Data Science and Machine Learning: Mathematical and Statistical Methods

"This textbook is a well-rounded, rigorous, and informative work presenting the mathematics behind modern machine learning techniques. It hits all the right notes: the choice of topics is up-to-date and perfect for a course on data science for mathematics students at the advanced undergraduate or early graduate level. This book fills a sorely-needed gap in the existing literature by not sacrificing depth for breadth, presenting proofs of major theorems and subsequent derivations, as well as providing a copious amount of Python code. I only wish a book like this had been around when I first began my journey!" -Nicholas Hoell, University of Toronto "This is a well-written book that provides a deeper dive into data-scientific methods than many introductory texts. The writing is clear, and the text logically builds up regularization, classification, and decision trees. Compared to its probable competitors, it carves out a unique niche. -Adam Loy, Carleton College The purpose of Data Science and Machine Learning: Mathematical and Statistical Methods is to provide an accessible, yet comprehensive textbook intended for students interested in gaining a better understanding of the mathematics and statistics that underpin the rich variety of ideas and machine learning algorithms in data science. Key Features: Focuses on mathematical understanding. Presentation is self-contained, accessible, and comprehensive. Extensive list of exercises and worked-out examples. Many concrete algorithms with Python code. Full color throughout. Further Resources can be found on the authors website: https://github.com/DSML-book/Lectures

Table of Contents:
Preface Notation 1. Importing, Summarizing, and Visualizing Data 1.1 Introduction 1.2 Structuring Features According to Type 1.3 Summary Tables 1.4 Summary Statistics 1.5 Visualizing Data 1.5.1 Plotting Qualitative Variables 1.5.2 Plotting Quantitative Variables 1.5.3 Data Visualization in a Bivariate Setting Exercises 2. Statistical Learning 2.1 Introduction 2.2 Supervised and Unsupervised Learning 2.3 Training and Test Loss 2.4 Tradeoffs in Statistical Learning 2.5 Estimating Risk 2.5.1 In-Sample Risk 2.5.2 Cross-Validation 2.6 Modeling Data 2.7 Multivariate Normal Models 2.8 Normal Linear Models 2.9 Bayesian Learning Exercises 3. Monte Carlo Methods 3.1 Introduction 3.2 Monte Carlo Sampling 3.2.1 Generating Random Numbers 3.2.2 Simulating Random Variables 3.2.3 Simulating Random Vectors and Processes 3.2.4 Resampling 3.2.5 Markov Chain Monte Carlo 3.3 Monte Carlo Estimation 3.3.1 Crude Monte Carlo 3.3.2 Bootstrap Method 3.3.3 Variance Reduction 3.4 Monte Carlo for Optimization 3.4.1 Simulated Annealing 3.4.2 Cross-Entropy Method 3.4.3 Splitting for Optimization3.4.4 Noisy Optimization Exercises 4. Unsupervised Learning 4.1 Introduction 4.2 Risk and Loss in Unsupervised Learning 4.3 Expectation–Maximization (EM) Algorithm 4.4 Empirical Distribution and Density Estimation 4.5 Clustering via Mixture Models 4.5.1 Mixture Models 4.5.2 EM Algorithm for Mixture Models 4.6 Clustering via Vector Quantization 4.6.1 K-Means 4.6.2 Clustering via Continuous Multiextremal Optimization 4.7 Hierarchical Clustering 4.8 Principal Component Analysis (PCA) 4.8.1 Motivation: Principal Axes of an Ellipsoid 4.8.2 PCA and Singular Value Decomposition (SVD) Exercises 5. Regression 5.1 Introduction 5.2 Linear Regression 5.3 Analysis via Linear Models 5.3.1 Parameter Estimation 5.3.2 Model Selection and Prediction 5.3.3 Cross-Validation and Predictive Residual Sum of Squares 5.3.4 In-Sample Risk and Akaike Information Criterion 5.3.5 Categorical Features 5.3.6 Nested Models 5.3.7 Coefficient of Determination 5.4 Inference for Normal Linear Models 5.4.1 Comparing Two Normal Linear Models 5.4.2 Confidence and Prediction Intervals 5.5 Nonlinear Regression Models 5.6 Linear Models in Python 5.6.1 Modeling 5.6.2 Analysis 5.6.3 Analysis of Variance (ANOVA) 5.6.4 Confidence and Prediction Intervals 5.6.5 Model Validation 5.6.6 Variable Selection 5.7 Generalized Linear Models Exercises 6. Regularization and Kernel Methods 6.1 Introduction 6.2 Regularization 6.3 Reproducing Kernel Hilbert Spaces 6.4 Construction of Reproducing Kernels 6.4.1 Reproducing Kernels via Feature Mapping 6.4.2 Kernels from Characteristic Functions 6.4.3 Reproducing Kernels Using Orthonormal Features 6.4.4 Kernels from Kernels 6.5 Representer Theorem 6.6 Smoothing Cubic Splines 6.7 Gaussian Process Regression 6.8 Kernel PCA Exercises 7. Classification 7.1 Introduction 7.2 Classification Metrics 7.3 Classification via Bayes’ Rule 7.4 Linear and Quadratic Discriminant Analysis 7.5 Logistic Regression and Softmax Classification 7.6 K-nearest Neighbors Classification 7.7 Support Vector Machine 7.8 Classification with Scikit-Learn Exercises 8. Decision Trees and Ensemble Methods 8.1 Introduction 8.2 Top-Down Construction of Decision Trees 8.2.1 Regional Prediction Functions 8.2.2 Splitting Rules 8.2.3 Termination Criterion 8.2.4 Basic Implementation 8.3 Additional Considerations 8.3.1 Binary Versus Non-Binary Trees 8.3.2 Data Preprocessing 8.3.3 Alternative Splitting Rules 8.3.4 Categorical Variables 8.3.5 Missing Values 8.4 Controlling the Tree Shape 8.4.1 Cost-Complexity Pruning 8.4.2 Advantages and Limitations of Decision Trees 8.5 Bootstrap Aggregation 8.6 Random Forests 8.7 Boosting Exercises 9. Deep Learning 9.1 Introduction 9.2 Feed-Forward Neural Networks 9.3 Back-Propagation 9.4 Methods for Training 9.4.1 Steepest Descent 9.4.2 Levenberg–Marquardt Method 9.4.3 Limited-Memory BFGS Method 9.4.4 Adaptive Gradient Methods 9.5 Examples in Python 9.5.1 Simple Polynomial Regression 9.5.2 Image Classification Exercises A. Linear Algebra and Functional Analysis A.1 Vector Spaces, Bases, and Matrices A.2 Inner Product A.3 Complex Vectors and Matrices A.4 Orthogonal Projections A.5 Eigenvalues and Eigenvectors A.5.1 Left- and Right-Eigenvectors A.6 Matrix Decompositions A.6.1 (P)LU Decomposition A.6.2 Woodbury Identity A.6.3 Cholesky Decomposition A.6.4 QR Decomposition and the Gram–Schmidt Procedure A.6.5 Singular Value Decomposition A.6.6 Solving Structured Matrix Equations A.7 Functional Analysis A.8 Fourier Transforms A.8.1 Discrete Fourier Transform A.8.2 Fast Fourier Transform B. Multivariate Differentiation and Optimization B.1 Multivariate Differentiation B.1.1 Taylor Expansion B.1.2 Chain Rule B.2 Optimization Theory B.2.1 Convexity and Optimization B.2.2 Lagrangian Method B.2.3 Duality B.3 Numerical Root-Finding and Minimization B.3.1 Newton-Like Methods B.3.2 Quasi-Newton Methods B.3.3 Normal Approximation Method B.3.4 Nonlinear Least Squares B.4 Constrained Minimization via Penalty Functions C. Probability and Statistics C.1 Random Experiments and Probability Spaces C.2 Random Variables and Probability Distributions C.3 Expectation C.4 Joint Distributions C.5 Conditioning and Independence C.5.1 Conditional Probability C.5.2 Independence C.5.3 Expectation and Covariance C.5.4 Conditional Density and Conditional Expectation C.6 Functions of Random Variables C.7 Multivariate Normal Distribution C.8 Convergence of Random Variables C.9 Law of Large Numbers and Central Limit Theorem C.10 Markov Chains C.11 Statistics C.12 Estimation C.12.1 Method of Moments C.12.2 Maximum Likelihood Method C.13 Confidence Intervals C.14 Hypothesis Testing D. Python Primer D.1 Getting Started D.2 Python Objects D.3 Types and Operators D.4 Functions and Methods D.5 Modules D.6 Flow Control D.7 Iteration D.8 Classes D.9 Files D.10 NumPy D.10.1 Creating and Shaping Arrays D.10.2 Slicing D.10.3 Array Operations D.10.4 Random Numbers D.11 Matplotlib D.11.1 Creating a Basic Plot D.12 Pandas D.12.1 Series and DataFrame D.12.2 Manipulating Data Frames D.12.3 Extracting Information D.12.4 Plotting D.13 Scikit-learn D.13.1 Partitioning the Data D.13.2 Standardization D.13.3 Fitting and Prediction D.13.4 Testing the Model D.14 System Calls, URL Access, and Speed-Up Bibliography Index