使用Wavelets和机器学习的高级数据分析

机器学习、数据驱动工程、小波分析、傅立叶变换和动力系统，欢迎来到我的机器学习和数据分析课程，这门课程将教你如何使用高级算法来解决数据的实际问题。我是伊曼纽，一名拥有高级算法博士学位的机械工程师，我将是您这门课的讲师。本课程由四个主要部分组成:第一部分:傅立叶分析和小波概述。您将学习这两个强大的数学工具的基础知识，用于分析不同领域的信号和图像。第2部分:傅立叶级数、变换和小波的数据分析。您将学习如何应用这些方法在时域和频域中高效地处理和探索数据。第3部分:机器学习方法。您将学习如何使用使计算机能够从数据中学习并做出智能预测或决策的技术，如线性回归、曲线拟合、最小二乘法、梯度下降、奇异值分解(等等)。第4部分:动力系统。您将学习如何使用数学方程来模拟和理解复杂的非线性现象，这些现象会随着时间的推移而变化。我们也将把机器学习技术应用于动力系统，例如SINDy算法。本课程结束时，您将能够:理解傅立叶分析和小波的原理和应用使用傅立叶级数和变换来分析各种领域的数据将机器学习方法应用于不同的问题使用小波从数据中提取特征理解自然数据稀疏性的重要性，以及具有现实示例的压缩感知的革命性概念。从时间序列数据中发现动力系统的控制方程(SINDy算法)。Advanced Data Analysis Using Wavelets And Machine Learning

我希望你喜欢这门课程，并发现它对你的个人和职业目标有用。让我们提供更多关于本课程主要部分的细节:第一部分是傅立叶和小波分析的初步介绍。重点是理解与这些基本主题最相关的概念。在第2部分中，介绍了傅立叶级数和傅立叶变换。虽然显示了最重要的数学公式，但重点不在数学上。这一部分的重点之一是展示傅立叶变换的一个可能的应用:光谱导数。然后，我们通过展示多分辨分析的一些应用来更详细地介绍小波的概念。这是用Matlab举例说明的，没有使用严格的数学公式。即使学生无法使用Matlab，他们也可以跟随并获得直觉。这一部分的另一个重要成果是对众所周知的计算FFT方法进行了简单而透彻的解释。还有一些关于小波逆变换和测不准原理的附加内容(这里我们看到更多的数学，但这是附加内容，如果你想跳过，就做吧)。第三部分介绍了一些机器学习技术:曲线拟合、梯度下降、线性回归、奇异值分解、分类、高斯混合模型(GMM)。这一部分的目的是展示一些实际应用，并阐明它们的有用性。我们还将关注稀疏性和压缩感知，这是信号处理中的相关概念。稀疏性是指信号可以用某个域中几个非零系数来表示，如频率或小波。压缩感知意味着，通过利用信号的稀疏性和使用优化技术，可以从比奈奎斯特-香农采样定理所需更少的测量值中重构信号。这些概念对于降低机器学习应用(如图像处理或雷达成像)中数据的维度和复杂性非常有用。第4部分是对动态模型的独立介绍。这一部分包含的模型有食饵-捕食者模型，流行病模型，人口增长的逻辑斯蒂模型。学生将学习如何使用免费的开源软件Scilab(与Matlab非常相似)来实现这些模型。与第4部分相关，有一个称为SINDy的机器学习技术的应用，这是非线性动力学稀疏识别的缩写。它是一种机器学习算法，可以从时间序列数据中发现动力系统的控制方程。主要思想是假设系统可以用一组稀疏的非线性函数来描述，然后使用一种促进稀疏性的回归技术来寻找这些函数中最适合数据的系数。通过这种方式，SINDy可以恢复复杂系统的可解释且简洁的模型。注意:对于课程的一些讲座，我受到了S.L. Brunton和J. N. Kutz的书《数据驱动的科学与工程》的启发。这本书是一个极好的信息来源，可以更深入地挖掘课程中讨论的大多数(尽管不是全部)主题。

MP4 |视频:h264，1280×720 |音频:AAC，44.1 KHz
语言:英语|大小:9 GB |时长:10h 5m

你会学到什么
理解傅立叶分析和Wavelets的原理和应用(重点是物理见解而不是数学)
使用傅立叶级数和变换来分析不同领域的数据
将机器学习方法应用于不同的问题
使用Wavelets从数据中提取特征
理解自然数据稀疏性的重要性
通过实际例子了解压缩传感的革命性概念。
从时间序列数据中发现动力系统的控制方程(SINDy算法)
用Matlab实现高效的机器学习算法
理解并应用奇异值分解(SVD)(我们甚至证明了！)
了解如何使用奇异值分解来近似图像
从实例中理解最小二乘法(LSM)
理解并应用快速傅立叶变换(FFT) -迄今为止发现的最重要的算法之一
理解并应用离散余弦变换(DCT)
学习如何推导小波逆变换
了解如何获得反离散余弦变换
学习如何推导傅立叶逆变换
学习如何推导测不准原理，以及这如何影响时间-频率分辨率

要求
熟悉一些线性代数将使这门课更容易跟上。
微积分可能有助于在更大程度上理解机器学习技术和小波。我的主要目的不是向你展示数学，但是有了一些数学背景，你将能够更彻底地欣赏内容

概观
第1部分:傅立叶和Wavelet分析概述

第1讲傅立叶分析概述

第二讲短时傅里叶变换的空间频率分辨率

第三讲小波和空间频率分辨率

第二部分:傅立叶级数和变换的数据分析

第四讲傅立叶级数和傅立叶变换概述

第5讲傅立叶变换的符号

第六讲函数导数的傅里叶变换

第7讲快速傅立叶变换(FFT)的重要性

第八讲光谱导数

第9讲小波和多分辨率分析

第10讲附加内容:为什么狄拉克δ有助于导出傅立叶逆变换

第11讲附加部分:小波逆变换的数学推导

第12讲附加部分:不确定性原理-数学证明

第三部分:机器学习的方法

第13讲曲线拟合

第14讲曲线拟合的例子-最小二乘法

第15讲梯度下降

第16讲奇异值分解

第17讲用奇异值分解逼近图像

第18讲监督机器学习-用奇异值分解和小波提取特征

第19讲线性回归:矩阵形式的最小二乘法

第20讲线性回归:对数据中异常值的敏感性

第21讲分类/决策树

第22讲高斯混合模型

第二十三讲高斯混合模型的例子

第4部分:稀疏性和压缩感知

第24讲稀疏性和压缩感知:稀疏性介绍

第25讲稀疏性和压缩感知:为什么“自然”信号是可压缩的

第26讲稀疏性和压缩感知:压缩感知介绍

第27讲压缩传感示例

第28讲离散余弦变换及其逆变换的定义

第29讲附加内容:对于求反离散余弦变换至关重要的公式

第五节:动力系统

第30讲数学模型部分介绍

第三十一讲纯食饵-捕食者模型

第32讲平衡点及其稳定性

第33讲食饵-捕食者模型中的平衡点

第34讲Scilab简介

第35讲用Scilab构建模型第1部分

第36讲用Scilab构建模型第2部分

第37讲参数如何影响模型的输出

第三十八讲捕鱼对模型的影响

第39讲模型中增加了逻辑术语

第40讲流行病进化模型

第41讲稳定性的数学分析

第42讲单种群逻辑斯蒂模型的模拟和数学

第六部分:机器学习在动力系统中的应用

第43讲动力系统和混沌:洛伦兹系统

第44讲机器学习寻找数据背后的动态模型(SYNDy算法)

第7节:奇异值分解的证明

第45讲介绍奇异值分解的证明

第46讲线性代数中的对角化定理

第47讲奇异值分解(SVD)背后的直觉

寻求加强对机器学习技术的理解并提高游戏水平的数据科学家，想要成为数据分析师或人工智能爱好者的人，人工智能工程师，软件开发人员，应用数学家，物理学家，研究人员，程序员，任何想学习如何使用高级算法解决数据实际问题的人。对于那些对机器学习和数据分析感兴趣的人来说尤其有用。

Machine Learning, Data-Driven Engineering, Wavelet Analysis, Fourier Transforms, and Dynamical Systems

What you’ll learn
Understand the principles and applications of Fourier analysis and wavelets (with emphasis on the physical insights rather than the mathematics)
Use Fourier series and transforms to analyze data in various domains
Apply machine learning methods to different problems
Extract features from data using wavelets
Understand the importance of sparsity of natural data
Understand the revolutionary concept of compressed sensing, with realistic examples.
Discover the governing equations of a dynamical system from time series data (SINDy algorithm)
Implement efficient Machine Learning algorithms with Matlab
Understand and apply the Singular Value Decomposition (SVD) (we even prove it!)
Learn how to use the SVD to approximate images
Understand the Least Squares Method (LSM) from practical examples
Understand and apply the Fast Fourier Transform (FFT) – one of the most important algorithms ever discovered
Understand and apply the Discrete Cosine Transform (DCT)
Learn how to derive the Inverse Wavelet Transform
Learn how to derive the Inverse Discrete Cosine Transform
Learn how to derive the Inverse Fourier Transform
Learn how to derive the Uncertainty Principle, and how this affects the time-frequency resolution

Requirements
Familiarity with some linear algebra will make the class easier to follow along with.
Calculus might be useful to understand machine learning techniques and wavelets to a greater degree. My primary aim is not to show you the mathematics, but with some mathematical background you would be able to appreciate the contents more thoroughly

Description
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. The focus will be 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), 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.

Overview
Section 1: Overview of Fourier and Wavelet Analysis

Lecture 1 Overview of Fourier Analysis

Lecture 2 Space-Frequency resolution for the Short Time Fourier Transform

Lecture 3 Wavelets and Space-Frequency resolution

Section 2: Data Analysis with Fourier Series and Transform

Lecture 4 Summary of Fourier Series and Fourier Transform

Lecture 5 Notation for the Fourier Transform

Lecture 6 Fourier Transform of the derivative of a function

Lecture 7 The importance of the Fast Fourier Transform (FFT)

Lecture 8 Spectral derivative

Lecture 9 Wavelets and Multiresolution Analysis

Lecture 10 Extra: Why the Dirac delta helps derive the Inverse Fourier Transform

Lecture 11 Extra: Mathematical derivation of the Inverse Wavelet Transform

Lecture 12 Extra: Uncertainty principle – mathematical proof

Section 3: Methods in Machine Learning

Lecture 13 Curve fitting

Lecture 14 Example of curve fitting – least squares method

Lecture 15 Gradient descent

Lecture 16 Singular Value Decomposition – SVD

Lecture 17 Approximation of images with the SVD

Lecture 18 Supervised machine learning – extraction of features with SVD and Wavelets

Lecture 19 Linear regression: least squares method in matrix form

Lecture 20 Linear regression: sensitivity to outliers in the data

Lecture 21 Classification/decision trees

Lecture 22 Gaussian Mixture Models

Lecture 23 Example of Gaussian mixture model

Section 4: Sparsity and Compressed Sensing

Lecture 24 Sparsity and compressed sensing: intro to sparsity

Lecture 25 Sparsity and compressed sensing: why “natural” signals are compressible

Lecture 26 Sparsity and compressed sensing: intro to compressed sensing

Lecture 27 Example of compressed sensing

Lecture 28 Definition of the Discrete Cosine Transform (DCT) and its inverse

Lecture 29 Extra: formula which is crucial to finding the Inverse Discrete Cosine Transform

Section 5: Dynamical systems

Lecture 30 Introduction to the section on mathematical models

Lecture 31 Pure prey-predator model

Lecture 32 Equilibrium points and their stability

Lecture 33 Equilibrium points in the prey-predator model

Lecture 34 Introduction to Scilab

Lecture 35 Constructing the model with Scilab part 1

Lecture 36 Constructing the model with Scilab part 2

Lecture 37 How parameters affect the output of the model

Lecture 38 Influence of fishing on the model

Lecture 39 Addition of logistic terms to the model

Lecture 40 Model on the evolution of epidemics

Lecture 41 Mathematical analysis of stability

Lecture 42 Simulation and mathematics of the logistic model with one population

Section 6: Machine learning applied to dynamical systems

Lecture 43 Dynamical systems and chaos: Lorenz system

Lecture 44 Machine learning to find dynamical models behind data (SYNDy algorithm)

Section 7: Proof of the SVD decomposition

Lecture 45 Introduction to this section on the proof of the SVD

Lecture 46 Diagonalization theorem in Linear Algebra

Lecture 47 Intuition behind the Singular Value Decomposition (SVD)

data scientists who seek to reinforce their understanding of Machine Learning techniques and step up their game,Wannabe data analysts or A.I. enthusiasts,ML engineers,software developers,applied mathematicians,physicists,Researchers,Programmers,Anyone who wants to learn how to use advanced algorithms to solve real problems with data. It is especially useful for those who are interested in machine learning and data analysis.