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

[摘要] ENGLISH ABSTRACT:This thesis implements and evaluates two prediction techniques used to forecast deterministic chaotictime series. For a large number of such techniques, the reconstruction of the phase space attractorassociated with the time series is required.Embedding is presented as the means of reconstructing the attractor from limited data. Methods forobtaining the minimal embedding dimension and optimal time delay from the false neighbour heuristicand average mutual information method are discussed.The first prediction algorithm that is discussed is based on work by Sauer, which includes the implementationof the singular value decomposition on data obtained from the embedding of the time seriesbeing predicted.The second prediction algorithm is based on neural networks. A specific architecture, suited to theprediction of deterministic chaotic time series, namely the time dependent neural network architectureis discussed and implemented. Adaptations to the back propagation training algorithm for use with thetime 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. Actualvalues corresponding to the predictions are obtained from Lorenz time series, which aid in evaluatingthe prediction accuracies. The predicted time series are evaluated in terms of two criteria, predictionaccuracy and qualitative behavioural accuracy. Behavioural accuracy refers to the ability of the algorithmto 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 minimumembedding 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 embeddingdimension satisfy the behavioural criterion adequately, as expected. Sauer's algorithm has thegreatest 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 forvalues 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 tothe optimal time delay and Sauer's algorithm has the best behavioural accuracy for small values of thetime delay.Matlab code is presented for both algorithms.

[发布机构] Stellenbosch University