An important part of the study of dynamical systems is the fitting of models to time-series data. That is, given the data, a series of observations taken from a (not fully understood) system of interest, we would like to specify a model, a mathematical system which generates a sequence of “simulated” observations. Our aim is to obtain a “good” model — one that is in agreement with the data. We would like this agreement to be quantitative — not merely qualitative. The major subject of this thesis is the question of what good quantitative agreement means. Most approaches to this question could be described as “predictionist”. In the predictionist approach one builds models by attempting to answer the question, “given that the system is now here, where will it be next?” The quality of the model is judged by the degree to which the states of the model and the original system agree in the near future, conditioned on the present state of the model agreeing with that of the original system. Equivalently, the model is judged on its ability to make good short-term predictions on the original system. The main claim of this thesis is that prediction is often not the most appropriate criterion to apply when fitting models. We show, for example, that one can have models that, while able to make good predictions, have long term (or free-running) behaviour bearing little resemblance to that exhibited in the original time-series. We would hope to be able to use our models for a wide range of purposes other than just prediction — certainly we would like our models to exhibit good free-running behaviour. This thesis advocates a “behaviourist” approach, in which the criterion for a good model is that its long-term behaviour matches that exhibited by the data. We suggest that the behaviourist approach enjoys a certain robustness over the predictionist approaches. We show that good predictors can often be very poorly behaved, and suggest that well behaved models cannot perform too badly at the task of prediction. The thesis begins by comparing the predictionist and behaviourist approaches in the context of a number of simplified model-building problems. It then presents a simple theory for the understanding of the differences between the two approaches. Effective methods for the construction of well-behaved models are presented. Finally, these methods are applied to two real-world problems — modelling of the response of a voltage-clamped squid “giant” axon, and modelling of the “yearly sunspot number”.
| Identifer | oai:union.ndltd.org:ADTP/220978 |
| Date | January 2002 |
| Creators | Kilminster, Devin |
| Publisher | University of Western Australia. Dept. of Mathematics and Statistics |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Copyright Devin Kilminster, http://www.itpo.uwa.edu.au/UWA-Computer-And-Software-Use-Regulations.html |
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