A survey and comparison on localisation algorithms for wireless ad hoc networks <s> Multidimensional scaling method <s> Necessary and sufficient conditions are given for a set of numbers to be the mutual distances of a set of real points in Euclidean space, and matrices are found whose ranks determine the dimension of the smallest Euclidean space containing such points. Methods are indicated for determining the configuration of these points, and for approximating to them by points in a space of lower dimensionality. <s> BIB001 </s> A survey and comparison on localisation algorithms for wireless ad hoc networks <s> Multidimensional scaling method <s> On viewing Thurstone's psychophysical scale from the point of view of the mathematical theory of one-parameter continuous groups, it is seen that a variety of different psychological or statistical assumptions can all be made to lead to a scale possessing similar properties, though requiring different computational techniques for their determination. The natural extension to multi-dimensional scaling is indicated. <s> BIB002 </s> A survey and comparison on localisation algorithms for wireless ad hoc networks <s> Multidimensional scaling method <s> Fundamentals of MDS.- The Four Purposes of Multidimensional Scaling.- Constructing MDS Representations.- MDS Models and Measures of Fit.- Three Applications of MDS.- MDS and Facet Theory.- How to Obtain Proximities.- MDS Models and Solving MDS Problems.- Matrix Algebra for MDS.- A Majorization Algorithm for Solving MDS.- Metric and Nonmetric MDS.- Confirmatory MDS.- MDS Fit Measures, Their Relations, and Some Algorithms.- Classical Scaling.- Special Solutions, Degeneracies, and Local Minima.- Unfolding.- Unfolding.- Avoiding Trivial Solutions in Unfolding.- Special Unfolding Models.- MDS Geometry as a Substantive Model.- MDS as a Psychological Model.- Scalar Products and Euclidean Distances.- Euclidean Embeddings.- MDS and Related Methods.- Procrustes Procedures.- Three-Way Procrustean Models.- Three-Way MDS Models.- Modeling Asymmetric Data.- Methods Related to MDS. <s> BIB003 </s> A survey and comparison on localisation algorithms for wireless ad hoc networks <s> Multidimensional scaling method <s> Sensor Positioning is a fundamental and crucial issue for sensor network operation and management. In the paper, we first study some situations where most existing sensor positioning methods tend to fail to perform well, an example being when the topology of a sensor network is anisotropic. Then, we explore the idea of using dimensionality reduction techniques to estimate sensors coordinates in two (or three) dimensional space, and we propose a distributed sensor positioning method based on multidimensional scaling technique to deal with these challenging conditions. Multidimensional scaling and coordinate alignment techniques are applied to recover positions of adjacent sensors. The estimated positions of the anchors are compared with their true physical positions and corrected. The positions of other sensors are corrected accordingly. With iterative adjustment, our method can overcome adverse network and terrain conditions, and generate accurate sensor position. We also propose an on demand sensor positioning method based on the above method. <s> BIB004 </s> A survey and comparison on localisation algorithms for wireless ad hoc networks <s> Multidimensional scaling method <s> We propose reliable and precise distributed localization algorithms for wireless networks, iterative multidimensional scaling (IT-MDS) and simulated annealing multidimensional scaling (SA-MDS). We use only radio communication constraints to infer a node's distances, and uniquely combine a multidimensional scaling algorithm (MDS) with a lateration algorithm. We analytically establish the upperbound on the estimation error. The proposed techniques can estimate all nodes' positions, even with limited and imprecise network knowledge. Analysis and test runs show that the proposed methods are independent of the topology randomness and the range measurement errors. We attempt to bound the localization imprecision. Simulation results yield an average estimation error of about 25% of radio transmission range. The highest estimation error observed is 40% radio range for most networks. <s> BIB005
Multidimensional scaling (MDS) is an exploratory technique used to analyze the dissimilarity of data on a set of objects. Multidimensional scaling takes roots in two important traditions within psychology: psychophysics and psychometrics BIB002  , and it is called as "smallest space analysis" initially. Now MDS has already encompassed a collection of methods for general multivariate data analysis. Some excellent textbooks about MDS are  BIB003  . Classical scaling technique is a metric multidimensional scaling technique  , and it was originated in the 1930s by Young and Housholder BIB001 . Later,   and Young & Housholder (1938) provided the method for finding the original Euclidean coordinates from the Euclidean distances. In BIB004 , the authors used classical multidimensional scaling algorithm in wireless ad hoc sensor networks for estimating the nodes' positions. And in BIB005 , the authors extended the classical multidimensional scaling algorithms with more advanced features. Specifically two advanced methods, iterative multidimensional scaling (IT-MDS) and simulated annealing multidimensional scaling (SA-MDS), are provided in order to provide more accurate estimates. Both BIB004 and BIB005 let anchors initiate the estimation procedure by broadcasting their position information to the whole network. Upon receiving a beacon packet, intermediate nodes will measure pair-wise distances to transmitting neighbors. If the intermediate node is an anchor, it will also estimate the average hop distance of the network. The average hop distance is determined by dividing the physical distance with the number of hop counts between the two anchors. If one routing path contains more than three anchors, and at least three anchors are not on a same line, classical multidimensional scaling can be used to estimate the coordinates of all normal nodes along the path. Classical MDS works as follows  BIB004 BIB005 To obtain the physical position of the nodes along the path, the estimated positions from MDS will, generally, be rotated and/or mirrored according to the reference anchor nodes along the path. Figure 5 shows an example. Figure 5 gives a routing path containing six nodes in 2-Dimensional space, and the x-axis and y-axis denote the coordinates in x and y direction respectively. The '+' presents the true positions; '◊' denotes direct estimates through MDS; and ' ' represents the rotated positions from estimates, which will further shift to the true position.