Acknowledgments
Introduction
1 Graph Theory
1.1 Basic Terminology
1.2 The Power of the Adjacency Matrix
1.3 Eigenvalues and Eigenvectors as Key Players
1.4 CASE STUDY: Applications in Sport Ranking
1.5 CASE STUDY: Gerrymandering
1.6 Exercises
2. Stochastic Processes
2.1 Markov Chain Basics
2.2 Hidden Markov Models
2.2.1 The Likelihood Problem
2.2.2 The Decoding Problem
2.2.3 The Learning Problem
2.3 CASE STUDY: Spread of Infectious Disease
2.4 CASE STUDY: Text Analysis and Autocorrect
2.5 CASE STUDY: Tweets and Time Series
2.6 Exercises
3. SVD and PCA
3.1 Vector and Inner Product Spaces
3.2 Singular Values
3.3 Singular Value Decomposition
3.4 Compression of Data Using Principal Component Analysis (PCA)
3.5 PCA, Covariance, and Correlation
3.6 Linear Discriminant Analysis
3.7 CASE STUDY: Digital Humanities
3.8 CASE STUDY: Facial Recognition Using PCA and LDA
3.9 Exercises
4. Interpolation
4.1 Lagrange Interpolation
4.2 Orthogonal Families of Polynomials
4.3 Newton’s Divided Difference
4.3.1 Newton’s interpolation via divided difference
4.3.2 Newton’s interpolation via the Vandermonde matrix
4.4 Chebyshev interpolation
4.5 Hermite interpolation
4.6 Least Squares Regression
4.7 CASE STUDY : Chebyshev Polynomials and Cryptography
4.8 CASE STUDY: Racial Disparities in Marijuana Arrests
4.9 CASE STUDY : Interpolation in Higher Education Data
4.10 Exercises
5. Optimization and Learning Techniques for Regression
5.1 Basics of Probability Theory
5.2 Introduction to Matrix Calculus
5.2.1 Matrix Differentiation
5.2.2 Matrix Integration
5.3 Maximum Likelihood Estimation
5.4 Gradient Descent Method
5.5 Introduction to Neural Networks
5.5.1 The Learning Process
5.5.2 Sigmoid Activation Functions
5.5.3 Radial Activation Functions
5.6 CASE STUDY: Handwriting Digit Recognition
5.7 CASE STUDY: Poisson Regression and COVID Counts
5.8 Exercises
6 Decision Trees and Random Forests
6.1 Decision Trees
6.1.1 Decision Trees Regression
6.2 Regression Trees
6.3 Random Decision Trees and Forests
6.4 CASE STUDY: Entropy of Wordle
6.5 CASE STUDY : Bird Call Identification
6.6 Exercises
7. Random Matrices and Covariance Estimate
7.1 Introduction to Random Matrices
7.2 Stability
7.3 Gaussian Orthogonal Ensemble
7.4 Gaussian Unitary Ensemble
7.5 Gaussian Symplectic Ensemble
7.6 Random Matrices and the Relationship to the Covariance
7.7 CASE STUDY: Finance and Brownian Motion
7.8 CASE STUDY: Random Matrices in Gene Interaction
7.9 Exercises
8. Sample Solutions to Exercises
8.1 Chapter 1
8.2 Chapter 2
8.3 Chapter 3
8.4 Chapter 4
8.5 Chapter 5
8.6 Chapter 6
8.7 Chapter 7
Github Links 349
Bibliography 351
Index 355
Biography
Dr. Crista Arangala is Professor of Mathematics and Chair of the Department of Mathematics and Statistics at Elon University in North Carolina. She has been teaching and researching in a variety of fields including inverse problems, applied partial differential equations, applied linear algebra, mathematical modeling and service learning education. She runs a traveling science museum with her Elon University students in Kerala, India. Dr. Arangala was chosen to be a Fulbright Scholar in 2014 as a visiting lecturer at the University of Colombo where she continued her projects in inquiry learning in Linear Algebra and began working with a modeling team focusing on Dengue fever research. Dr. Arangala has published several textbooks that implore inquiry learning techniques including Exploring Linear Algebra: Labs and Projects with MATLAB® and Mathematical Modeling: Branching Beyond Calculus.
