Advanced Data Analysis using Wavelets and Machine Learning

Machine Learning, Data Analysis, Fourier Methods and Dynamical SystemsThis course brings together several tools that are often studied separately: Fourier analysis, wavelets, data analysis, machine learning, compressed sensing, and dynamical systems.The common thread is simple: we want to understand how mathematical methods can help us extract structure from data.Some parts of the course are more mathematical, especially those involving Fourier transforms, wavelets, sparsity, and compressed sensing. Other parts are more computational and practical, with examples involving regression, classification, feature extraction, and dynamical models.My background is in mechanical engineering, research, signal processing, and mathematical modelling, so the course is taught from that perspective: not as a collection of isolated algorithms, but as a guided path through the ideas that make these methods useful.Course StructureThe course is divided into four main parts.Part 1: Fourier Analysis and WaveletsWe begin with an introduction to Fourier analysis and wavelets.The goal is to build intuition for two of the most important tools used in signal processing, data analysis, and applied mathematics. We discuss how signals can be represented in different domains, why frequency information is useful, and why wavelets are especially valuable when the structure of a signal changes over time or space.This part is meant to prepare the ground for the more applied sections of the course.Part 2: Data Analysis with Fourier Series, Fourier Transforms and WaveletsIn the second part, we apply Fourier series, Fourier transforms, and wavelets to data analysis.The emphasis is not on abstract mathematical rigor, although the main formulas are introduced when needed. The focus is instead on understanding what these tools do and how they can be used in practice.One important exampl