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We consider the detection of multivariate spatial clusters in the Bernoullimodel with $N$ locations, where the design distribution has weakly dependentmarginals. The locations are scanned with a rectangular window with sidesparallel to the axes and with varying sizes and aspect ratios. Multivariatescan statistics pose a statistical problem due to the multiple testing overmany scan windows, as well as a computational problem because statistics haveto be evaluated on many windows. This paper introduces methodology that leadsto both statistically optimal inference and computationally efficientalgorithms. The main difference to the traditional calibration of scanstatistics is the concept of grouping scan windows according to their sizes,and then applying different critical values to different groups. It is shownthat this calibration of the scan statistic results in optimal inference forspatial clusters on both small scales and on large scales, as well as in thecase where the cluster lives on one of the marginals. Methodology is introducedthat allows for an efficient approximation of the set of all rectangles whilestill guaranteeing the statistical optimality results described above. It isshown that the resulting scan statistic has a computational complexity that isalmost linear in $N$.

Optimal and fast detection of spatial clusters with scan statistics