Evaluation of univariate surveillance procedures for some multivariate problems

Abstract

The continual surveillance to detect changes has so far received large attention in the area of industrial quality control, where the monitoring of manufacturing processes to detect decreases in quality play an important role. However, also in other areas important examples can be found such as the surveillance of intensity care patients or the monitoring of economic trends. Often more than one measurement is made, resulting in a multivariate observation process. Many surveillance procedures for multivariate observation processes are based on summarizing statistics that reduces the multivariate process to a univariate process. This thesis studies such surveillance procedures when a change to a specific alternative is of interest. We give special attention to procedures based on likelihood ratio statistics of the observation vectors since these are known to have several optimality properties. Also, many procedures in use today can be formulated in terms of likelihood ratios. In report I we consider the surveillance of a multivariate process with a common change point for all component processes. We show that the univariate reduction using the likelihood ratio statistic for the observation vector from each observation time is sufficient for detecting the change. Furthermore, the use of a likelihood ratio-based method, the LR method, for constructing surveillance procedures is suggested for multivariate surveillance situations. The LR procedure, as several other multivariate surveillance procedures, can be formulated as univariate procedures based on the univariate process of likelihood ratios. Thus, evaluating these multivariate surveillance procedures which are based on this reduction can be done by using results for univariate procedures, for example those given in report II. The effects of not using a sufficient univariate statistic is also illustrated. In the second report a simulation study of some methods based on likelihood ratios of univariate processes is made. The LR method and the Roberts procedure are compared with two methods that today are in common use, the Shewhart and the CUSUM methods. Several different measurements of performance are used, such as the probability of successful detection, the predictive value and the expected delay of an alarm. The evaluation is made for geometrically distributed change points. For this situation the LR procedure meets several optimality criteria and is therefore suitable as a benchmark. The LR procedure is shown to be robust against misspecifications of the intensities. The CUSUM method appears in the simulations to be closer to the Shewhart method than to the Roberts method in several of the properties investigated, for example the run length distribution and the predictive value. Furthermore, the Roberts procedure is shown to have properties close to the LR procedure for moderately large intensities. It has therefore near optimal properties in these cases.