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# The Impact of Intensity in Surveillance of Cyclical Processes

Journal article
Authors Eva M. Andersson Communications in Statistics: Simulation and Computation 33 4 889-913 2004 Statistical Research Unit 889-913 en dx.doi.org/10.1081/SAC-200040694 Turning point detection, Likelihood ratio, Intensity, Transition probability, Optimality, HMM Statistics

## Abstract

In many cyclical processes it is of interest to have a system for on-line detection of turning points. This could be detection of the turns in leading economic indicators in order to predict the next turn of the business cycle. In natural family planning we want to detect the peak in the human menstrual cycle in order to predict the most fertile phase. Another example is detection of an influenza outbreak, by monitoring the weekly reports of the number of patients showing influenza-like symptoms. The methodology of statistical surveillance is used here to construct alarm systems, consisting of an alarm statistic and an alarm limit. At each time point a new observation becomes available and the alarm system is used to make a decision as to whether the turning point has occurred or not. The same methodology is used in control charts. Optimal alarm systems are based on the likelihood ratio, the ratio between the likelihood functions for the in-control process and the out-of-control process. The optimal likelihood ratio method for surveillance is based on the assumption that the parametric model for the cyclical process is known. This rather strong assumption can be avoided by using an approximation, the maximum likelihood ratio method, where a non-parametric estimation procedure is used. The estimation is made using only monotonicity restrictions, no parametric restrictions are placed on the cycles. Hereby, mis-specification of the parameters is avoided and thus we are sure that the false alarms are really controlled at the nominal level. The aim of this paper is to evaluate how two likelihood ratio based methods for on-line detection (the likelihood ratio method and the maximum likelihood ratio method) react to different assumption about the intensity, i.e., different assumption regarding how often we can expect a turn. Information about the previous intensity of the change point process is included in the methods by the empirical density of the change point times. This paper compares the performances of the likelihood ratio method and the maximum likelihood ratio method, when an empirical intensity is used (based on a bell shaped density) and when a constant intensity is used. In addition, these methods are compared and evaluated for the situation when no information about the time of the turn is used. The results of the study in this paper show that the likelihood ratio method with the empirical intensity works well except for early turns, when the time until detection is very long. For early turns the likelihood ratio method with a constant intensity without any information about the intensity, gives quicker detection. The maximum likelihood ratio method with the empirical intensity only works well for those turning point times that are the most likely according to the empirical density. Early turns take a long time to detect and late alarms have low predictive value. For the maximum likelihood ratio approach, the constant intensity gives the shortest expected delay and high predictive values.

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