{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
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
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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. Orthonormal : Two vectors v 1 and v 2 are orthonormal if they are orthogonal and of unit length. Span : The span of subspace generated by vectors v 1 and v 2 V is span( v 1 ,v 2 ) ≡ { rv 1 + sv 2 | r,s R } .
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern