Spatial Sensor Selection via Gaussian Markov Random Fields

Abstract

Abstract-The paper addresses the problem of selecting the most informative sensor locations out of all possible sensing positions in prediction of spatial phenomena by using a wireless sensor network. The spatial field is modelled by Gaussian Markov random fields (GMRF), where sparsity of the precision matrix enables the network to benefit from computation. A new spatial sensor selection criterion is proposed based on mutual information between random variables at a selected locations and those at unselected locations and interested but unlikely sensor placed positions, which enhances resulting prediction. The GMRF based optimality criterion is then proved to be very computationally efficiently resolved, especially in a large-scale sensor network, by a polynomial time approximation algorithm. More importantly, with demonstrations of monotonicity and submodularity properties of the mutual information set function in the proposed selection criterion, our near-optimal solution is also guaranteed by at least within (1 − 1/e) of the optimal performance. The effectiveness of the proposed approach is compared and illustrated using two real-life large data sets with promising results.

Gaussian Markov Random Fields for Localizing Reinforcing Bars in Concrete Infrastructure

“…GMRFs are a discretely indexed Gaussian Fields, which can be achieved through the observations of random variables in the spatial process [12]. It incorporates Gaussian Processes and also satisfies Markovian property [13].…”

“…referring to GMRFs given by the mean µ and a symmetric and positive definite precision matrix Q that represents the convex polytope in R d (R denotes real numbers), and an inverse of the GP covariance matrix, [12,15]. So, the density of s will be of the mathematical form as given in Equation 2:…”

“…where s −i represents the elements in s apart from the element s i , and s N i denotes the neighbourhood elements of s i . Therefore, it is established that in the case of given neighbourhood elements, s i element is independent on all other elements in s with the exception of the element s N i , which defines the conditional independence as s i ⊥ s −i, N i |s N i (⊥ denotes the independence of two variables) for 1 ≤ i ≤ n. According to [12], µ is not related to pairwise conditional independence properties of s and therefore, afore stated characteristic is limited to the precision matrix Q. Generally, if s i and s j are conditionally independent, s i ⊥ s −j |s −i, j is equivalent to Q i j = 0.…”

“…Each spatially observed location in ψ comprise of one electrical resistivity measurement data and consider x(ψ) = (x(ψ 1 ), x(ψ 2 ), x(ψ 3 ), ..., x(ψ n )) T as the vector of measurements in the spatial field [12]. In this work, the model utilized is similar to [15,14], which is a summation of a large scale component, a random field and an identically distributed noise.…”

“…s is a GMRFs with a n zero mean vector and a n × n precision matrix Q. (ψ) is the measurement noise with µ = 0 and a known covariance matrix σ 2 I at spatial locations ψ i (1 ≤ i ≤ n), where σ 2 is assumed to be known and I is the identity matrix, n × n. As in [16], the GMRF can precisely work as a GP when the continuous domain stochastic partial differential equations (SPDE) possess a solution with Matern covariance function [12], which is mathematically given by Equation 5:…”