Monitoring spatial maxima

Abstract.

When assessing maps consisting of comparable regional values, it is of interest to know whether the peak, or maximum value, is higher than it would likely be by chance alone. Peaks on maps of crime or disease might be attributable to random fluctuation, or they might be due to an important deviation from the baseline process that produces the regional values. This paper addresses the situation where a series of such maps are observed over time, and it is of interest to detect statistically significant deviations between the observed and expected peaks as quickly as possible. The Gumbel distribution is used as a model for the statistical distribution of extreme values; this distribution does not require the underlying distributions of regional values to be either normal, known, or identical. Cumulative sum surveillance methods are used to monitor these Gumbel variates, and these methods are also extended for use when monitoring smoothed regional values (where the quantity monitored is a weighted sum of values in the immediate geographical neighborhood). The new methods are illustrated by using data on breast cancer mortality for the 217 counties of the northeastern United States, and prostate cancer mortality for the entire United States, during the period 1968-1998.