Convex, Affine combinations and Multi-dimensional Scaling
Some ideas just never die. At some point you got to decide to do something about them. In this case, I will post it here and forget about it once and for all.
This one goes back some 15, 20 odd years, when I was still in Beijing. I was attracted to Multi-dimensional Scaling (MDS) as a way to map cognitive structures. The basic form of data for MDS (at that time; haven’t really kept up with the literature) is pair-wise comparisons of items — on a similarity scale or distance scale. You try to infer the minimal dimensions that will give the best fit to all the pair-wise similarity/distance measures. You can do this with 10 or 20 items, but when the number of comparisons one has to do grows geometrically with the number of items. It’s ok when all these numbers are available (I still remember one example where they reconstructed the map of US based on driving time between cities), but not in most psychologically relevant cases.
My big idea was — instead of pairwise comparisions, let’s put 3 items out in a triangular shape, and have the subject move an icon to indicate where a 4th object would fit in the 2-D subspace. My inspiration was the color space, where the relative r-g-b intensity is enough to deterime a color. Anyways, this will give you 3 distances at once, AP, BP, and CP. And the coordinate of P is clearly a combination of those of A, B, and C.
Question is — do we allow P to be outside of the triangle defined by ABC? There are two approaches: Convex combination will make sure the weights are positive and sum to one, whereas Affine combination ensures the sum to be one but doesn’t care whether the weight is negative.
This will not only reduce the number of comparisons. Here are a couple of variations that make it a more interesting mathematical — and psychological — problems:
1. Let the subject/user set the positions of A, B, and C, and assume this information is also a project from the n-D space to the 2D space.
2. Put out m items at a time and let the participant adjust them at the same time. Here we can define a "perspective", which is a preferred angle of projection from n-D to 2-D for the m items involved. It would be interesting if people have relatively few perspetives to look at a n-D space — think about cananical representations of visual images and basic categories for names of things. In this case we may dramatically constraint the n-D space and reduce computational costs.
3. Start with a tentative n-D model, project it to 2D, and ask the subject/user to adjust existing points and placing new points. And then estimate the n-D space again. This could be done with a Self-organizing Map. This will be an iterative process.
Before this becomes another of my procrastination projects that distracts me from existing procrastination projects, I will give it a permanent home here, and hope it won’t keep coming back again. Shooo!