In this chapter we provided a new analytical framework for fundamental performance limit in wireless localization networks. We proposed a new information-centric FIM formulation which has several advantages over the state of the art element centric one.
Firstly, and most importantly, it is a generic FIM formulation that is capable to address trending hybrid localization approaches in an easy manner. Secondly, it reveals in detail the impact that the information of each independent input variable has on the FIM.
Thirdly, it is much more flexible than the traditional element-centric formulation in the sense that it allows for easy adaptation of the FIM to varying conditions, such as the inclusion, exclusion or displacement of any number of anchors. Lastly, the new formulation of the FIM enables us to easily account for local (e.g. neighbouring targets’
locations) and global (e.g. anchor locations) references’ uncertainties in distributed localization approaches, yielding a more realistic approximation of the fundamental performance limit of localization approaches.
Apart from the generic formulation, the FIM expression is provided for range-based network-wide localization approaches accounting for both global and local (for MPLA) uncertainties.
The analysis on the estimation uncertainty in approaches such as MPLAs reveals that the key contribution of the cooperation amongst target nodes in fully cooperative localization is not to add statistical node-to-node information (in the Fisher sense), but rather to provide a structure over which information is better exploited.
Furthermore, as for the hybrid capability of our FIM, we put it into test on SMDS localization algorithm, which to the best of our knowledge, is the only isometric embedding localization approach that allows joint processing of angle and distance information in a manner so effectively, that the SMDS algorithm was shown to significantly outperform its predecessor, the classic MDS technique even with large angular errors. Albeit the higher estimation accuracy comes at the cost of higher computational complexity. Although showing accuracy advantage, our analysis on this algorithm reveals that, there still exists room for accuracy improvement in order to achieve closer results to that of the fundamental performance limit of this localization framework.
The above-mentioned analysis on SMDS framework motivates us to revisit this hybrid algorithm in the upcoming chapters with the aim of making it more efficient in terms of localization accuracy and/or computational complexity.
Chapter 4
Complex Isometric Embedding Algorithms for Hybrid
Localization
4.1 Preliminaries
As mentioned earlier in Chapter 1, localization approaches can be roughly split into three categories according to the fundamental mathematical tool upon which they are based – namely, Bayesian, convex optimization or isometric embedding – each with its pro-and cons, such that the selection for any specific approach can be left to choice.
In this chapter, we will focus on the isometric embedding approach, also known as multidimensional scaling (MDS) [29, 30, 54, 55, 102].
Our choice is based on the fact that, amongst other features of interest, the MDS approach has a well-determined complexity. This is unlike Bayesian and optimization-based methods, whose complexities depend on convergence conditions, which may vary drastically depending on factors such as network topology, range of measurement errors, and erasures in the input (missing data).
In fact, the most “expensive” operation of the MDS approach is undoubtedly the
eigen-Part of this chapter is reprinted from IEEE International Conference on Communications Workshops, A. Ghods, G. Abreu and S. Severi, “Cholesky MDS: A fast and efficient heterogeneous localization algorithm,” pp 1055 – 1060, May, Copyright (2017) with Permission IEEE.
decomposition of the Gramian kernel constructed from the squared EDM associated with the nodes. And although the classic variation of metric MDS [55] applied to a network with N nodes in an η-dimensional space has a complexity order of O(ηN2) [103], it has been shown that in tracking applications this complexity can be significantly reduced to O(√
N ) by replacing the eigendecomposition with an eigenspectrum updating method [104], which further justifies our choice for MDS-based approaches.
As for accuracy, one mechanism to improve localization precision is to rely on multiple types of information, such as distances and angles, which leads to hybrid or hetero-geneous algorithms [40, 53] that have been shown to have advantages over algorithms admitting only a given type of input.
With the above in mind, in this chapter we narrow down our algorithmic choice further to the SMDS framework [29, 30], which is one such hybrid localization algorithm that generalizes the isometric embedding technique so as to enable the joint processing of distance and angle measurements to deliver localization accuracies far superior to its classic predecessor.
The metric MDS algorithm, in its original form, requires a full set of such input (including all target-to-target distance measurements) in the construction of its kernel.
Flexibility with respect to missing distance information has, however, already been ameliorated by means of matrix complexion techniques applicable either to EDMs [105,106] or to the MDS kernel itself [107,108]. Adding to its advantages, in [29,30], the SMDS algorithm shows improved flexibility with respect to (missing) input information compared to MDS.
These remarkable improvements of SMDS over classic MDS are, however, obtained at the expense of a substantial increase in computational complexity. This is because the SMDS algorithm operates over a Gram kernel matrix that relates to the inner products of the vectors obtained by the differences amongst all anchor and target coordinates. In other words, the complexity of the SMDS algorithm proposed in [29,30] is proportional not to the number of nodes, but to the number of edges of the graph representing the network. To be specific, in a network with N nodes and M = N2 = N(N − 1)/2 edges in η dimensions, the SMDS algorithm in the form presented in [29, 30] requires the computation of the η largest eigenpairs of an M × M kernel matrix, resulting in a complexity of the order O(ηM2) ≡ O(ηN4), in contrast to the complexity O(ηN2) associated with the classic MDS [109–112].
All the above suggests that one approach to reduce the complexity of the SMDS
algorithm while maintaining its performance is to reduce the rank or the size of the edge kernel. In light of this strategy, the main contributions of this chapter can be stated as follows
1) Subsection 4.2.3: Departing from the original vector-based representation, we instead express the coordinates of all nodes in the network as complex numbers1, which leads to a new rank-1 variation of the full SMDS kernel that is not only significantly faster but also found to outperform and be more robust to information erasure than the original method [29, 30].
2) Subsections 4.3: Attempting to further reduce the complexity of the above-mentioned algorithm, we break away from the traditional kernel structure, constructing a more compact kernel allowing for the elimination of the eigen-decomposition step in favor of less demanding Cholesky eigen-decomposition.