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所在平台: Udemy |
课程主页: https://www.udemy.com/course/advanced-data-analysis-using-wavelets-and-machine-learning/
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Coursera课程《高级数据分析——小波与机器学习》概要 本课程旨在教授学员如何运用先进的算法解决实际数据问题。课程内容涵盖了四大主要部分： **第一部分：傅里叶分析与小波概述** 本部分将介绍傅里叶分析和小波分析这两个强大的数学工具。学员将学习它们在不同域中分析信号和图像的基础知识。AI领域资深专家Emanuele将作为导师，带领大家深入理解这些核心概念。 **第二部分：基于傅里叶级数、变换和小波数据分析** 本部分将演示如何利用傅里叶级数、傅里叶变换和小波方法，在时域和频域高效地处理和探索数据。课程将展示傅里叶变换在频谱导数等方面的应用，并通过多分辨率分析的实例深入浅出地介绍小波概念。值得注意的是，课程将通过Matlab演示，即使没有Matlab使用经验的学员也能轻松掌握，并会详细解释FFT（快速傅里叶变换）的计算原理。此外，还将额外介绍小波逆变换与不确定性原理，提供更深入的数学探索（该部分可选择性学习）。 **第三部分：机器学习方法** 本部分将介绍一系列关键的机器学习技术，包括曲线拟合、梯度下降、线性回归、奇异值分解（SVD）、特征提取、分类以及高斯混合模型（GMM）等。课程重点在于展示这些方法的实际应用及其价值。特别地，课程将深入讲解“稀疏性”和“压缩传感”这两个在信号处理领域至关重要的概念。“稀疏性”指的是信号在特定域（如频域或小波域）可以通过少数非零系数表示，而“压缩传感”则是在利用信号稀疏性的基础上，通过优化技术，仅需少量测量值即可重构信号，这对于机器学习应用（如图像处理）中降低数据维度和复杂度至关重要。 **第四部分：动力系统** 本部分将独立介绍动力学模型，如捕食者-猎物模型、传染病模型和人口增长的逻辑斯蒂模型。学员将学习如何使用免费开源软件Scilab（与Matlab类似）来实现这些模型。作为与第四部分相关的应用，课程将介绍一种名为SINDy（Sparse Identification of Nonlinear Dynamics，非线性动力学的稀疏识别）的机器学习算法。SINDy能够从时间序列数据中发现描述动力系统的控制方程，其核心思想是假设系统可以由少数非线性函数稀疏表示，然后利用稀疏回归技术找到最优系数来拟合数据，从而恢复出可解释的、简洁的复杂系统模型。 **课程目标** 通过本课程的学习，学员将能够： * 理解傅里叶分析和小波分析的原理及其应用。 * 运用傅里叶级数和变换在不同域中分析数据。 * 将机器学习方法应用于各种问题。 * 利用小波进行特征提取。 * 理解自然数据稀疏性的重要性，以及压缩传感的革命性概念，并通过实际案例进行理解。 * 从时间序列数据中识别（发现）动力系统的控制方程（SINDy算法）。 **参考资料** 本课程部分内容受到S.L. Brunton和J. N. Kutz合作的《Data-Driven Science and Engineering》一书的启发，该书为深入学习课程主题提供了极佳的参考。 本课程适合希望提升数据分析和机器学习能力的工程师、科学家及对该领域感兴趣的学习者。
Welcome to my course on Machine Learning and Data Analysis, a course that will teach you how to use advanced algorithms to solve real problems with data. I am Emanuele, a mechanical engineer with a PhD in advanced algorithms, and I will be your instructor for this course.This course consists of four main parts:Part 1: Overview on Fourier Analysis and Wavelets. You will learn the basics of these two powerful mathematical tools for analyzing signals and images in different domains.Part 2: Data Analysis with Fourier Series, Transforms and Wavelets. You will learn how to apply these methods to process and explore data efficiently and effectively, both in time and frequency domains.Part 3: Machine Learning Methods. You will learn how to use techniques that enable computers to learn from data and make intelligent predictions or decisions, such as linear regression, curve fitting, least squares, gradient descent, Singular Value Decomposition (and more).Part 4: Dynamical Systems. You will learn how to model and understand complex and nonlinear phenomena that change over time, using mathematical equations. We will also apply machine learning techniques to dynamical systems, such as the SINDy algorithm.By the end of this course, you will be able to:Understand the principles and applications of Fourier analysis and waveletsUse Fourier series and transforms to analyze data in various domainsApply machine learning methods to different problemsExtract features from data using waveletsUnderstand the importance of sparsity of natural data, as well as the revolutionary concept of compressed sensing, with realistic examples.Discover the governing equations of a dynamical system from time series data (SINDy algorithm).I hope you enjoy this course and find it useful for your personal and professional goals.------------------------------------------------------------------Let's provide some more details about the main parts of this course: Part 1 constitutes a preliminary introduction to Fourier and Wavelet Analysis. Special focus will be put on understanding the most relevant concepts related to these fundamental topics.In part 2, the Fourier series and the Fourier Transform are introduced. Although the most important mathematical formulae are shown, the focus is not on the mathematics. One of the key points of this part is to show one possible application of the Fourier Transform: the spectral derivative. Then, we introduce the concept of Wavelets more in detail by showing some applications of Multiresolution Analysis.This is exemplified with Matlab, without using rigorous mathematical formulae. The student can follow and get the intuition even if they have no access to Matlab.Another important achievement of this part is to convey a simple but thorough explanation of the well-known computational FFT method.There are also some extras on the Inverse Wavelet Transform and the Uncertainty principle (here we see more mathematics, but this is an extra, if you want to skip it, just do it).In part 3, some machine learning techniques are introduced: the methods of curve-fitting, gradient descent, linear regression, Singular Value Decomposition (SVD), feature extraction, classification, Gaussian Mixture Model (GMM). The objective in this part is to show some practical applications and cast light on their usefulness.We will also focus on sparsity and compressed sensing, which are related concepts in signal processing. Sparsity means that a signal can be represented by a few non-zero coefficients in some domain, such as frequency or wavelet. Compressed sensing means that a signal can be reconstructed from fewer measurements than the Nyquist-Shannon sampling theorem requires, by exploiting its sparsity and using optimization techniques. These concepts are useful for reducing the dimensionality and complexity of data in machine learning applications, such as image processing or radar imaging.Part 4 is a self-contained introduction to dynamical models. The models contained in this part are the prey-predator model, the model of epidemics, the logistic model of population growth.The student will learn how to implement these models using free and open-source software called Scilab (quite similar to Matlab).Related to Part 4, there is an application of machine learning technique called SINDy, which is an acronym for Sparse Identification of Nonlinear Dynamics. It is a machine learning algorithm that can discover the governing equations of a dynamical system from time series data. The main idea is to assume that the system can be described by a sparse set of nonlinear functions, and then use a sparsity-promoting regression technique to find the coefficients of these functions that best fit the data. This way, SINDy can recover interpretable and parsimonious models of complex systems.Note: For some of the lectures of the course, I was inspired by S.L. Brunton and J. N. Kutz's book titled "Data-Driven Science and Engineering". This book is an excellent source of information to dig deeper on most (although not all) of the topics discussed in the course.
