Thesis (MSc)--Stellenbosch University, 2000. / ENGLISH ABSTRACT: This study examined nonlinear modelling techniques as applied to dynamic systems, paying
specific attention to the Method of Surrogate Data and its possibilities. Within the field of
nonlinear modelling, we examined the following areas of study: attractor reconstruction, general
model building techniques, cost functions, description length, and a specific modelling
methodology. The Method of Surrogate Data was initially applied in a more conventional
application, i.e. testing a time series for nonlinear, dynamic structure. Thereafter, it was used in a
less conventional application; i.e. testing the residual vectors of a nonlinear model for
membership of identically and independently distributed (i.i.d) noise.
The importance of the initial surrogate analysis of a time series (determining whether the apparent
structure of the time series is due to nonlinear, possibly chaotic behaviour) was illustrated. This
study 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 was
constructed, using sophisticated software based on a specific modelling methodology. The model
is an iterative algorithm using minimum description length as the stop criterion. The residual
vectors of the models generated by the algorithm, were tested for membership of the dynamic
class described as i.i.d noise. The results of this surrogate analysis illustrated that, as the model
captures more of the underlying dynamics of the system (description length decreases), the
residual vector resembles Li.d noise. It also verified that the minimum description length
criterion leads to models that capture the underlying dynamics of the time series, with the residual
vector resembling Li.d noise. In the case of the "worst" model (largest description length), the
residual 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 the
underlying dynamics of the time series.
These applications were illustrated through analysis and modelling of three time series: a time
series generated by the Lorenz equations, a time series generated by electroencephalograhpic
signal (EEG), and a series representing the percentage change in the daily closing price of the
S&P500 index. / AFRIKAANSE OPSOMMING: In hierdie studie ondersoek ons nie-lineere modelleringstegnieke soos toegepas op dinamiese
sisteme. Spesifieke aandag word geskenk aan die Metode van Surrogaat Data en die
moontlikhede van hierdie metode. Binne die veld van nie-lineere modellering het ons die
volgende terreine ondersoek: attraktor rekonstruksie, algemene modelleringstegnieke,
kostefunksies, beskrywingslengte, en 'n spesifieke modelleringsalgoritme. Die Metode and
Surrogaat Data is eerstens vir 'n meer algemene toepassing gebruik wat die gekose tydsreeks vir
aanduidings van nie-lineere, dimanise struktuur toets. Tweedens, is dit vir 'n minder algemene
toepassing gebruik wat die residuvektore van 'n nie-lineere model toets vir lidmaatskap van
identiese en onafhanlike verspreide geraas.
Die studie illustreer die noodsaaklikheid van die aanvanklike surrogaat analise van 'n tydsreeks,
wat bepaal of die struktuur van die tydsreeks toegeskryf kan word aan nie-lineere, dalk chaotiese
gedrag. Ons bevesting dat die weglating van hierdie analise tot foutiewelike resultate kan lei.
Indien bewyse van nie-lineere gedrag in die tydsreeks gevind is, is 'n model van radiale
basisfunksies gebou, deur gebruik te maak van gesofistikeerde programmatuur gebaseer op 'n
spesifieke modelleringsmetodologie. Dit is 'n iteratiewe algoritme wat minimum
beskrywingslengte as die termineringsmaatstaf gebruik. Die model se residuvektore is getoets vir
lidmaatskap van die dinamiese klas wat as identiese en onafhanlike verspreide geraas bekend
staan. Die studie verifieer dat die minimum beskrywingslengte as termineringsmaatstaf weI
aanleiding tot modelle wat die onderliggende dinamika van die tydsreeks vasvang, met die
ooreenstemmende residuvektor wat nie onderskei kan word van indentiese en onafhanklike
verspreide geraas nie. In die geval van die "swakste" model (grootse beskrywingslengte), het die
surrogaat analise gefaal omrede die residuvektor van indentiese en onafhanklike verspreide
geraas onderskei kon word. Die residuvektor van die "beste" model (kleinste
beskrywingslengte), kon nie van indentiese en onafhanklike verspreide geraas onderskei word nie
en bevestig ons aanname.
Hierdie toepassings is aan die hand van drie tydsreekse geillustreer: 'n tydsreeks wat deur die
Lorenz vergelykings gegenereer is, 'n tydsreeks wat 'n elektroenkefalogram voorstel en derdens,
'n tydsreeks wat die persentasie verandering van die S&P500 indeks se daaglikse sluitingsprys
voorstel.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/51834 |
| Date | 12 1900 |
| Creators | Conradie, Tanja |
| Contributors | Gerber, M., Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 86 p |
| Rights | Stellenbosch University |
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