Veitch - Wavelet Neural Networks and their application in...

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Unformatted text preview: Wavelet Neural Networks and their application in the study of dynamical systems David Veitch Dissertation submitted for the MSc in Data Analysis, Networks and Nonlinear Dynamics. Department of Mathematics University of York UK August 2005 1 Abstract The main aim of this dissertation is to study the topic of wavelet neural networks and see how they are useful for dynamical systems applications such as predicting chaotic time series and nonlinear noise reduction. To do this, the theory of wavelets has been studied in the first chapter, with the emphasis being on discrete wavelets. The theory of neural networks and its current applications in the modelling of dynamical systems has been shown in the second chapter. This provides sufficient background theory to be able to model and study wavelet neural networks. In the final chapter a wavelet neural network is implemented and shown to accurately estimate the dynamics of a chaotic system, enabling prediction and enhancing methods already available in nonlinear noise reduction. Contents Notations 4 Chapter 1. Wavelets 6 1.1. Introduction 7 1.2. What is a Wavelet? 7 1.3. Wavelet Analysis 8 1.4. Discrete Wavelet Transform Algorithm 12 1.5. Inverse Discrete Wavelet Transform 16 1.6. Daubechies Discrete Wavelets 17 1.7. Other Wavelet Definitions 19 1.8. Wavelet-Based Signal Estimation 21 Chapter 2. Neural Networks 28 2.1. Introduction - What is a Neural Network? 29 2.2. The Human Brain 29 2.3. Mathematical Model of a Neuron 29 2.4. Architectures of Neural Networks 31 2.5. The Perceptron 33 2.6. Radial-Basis Function Networks 38 2.7. Recurrent Networks 41 Chapter 3. Wavelet Neural Networks 50 3.1. Introduction 51 3.2. What is a Wavelet Neural Network? 51 3.3. Learning Algorithm 54 3.4. Java Program 57 3.5. Function Estimation Example 59 3.6. Missing Sample Data 61 3.7. Enhanced Prediction using Data Interpolation 62 3.8. Predicting a Chaotic Time-Series 65 3.9. Nonlinear Noise Reduction 65 3.10. Discussion 69 Appendix A. Wavelets - Matlab Source Code 71 A.1. The Discrete Wavelet Transform using the Haar Wavelet 71 2 CONTENTS 3 A.2. The Inverse Discrete Wavelet Transform using the Haar Wavelet 72 A.3. Normalised Partial Energy Sequence 73 A.4. Thresholding Signal Estimation 73 Appendix B. Neural Networks - Java Source Code 75 B.1. Implementation of the Perceptron Learning Algorithm 75 Appendix C. Wavelet Neural Networks - Source Code 79 C.1. Function Approximation using a Wavelet Neural Network 79 C.2. Prediction using Delay Coordinate Embedding 87 C.3. Nonlinear Noise Reduction 88 Appendix. Bibliography 89 NOTATIONS 4 Notations Orthogonal : Two elements v 1 and v 2 of an inner product space E are called orthogonal if their inner product ( v 1 ,v 2 ) is 0....
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Veitch - Wavelet Neural Networks and their application in...

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