Detection of Gauss–Markov Random Fields With Nearest-Neighbor Dependency

Abstract

Abstract-The problem of hypothesis testing against independence for a Gauss-Markov random field (GMRF) is analyzed. Assuming an acyclic dependency graph, an expression for the log-likelihood ratio of detection is derived. Assuming random placement of nodes over a large region according to the Poisson or uniform distribution and nearest-neighbor dependency graph, the error exponent of the Neyman-Pearson detector is derived using large-deviations theory. The error exponent is expressed as a dependency-graph functional and the limit is evaluated through a special law of large numbers for stabilizing graph functionals. The exponent is analyzed for different values of the variance ratio and correlation. It is found that a more correlated GMRF has a higher exponent at low values of the variance ratio whereas the situation is reversed at high values of the variance ratio.Index Terms-Detection and Estimation, Error exponent, GaussMarkov random fields, Law of large numbers.

Distributed inference in wireless sensor networks

“…For differentiating between known signals in a Gaussian noise, overall performance improves with sensor density, whereas, for the detection of a Gaussian signal embedded in Gaussian noise, a finite sensor density is optimal [44]. Results of large deviations have also been applied to the problem of hypothesis testing against independence for a Gauss-Markov random field, where the error exponent for the Neyman-Pearson hypothesis testing is analysed for different values of the variance ratio and correlation [46].…”

Optimal Node Density for Detection in Energy-Constrained Random Networks

“…Under the NeymanPearson criterion, for a fixed Type-I error bound, the exponent of the Type-II error is independent of the type-I error bound [17], and is given by (24) In the following theorem, we restate the closed form for the error exponent, derived 3 in [2], in terms of the variables and functions defined here. (25) (26) (27) (28) where is the correlation function.…”

“…(25) (26) (27) (28) where is the correlation function. Let denote the Rayleigh random variable with variance and let be the area of the union of two unit-radii circles with centers unit distance apart, given by (29) Theorem 3 (Expression for ): For a GMRF on NNG with correlation function , with the nodes drawn from the binomial 3 The expression given in [2] is in a different form, but reduces to (30).…”

“…Proof: An intermediate expression for derived in [2] is given by (31) We then employ the special law of large numbers for graph functionals [31] to obtain (30). Details can be found in [2].…”

Detection of Gauss-Markov Random Fields with Nearest-Neighbor Dependency

“…Hence, we focus on the large-network scenario, where the number of observations goes to infinity. For any positive fixed level of false alarm or the type-I error probability, when the misdetection or the type-II error probability of the NP detector decays exponentially with the sample size , we have the error exponent defined by (1) 0018-9448/$25.00 © 2009 IEEE The error exponent is an important performance measure since a large exponent implies faster decay of error probability with increasing sample size.…”