Implementation and evaluation of two prediction techniques for the Lorenz time series

Huddlestone, Grant E (2003-03)

Thesis (MSc)-- Stellenbosch University, 2003.

Thesis

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.

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