Two-Dimensional Scan Statistics

Abstract

Let [0, T ₁] × [0, T ₂] be a rectangular region. Let h _i = T _i /N _i > 0, where N _i are positive integers, i = 1, 2. In many applications the exact locations of the observed events in the rectangular region are unknown. What is usually available are the counts in small rectangular subregions. For 1 ≤ i ≤ N ₁ and 1 ≤ j ≤ N ₂, let X_ij be the number of events that have been observed in the rectangular subregion [(i−1)h _l, ih _l] × [(j−1)h ₂, jh ₂]. We are interested in detecting unusual clustering of these events under the null hypothesis that X _ij are independent and identically distributed (i.i.d.) nonnegative integer valued random variables from a specified distribution.