[R] isoMDS

[Hanke, Alex]

Distances cannot always be constructed from similarities. This can be done
only if the matrix of similarities is nonnegative definite. With the
nonnegative definite condition, and with the maximum similarity scaled so
that s_ii=1, d_ik=(2*(1-s_ik))^-.5

Check out the vegan package.
Alex

-----Original Message-----
From: Doran, Harold [mailto:HDoran at air.org] 
Sent: September 8, 2004 10:00 AM
To: r-help at stat.math.ethz.ch
Cc: Doran, Harold
Subject: [R] isoMDS


Dear List:

I have a question regarding an MDS procedure that I am accustomed to
using. I have searched around the archives a bit and the help doc and
still need a little assistance. The package isoMDS is what I need to
perform the non-metric scaling, but I am working with similarity
matrices, not dissimilarities. The question may end up being resolved
simply.

Here is a bit of substantive background. I am working on a technique
where individuals organize items based on how similar they perceive the
items to be. For example, assume there are 10 items. Person 1 might
group items 1,2,3,4,5 in group 1 and the others in group 2. I then turn
this grouping into a binomial similarity matrix. The following is a
sample matrix for Person 1 based on this hypothetical grouping. The off
diagonals are the similar items with the 1's representing similarities. 
  a b c d e f g h i j
a 1 1 1 1 1 0 0 0 0 0
b 1 1 1 1 1 0 0 0 0 0
c 1 1 1 1 1 0 0 0 0 0
d 1 1 1 1 1 0 0 0 0 0
e 1 1 1 1 1 0 0 0 0 0
f 0 0 0 0 0 1 1 1 1 1
g 0 0 0 0 0 1 1 1 1 1
h 0 0 0 0 0 1 1 1 1 1
i 0 0 0 0 0 1 1 1 1 1
j 0 0 0 0 0 1 1 1 1 1


Each of these individual matrices are summed over individuals. So, in
this summed matrix diagonal elements represent the total number of
participants and the off-diagonals represent the number of times an item
was viewed as being similar by members of the group (obviously the
matrix is symmetric below the diagonal). So, a "4" in row 'a' column 'c'
means that these items were viewed as being similar by 4 people. A
sample total matrix is at the bottom of this email describing the
perceived similarities of 10 items across 4 individuals.

It is this total matrix that I end up working with in the MDS. I have
previously worked in systat where I run the MDS and specify the matrix
as a similarity matrix. I then take the resulting data from the MDS and
perform a k-means cluster analysis to identify which items belong to a
particular cluster, centroids, etc.

So, here are my questions. 

1)	Can isoMDS work only with dissimilarities? Or, is there a way
that it can perform the analysis on the similarity matrix as I have
described it?
2)	If I cannot perform the analysis on the similarity matrix, how
can I turn this matrix into a dissimilarity matrix necessary? I am less
familiar with this matrix and how it would be constructed?

Thanks for any help offered,

Harold 


  a b c d e f g h i j
a 4 2 4 3 3 2 0 0 0 0
b 2 4 2 3 1 0 2 2 2 2
c 4 2 4 3 3 2 0 0 0 0
d 3 3 3 4 2 1 1 1 1 1
e 3 1 3 2 4 3 1 1 1 1
f 2 0 2 1 3 4 2 2 2 2
g 0 2 0 1 1 2 4 4 4 4
h 0 2 0 1 1 2 4 4 4 4
i 0 2 0 1 1 2 4 4 4 4
j 0 2 0 1 1 2 4 4 4 4

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