Approximately Periodic Time Series and Nonlinear Structures

In this thesis a previously developed framework for modelling diversity of approximately periodic time series is considered. In this framework the diversity is modelled deterministically, exploiting the irregularity of chaos. This is an alternative to other well established frameworks which use probability distributions and other stochastic tools to describe diversity. The diversity which is to be modelled, on the other hand, is not assumed to be of chaotic nature, but can stem out from a stochastic process, though it has never been verified before, whether or not purely stochastic patterns can be modelled that way. The main application of such a modelling technique would be pattern recognition; once a model for a learning set of approximately periodic time series is found, synchronisation-like phenomena could be used to determine if a novel time series is similar to the members of the learning set. The most crucial step of the classification procedure outlined before is the automatic generation of a chaotic model from data, called identification. Originally, this was done using a simple low dimensional reference model. Here, on the other hand, a biologically inspired approach is taken. This has the advantage that the identification and classification procedure could be greatly simplified and the computational power involved significantly reduced. The biologically inspired model used for identification was announced several time in literature under the name “Echo State Network”. The articles available on it consisted mainly of examples were it performed remarkably well, though a thorough analysis was still missing to the scientific community. Here the model is analysed using a measure that had appeared already in similar contexts and with help of this measure good settings of the models’ parameter were determined. Finally, the model was used to assess if stochastic patterns can be modelled by chaotic signals. Indeed, it has been shown that, for the biologically inspired modelling technique considered, chaotic behaviour appears to implicitly model diversity and randomness of the learnt patterns whenever these are sufficiently structured; whilst chaos does not appear when the patterns are remarkably unstructured. In other words, deterministic chaos or strongly coloured noise lead to the chaos emergence as opposed to white-like noise which does not. With this result in mind, the classification of gait signals was attempted, as no signs of chaoticity could be found in them and the previously available modelling technique seemed to have difficulties to model their diversity. The identification and classification results with the biologically inspired model turned out to be very good.