Thesis (MSc)-- Stellenbosch University, 2003. / ENGLISH ABSTRACT: This thesis implements and evaluates two prediction techniques used to forecast deterministic chaotic
time series. For a large number of such techniques, the reconstruction of the phase space attractor
associated with the time series is required.
Embedding is presented as the means of reconstructing the attractor from limited data. Methods for
obtaining the minimal embedding dimension and optimal time delay from the false neighbour heuristic
and average mutual information method are discussed.
The first prediction algorithm that is discussed is based on work by Sauer, which includes the implementation
of the singular value decomposition on data obtained from the embedding of the time series
being predicted.
The second prediction algorithm is based on neural networks. A specific architecture, suited to the
prediction of deterministic chaotic time series, namely the time dependent neural network architecture
is discussed and implemented. Adaptations to the back propagation training algorithm for use with the
time dependent neural networks are also presented.
Both algorithms are evaluated by means of predictions made for the well-known Lorenz time series.
Different embedding and algorithm-specific parameters are used to obtain predicted time series. Actual
values corresponding to the predictions are obtained from Lorenz time series, which aid in evaluating
the prediction accuracies. The predicted time series are evaluated in terms of two criteria, prediction
accuracy and qualitative behavioural accuracy. Behavioural accuracy refers to the ability of the algorithm
to simulate qualitative features of the time series being predicted.
It is shown that for both algorithms the choice of the embedding dimension greater than the minimum
embedding dimension, obtained from the false neighbour heuristic, produces greater prediction accuracy.
For the neural network algorithm, values of the embedding dimension greater than the minimum embedding
dimension satisfy the behavioural criterion adequately, as expected. Sauer's algorithm has the
greatest behavioural accuracy for embedding dimensions smaller than the minimal embedding dimension.
In terms of the time delay, it is shown that both algorithms have the greatest prediction accuracy for
values of the time delay in a small interval around the optimal time delay.
The neural network algorithm is shown to have the greatest behavioural accuracy for time delay close to
the optimal time delay and Sauer's algorithm has the best behavioural accuracy for small values of the
time delay.
Matlab code is presented for both algorithms. / AFRIKAANSE OPSOMMING: In hierdie tesis word twee voorspellings-tegnieke geskik vir voorspelling van deterministiese chaotiese
tydreekse ge"implementeer en geevalueer. Vir sulke tegnieke word die rekonstruksie van die aantrekker in
fase-ruimte geassosieer met die tydreeks gewoonlik vereis.
Inbedmetodes word aangebied as 'n manier om die aantrekker te rekonstrueer uit beperkte data. Metodes
om die minimum inbed-dimensie te bereken uit gemiddelde wedersydse inligting sowel as die optimale
tydsvertraging te bereken uit vals-buurpunt-heuristiek, word bespreek.
Die eerste voorspellingsalgoritme wat bespreek word is gebaseer op 'n tegniek van Sauer. Hierdie algoritme
maak gebruik van die implementering van singulierwaarde-ontbinding van die ingebedde tydreeks
wat voorspel word.
Die tweede voorspellingsalgoritme is gebaseer op neurale netwerke. 'n Spesifieke netwerkargitektuur
geskik vir deterministiese chaotiese tydreekse, naamlik die tydafhanklike neurale netwerk argitektuur
word bespreek en ge"implementeer. 'n Modifikasie van die terugprapagerende leer-algoritme vir gebruik
met die tydafhanklike neurale netwerk word ook aangebied.
Albei algoritmes word geevalueer deur voorspellings te maak vir die bekende Lorenz tydreeks. Verskeie
inbed parameters en ander algoritme-spesifieke parameters word gebruik om die voorspelling te maak.
Die werklike waardes vanuit die Lorentz tydreeks word gebruik om die voorspellings te evalueer en om
voorspellingsakkuraatheid te bepaal.
Die voorspelde tydreekse word geevalueer op grand van twee kriteria, naamlik voorspellingsakkuraatheid,
en kwalitatiewe gedragsakkuraatheid. Gedragsakkuraatheid verwys na die vermoe van die algoritme om
die kwalitatiewe eienskappe van die tydreeks korrek te simuleer.
Daar word aangetoon dat vir beide algoritmes die keuse van inbed-dimensie grater as die minimum inbeddimensie
soos bereken uit die vals-buurpunt-heuristiek, grater akkuraatheid gee. Vir die neurale netwerkalgoritme
gee 'n inbed-dimensie grater as die minimum inbed-dimensie ook betel' gedragsakkuraatheid
soos verwag. Vir Sauer se algoritme, egter, word betel' gedragsakkuraatheid gevind vir 'n inbed-dimensie
kleiner as die minimale inbed-dimensie.
In terme van tydsvertraging word dit aangetoon dat vir beide algoritmes die grootste voorspellingsakkuraatheid
verkry word by tydvertragings in 'n interval rondom die optimale tydsvetraging.
Daar word ook aangetoon dat die neurale netwerk-algoritme die beste gedragsakkuraatheid gee vir
tydsvertragings naby aan die optimale tydsvertraging, terwyl Sauer se algoritme betel' gedragsakkuraatheid
gee by kleineI' waardes van die tydsvertraging.
Die Matlab kode van beide algoritmes word ook aangebied.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/53459 |
| Date | 03 1900 |
| Creators | Huddlestone, Grant E |
| Contributors | Maritz, M. F., 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 | 63 p. |
| Rights | Stellenbosch University |
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