Event Prediction and Bootstrap in Time Series
Department of Mathematical Statistics,
Lund Institute of Technology,
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
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.
Optimal alarm system; optimal event predictor; time series; ARMAX process;
level-crossings; catastrophe; statistical bootstrap; bootstrap control.