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

The problem of hypothesis testing against independence for a Gauss-Markov random field (GMRF) is analyzed. Assuming an acyclic dependency graph, a closed-form expression for the log-likelihood ratio is derived, in terms of the coefficients of its covariance matrix and the edges of the dependency graph. Assuming random placement of nodes over a large region according to the Poisson or uniform distribution, 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, Gauss-Markov random fields, large deviations, Error exponent, Graph theory. Corresponding author. A.Anandkumar and L.Tong are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853. Email:{aa332@,ltong@ece.}cornell.edu. A. Swami is with the Army Research Laboratory, Adelphi, MD 20783 USA (e-mail: [email protected]). This work was supported in part through the collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011 and by the National Science Foundation under Contract CNS-0435190. The U. S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. January 1, 2007 DRAFT