Mathematical Methods in Data Science

Mathematical Methods in Data Science

1st Edition - January 2, 2023

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  • Authors: Jingli Ren, Haiyan Wang
  • Paperback ISBN: 9780443186790

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Description

Mathematical Methods in Data Science introduces a new approach based on network analysis to integrate big data into the framework of ordinary and partial differential equations for data analysis and prediction. The mathematics is accompanied with examples and problems arising in data science to demonstrate advanced mathematics, in particular, data-driven differential equations used. Chapters also cover network analysis, ordinary and partial differential equations based on recent published and unpublished results. Finally, the book introduces a new approach based on network analysis to integrate big data into the framework of ordinary and partial differential equations for data analysis and prediction. There are a number of books on mathematical methods in data science. Currently, all these related books primarily focus on linear algebra, optimization and statistical methods. However, network analysis, ordinary and partial differential equation models play an increasingly important role in data science. With the availability of unprecedented amount of clinical, epidemiological and social COVID-19 data, data-driven differential equation models have become more useful for infection prediction and analysis.

Key Features

  • Combines a broad spectrum of mathematics, including linear algebra, optimization, network analysis and ordinary and partial differential equations for data science
  • Written by two researchers who are actively applying mathematical and statistical methods as well as ODE and PDE for data analysis and prediction
  • Highly interdisciplinary, with content spanning mathematics, data science, social media analysis, network science, financial markets, and more
  • Presents a wide spectrum of topics in a logical order, including probability, linear algebra, calculus and optimization, networks, ordinary differential and partial differential equations

Readership

Advanced undergraduate students, graduate students and researchers. Background preparations and necessary references

Table of Contents

  • 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

Product details

  • No. of pages: 255
  • Language: English
  • Copyright: © Elsevier 2023
  • Published: January 2, 2023
  • Imprint: Elsevier
  • Paperback ISBN: 9780443186790

About the Authors

Jingli Ren

She received the Ph.D. degree in applied mathematics from Beijing Institute of Technology, Beijing, China, in 2004. Her research interests include data science, applied mathematics, and applied statistics. She conducted five Projects of National Nature Science Foundation of China, one Alexander von Humboldt Fellowship for Experienced Researcher, and five Provincial Projects. She has published numerous articles in scholarly journals, such as Acta Mater.、Appl. Phys. Lett.、IEEE Trans. SMC、Infor. Sci.、J. Stat. Phys.、J. Nonlinear Sci.、 Phys. Rev. B、Phys. Rev. E、Sci. China Math.、Sci. China Phys. and Sci. China Mater., etc.

Affiliations and Expertise

Professor, Zhengzhou University, China

Haiyan Wang

He completed his doctorate in mathematics, while also earning a master's degree in computer science at Michigan State University in 1997. He worked as a full-time software engineer in industry for almost ten years before joining Arizona State University. Dr. Wang’s research interests include applied mathematics, data science, differential equations, online social networks. He has published numerous articles in scholarly journals and a book entitled, “Modeling Information Diffusion in Online Social Networks with Partial Differential Equations”, Springer, 2020. Recently he developed and taught a course, Mathematical Methods in Data Science, at Arizona State University.

Affiliations and Expertise

Arizona State University, USA

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