Mathematical Methods in Data Science

Chapter 1 Linear Algebra

1.1 Introduction

1.2 Elements of Linear Algebra

1.2.1 Linear Spaces

1.2.2 Orthogonality

1.2.3 Gram-Schmidt Process

1.2.4 Eigenvalues and Eigenvectors

1.3 Linear Regression

1.3.1 QR Decomposition

1.3.2 Least-squares Problems

1.3.3 Linear Regression

1.4 Principal Component Analysis

1.4.1 Singular Value Decomposition

1.4.2 Low-Rank Matrix Approximations

1.4.3 Principal Component Analysis

Chapter 2 Probability

2.1 Introduction

2.2 Probability Distribution

2.2.1 Probability Axioms

2.2.2 Conditional Probability

2.2.3 Discrete Random Variables

2.2.4 Continues Random Variables

2.3 Independent Variables and Random Samples

2.3.1 Joint Probability Distributions

2.3.2 Correlation and Dependence

2.3.3 Random Samples

2.4 Maximum Likelihood Estimation

2.4.1 MLE for Random Samples

2.4.2 Linear Regression

Chapter 3 Calculus and Optimization

3.1 Introduction

3.2 Continuity and Differentiation

3.2.1 Limits and Continuity

3.2.2 Derivatives

3.2.3 Taylor’s Theorem

3.3 Unconstrained Optimization

3.3.1 Necessary and Suffcent Conditions of Local Minimizers

3.3.2 Convexity and Global Minimizers

3.3.3 Gradient Descent

3.4 Logistic Regression

3.5 K-means

3.6 Support Vector Machine

3.7 Neural Networks

3.7.1 Mathematical Formulation

3.7.2 Activation Functions

3.7.3 Cost Function

3.7.4 Backpropagation

3.7.5 Backpropagation Algorithm

Chapter 4 Network Analysis

4.1 Introduction

4.1.1 Graph Models

4.1.2 Laplacian Matrix

4.2 Spectral Graph Bipartitioning

4.3 Network Embedding

4.4 Network Based Influenza Prediction

4.4.1 Introduction

4.4.2 Data Analysis with Spatial Networks

4.4.3 ANN Method for Prediction

Chapter 5 Ordinary Differential Equations

5.1 Introduction

5.1.1 Logistic Differential Equations

5.2 Epidemical Model

5.3 Prediction of Daily PM2.5 Concentration

5.3.1 Introduction

5.3.2 Genetic Programming for ODE

5.3.3 Experimental Results and Prediction Analysis

5.4 Analysis of COVID-19

5.4.1 Introduction

5.4.2 Modeling and Parameter Estimation

5.4.3 Model Simulations

5.4.4 Conclusion and Perspective

5.5 Analysis of COVID-19 in Arizona

5.5.1 Introduction

5.5.2 Data Sources and Collection

5.5.3 Model Simulations

5.5.4 Remarks

Chapter 6 Partial Differential Equations

6.1 Introduction

6.1.1 Formulation of Partial Differential Equation Models

6.2 Bitcoin Price Prediction

6.2.1 Network Analysis for Bitcoin

6.2.2 PDE Modeling

6.2.3 Bitcoin Price Prediction

6.2.4 Remarks

6.3 Prediction of PM2.5 in China

6.3.1 Introduction

6.3.2 PDE model for PM2.5

6.3.3 Data Collection and Clustering

6.3.4 PM2.5 Prediction

6.3.5 Remarks

6.4 Prediction of COVID-19 in Arizona

6.4.1 Introduction

6.4.2 Arizona COVID Data

6.4.3 PDE Modeling of Arizona COVID-19

6.4.4 Model Prediction

6.4.5 Remarks

6.5 Compliance with COVID-19 Mitigation Policies in the US

6.5.1 Introduction

6.5.2 Dataset Sources and Collection

6.5.3 PDE Model for Quantifying Compliance with COVID-19 Policies

6.5.4 Model Prediction

6.5.5 Analysis of Compliance with the US COVID-19 Mitigation Policy

6.5.6 Remarks