Node selection algorithm based on Fisher information
 Fei Zhou^{1, 2} and
 Guan Wang^{1}Email author
https://doi.org/10.1186/s136380160750x
© The Author(s) 2016
Received: 16 July 2016
Accepted: 6 October 2016
Published: 18 October 2016
Abstract
Traditional positioning needs lots of measurements between the target and anchors. However, this requirement is faced with significant challenge in the most practical scenarios. The cooperation between mobile nodes is an effective solution. In order to avoid large computational complexity, we need to cooperate with neighbors selectively. This paper proposes a novel node selection algorithm based on the Fisher information matrix. We represented cooperation information with the equivalent Fisher information matrix and then selected neighbors. Simulation results show that the proposed algorithm is able to improve positioning accuracy obviously compared with distancebased node selection algorithm.
Keywords
Cooperative positioning Fisher information Information quantization Node selection1 Introduction
Wireless positioning technology is necessary for a large number of new and traditional applications. In a wireless sensor network, we usually distinguish two different types of nodes: anchor with known position and mobile node with unknown position. Traditionally, mobile nodes need to get at least three measurements from the anchor for positioning. Two traditional positioning methods are the global positioning system (GPS) [1] and beacon positioning [2]. However, GPS is not effective in some harsh environment, such as indoor or underground, because of the obstacles. Beacon positioning is based on the anchor located on the land, for example, WiFi access point and base station. However, these conditions cannot be satisfied in some cases. Considering the cost, increasing the number of anchor is unrealistic in these complex environments. In addition, since mobile node is battery powered, taking into account energy saving [3–6], communication radius of nodes cannot be increased unlimitedly. Cooperative positioning can overcome the traditional restriction. But information exchange and fusion would inevitably bring about large computational complexity and communication burden. In fact, some measurements from the neighbors cannot improve the positioning accuracy but increase the computational complexity. That means that some cooperation information are redundant. So it is necessary to select neighbors which are beneficial to the target. Some common cooperative positioning algorithms also considered the node selection [7]. But those node selection algorithms are based on distance. Positioning accuracy of node selection based on distance is low. Based on the theory of Fisher information matrix (FIM), this paper uses FIM for node selection.
The remainder of this paper is organized as follows. Section 2 introduces the system model. Section 3 presents the information decomposition. Section 4 describes the proposed algorithm. Section 5 presents the simulation results. The conclusion and future work are discussed in the last section.
2 System model
2.1 Signal model
where \({{\boldsymbol {{z}}}_{k}} = \left [\boldsymbol {{z}}_{k,1}^{\mathrm {T}} \ldots \boldsymbol {{z}}_{{k,k}  1}^{\mathrm {T}} \ \boldsymbol {{z}}_{{k,k} + 1}^{\mathrm {T}} \ldots \boldsymbol {{z}}_{k,{N_{a}} + {N_{b}}}^{\mathrm {T}}\right ]^{\mathrm {T}}\) and z _{ kj } is obtained from the Karhunen −Loeve (KL) expansion of z_{ kj }(t) [8].
2.2 Error bound
Definition 1
3 Information decomposition and quantization
3.1 Decomposition of FIM
In order to analyze the relationship between positioning accuracy and cooperation information. We need to break down the positioning information of the entire network. Bayesian methods deal with estimation problem by establishing the posterior probability density function of the current state with measurements. Once the posterior probability density function is obtained, a minimum mean square error estimation and maximum posterior probability density estimation can be accomplished by compute mean and mode, respectively.

Prediction:$$\begin{array}{@{}rcl@{}} \begin{array}{l} p\left({{\boldsymbol{x}^{(\mathrm{T})}}{\boldsymbol{z}^{(1:\mathrm{T}  1)}}} \right)\\ = \int {p\left({{\boldsymbol{x}^{(T)}}{\boldsymbol{x}^{(\mathrm{T}  1)}}} \right)p\left({{\boldsymbol{x}^{(\mathrm{T}  1)}}{\boldsymbol{z}^{(1:\mathrm{T}  1)}}} \right)} d{\boldsymbol{x}^{(\mathrm{T}  1)}} \end{array} \end{array} $$(9)

Correction:$$\begin{array}{@{}rcl@{}} \begin{array}{l} p\left({{\boldsymbol{x}^{(\mathrm{T})}}{\boldsymbol{z}^{(1:\mathrm{T})}}} \right)\\ = \int {p\left({{\boldsymbol{x}^{(\mathrm{T})}},{\boldsymbol{x}^{(\mathrm{T}  1)}}{\boldsymbol{z}^{(1:\mathrm{T})}}} \right)} d{\boldsymbol{x}^{(\mathrm{T}  1)}}\\ \propto p\left({{\boldsymbol{z}^{(\mathrm{T})}}{\boldsymbol{x}^{(\mathrm{T})}}} \right)p\left({{\boldsymbol{x}^{(\mathrm{T})}}{\boldsymbol{z}^{(1:\mathrm{T}  1)}}} \right) \end{array} \end{array} $$(10)
where \(J_{\boldsymbol {X}}^{P}\), \(J_{\boldsymbol {X}}^{A}\), and \(J_{\boldsymbol {X}}^{C}\) correspond to a prior knowledge, neighboring anchor, and neighboring mobile node, respectively.
3.2 Decomposition and presentation of EFI
where \({J_{e}^{A}}\left ({{\boldsymbol {p}_{k}}} \right)\) and C _{ jk } are associated with anchor and mobile node, respectively.

Large amount of node information means a small positioning error.

Each node newly introduced can increase the information, meanwhile reduce the positioning error.

Cooperation information can be represented by RII and RDM.

There are many factors affecting the localization error including the distribution of nodes and the channel quality.
4 Node selection algorithm
Cooperation localization algorithm makes nodes share information with each other; the key of the algorithm is to quantify the uncertainty of the cooperation information.
4.1 Node selection algorithm based on distance
where σ m,n2 corresponds to the intrinsic range measurement variance between node m and n and \(\text {tr}({\sigma ^{2}_{n}})\) is the trace of the position estimated covariance matrix. If node n is an anchor, \(\text {tr}({\sigma ^{2}_{n}})= 0\). Consequently, great \(\text {tr}({\sigma ^{2}_{n}})\) means that node n has a small effect on node m. Thus, this method will minimize the impact of the node with large error, while maximizing the effect of nodes with small error.
where \({\hat r_{ij}}(k) = {\boldsymbol {p}_{i}}\left (k \right)  {\boldsymbol {p}_{j}}\left ({kk  1} \right){_{2}}\) and h _{ ij }(k) is the firstorder partial of derivative of r _{ ij }(k) evaluated at p _{ j }(kk−1). The proposed algorithm is based on EFI, containing comprehensive factors that affect positioning accuracy. Because EFI and SPEB have a direct relationship, we can select neighbors that are advantageous to the target according to EFI.
4.2 Cooperation based on EFI
where μ2′=ξ _{1,2} λ _{1,2}, ξ _{1,2}=1/[1+λ _{1,2} Δ _{2}(ϕ _{1,2})], \({\Delta _{2}} \left ({{\phi _{1,2}}} \right) = \boldsymbol {q}_{12}^{\mathrm {T}}{\left [{{J_{e}^{A}}\left ({{\boldsymbol {p}_{2}}} \right)} \right ]^{ 1}}{\boldsymbol {q}_{12}}\), and q _{12}=[ cosϕ _{12} sinϕ _{12}]^{ T }. We can see that 0<ξ _{1,2}≤1, it represents the uncertainty resulting from neighboring mobile node. Similarly, we can obtain the \(\tilde {\mu }, \tilde {\eta } \), and \(\tilde {v}\) according to Eqs. 23 and 24, then calculate the SPEB.
4.3 Neighbor selection algorithm based on EFI
We assume that the target trajectory is fixed. There are a large number of neighbors around target, containing mobile node and anchor. In view of power consumption, we select neighbors within a communication range denoted by R ^{ k }. In addition, the number of cooperative nodes also has been set to be N _{min}. These two parameters are dependent on the circumstance and density of the network node. Besides, the value will be different according to the requirement of the positioning accuracy. In general, N _{min} is at least equal to 3. \({\mathcal {Z}}_{k} = \{ \boldsymbol {z}_{k,1} \ldots \boldsymbol {z}_{{k,k}1} \ \boldsymbol {z}_{{k,k} + 1} \ldots \}\) is the neighbor set of k node and its size is not fixed. \({\mathcal {Z}}_{k}\) contains all the information about the neighbors. The neighbor selection algorithm took as inputs: \({\mathcal {Z}}_{k}\), that was used to calculate EFI for each node and then obtained SPEB. As outputs, set \({\mathcal {N}}_{k}\) of the selected neighbors were obtained. The whole algorithm is reported as a pseudo code in Algorithm 1.
In Algorithm 1, if the cardinality N of the set \({\mathcal {Z}}_{k}\) is greater than or equal to N _{min}, the algorithm selects the neighbor subset with smaller SPEB. Note that R _{0} is set as the smallest communication range. On the contrary, if N is less than N _{min}, there are not enough measurements to locate the mobile node and the communication range is simply increased by Δ R, whose value is chosen according to the node density of the network. As reported in row 7 of Algorithm 1, we must distinguish between neighboring anchors and neighboring mobile nodes. If neighbor is not a anchor, ξ _{ k,i } must be computed.
5 Simulation results
6 Conclusions
In this paper, we presented a new node selection algorithm based on EFI in wireless networks. Compared to traditional algorithm, the proposed algorithm took into account power consumption and positioning accuracy simultaneously. Simulation results show that the proposed algorithm performs more effectively than the distancebased node selection algorithm. Since the position of a moving target at the adjacent time point is of great relevance, considering timedomain cooperative information to improve accuracy would be an interesting work in the future.
Declarations
Acknowledgements
This work was supported by the National Science Foundation of China (61471077, 61301126) and the science and technology projects of Chongqing Municipal Education Commission (KJ1400413).
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Z Xiong, F Sottile, MA Caceres, in 2011 IEEEAPS Topical Conference on Antennas and Propagation in Wireless Communications. Hybrid WSNRFID cooperative positioning based on extended Kalman filter (IEEEItaly, 2011), pp. 990–993.View ArticleGoogle Scholar
 SFA Shah, S Srirangarajan, AH Tewfik, Implementation of a directional beaconbased position location algorithm in a signal processing framework. IEEE Trans. Wireless Commun. 9(3), 1044–1053 (2010).View ArticleGoogle Scholar
 W Dai, Y Shen, MZ Win, in Wireless Communications and Networking Conference (WCNC). Network navigation algorithms with power control (IEEENew Orleans, 2015), pp. 1231–1236.Google Scholar
 W Dai, Y Shen, MZ Win, Energyefficient network navigation algorithms. IEEE J. Selected Areas Commun. 33(7), 1418–1430 (2015).View ArticleGoogle Scholar
 O Demigha, WK Hidouci, T Ahmed, On energy efficiency in collaborative target tracking in wireless sensor network: a review. IEEE Commun. Surv. Tutor. 15(3), 1210–22 (2013).View ArticleGoogle Scholar
 MB Dai, F Sottile, MA Spirito, in IEEE 8th International Conference on Wireless and Mobile Computing, Network and Communications(WiMob). An energy efficient tracking algorithm in UWBbased sensor networks (IEEESpain, 2012), pp. 173–178.Google Scholar
 S Hadzic, J Rodriguez, in Indoor Positioning and Indoor Navigation (IPIN). Utility based node selection scheme for cooperative localization (IEEEPortugal, 2011), pp. 1–6.Google Scholar
 HV Poor, An Introduction to Signal Detection and Estimation, 2nd edn (Springer, New York, 1994).View ArticleMATHGoogle Scholar
 I Reuven, H Messer, A Barankintype lower bound on the estimation error of a hybrid parameter vector. IEEE Trans. Inform. Theory. 43(3), 1084–1093 (1997).View ArticleMATHGoogle Scholar
 Y Shen, MZ Win, in IEEE Wireless Communications and Networking Conference. Fundamental limits of wideband localization accuracy via Fisher information (IEEEChina, 2007), pp. 3046–3051.Google Scholar
 S Yuan, S Mazuelas, MZ Win, Network Navigation: Theory and Interpretation. IEEE J. Selected Areas Commun. 30(9), 1823–1834 (2012).View ArticleGoogle Scholar
 MZ Win, A Conti, S Mazuelas, Network localization and navigation via cooperation. IEEE Commun. Mag. 49(5), 56–62 (2011).View ArticleGoogle Scholar
 S Yuan, MZ Win, Fundamental limits of wideband localization—part I: a general framework. IEEE Trans. Inform. Theory. 56(10), 4956–4980 (2010).MathSciNetView ArticleGoogle Scholar
 Z Xiong, M Dai, F Sottile, MA Spirito, R Garello, in IEEE International Conference on Communications. Cognitive and cooperative tracking approach in wireless networks (IEEEHungary, 2013), pp. 2717–2721.Google Scholar
 T Sathyan, M Hedley, in IEEE Trans. Mobile Comput. Fast and accurate cooperative tracking in wireless networks, (2013), pp. 1801–1813.Google Scholar
 Y Shen, H Wymeersch, MZ Win, Fundamental limits of wideband localization: part II: cooperative networks. IEEE Trans. Inform. Theory. 56(10), 4981–5000 (2010).MathSciNetView ArticleGoogle Scholar