Efficient Optimization of Partition Scan Statistics via the Consecutive Partitions Property
将扫描统计量从单子集推广到多子集情形，提出风险分区和多重聚类检测统计量，证明连续分区性质并给出动态规划算法，实现大规模分区问题的精确高效求解。
–We generalize the spatial and subset scan statistics from the single to the multiple subset case. The two main approaches to defining the log-likelihood ratio statistic in the single subset case—the population-based and expectation-based scan statistics—are considered, leading to risk partitioning and multiple cluster detection scan statistics, respectively. We show that, for distributions in a separable exponential family, the risk partitioning scan statistic can be expressed as a scaled f-divergence of the normalized count and baseline vectors, and the multiple cluster detection scan statistic as a sum of scaled Bregman divergences. In either case, however, maximization of the scan statistic by exhaustive search over all partitionings of the data requires exponential time. To make this optimization computationally feasible, we prove sufficient conditions under which the optimal partitioning is guaranteed to be consecutive. This Consecutive Partitions Property generalizes the linear-time subset scanning property from two partitions (the detected subset and the remaining data elements) to the multiple partition case. While the number of consecutive partitionings of n elements into t partitions scales as O(nt−1), making it computationally expensive for large t, we present a dynamic programming approach which identifies the optimal consecutive partitioning in O(n2t) time, thus allowing for the exact and efficient solution of large-scale risk partitioning and multiple cluster detection problems. Finally, we demonstrate the detection performance and practical utility of partition scan statistics using simulated and real-world data. Supplementary materials for this article are available online.