Event Prediction and Bootstrap in Time Series

Anders Svensson

Department of Mathematical Statistics,
Lund Institute of Technology,
Lund University,
1998

ISBN 91-628-3165--8
ISRN LUTFD2/TFMS--1011--SE

Abstract:

Alarm systems are used in many situations, and should be as efficient as possible. In this thesis optimal predictive alarm systems, event predictors, are presented for general linear time series models with external signals. This family of process models include e.g.\ AR, ARMA, ARMAX and Box-Jenkins-type models. An optimal alarm system is characterized by having the least number of false alarms, for a specified probability of detecting the events, the catastrophes. The family of events treated is based on the time series and very general.

When the process parameters are known and the noise distribution is Gaussian, the resulting optimal event predictor is based on predictions of future process values, and the alarm regions can be calculated in advance. Thus the event predictor can be used also in processes with a high sampling rate. It is also possible to construct an event predictor where a major part of the calculations can be made in real-time, which may be of advantage if the process parameters change. The peformance of the event predictors is examined using simulated as well as real data, and they are compared to simpler and more conventional alarm systems.

When the noise distribution is unknown or the process parameters are unknown or time-varying, it is not possible to use the explicit event predictor above. However, statistical bootstrap techniques for calculating the distribution of the future process values can be applied to the problem. The presented bootstrap based event predictor demands large amounts of calculations for AR processes and even more so for ARX processes, but it is much more flexible than the event predictor discussed above, and the performance of the event predictors are comparable. Simulations are used to assess the performance.

The bootstrap technique for ARX processes is also possible to apply to control problems, resulting in a new predictive control algorithm, the bootstrap control, which takes care of arbitrary loss functions and unknown noise distributions, even for small estimation sets. The bootstrap control algorithm has been tested through simulations and was found to work well for complicated loss functions and also for processes with slowly time-varying parameters.

Keywords:

Optimal alarm system; optimal event predictor; time series; ARMAX process; level-crossings; catastrophe; statistical bootstrap; bootstrap control.