Re: MDS with different matrices

Aleks Jakulin wrote:
> "Rich Ulrich" <[EMAIL PROTECTED]> wrote:
>
> > Has somebody cared enough about MDS  to update the
> > computer programs?  It's long been my impression that
> > 'marketing' was using MDS.  From google, it also seems
> > like MDS  sometimes is included in the tools of data mining.
>
>
> MDS has become slightly outdated, but there has been some good work into
> the same direction recently, primarily in the direction of locally linear
> embedding. The problem with MDS is globality: in most stress functions
> short dissimilarities are approximated with a similar precision as large
> dissimilarities. In reality, what a human analyst expects from MDS is more
> a structure alike clustering: one that would identify groups of similar
> objects.
>
> The 'old' solution was Shepard's non-metric MDS. NMDS attempts to modify
> the dissimilarity matrix so that the distance rankings are maintained,
> rather than metric deviations. For example, we are not interested in the
> exact distances between A, B and C, but we do want distance A-B to be
> greater than A-C if C is more dissimilar from A than is B. It does not work
> well in practice, unfortunately. Today, new methods have emerged. For
> example, locally linear embedding instead only evaluates the distances from
> an object to K of its nearest neighbors. This yields nice results.
>
> Good starting points for further exploration are
> http://www.cs.toronto.edu/~roweis/lle/
> http://basis.stanford.edu/carrie-web/
>
> Aleks

Forrest Young still distributes the Fortran code and Dos executable of 
Alscal, the descendent of KYST, the Bell Labs NMDS program.  MDS, for 
some reason, is often used as an acronym for NMDS. I follow those who 
restrict MDS to Shepard's metric scaling, which is identical to Gower's 
Principal coordinates analysis.
http://forrest.psych.unc.edu/

MDS still has huge advantages, if you have access to the case by 
variables matrix. There are a variety of transformations that can be 
used to transform the data matrix prior to doing the metric scaling. 
Pierre Legendre & I wrote a paper in 2001 that describes a number of 
these transformations. Pierre provides code on his web page for mac & pc 
for doing the transformations and PCA's, and I provide the Matlab 4 & 
Matlab 6 code for doing the same:
http://www.es.umb.edu/edgwebp.htm#LegGallMat6
  Strictly speaking the MDS model (not the MDS model) can have problems 
with non-metric distances, producing negative eigenvalues. Legendre & 
Legendre (1998) Numerical Ecology, 2nd ed. review 3 solutions to this 
problem, and their algorithms are programmed in Pierre & my programs.
  Often it is not the ordination of the cases that is important but 
explaining why the cases take the positions they do in low-dimension 
space.  The Gabriel Euclidean distance biplot and correlation biplot are 
important tools for interpreting the low-dimension ordination. Gower's 
book 'biplots' is the best overall description of the process.
  MDS has also been revived by the use of constrained ordination 
techinques. Canonical correspondence analysis can impose conditions that 
either the distances among cases in low-dimension space must be linear 
functions of a set of external explanatory variables, or uncorrelated 
with a set of covariates (partial canonical correspondence analysis). 
When preserving Euclidean distances, as in the MDS model, the technique 
is called redundancy analysis. Both redundancy analysis and canonical 
correspondence analysis (not to be confused with canonical correlation 
analysis) are available in the CANOCO package. Using algorithms 
presented by ter Braak or by Legendre & Legendre, the basic CANOCO or 
redundancy analysis models can be programmed in languages such as Matlab.
  BTW, you can perform an MDS, aka Gower Principal coordinates 
analysis, on a correlation matrix after converting it to a distance 
matrix. This distance matrix will not be Euclidean, so anticipate 
negative eigenvalues. The standard NMDS programs have an option to 
specify whether the matrix entered is a distance or similarity matrix.
Gene Gallagher