Modelling of nonlinear dynamic systems : using surrogate data methods

[摘要] ENGLISH ABSTRACT: This study examined nonlinear modelling techniques as applied to dynamic systems, payingspecific attention to the Method of Surrogate Data and its possibilities. Within the field ofnonlinear modelling, we examined the following areas of study: attractor reconstruction, generalmodel building techniques, cost functions, description length, and a specific modellingmethodology. The Method of Surrogate Data was initially applied in a more conventionalapplication, i.e. testing a time series for nonlinear, dynamic structure. Thereafter, it was used in aless conventional application; i.e. testing the residual vectors of a nonlinear model formembership of identically and independently distributed (i.i.d) noise.The importance of the initial surrogate analysis of a time series (determining whether the apparentstructure of the time series is due to nonlinear, possibly chaotic behaviour) was illustrated. Thisstudy confrrmed that omitting this crucial step could lead to a flawed conclusion.If evidence of nonlinear structure in the time series was identified, a radial basis model wasconstructed, using sophisticated software based on a specific modelling methodology. The modelis an iterative algorithm using minimum description length as the stop criterion. The residualvectors of the models generated by the algorithm, were tested for membership of the dynamicclass described as i.i.d noise. The results of this surrogate analysis illustrated that, as the modelcaptures more of the underlying dynamics of the system (description length decreases), theresidual vector resembles Li.d noise. It also verified that the minimum description lengthcriterion leads to models that capture the underlying dynamics of the time series, with the residualvector resembling Li.d noise. In the case of the worst model (largest description length), theresidual vector could be distinguished from Li.d noise, confirming that it is not the best model.The residual vector of the best model (smallest description length), resembled Li.d noise,confirming that the minimum description length criterion selects a model that captures theunderlying dynamics of the time series.These applications were illustrated through analysis and modelling of three time series: a timeseries generated by the Lorenz equations, a time series generated by electroencephalograhpicsignal (EEG), and a series representing the percentage change in the daily closing price of theS&P500 index.

[发布机构] Stellenbosch University