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January 2003

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Subject:
From:
Patrick Groenen <[log in to unmask]>
Reply To:
Classification, clustering, and phylogeny estimation
Date:
Tue, 28 Jan 2003 10:23:38 +0100
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Hi Martin,

Below I will give you my opinion on your questions.

>         Recently I am learning about multi-dimensional scaling myself.
> May I ask about pointers to the literature on the following questions?
> As I am pretty new to MDS, please accept my apology if they sound standard
> to you. I am mostly interested in metric MDS, in particular metric least
> square scaling.
>
> 1.      What is the state of the art for the additive constant problem?

Do interval MDS, so that you do not have to worry about the additive
constant.

> 2.      What is the best way to convert a similarity matrix to a
>         dissimilarity matrix? In addition to the additive constant
>         problem, what else can go wrong?

There is no best way to do this. It all depends on your data. Most MDS
programs (I favor SPSS Proxscal) allow you to specify an interval
transformation for similarities. Then automatically large distances will
correspond to small similarities and small distances to large similarities.

> 3.      Suppose I have performed MDS on a set of n by n dissimilarity
>         matrix. Now I have a new item $z$, and I know its
> dissimilarity with
>         all existing n items. If I want to do MDS on these (n+1) items,
>         must I recalculate everything from scratch? Or is there any
>         approximation method to find the co-ordinate for $z$?

There is. It is called external unfolding. Proxscal allows you to fix all
coordinates and leave one free. Alternatively, you could use the Prefmap
program.

> 4.      Has anyone studied the "inverse-mapping" problem of MDS, that is,
>         given an arbitrary point in the projected space, we try to
>         recover the corresponding item? Obviously, this requires
>         knowledge on the space the items are located, in addition to
>         the dissimilarity matrix.

If I understand it correctly, your problem is a simple one if all points are
represented in the space: simply select the one at closest distance. If some
points are not (yet) are represented in the space, you can do external
unfolding for all points that are not yet represented and select the point
that is closest to your original arbitrary space.

For the case that you dissimilarities are derived from a rectangular data
matrix, I refer you to the Biplot book of Gower and Hand (1995). They
formulated an entire theory of how this can be done.

I hope this helps,  Patrick Groenen
-------------------------------------------------
Prof. dr. Patrick J.F. Groenen
Econometric Institute
Erasmus University Rotterdam
Room H11.23
P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
tel:    ++ 31 10 408 1281
fax:    ++ 31 10 408 9162
e-mail: [log in to unmask]
-------------------------------------------------

>         Thank you for your advice.
>
> Best regards,
>
> Martin
>

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