582 Notes - AMATH 582 Computation Methods for Data Analysis...

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Unformatted text preview: AMATH 582 Computation Methods for Data Analysis ∗ J. Nathan Kutz † March 14, 2010 Abstract This course is a survey of computational methods used for extracting meaningful results out of experimental or computational data. The cen- tral focus is on using a combination of spectral methods, statistics and linear algebra to analyze data and determine trends which are statistically significant. * These notes are intended as the primary source of information for AMATH 582. The notes are incomplete and may contain errors. Any other use aside from classroom purposes and personal research please contact me at [email protected] . c circlecopyrt J.N.Kutz, Winter 2010 (Version 1.0) † Department of Applied Mathematics, Box 352420, University of Washington, Seattle, WA 98195-2420 ( [email protected] ). 1 AMATH 582 ( c circlecopyrt J. N. Kutz) 2 Contents 1 Statistical Methods and Their Applications 3 1.1 Basic probability concepts . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Random variables and statistical concepts . . . . . . . . . . . . . 10 1.3 Hypothesis testing and statistical significance . . . . . . . . . . . 19 2 Time-Frequency Analysis: Fourier Transforms and Wavelets 25 2.1 Basics of Fourier Series and the Fourier Transform . . . . . . . . 26 2.2 FFT Application: Radar Detection and Filtering . . . . . . . . . 34 2.3 FFT Application: Radar Detection and Averaging . . . . . . . . 41 2.4 Time-Frequency Analysis: Windowed Fourier Transforms . . . . 48 2.5 Time-Frequency Analysis and Wavelets . . . . . . . . . . . . . . 55 2.6 Multi-Resolution Analysis and the Wavelet Basis . . . . . . . . . 63 2.7 Spectrograms and the G´ abor transforms in MATLAB . . . . . . 68 2.8 MATLAB Filter Design and Wavelet Toolboxes . . . . . . . . . . 74 3 Image Processing and Analysis 80 3.1 Basic concepts and analysis of images . . . . . . . . . . . . . . . 81 3.2 Linear filtering for image denoising . . . . . . . . . . . . . . . . . 88 3.3 Diffusion and image processing . . . . . . . . . . . . . . . . . . . 94 4 Linear Algebra and Singular Value Decomposition 100 4.1 Basics of The Singular Value Decomposition (SVD) . . . . . . . . 101 4.2 The SVD in broader context . . . . . . . . . . . . . . . . . . . . . 106 4.3 Introduction to Principle Component Analysis (PCA) . . . . . . 112 4.4 Principal Components, Diagonalization and SVD . . . . . . . . . 117 4.5 Principal Components and Proper Orthogonal Models . . . . . . 120 5 Independent Component Analysis 126 5.1 The concept of independent components . . . . . . . . . . . . . . 126 5.2 Image separation problem . . . . . . . . . . . . . . . . . . . . . . 134 5.3 Image separation and MATLAB . . . . . . . . . . . . . . . . . . 139 6 Image Recognition 145 6.1 Recognizing dogs and cats . . . . . . . . . . . . . . . . . . . . . . 145 6.2 The SVD and Linear Discrimination Analysis . . . . . . . . . . . 153 6.3 Implementing cat/dog recognition in MATLAB . . . . . . . . . . 161 7 Equation Free Modeling 165 7.1 Multi-scale physics: an equation-free approach . . . . . . . . . . 1657....
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This note was uploaded on 03/23/2012 for the course AMATH 582 taught by Professor N.k during the Winter '11 term at University of Washington.

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582 Notes - AMATH 582 Computation Methods for Data Analysis...

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