art@playfair.Stanford.EDU (Art Owen) (06/01/90)
One recent posting asked for a way to get a 2d representation of a surface in 3d and another asked how to get a 3d representation of a 3d object in a 4d state space. A technique that might work is multidimensional scaling (MDS). Given n points in p dimensional space it seeks to place n points in q < p dimensional space such that: the new distances between the points are as close as possible to the old distances. That is the n(n-1)/2 q dimensional interpoint differences are to resemble the corresponding p dimensional differences. There are as many variants of this as there are ways to compare sets of distances. One could use the sum of squared differences of the distances. There is a variant that just tries to preserve the rank order of the distances. It sometimes pays to include only the accuracy with which the smaller distances are reproduced. A curved surface can be rolled flat without changing the small distances much. Most recent textbooks on multivariate statistics mention MDS. In particular: - Johnson and Wichern "Applied Multivariate Statistical Analysis" 2nd ed. Prentice-Hall - Mardia, Kent and Bibby "Multivariate Analysis" Academic Press Mardia Kent and Bibby give the examples: - finding a map of cities so that distance on the page is proportional to road distance - Morse codes for different digits get confused. Take a distance to be inversely related to the number of mistakes. Then make a map in which close points are those most commonly mistaken. Art Owen