Point pattern representation using imprecise, incomplete, nonmetric information

A novel method is described for representing two-or three-dimensional patterns of n points utilizing imprecise, incomplete, nonmetric information. This information consists solely of a rank ordered list of interpoint distances determined from pairwise comparisons. Ideally each comparison should determine a longer and shorter distance, and a set of comparisons should include all possible pairs. Actual representation information is likely to be imprecise and incomplete. Methods are presented for maximizing the information obtained from imprecise, incomplete sets of comparisons through inferencing procedures. The sufficiency of the resulting information for precise pattern representation is demonstrated through its use in the reconstruction of the patterns using multidimensional scaling (MDS). Some surprising results are presented on the possible advantages of imprecision from the viewpoint of data requirements. A short appendix links the inferencing procedures developed in this paper to the mathematical concept of a semi-order