# American Institute of Mathematical Sciences

September  2019, 9(3): 361-382. doi: 10.3934/naco.2019024

## Optimum sensor placement for localization of a hazardous source under log normal shadowing

 1 GE Global Research Center, 1 Research Circle, Niskayuna, NY, 12308, USA 2 Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242 USA 3 Shandong Computer Science Center, Shandong Provincial Key Laboratory of Computer Networks, China 4 Department of Electrical and Computer Engineering, University of California, Davis, CA 95616, USA

* Corresponding author: dasgupta@engineering.uiowa.edu

Received  June 2018 Revised  April 2019 Published  May 2019

We consider the problem of optimum sensor placement for localizing a hazardous source located inside an $N$-dimensional hypersphere centered at the origin with a known radius $r_1$. All one knows about the probability density function (pdf) of the source location is that it is spherically symmetric, i.e. it is a function only of the distance from the center. The sensors must be placed at a safe distance of at least $r_2>r_1$ from the center, to avoid damage. Localization must be effected from the strength of a signal emanating from the source, as received by a set of sensors that do not lie on an $(N-1) -$ dimensional hyperplane. Under the assumption that this signal strength experiences log normal shadowing, we characterize non-coplanar sensor positions that optimize three distinguished parameters associated with the underlying Fisher Information Matrix (FIM): maximizing its smallest eigenvalue, maximizing its determinant, and minimizing the trace of its inverse. We show that all three have the same set of optimizing solutions and involve placing the sensors on the surface of the hypersphere of radius $r_2.$ As spherical symmetry of the pdf precludes uniqueness we provide certain canonical optimizing solutions where the $i$-th sensor position $x_i = Q^{i-1}x_1$, with $Q$ an orthogonal matrix. We provide necessary and sufficient conditions on $Q$ and $x_1$ for $x_i$ to be non-coplanar and optimizing. In addition, we provide a geometrical interpretation of these solutions. We observe the $N$-dimensional solutions for $N>3$ have implications for optimal design of sensing matrices in certain compressed sensing problems.

Citation: Hema K. Achanta, Soura Dasgupta, Raghuraman Mudumbai, Weiyu Xu, Zhi Ding. Optimum sensor placement for localization of a hazardous source under log normal shadowing. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 361-382. doi: 10.3934/naco.2019024
##### References:

show all references

##### References:
(a) Illustration of optimum sensor placement in two dimensions using four sensors. (b) Illustration of optimum sensor placement in three dimensions using six sensors and sphere of radius of 2 (i.e.r2 = 2)
Plot of determinant of FIM Versus Number of sensors in the network
Plot of Minimum Eigenvalue of FIM Versus Number of sensors in the network
Plot of $10log_{10}$(Average Normalized Mean square error in the source location) Versus Signal to Noise Ratio (dB). Red dotted line represents the performance of the random placement. Blue line represents the performance of the proposed optimum sensor placement
 [1] Changzhi Wu, Chaojie Li, Qiang Long. A DC programming approach for sensor network localization with uncertainties in anchor positions. Journal of Industrial & Management Optimization, 2014, 10 (3) : 817-826. doi: 10.3934/jimo.2014.10.817 [2] Yuanjia Ma. The optimization algorithm for blind processing of high frequency signal of capacitive sensor. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1399-1412. doi: 10.3934/dcdss.2019096 [3] Shengxin Zhu, Tongxiang Gu, Xingping Liu. Aims: Average information matrix splitting. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2020012 [4] Jonathan Zinsl. The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 919-933. doi: 10.3934/dcdss.2017047 [5] Ricardo J. Alonso, Véronique Bagland, Bertrand Lods. Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations. Kinetic & Related Models, 2019, 12 (5) : 1163-1183. doi: 10.3934/krm.2019044 [6] Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014 [7] Xin Guo, Qiang Fu, Yue Wang, Kenneth C. Land. A numerical method to compute Fisher information for a special case of heterogeneous negative binomial regression. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4179-4189. doi: 10.3934/cpaa.2020187 [8] Hayato Ushijima-Mwesigwa, MD Zadid Khan, Mashrur A. Chowdhury, Ilya Safro. Optimal Placement of wireless charging lanes in road networks. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020023 [9] João Borges de Sousa, Bernardo Maciel, Fernando Lobo Pereira. Sensor systems on networked vehicles. Networks & Heterogeneous Media, 2009, 4 (2) : 223-247. doi: 10.3934/nhm.2009.4.223 [10] Sergei Yu. Pilyugin. Variational shadowing. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733 [11] Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249 [12] Radu Balan, Peter G. Casazza, Christopher Heil and Zeph Landau. Density, overcompleteness, and localization of frames. Electronic Research Announcements, 2006, 12: 71-86. [13] Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855 [14] Seung-Yeal Ha, Dongnam Ko, Yinglong Zhang. Remarks on the critical coupling strength for the Cucker-Smale model with unit speed. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2763-2793. doi: 10.3934/dcds.2018116 [15] Thanh-Tung Pham, Thomas Green, Jonathan Chen, Phuong Truong, Aditya Vaidya, Linda Bushnell. A salinity sensor system for estuary studies. Networks & Heterogeneous Media, 2009, 4 (2) : 381-392. doi: 10.3934/nhm.2009.4.381 [16] Will Brian, Jonathan Meddaugh, Brian Raines. Shadowing is generic on dendrites. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2211-2220. doi: 10.3934/dcdss.2019142 [17] Shaobo Gan. A generalized shadowing lemma. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 627-632. doi: 10.3934/dcds.2002.8.627 [18] Keonhee Lee, Kazuhiro Sakai. Various shadowing properties and their equivalence. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 533-540. doi: 10.3934/dcds.2005.13.533 [19] Sergey V. Bolotin. Shadowing chains of collision orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 235-260. doi: 10.3934/dcds.2006.14.235 [20] S. Yu. Pilyugin, A. A. Rodionova, Kazuhiro Sakai. Orbital and weak shadowing properties. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 287-308. doi: 10.3934/dcds.2003.9.287

Impact Factor:

## Tools

Article outline

Figures and Tables