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E-grāmata: Practical Linear Algebra for Data Science: From Core Concepts to Applications Using Python

4.40/5 (20 ratings by Goodreads)

Mike X Cohen

Formāts: 328 pages
Izdošanas datums: 06-Sep-2022
Izdevniecība: O'Reilly Media
Valoda: eng
ISBN-13: 9781098120573

Citas grāmatas par šo tēmu:

Machine learning

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Formāts: 328 pages
Izdošanas datums: 06-Sep-2022
Izdevniecība: O'Reilly Media
Valoda: eng
ISBN-13: 9781098120573

Citas grāmatas par šo tēmu:

Machine learning

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If you want to work in any computational or technical field, you need to understand linear algebra. As the study of matrices and operations acting upon them, linear algebra is the mathematical basis of nearly all algorithms and analyses implemented in computers. But the way it's presented in decades-old textbooks is much different from how professionals use linear algebra today to solve real-world modern applications.

This practical guide from Mike X Cohen teaches the core concepts of linear algebra as implemented in Python, including how they're used in data science, machine learning, deep learning, computational simulations, and biomedical data processing applications. Armed with knowledge from this book, you'll be able to understand, implement, and adapt myriad modern analysis methods and algorithms.

Ideal for practitioners and students using computer technology and algorithms, this book introduces you to:

The interpretations and applications of vectors and matrices
Matrix arithmetic (various multiplications and transformations)
Independence, rank, and inverses
Important decompositions used in applied linear algebra (including LU and QR)
Eigendecomposition and singular value decomposition
Applications including least-squares model fitting and principal components analysis

Preface

1 Introduction

(6)

What Is Linear Algebra and Why Learn It?

(1)

About This Book

(1)

Prerequisites

(2)

Math

(1)

Attitude

(1)

Coding

(1)

Mathematical Proofs Versus Intuition from Coding

(1)

Code, Printed in the Book and Downloadable Online

(1)

Code Exercises

(1)

How to Use This Book (for Teachers and Self Learners)

(1)

2 Vectors, Part 1

(26)

Creating and Visualizing Vectors in NumPy

(4)

Geometry of Vectors

(1)

Operations on Vectors

(6)

Adding Two Vectors

(1)

Geometry of Vector Addition and Subtraction

(1)

Vector-Scalar Multiplication

(1)

Scalar-Vector Addition

(1)

Transpose

(1)

Vector Broadcasting in Python

(1)

Vector Magnitude and Unit Vectors

(1)

The Vector Dot Product

(4)

The Dot Product Is Distributive

(1)

Geometry of the Dot Product

(1)

Other Vector Multiplications

(2)

Hadamard Multiplication

(1)

Outer Product

(1)

Cross and Triple Products

(1)

Orthogonal Vector Decomposition

(4)

Summary

(1)

Code Exercises

(4)

3 Vectors, Part 2

(16)

Vector Sets

(1)

Linear Weighted Combination

(1)

Linear Independence

(3)

The Math of Linear Independence

(1)

Independence and the Zeros Vector

(1)

Subspace and Span

(3)

Basis

(5)

Definition of Basis

(2)

Summary

(1)

Code Exercises

(3)

4 Vector Applications

(12)

Correlation and Cosine Similarity

(3)

Time Series Filtering and Feature Detection

(1)

k-Means Clustering

(4)

Code Exercises

(4)

Correlation Exercises

(1)

Filtering and Feature Detection Exercises

(2)

k-Means Exercises

(1)

5 Matrices, Parti

(20)

Creating and Visualizing Matrices in NumPy

(4)

Visualizing, Indexing, and Slicing Matrices

(2)

Special Matrices

(2)

Matrix Math: Addition, Scalar Multiplication, Hadamard Multiplication

(2)

Addition and Subtraction

(1)

"Shifting" a Matrix

(1)

Scalar and Hadamard Multiplications

(1)

Standard Matrix Multiplication

(5)

Rules for Matrix Multiplication Validity

(1)

Matrix Multiplication

(1)

Matrix-Vector Multiplication

(2)

Matrix Operations: Transpose

(1)

Dot and Outer Product Notation

(1)

Matrix Operations: LIVE EVIL (Order of Operations)

(1)

Symmetric Matrices

(1)

Creating Symmetric Matrices from Nonsymmetric Matrices

(1)

Summary

(1)

Code Exercises

(5)

6 Matrices, Part 2

(32)

Matrix Norms

(2)

Matrix Trace and Frobenius Norm

(1)

Matrix Spaces (Column, Row, Nulls)

(7)

Column Space

(4)

Row Space

(1)

Null Spaces

(3)

Rank

(8)

Ranks of Special Matrices

(2)

Rank of Added and Multiplied Matrices

(1)

Rank of Shifted Matrices

(1)

Theory and Practice

(1)

Rank Applications

(2)

In the Column Space?

(1)

Linear Independence of a Vector Set

100

(1)

Determinant

101

(5)

Computing the Determinant

102

(1)

Determinant with Linear Dependencies

103

(1)

The Characteristic Polynomial

104

(2)

Summary

106

(1)

Code Exercises

107

(6)

7 Matrix Applications

113

(16)

Multivariate Data Covariance Matrices

113

(3)

Geometric Transformations via Matrix-Vector Multiplication

116

(4)

Image Feature Detection

120

(4)

Summary

124

(1)

Code Exercises

124

(5)

Covariance and Correlation Matrices Exercises

124

(2)

Geometric Transformations Exercises

126

(1)

Image Feature Detection Exercises

127

(2)

8 Matrix Inverse

129

(18)

The Matrix Inverse

129

(1)

Types of Inverses and Conditions for Invertibility

130

(1)

Computing the Inverse

131

(7)

Inverse of a 2 × 2 Matrix

131

(2)

Inverse of a Diagonal Matrix

133

(1)

Inverting Any Square Full-Rank Matrix

134

(2)

One-Sided Inverses

136

(2)

The Inverse Is Unique

138

(1)

Moore-Penrose Pseudoinverse

138

(1)

Numerical Stability of the Inverse

139

(2)

Geometric Interpretation of the Inverse

141

(1)

Summary

142

(1)

Code Exercises

143

(4)

9 Orthogonal Matrices and QR Decomposition

147

(12)

Orthogonal Matrices

147

(2)

Gram-Schmidt

149

(1)

QR Decomposition

150

(4)

Sizes of Q and R

151

(3)

QR and Inverses

154

(1)

Summary

154

(1)

Code Exercises

155

(4)

10 Row Reduction and LU Decomposition

159

(16)

Systems of Equations

159

(4)

Converting Equations into Matrices

160

(1)

Working with Matrix Equations

161

(2)

Row Reduction

163

(6)

Gaussian Elimination

165

(1)

Gauss-Jordan Elimination

166

(1)

Matrix Inverse via Gauss-Jordan Elimination

167

(2)

LU Decomposition

169

(2)

Row Swaps via Permutation Matrices

170

(1)

Summary

171

(1)

Code Exercises

172

(3)

11 General Linear Models and Least Squares

175

(18)

General Linear Models

176

(2)

Terminology

176

(1)

Setting Up a General Linear Model

176

(2)

Solving GLMs

178

(5)

Is the Solution Exact?

179

(1)

A Geometric Perspective on Least Squares

180

(1)

Why Does Least Squares Work?

181

(2)

GLM in a Simple Example

183

(4)

Least Squares via QR

187

(1)

Summary

188

(1)

Code Exercises

188

(5)

12 Least Squares Applications

193

(20)

Predicting Bike Rentals Based on Weather

193

(7)

Regression Table Using statsmodels

198

(1)

Multicollinearity

199

(1)

Regularization

199

(1)

Polynomial Regression

200

(4)

Grid Search to Find Model Parameters

204

(2)

Summary

206

(1)

Code Exercises

206

(7)

Bike Rental Exercises

206

(1)

Multicollinearity Exercise

207

(1)

Regularization Exercise

208

(2)

Polynomial Regression Exercise

210

(1)

Grid Search Exercises

210

(3)

13 Eigendecomposition

213

(28)

Interpretations of Eigenvalues and Eigenvectors

214

(3)

Geometry

214

(1)

Statistics (Principal Components Analysis)

215

(1)

Noise Reduction

216

(1)

Dimension Reduction (Data Compression)

217

(1)

Finding Eigenvalues

217

(3)

Finding Eigenvectors

220

(2)

Sign and Scale Indeterminacy of Eigenvectors

221

(1)

Diagonalizing a Square Matrix

222

(2)

The Special Awesomeness of Symmetric Matrices

224

(3)

Orthogonal Eigenvectors

224

(2)

Real-Valued Eigenvalues

226

(1)

Eigendecomposition of Singular Matrices

227

(1)

Quadratic Form, Definiteness, and Eigenvalues

228

(4)

The Quadratic Form of a Matrix

228

(2)

Definiteness

230

(1)

ATA Is Positive (Semi)definite

231

(1)

Generalized Eigendecomposition

232

(1)

Summary

233

(1)

Code Exercises

234

(7)

14 Singular Value Decomposition

241

(14)

The Big Picture of the SVD

241

(2)

Singular Values and Matrix Rank

243

(1)

SVD in Python

243

(1)

SVD and Rank-1 "Layers" of a Matrix

244

(2)

SVD from EIG

246

(3)

SVD of ATA

247

(1)

Converting Singular Values to Variance, Explained

247

(1)

Condition Number

248

(1)

SVD and the MP Pseudoinverse

249

(1)

Summary

250

(1)

Code Exercises

251

(4)

15 Eigendecomposition and SVD Applications

255

(24)

PCA Using Eigendecomposition and SVD

255

(5)

The Math of PCA

256

(3)

The Steps to Perform a PCA

259

(1)

PCA via SVD

259

(1)

Linear Discriminant Analysis

260

(2)

Low-Rank Approximations via SVD

262

(1)

SVD for Denoising

263

(1)

Summary

263

(1)

Exercises

264

(15)

PCA

264

(5)

Linear Discriminant Analyses

269

(3)

SVD for Low-Rank Approximations

272

(3)

SVD for Image Denoising

275

(4)

16 Python Tutorial

279

(24)

Why Python, and What Are the Alternatives?

279

(1)

IDEs (Interactive Development Environments)

280

(1)

Using Python Locally and Online

280

(2)

Working with Code Files in Google Colab

281

(1)

Variables

282

(3)

Data Types

283

(1)

Indexing

284

(1)

Functions

285

(5)

Methods as Functions

286

(1)

Writing Your Own Functions

287

(1)

Libraries

288

(1)

NumPy

289

(1)

Indexing and Slicing in NumPy

289

(1)

Visualization

290

(3)

Translating Formulas to Code

293

(3)

Print Formatting and F-Strings

296

(1)

Control Flow

297

(4)

Comparators

297

(1)

If Statements

297

(2)

For Loops

299

(1)

Nested Control Statements

300

(1)

Measuring Computation Time

301

(1)

Getting Help and Learning More

301

(1)

What to Do When Things Go Awry

301

(1)

Summary

302

(1)

Index

303

Mike is an associate professor of neuroscience at the Donders Institute (Radboud University Medical Centre) in the Netherlands. He has over 20 years experience teaching scientific coding, data analysis, statistics, and related topics, and has authored several online courses and textbooks. He has a suspiciously dry sense of humor and enjoys anything purple.

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