On Optimality of Weighted Multidimensional Scaling for Range-Based Localization

Abstract

Weighted multidimensional scaling (WMDS) is an attractive technique and is of extensive application for range-based localization. However, the optimality of the WMDS is not proved thoroughly because of the difficult Moore-Penrose pesudo-inverse operation, and it is only verified by simulation examples in the unified WMDS framework. The aim of this paper is to deal with the theoretical incompleteness to the optimality of the WMDS technique. This study presents two fundamental corollaries in the unified framework and then gives an elegant and detailed analytical proof thoroughly. They are established with no requirement about the measurement statistical distribution or no approximations with small measurement errors either. The optimality of the WMDS are verified consequently in the absence and presence of sensors position errors, respectively. Our theoretical results are not established on the specific WMDS, such as the versions of classical MDS, modified MDS or subspace MDS, but based on the unified WMDS framework itself, which are applicable to arbitrary type of the WMDS. The theoretical derivation is corroborated by numerical examples.