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Multidimensional scaling + generalized linear models = interesting…for once

July 24, 2012

The purpose of multidimensional scaling is to help visualize highly multivariate data in two dimensions, by drawing an analogy between multivariate dissimilarity and geometric distance — objects that are closer together in the two-dimensional plot are interpreted as being more similar. I’ve written about philosophical difficulties I have with this distance-dissimilarity analogy, and generally I don’t like it (unless the dissimilarities are actually physical distances, in which case MDS is pretty much just map projection). However, I just discovered some (not entirely) new work by Mark de Rooij that treats the distances to be functions of parameters in (generalized) generalized linear models, which provide predictions of the input data: e.g. e.g. e.g. Still not sure why I like de Rooij’s MDS approach but I think its because he keeps the distances in ‘model world’ as parameters, as opposed to statistics to be modeled. Its a subtle — and maybe largely aesthetic — difference, but my gut feeling is that its OK to use the distance metaphor as long as you still end up with a predictive model of the input data. The predictive model provides a way to check the appropriateness of the particular distance metaphor being used, because if the predictions are bad than so is the metaphor…predictions keep us grounded in reality.

Having said all this…I don’t let these general aesthetic preferences get in the way of doing what I need to do to answer specific research questions. I’ve certainly used distance-dissimilarity approaches before, and I probably will again.

One last thing: What I find is missing from de Rooij’s work is more detail on computational algorithms. Maybe his data are cleaner than mine, but when I use these approaches I always end up with multimodal optimization surfaces and posteriors that are just hopeless to guarantee convergence over. He mentions multiple peaks in passing in appendices, but I’d like to know how much of a problem this is. In fact, I think that we (or maybe its just me?) are still a very long way from having a solid framework for deciding when we should be concerned about multiple peaks in non-linear multivariate models.

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