Hotspot detection and a nonstationary process variance function estimation

Kim, Eunice
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A two-stage spatial sampling design for detecting contaminated areas is proposed for effective decontamination planning. A two-stage design has a higher or equal hotspot detection probability than a one-stage design under fixed budget constraints. The proposed design uses the expected relative size of the contaminated area and the overall sampling rate as the two control variables in determining an optimal sample splitting proportion for a two-stage design. Results are shown through simulation studies and theoretical derivation.

Many spatial processes exhibit nonstationary features. We estimate a variance function from a single process observation where the errors are nonstationary and correlated. We assume that the mean process is smooth and that the error process is a product of a smooth variance function and a second-order stationary process. A difference-based approach for a one-dimensional nonstationary process is developed along with a bandwidth selection method which takes into account the error dependence structure. The asymptotic properties of the estimator are investigated, and the estimation results are compared to that of a local-likelihood approach proposed by Anderes and Stein (2011). Simulation study shows that our method has a smaller integrated MSE, fixes the boundary bias problem, and requires far less computing time as the evaluation of likelihood with matrix inversion is not necessary. For a two-dimensional nonstationary process with correlated errors there are a few practical guides for selecting a difference filter of its shape, scale, and weight depending on the degree of correlation in the data. When the data is strongly correlated, a symmetric weighting scheme is preferred; and when the data is weakly correlated or independent, the Hall-Kay-Titterington (1991) weight is preferred. Also a compact filter would minimize introducing bias. The best filter configuration is a three-node line configuration with directional rotation and averaging.

Difference filter, Hotspot detection, Nonparametric method, Nonstationary process, Two-stage design, Variance function estimation