RE: measuring distance in grids

[Darren Schreiber]

My political party formation model actually uses a weighted Euclidean 
distance metric.  I designed this so that I could explore the effect 
of changes in issue salience and separability.  For instance, I 
usually run the model with the weight matrix as the identity matrix:

        1 0
        0 1

So, all issues are equally salient and totally independent.

But, to simulate the effect of the introduction of a new issue, I can 
run the model with the weight matrix

        1 0
        0 0

And then change the weight matrix after some number of iterations:

        1 0
        0 a

Where a is the salience of the new issue.  I used this to estimate 
the conditions under which each of Sundquist's types of political 
realignment occur.

Furthermore, I can demonstrate the development of ideology (what 
political issue goes with what), by changing the 0's and making the 
issues connected.  Geometrically, this makes the preferences look 
like a diagonal ellipse.

Another feature of the weighed Euclidean distance methods that I am 
using is that every single agent in my model can have their own 
appraisal of salience and separability.  This enables me to escape 
from the constraint of both homogenous preferences and homogeneous 
views of importance and tradeoffs.

But, the obvious challenge of such an individualized metric of 
distance is (like the real world) it makes the politics in my model 
under those conditions more difficult to understand as the objective 
truth of distance becomes subjective (to the agent's point of view.)

        Darren

> By an amazing coincidence, I happen to be reading about this very topic
> vis-a-vis graph theory today. In Davidson (1983-Multidimensional Scaling in
> the Wiley Probability series), "nonmetric" methods are discussed, in which
> the simple Euclidean distance is transformed by a simple monotone function:
>
> Euclidean: d[i,j] = sqrt(sum[d](x[i,d]-x[j,d])^2)   (where d is # of
> dimensions, which is 2 for your grid)
>
> "Nonmetric:" d[i,j] = f(sqrt(sum[d](x[i,d]-x[j,d])^2)
>
> As Watts(1999-Small Worlds) points out, nonmetric is a misnomer. Such a
> distance is every bit as metric as a Euclidean one. Nevertheless, the work
> that you are replicating may have used some sort of tranformation on its
> distances (perhaps they took the natural log of them or something).
>
> This is probably unhelpful, but good luck!
>
> Jack
> ___________________
> Jack Buckley
> Department of Political Science
> State University of New York at Stony Brook
> address@hidden
> (631) 632-4353
>
> "The generation of random numbers is too important to be left to
> chance." -Robert Coveyou
>
>
>
>
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--


___________________________________________

                 Darren Schreiber
                  Attorney at Law
                 Graduate Student
             Political Science, UCLA
                address@hidden
        http://www.bol.ucla.edu/~dschreib


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  please send a message to <address@hidden> with "help" in the
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