From: tayloe@uinpla.npl.uiuc.edu (Rex Tayloe)
Newsgroups: sci.math.num-analysis,sci.math.stat,sci.physics
Subject: HELP: A least-squares problem with nonlinear constraints
Date: 10 Sep 92 01:06:53 GMT
Reply-To: tayloe@uinpla.npl.uiuc.edu (Rex Tayloe)
Organization: Nuclear Physics Laboratory, University of Illinois, Urbana-Champaign


   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
 |_|/ \ _|/ |_|/      |_____|/