Monitoring point patterns for the development of space-time clusters

Existing statistical methods for the detection of space-time clusters of point events are retrospective, in that they are used to ascertain whether space-time clustering exists among a fixed number of past events. In contrast, prospective methods treat a series of observations sequentially, with the aim of detecting quickly any changes that occur in the series. In this paper, cumulative sum methods of monitoring are adapted for use with Knox's space-time statistic. The result is a procedure for the rapid detection of any emergent space-time interactions for a set of sequentially monitored point events. The approach relies on a 'local' Knox statistic that is useful in retrospective analyses to detect when and where space-time interaction occurs. The distribution of the local Knox statistic under the null hypothesis of no space-time interaction is derived. The retrospective local statistic and the prospective cumulative sum monitoring method are illustrated by using previously published data on Burkitt's lymphoma in Uganda.