |_____|/
Re: HELP: A least-squares problem with nonlinear constraints
Author: Jim Carr
Date: Sun, 13 Sep 1992 18:40
Date: Sun, 13 Sep 1992 18:40
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I am far from expert in your particular problem, but you seem to be encountering a common problem in numerical analysis. Perhaps my answer will trigger some other replies to your inquiry. In article <0096060C.3C725B20@uinpla.npl.uiuc.edu> tayloe@uinpla.npl.uiuc.edu (Rex Tayloe) writes: > [... details of actual thesis problem omitted ...] > 1) Is anyone familiar with the method on p.308 of "Probability and >Statistics in Particle Physics" by Frodesen et. al? (I dont know the proper >name of the method). It works well for most events using the LL hypothesis but >fails for the LS hypothesis due to the fact that one of the matricies is very >close to singular and cannot be inverted. What am I doing wrong? If it is "close" to singular you should get an answer; the problem is with its reliability and the warning flags it should raise. There are two possible problems: your fit parameters in the LS case are not really independent of one another or you have encountered a standard problem with large least squares systems. (A large least-squares matrix tends to resemble the Hilbert matrix, which has an inverse but has the property of being almost indeterminant when you try to solve it with any numerical method on machine with finite precision.) Everything you say seems to argue against the latter problem, unless I missed something important. A look at the error matrix might give some guidance. Your trouble with other methods might also be due to a correlation, since I gather you use the same description of the fit function in all cases. > 4) Are there guidelines for a reasonable covariance matrix? I find myself >looking at the darn thing not being able to tell if it is right (other than the >diagonal terms are the error**2). Should it be positive definite or something >like that? Mine always have a fairly large condition number (~10000-100000) Is >this a signal of something wrong? The off-diagonal entries are telling you about correlations in your fit. If they are large, it is because you are trying to fit parameters that are not entirely independent of one another. Your problem in (1) above would imply that such a result is likely. If there is a single large off-diagonal entry, that is where your correlation lies. You might also try looking at the Chi-square surface in the space of such correlated parameters for guidance; a rotation to a better parameterization could be all you need to get a fit. Hope this helps. -- J. A. Carr | "The New Frontier of which I jac@gw.scri.fsu.edu | speak is not a set of promises Florida State University B-186 | -- it is a set of challenges." Supercomputer Computations Research Institute | John F. Kennedy (15 July 60)
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