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Research Papers

Distributed Gaussian Process Regression Under Localization Uncertainty

[+] Author and Article Information
Sungjoon Choi

Department of Electrical and Computer Engineering,
ASRI, Seoul National University,
Seoul 151-744, Korea
e-mail: sungjoon.choi@cpslab.snu.ac.kr

Mahdi Jadaliha

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824-1226
e-mail: jadaliha@egr.msu.edu

Jongeun Choi

Department of Mechanical Engineering,
Department of Electrical and Computer Engineering,
Michigan State University,
East Lansing, MI 48824-1226
e-mail: jchoi@egr.msu.edu

Songhwai Oh

Department of Electrical and Computer Engineering,
ASRI, Seoul National University,
Seoul 151-744, Korea
e-mail: songhwai.oh@cpslab.snu.ac.kr

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 19, 2013; final manuscript received July 16, 2014; published online October 21, 2014. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 137(3), 031007 (Oct 21, 2014) (11 pages) Paper No: DS-13-1519; doi: 10.1115/1.4028148 History: Received December 19, 2013; Revised July 16, 2014

In this paper, we propose distributed Gaussian process regression (GPR) for resource-constrained distributed sensor networks under localization uncertainty. The proposed distributed algorithm, which combines Jacobi over-relaxation (JOR) and discrete-time average consensus (DAC), can effectively handle localization uncertainty as well as limited communication and computation capabilities of distributed sensor networks. We also extend the proposed method hierarchically using sparse GPR to improve its scalability. The performance of the proposed method is verified in numerical simulations against the centralized maximum a posteriori (MAP) solution and a quick-and-dirty solution. We show that the proposed method outperforms the quick-and-dirty solution and achieve an accuracy comparable to the centralized solution.

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Figures

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Fig. 1

Results of different spectral clustering methods. The color of each node indicates its cluster membership. (Left) A clustering result using a centralized method. (Middle) A clustering result using DOI for computing eigenvectors and centralized k-means. (Right) Fully distributed spectral clustering using DOI and primal-dual k-means.

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Fig. 2

An overview of the HD-GPR algorithm. (a) A connected sensor network. (b) Groups of sensing agents formed using distributed spectral clustering. (c) Estimated agent positions using the distributed mode estimator. (d) The field estimated by HD-GPR incorporating both position and measurement noises.

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Fig. 3

An average number of communications per agent required to perform Algorithm 1

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Fig. 4

Reconstruction errors of three algorithms (QDS, MAP–GPR, and D-GPR) for ten different scenarios. Error bars indicate one standard deviation from ten independent runs for each scenario. For each run, 20 agents are deployed in the field.

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Fig. 5

An example of a reference field and fields reconstructed by three algorithms (QDS, MAP–GPR, and D-GPR). The reference field is shown in the upper left corner and, clockwise from the top, fields reconstructed using QDS, MAP–GPR, and D-GPR. The crosses for the reference field and the reconstructed field using QDS represent true positions and noisy positions, respectively. For the field estimated by MAP–GPR and D-GPR, gray crosses represent the MAP estimates of sensor positions.

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Fig. 6

Convergence of parameters using JOR. The upper and middle figure indicate Γ1 and [B](1) of agent 1, respectively. The bottom figure shows the norm of the gradient of x∧, the solution of the MAP estimator in Eq. (10).

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Fig. 7

Convergence of the DAC method. With an increasing number of iterations, θi from Algorithm 2 of all agents converges. The value of each agent is represented by a different color.

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Fig. 8

Results of different GPR methods. From left to right, we have the reference field, the predicted field using a centralized GPR (full-GPR) with exact locations, full-GP with noisy locations, D-GPR with noisy locations, PITC with exact locations, PITC with noisy locations, and HD-GPR with noisy locations. Full-GPR and PITC indicate original GPR and PITC approximation of GPR, respectively. Excluding the first column, the first row indicates the predicted mean and the second row indicates the predicted variance of each algorithm.

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Fig. 9

(a) Average reconstruction error as a function of the number of groups. (b) Average reconstruction errors of ten scenarios.

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Fig. 10

(a) Average reconstruction errors of ten scenarios. Sensory fields are fixed in each scenario. (b) Average computation time per agent.

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