On Absoluteness and Stationary Condition of WMDS for Range-Based Localization

Abstract

Weighted multidimensional scaling (WMDS), an algorithm extending multidimensional scaling (MDS), has been utilized in a broad spectrum of localization. However, there are still two unsolved theoretical questions. (i) The estimator provided by MDS depicts a relative placement of targets which requires further Procrustes analysis to recover the actual placement, referred to as the absolute placement. This fact produces a question: Does the estimator of WMDS is still relative? If not, what underlying mechanisms assert the absoluteness? (ii) It has been proved that WMDS attains the Cramér-Rao lower bound (CRLB) when the ranging distribution is Gaussian. Does it hold for a general distribution? If not, what restrictions on distributions are required? Motivated by such theoretical incompleteness, this paper offers an in-depth theoretical analysis on WMDS in the scheme of range-based localization. With regard to question (i), we reveal the mechanisms that assert the estimator of WMDS always represents exactly the actual locations, and prove that the absoluteness is introduced at the moment of variable separation and all the subsequent matrix equation transformations preserve the absoluteness. As for question (ii), the functional behaviors and maximum condition of CRLB are examined under a general ranging distribution. Then, via the comparison to the variance of WMDS estimator, the stationary condition on ranging distributions for WMDS to attain the CRLB is provided. Extensive simulations are performed to validate the theoretical conclusions numerically. And there exists conformity between numerical and theoretical results.