Brief Contents

• Chapter 1: Basic Matrix Operations
• Chapter 2: Matrix Theory and Visualization
• Chapter 3: Matrix Applications to Machine Learning
• Chapter 4: Matrix Applications to Regression Models
• Appendix A: A Tutorial of R and RStudio
• Appendix B: Visualization of Matrices

Full Table of Contents

• Preface viii
• Acknowledgements xiii
• 1 Basic Matrix Operations 1
  • 1.1 Matrix as a data array 1
  • 1.2 Matrix Algebra 3
    • 1.2.1 Matrix equality, addition and subtraction 3
    • 1.2.2 Matrix multiplication 4
  • 1.3 Some Useful Operations for Matrices in Data Science but Not Covered in Traditional Linear Algebra Books 10
    • 1.3.1 Delete a Row or Column or Both; a Sub-Matrix 10
    • 1.3.2 Insert Rows or/and Columns to a Matrix 11
    • 1.3.3 Statistics of the row or column data 12
    • 1.3.4 Sweep a martrix by a vector 12
    • 1.3.5 Conversions between a Vector and a Matrix 14
    • 1.3.6 Reduce Dimensions of an n-Dimensional Array 14
  • 1.4 A Set of Linear Equations 15
  • 1.5 Eigenvalues and eigenvectors of a square space matrix 16
    • 1.5.1 Matrices of data anomalies, standardized anomalies, covariance, and correlation 17
    • 1.5.2 Eigenvectors and their corresponding eigenvalues 19
  • 1.6 An SVD representation model for space-time data 20
    • 1.6.1 Space-Time Data Matrix and Its decomposition 20
    • 1.6.2 SVD of the Space-Time Anomaly Data Matrix and Eigenvalue Problem of a Covariance Matrix 22
  • 1.7 Image analysis using SVD and R 25
    • 1.7.1 Data for a color photo and its grayscale figure 25
    • 1.7.2 SVD of the grayscale photo data 26
    • 1.7.3 Reconstructing a color photo from the data of monotone photos 29
  • 1.8 Mass balance for chemical equations in marine chemistry 31
  • 1.9 Multivariate linear regression using matrix notations 32
  • 1.10 Chapter summary 34
  • References and Further Readings 37
  • Exercises 38


• 2 Matrix Theory and Visualization 41
  • 2.1 Matrix Definitions 41
  • 2.2 Fundamental Properties of Matrices 44
  • 2.3 Some basic concepts and theories of linear algebra 49
    • 2.3.1 Linear equations 49
    • 2.3.2 Linear transformations 50
    • 2.3.3 Linear independence 51
    • 2.3.4 Determinants 52
    • 2.3.5 Rank of a matrix 53
  • 2.4 Eigenvectors and eigenvalues 56
    • 2.4.1 Definition of eigenvectors and eigenvalues 56
    • 2.4.2 Properties of eigenvectors and eigenvalues for a symmetric matrix 60
  • 2.5 Hadammard and Other Matrix Multiplications 62
    • 2.5.1 Hadamard Product of Two Matrices of the Same Dimensions 63
    • 2.5.2 Jordan Product of Two Matrices of the Same Dimensions 63
    • 2.5.3 Commutator of Two Matrices of the Same Dimensions 63
    • 2.5.4 Outer Product of Two Vectors and Two Matrices 64
    • 2.5.5 Kronecker Product of Two Matrices of the Same Dimensions 66
  • 2.6 Direct Sum of Two Matrices 67
  • 2.7 Visualization of Eigenvalues and Eigenvectors for a Covariance Matrix 67
  • 2.8 Singular Value Decomposition 68
    • 2.8.1 SVD formula and a simple SVD examlpe 68
  • 2.9 SVD for the standardized sea level pressure data of Tahiti and Darwin 74
  • 2.10 Chapter Summary 76
  • References and Further Readings 78
  • Exercises 79


• 3 Matrix Applications to Machine Learning 83
  • 3.1 K-means clustering 83
    • 3.1.1 K-means set up and trivial examples 84
    • 3.1.2 The number of exhaustive K clusters from N points 90
    • 3.1.3 A K-means algorithm 91
    • 3.1.4 K-means clustering for the daily Miami weather data 93
  • 3.2 Support vector machine 103
    • 3.2.1 SVM for a system of three points labeled in two categories 108
    • 3.2.2 SVM mathematical formulation for a system of many points in two categories 112
  • 3.3 Random forest method for classification and regression 118
    • 3.3.1 RF flower classification for a benchmark iris dataset 118
    • 3.3.2 RF regression for the daily ozone data of New York City 125
    • 3.3.3 What does a decision tree look like? 129
  • 3.4 Neural network and deep learning 132
    • 3.4.1 An NN model for an automized decision system 132
    • 3.4.2 An NN prediction of iris species 139
  • 3.5 Chapter Summary 143
  • References and Further Readings 144
  • Exercises 145


• 4 Matrix Applications to Regression Models 148
  • 4.1 Simple linear regression 148
    • 4.1.1 Temperature lapse rate and an approximately linear model 148
    • 4.1.2 Assumptions and formula derivations of the single variate linear regression 152
    • 4.1.3 Statistics of slope and intercept: Distributions, confidence intervals, and inference 163
  • 4.2 Multiple linear regression 176
    • 4.2.1 Calculating the Colorado TLR when taking location coordinates into account 176
    • 4.2.2 Formulas for estimating parameters in the multiple linear regression 179
  • 4.3 Nonlinear fittings using the multiple linear regression 181
    • 4.3.1 Diagnostics of linear regression: An example of global temperature 181
    • 4.3.2 Fit a third order polynomial 186
  • 4.4 Linear Regression by Weighted Least Squares 189
  • 4.5 Chapter Summary 190
  • References and Further Readings 191
  • Exercises 192


• Appendix A A Tutorial of R and R Studio 195
  • A.1 Download and install R and R-Studio 195
  • A.2 R Tutorial 196
    • A.2.1 R as a smart calculator 197
    • A.2.2 Define a sequence in R 198
    • A.2.3 Define a function in R 199
    • A.2.4 Plot with R 199
    • A.2.5 Symbolic calculations by R 200
    • A.2.6 Vectors and matrices 201
    • A.2.7 Simple statistics by R 204
  • A.3 Online Tutorials 205
    • A.3.1 Youtube tutorial: for true beginners 206
    • A.3.2 Youtube tutorial: for some basic statistical summaries 206
    • A.3.3 Youtube tutorial: Input data by reading a csv file into R 206
  • A.4 Chapter Summary 208
  • References and Further Readings 210
  • Exercises 211


• Appendix B Visualization of Matrices 214
  • References and Further Readings 246


• Index 251