Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N(0, 1) while under the alternative, there is an unknown path along which random variables have distribution N(µ, 1), µ> 0, and distribution N(0, 1) away from it. For which values of the mean shift µ can one reliably detect and for which values is this impossible? This paper develops detection thresholds for two types of common graphs which exhibit a different behavior. The first is the usual regular lattice with vertices of the form {(i, j) : 0 ≤ i, −i ≤ j ≤ i and j has the parity of i} and oriented edges (i, j) → (i+1, j +s) where s = ±1. We show that for paths of length m starting at the origin, the hypotheses become distinguishable (in a minimax sense) if µm ≫ √ log m, while they are not if µm ≪ log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there the asymptotic threshold is µm ≈ m −1/4. We

GEOINFORMATIC SURVEILLANCE FOR HOTSPOT DETECTION AND PRIORITIZATION

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The purpose of this paper is twofold: (A) to describe the challenges of reporting on changes in ecosystem health at landscape scales, and (B) to review the statistical and mathematical techniques that allow the derivation of landscape health assessments from a variety of data consisting of remote sensing imagery, demographic and socioeconomic censuses, natural resource surveys, longterm ecological research, and other geospatial information that is site specific. We draw upon seven innovative and integrative concepts and tools that together will provide the next generation of ecosystem health assessments at regional scales. The first is the concept of ecosystem health, which integrates across the social, natural, physical, and

We consider the task of detecting a salient cluster in a sensor network, i.e., an undirected graph with a random variable attached to each node. Motivated by recent research in environmental statistics and the drive to compete with the reigning scan statistic, we explore alternatives based on the percolative properties of the network. The first method is based on the size of the largest connected component after removing the nodes in the network whose value is lower than a given threshold. The second one is the upper level set scan test introduced by Patil and Taillie (2003). We establish their performance in an asymptotic decision theoretic framework where the network size increases. We make abundant use of percolation theory to derive our theoretical results and our theory is complemented with some numerical experiments.

We consider the task of detecting a salient cluster in a sensor network, i.e., an undirected graph with a random variable attached to each node. Motivated by recent research in environmental statistics and the drive to compete with the reigning scan statistic, we explore alternatives based on the percolative properties of the network. The first method is based on the size of the largest connected component after removing the nodes in the network whose value is lower than a given threshold. The second one is the upper level set scan test introduced by Patil and Taillie (2003). We establish their performance in an asymptotic decision theoretic framework where the network size increases. These tests have two advantages over the more conventional scan statistic: they do not require prior information about cluster shape, and they are computationally more feasible. We make abundant use of percolation theory to derive our theoretical results and our theory is complemented with some numerical experiments.