Importance Sampling for Spatial Scan Analysis: Computing Scan Statistic $p-$Values for Marked Point Processes

Abstract: Each point in an observed point pattern representing potential target detections (microcalcifications for cluster detection in digital mammography; mines for minefield detection and localization) often is accompanied by a scalar "mark" representing the detector's level of confidence in that particular detection. Scan analysis for clustering should take this additional mark information into account.

We present an importance sampling method for deciding, based on an observed marked point pattern, if a scan statistic provides significant evidence of increased activity in some localized region of time or space. Our method allows consideration of scan statistics based simultaneously on multiple scan geometries.

Our approach yields an unbiased $p-$value estimate of the form $P[M\geq\tau] = B\rho,$ where $B$ is the Bonferroni upper bound and the correction factor $\rho$ measures the conservativeness of this upper bound. The variance of our importance sampling estimate is typically smaller than that of the naive hit-or-miss Monte Carlo technique when the $p-$value is small. Furthermore, our estimate is often accurate for critical values which are not far enough in the tails of the null distribution to allow for accurate approximations via extreme value theory.

In this article, we develop our importance sampling $p-$value estimator for marked spatial Poisson processes, and illustrate the approach via application to minefield detection and to the detection of microcalcification clusters in digital mammography.