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 Subject: Re: MDS with a 400X400 Matrix From: "F. James Rohlf" <[log in to unmask]> Reply To: Classification, clustering, and phylogeny estimation Date: Wed, 13 Nov 2002 12:29:02 -0500 Content-Type: text/plain Parts/Attachments: text/plain (447 lines)
```I repeat (with emphasis added) "... I don't know how to tell whether the
nonmetric-MDS will give a _much_ better solution unless I try it ...". The
relative performance of different methods must be data dependent. My
experience may be different than others because my data sets usually involve
variables that have much higher correlations than usually found in fields
where these methods are typically used.

My module in NTSYSpc was based originally on the ideas in mdscal (Kruskal's
/ Carmone's software) but drifted independently to include some kyst ideas.
It does allow for both a principal coordinates analysis (=classical scaling)
as a starting configuration or a user provided configuration. A user can
specify stopping after 0 iterations but one does not need to as it displays
the history of stress values as it iterates. It has both monotone and linear
regression options. Its present size limitation is just due to my keeping
the number of elements in the half dissimilarity matrix under 64K. To
process larger problems I would have to change the width of the variable
used to hold the indices for the various matrices used in the computations
but that is not difficult - no one has asked to do it until now.

For my purposes both regression methods usually give much more satisfactory
solutions than the initial principal coordinates analysis because prin.
coords. (like principal components) gives so much weight to the larger
distances. In my applications I am also interested in fitting the near
neighbor relationships.

The only times I have seen the monotone regression option really fail is
when the clusters are too good (gaps between clusters are larger than any
within-cluster distances). In such cases stress can be made nearly perfect
by moving the clusters far enough apart. In such cases the solution provided
by the linear regression option does not degenerate in the same way.

Jim Rohlf

> -----Original Message-----
> Sent: Tuesday, November 12, 2002 11:52 PM
> To: Classification, clustering, and phylogeny estimation;
> Cc: J. Douglas Carroll
> Subject: Re: MDS with a 400X400 Matrix
>
>
> To Jim Rohlf and whomever else it may concern,
>
> If you are concerned about the fit via a monotonic instead of a
> merely linear
> function, what you can do is to fit the configuration (say in 2
> dimensions) via
> (presumably the classical) METRIC MDS procedure and then compute
> the STRESS for
> THAT configuration.  One way to do this is to use this as the starting
> configuration
> for KYST or another nonmetric (two-way) MDS program, in which the initial
> STRESS
> (called, in KYST, KYST2 or KYST2-A, and other earlier versions of
> Kruskal's
> nonmetric
> MDS program (i.e., some of the later versions of MDSCAL) the
> STRESS for the
> "zeroth iteration" if I recall this correctly.  (I believe there
> is also an
> option
> in the various versions of KYST to simply compute that "zeroth iteration"
> STRESS
> value-- using either STRESSFORM1 or STRESSFORM2-- but I assume
> the first of
> these,
> which is generally most appropriate for analyzing standard
> symmetric two-way
> matrices of proximity data).  I believe if you do this, you would
> find that the
> STRESS(Formula 1, which I will henceforth presume is used) computed this
> way for
> the METRIC solution will be almost the same (perhaps slightly larger, but
> probably
> not substantially so, and not enough to make up for the effective extra
> parameters
> being fit in the nonmetric 2-dimensional analysis) as that for
> the NONMETRIC
> solution in the same dimensionality.  Assuming you have obtained
> the global
> minimum in the nonmetric analysis,  that solution will
> necessarily yield at
> least as low, and almost certainly a LOWER, STRESS computed this
> way for the
> solution obtained via the metric 2-dimensional analysis-- but there is no
> theoretically rigorous way to determine the number of effective additional
> parameters the nonmetric analysis provides-- but it clearly DOES add what
> amounts
> to extra parameters to the overall model being fit, so that the
> STRESS will
> necessarily be equal to or less than that of the metric solution
> used as the
> "zeroth iteration" configuration in this way.  One way to verify this
> empirically
> is to note that the various versions of KYST actually use what
> amounts to a
> form
> of metric analysis-- but one I believe to be inferior in various
> ways to the
> full classical metric analysis a la Torgerson et al-- and the final
> solution obtained
> always has a lower STRESS value.  A number of people have
> experimented with
> using
> the full classical two-way metric MDS solution as a starting
> point for KYST
> analysis,
> and always find a similar result-- which is empirical proof, I
> would argue,
> that the
> resulting nonmetric model with all parameters fit simultaneously
> necessarily yields
> lower STRESS than when the model is fit sequentially by first doing the
> metric analysis
> and then using the MFIT algorithm to fit the best OLS monotone function,
> CONDITIONAL on
> the distances being computed based on that metrically fitted
> configuration.  These two
> facts together imply that there are effectively at least SOME additional
> parameters
> implicitly being allowed via the nonmetric analysis-- which can only be
> attributed, I
> believe, to the addition of the monotone function as an integral part of
> the nonmetric
> model thus being fitted.  (If an unconstrained nonlinear function were
> being fitted
> to approximate the distances as a function of the proximity data,
> and this
> function itself
> had p parameters, then you could definitively state that p extra
> parameters/degrees of
> freedom are being added by allowing this nonlinear function.  When,
> however, the function
> being fit, while generally nonlinear, is defined simply by a
> constraint to
> (weak) monotonicity,
> NOONE (to my knowledge) has yet been able to come up with a definitive
> theoretical OR
> empirical way to assess the number of effectively added
> parameters.  Noone
> questions the
> model being
> fitted, but
> I know of noone who has ever claimed to have a justifiable answer to the
> question of how
> many extra parameters this involves!)
>
> While I've assumed above that one of the versions of KYST or
> another of the
> Kruskal
> (et al) MDS programs is used for the nonmetric analysis, largely because
> I'm not
> familiar with the specific software for MDS Jim mentioned in his earliest
> e-mail
> on this topic (the "NTSYSpc  software"), which he indicates, among many
> other options,
> allows some form of nonmetric or metric MDS.  Since I don't know exactly
> what algorithm
> Jim uses for MDS in this software, I cannot comment on using IT, or other
> software, such
> as Willem Heiser's PROXSCAL, recently incorporated into the professional
> version of
> SPSS (but not in the student version, nor, I assume in any of the earlier
> versions of
> any kind-- which, I believe, allow only ALSCAL, which does NOT optimize
> STRESS, but a
> different loss function called "SSTRESS", which many people-- myself
> included-- feel is
> less desirable than an algorithm for two-way metric or nonmetric MDS
> optimizing STRESS;
> the three-way case becomes even more complicated vis a vis comparison of
> different
> algorithms, but suffice it to say that I personally feel ALSCAL has some
> especially serious
> problems when conducting a three-way MDS analysis).  IF Jim Rohlf's
> software DOES optimize
> Kruskal's STRESS (in the two-way case-- there are MANY different ways to
> generalize the
> STRESS measure to the three-way case, so, strictly speaking, I
> cannot speak
> to it in that
> case-- but the analysis we are now discussing is strictly limited to the
> simplest and most
> straightforward case of two-way MDS, so it is quite easy to determine
> whether or not it is
> an algorithm optimizing Kruskal's STRESS in this case), then it
> is possible
> that the same
> approach (to computing STRESS for the configuration obtained as the
> solution to the two-way
> metric MDS program, whether based on the classical method associated with
> Torgerson or a
> different type of two-way metric MDS analysis).  In practice, it may come
> down to very
> detailed program features, such as whether there is a provision to use a
> user provided
> configuration as the starting configuration, and whether in that case the
> best fitting OLS monotonic function relating data to distances is
> computed
> (and provided as output) for that
> starting configuration-- and then STRESS computed for this "zeroth
> iteration" solution (or,
> if this is not ordinarily done, whether there is an option to LIMIT the
> total number of iterations to zero, which really means simply
> computing the
> monotonic function and then the
> STRESS for that input configuration ONLY, and not iterating the gradient
> based algorithm at
> all beyond that point-- effectively simply computing the STRESS value for
> the user provided
> configuration and nothing more.  IF the NTSYSpc software BOTH uses an
> algorithm essentially
> equivalent to Kruskal's in that it optimizes (minimizes) STRESS, not some
> other loss function,
> AND if it allows one or both of these options, then what I
> suggest COULD be
> done using this
> software Jim Rohlf provides as well as by using some version of KYST
> (say).  It would also be
> fairly easy to write a standalone program, if one has Kruskal's MFIT
> algorithm available as a
> subroutine-- or is willing and able to reproduce it-- that would
> go through
> these steps and
> thus calculate the STRESS for ANY configuration provided as input.
>
> In short-- T\there are many ways to implement what I suggest
> (although not
> all would work for
> a matrix as large as 400x400-- none of Kruskal's algorithms used
> in the way
> described earlier
> could manage matrices that large for example-- but I'm sure this
> goal could
> be achieved one
> way or another easily enough.  If it were desired to do this
> numerous times
> (or even to devise
> a new approach consisting of analyzing the data via a metric MDS method--
> say the classical
> two-way metric MDS method-- and then computing the STRESS via use of a
> procedure, equivalent
> to the standalone program mentioned earlier, designed only to carry out
> that specific operation!
>
> (More than) enuf for now,
>
> Doug Carroll.
>
> At 07:58 PM 11/12/2002 -0500, F. James Rohlf wrote:
> >I agree with Doug's observations but the problem is that I don't
> know how to
> >tell whether the nonmetric-MDS will give a much better solution
> unless I try
> >it and see what the stress value is. Linear-MDS takes less computing and
> >avoids some of the degeneracy problems that Doug alludes to. I think they
> >are usually worth a try as 2D nonmetric-MDS can sometimes give a
> better fit
> >than a 3D metric-MDS. PCA, PCoord (classical scaling) can be
> much worse for
> >relationships among close points.
> >
> >Jim Rohlf
> >
> > > -----Original Message-----
> > > From: Classification, clustering, and phylogeny estimation
> > > Sent: Tuesday, November 12, 2002 9:40 PM
> > > Subject: Re: MDS with a 400X400 Matrix
> > >
> > >
> > > To whomever it may concern (especially Jim Rohlf and David Dubin),
> > >
> > > I sent an e-mail response to this same broadcast e-mail sent
> out via the
> > > CLASS-L system, but addressed it directly to Jim Rohlf,
> since, for some
> > > reason I misinterpreted this as being an inquiry initiated by him, not
> > > one from someone else (Dirk Meurer, I now see) simply passed on by Jim
> > > as administrator of CLASS-L.  I had thought at the time that perhaps
> > > David and Mark Rorvig, prior to Rorvig's tragic and untimely
> death last
> > > year, had put together such a program.  While I gather from
> this message
> > > from David that, instead, Michael Trosset (who was also involved
> > > in the same
> > > DIMACS Workshop on MDS Algorithms I mentioned in that earlier
> > > to Jim) has done so.  I would therefore endorse David's
> > > recommendation that
> > > this software developed by Trosset be used for this purpose.
> I personally
> > > think that, in practice, it seldom makes a great difference
> > > whether one uses
> > > an MDS procedure that is purely "metric" (as long as you're
> > > reasonably careful
> > > to make sure the data are transformed, if necessary, to a form as
> > > consistent
> > > as possible with the assumptions that the procedure is based on-- in
> > > particular,
> > > most likely that they be DISsimilarities-- which may simply
> > > entail reversing
> > > the scale, if data are similarities, by, say, subtracting all
> > > values from the
> > > largest similarity value in the matrix, and possibly, if the
> > > particular method
> > > assumes RATIO SCALE instead of merely INTERVAL SCALE
> dissimilarities, that
> > > these
> > > dissimilarity values are transformed so that it is reasonable to
> > > assume that a
> > > dissimlarity of zero (0) corresponds to a DISTANCE of zero (0) in the
> > > recovered
> > > multidimensional representation-- or "nonmetric" (in which the
> > > similarities or
> > > dissimilarities are assumed only to be monotonic with the
> proximities--
> > > monotone
> > > non-increasing or monotone non-decreasing respectively for
> the two types--
> > > sim.'s
> > > or dissim.'s-- of proximity data).  The metric MDS procedures are
> > > sufficiently
> > > robust under even  fairly severe nonlinearities in the
> monotonic "distance
> > > function"
> > > transforming proximities into distances in the underlying
> representation
> > > that the
> > > solutions are usually almost totally indistinguishable (as long
> > > as the same
> > > dimensionality is assumed in two analyses, of course), although,
> > > as is well
> > > known,
> > > an orthogonal rotation of a two-way MDS solution (based, as
> almost all MDS
> > > algorithms are, on assumption of Euclidean metric in the
> > > underlying space), no
> > > matter HOW obtained, is necessary to bring even what are
> really identical
> > > solutions
> > > into exact congruence (since Euclidean spaces are defined only up to
> > > similarity
> > > transforms-- including an orthogonal rotation, as well as
> translation of
> > > the origin
> > > and, in some cases a possible uniform dilation resulting in
> multiplication
> > > of all
> > > distances by a positive constant-- but the latter two types of
> > > transformations are
> > > typically resolved by normalization conventions, so generally do
> > > not need to be
> > > considered).  In fact, not only are metric solutions (in the same
> > > dimensionality)
> > > usually as good as nonmetric ones, there is considerable empirical and
> > > theoretical
> > > evidence that, for certain types of data, they are in fact
> > > BETTER-- being less
> > > susceptible to various degeneracies and quasi-degeneracies
> that can affect
> > > many
> > > nonmetric MDS analyses, if the data exhibit certain
> characteristics that
> > > are not at
> > > all unusual in the case of realistic proximity data from
> various domains.
> > >
> > > Best,
> > >
> > > Doug Carroll.
> > >
> > > At 03:52 PM 11/12/2002 -0600, David Dubin wrote:
> > > >Michael Trosset wrote a program for classical metric MDS
> that Mark Rorvig
> > > >and I used with success on matrices much larger than 400 by
> 400. But it
> > > >doesn't do nonmetric MDS. The program is written in Fortran and
> > > requires the
> > > >ARPACK libraries. I was able to compile with g77 on Linux  and
> > > Solaris with
> > > >little trouble.
> > > >
> > > >Dave Dubin
> > > >
> > > >
> > > > > Dear Listmembers,
> > > > > I would like to do a MDS with a 400X400 square, symetric matrix of
> > > > > (dis)similaritys. Most MDS-Software is limited to a much
> > > smaller amount
> > > > > of variables though. I have been told, that SAS might be able
> > > to process
> > > > > the analyis I need, but this is quite inconvenient for me
> in terms of
> > > > > access to the software, hardware-requirements etc. Could
> anybody tell
> > > > > me, if there is a stand-alone program that can do MDS (preferably
> > > > > nonmetric) with a matrix of that size?
> > > > >
> > > > > Thanks a lot for your help,
> > > > > dirk
> > > > >
> > > > > P.S: My Matrix is not suitable for factor analysis and
> clustering does
> > > > > not produce the results I need.
> > > > >
> > >
> > >
> > >
> > >
> ######################################################################
> > >    # J. Douglas Carroll, Board of Governors Professor of
> Management and #
> > >    #Psychology, Rutgers University, Graduate School of
> Management,      #
> > >    #Marketing Dept., MEC125, 111 Washington Street, Newark,
> New Jersey  #
> > >    #07102-3027.  Tel.: (973) 353-5814, Fax: (973) 353-5376.
>           #
> > >    # Home: 14 Forest Drive, Warren, New Jersey 07059-5802.
>           #
> > >    # Home Phone: (908) 753-6441 or 753-1620, Home Fax: (908)
> 757-1086.  #
>           #
> > >
> ######################################################################
> > >
>
>
>
>    ######################################################################
>    # J. Douglas Carroll, Board of Governors Professor of Management and #
>    #Psychology, Rutgers University, Graduate School of Management,      #
>    #Marketing Dept., MEC125, 111 Washington Street, Newark, New Jersey  #
>    #07102-3027.  Tel.: (973) 353-5814, Fax: (973) 353-5376.             #
>    # Home: 14 Forest Drive, Warren, New Jersey 07059-5802.              #
>    # Home Phone: (908) 753-6441 or 753-1620, Home Fax: (908) 757-1086.  #