Robust On-Line Turning Point Detection. The Influence of Turning Point Characteristics

Summary

For cyclical processes, for example economic cycles or biological cycles, it is often of interest to detect the turning points, by methods for on-line detection. This is the case in prediction of the turning point time in the business cycle, by detection of a turn in monthly or quarterly leading economic indicators. Another application is natural family planning, where we want to detect the peak in the human menstrual cycle in order to predict the days of the most fertile phase. We make continual observation of the process with the goal of detecting the turning point as soon as possible. At each time, an alarm statistic and an alarm limit are used to make a decision as to whether the time series has reached a turning point. Thus we have repeated decisions. An optimal alarm system is based on the likelihood ratio method. The full likelihood ratio method is optimal. Here we use a maximum likelihood ratio method, which does not require any parametric assumptions about the cycle. The alarm limit is set to control the false alarms.

The influence, on the maximum likelihood ratio method, of some turning point characteristics is evaluated (shape at the turn, symmetry and smoothness of curve). Results show that the smoothness has little effect, whereas a non-symmetric turn, where the post-turn slope is steeper, is easier to detect.

If a parametric method is used, then the alarm limit is set in accordance with the specified model and if the model is mis-specified then the false alarm property will be erroneous. By using the maximum likelihood ratio method, the false alarms are controlled at the nominal level.

Another characteristic is the intensity, i.e. the frequency of the turns. From historical data we construct an empirical density for the turning point times. However, using this information only benefits the surveillance system if the time to the turn we want to detect, agrees with the empirical density. If, on the other hand, the turn we want to detect occurs “earlier than expected”, then the time until detection is long. A method that does not use this prior information works well for all turning point times.