数据科学是建立在数学之上的。在本书中，我们将涵盖数据科学中广泛使用的数学工具，包括微积分、线性代数、优化、网络分析、概率和微分方程。特别地，本书介绍了一种基于网络分析的新方法，将大数据集成到常微分方程和偏微分方程的框架中进行数据分析和预测。本书中，我们把数学与数据科学中出现的示例和问题相结合，并展示高等数学，特别是数据驱动的微分方程在数据科学中的应用。

Preface
Acknowledgments
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
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
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 sufficient 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
4.Network analysis
4.1.Introduction
4.2.Graph modeling
4.3.Spectral graph bipartitioning
4.4.Network embedding
4.5.Network based influenza prediction
4.5.1.Introduction
4.5.2.Data analysis with spatial networks
4.5.3.ANN method for prediction
5.Ordinary differential equations
5.1.Introduction
5.2.Basic differential equation models
5.2.1.Logistic differential equations
5.2.2.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 COVD-19 in Arizona
5.5.1.Introduction
5.5.2.Data sources and collection
5.5.3.Model simulations
5.5.4.Rermarks
6.Partial differential equations
6.1.Introduction
6.2.Formulation of partial differential equation models
6.3.Bitcoin price prediction
6.3.1.Network analysis for bitcoin
6.3.2.PDE modeling
6.3.3.Bitcoin price prediction
6.3.4.Remarks
6.4.Prediction of PM2.5 In China
6.4.1.Introduction
6.4.2.PDE model for PM2.5
6.4.3.Data collection and clustering
6.4.4.PM2.5 prediction
6.4.5.Remarks
6.5.Prediction of COVD-19 in Arizona
6.5.1.Introduction
6.5.2.Arizona COVD data
6.5.3.PDE modeling of Arizona COVID-19
6.5.4.Model prediction
6.5.5.Remarks
6.6.Compliance with COVID-19 mitigation policies in the US
6.6.1.Introduction
6.6.2.Data set sources and collection
6.6.3.PDE model for quantlfying compliance with COVID-19 policies
6.6.4.Model prediction
6.6.5.Analysis of compliance with the US COVID-19 mitigation policy
6.6.6.Remarks
Bibllography
Index