Table of Contents

Preface　ix
Acknowledgments　xi
1.　Linear algebra　1
1.1.　Introduction　1
1.2.　Elements of linear algebra　2
1.2.1.　Linear spaces　2
1.2.2.　Orthogonality　5
1.2.3.　Gram-Schmidt process　10
1.2.4.　Eigenvalues and eigenvectors　11
1.3.　Linear regression　17
1.3.1.　QR decomposition　17
1.3.2.　Least-squares problems　18
1.3.3.　Linear regression　20
1.4.　Principal component analysis　21
1.4.1.　Singular value decomposition　21
1.4.2.　Low-rank matrix approximations　24
1.4.3.　Principal component analysis　25
2.　Probability　31
2.1.　Introduction　31
2.2.　Probability distribution　32
2.2.1.　Probability axioms　32
2.2.2.　Conditional probability　33
2.2.3.　Discrete random variables　34
2.2.4.　Continues random variables　38
2.3.　Independent variables and random samples　41
2.3.1.　Joint probability distributions　41
2.3.2.　Correlation and dependence　44
2.3.3.　Random samples　47
2.4.　Maximum likelihood estimation　48
2.4.1.　MLE for random samples　48
2.4.2.　Linear regression　49
3.　Calculus and optimization　51
3.1.　Introduction　51
3.2.　Continuity and differentiation　52
3.2.1.　Limits and continuity　52
3.2.2.　Derivatives　54
3.2.3.　Taylor's theorem　62
3.3.　Unconstrained optimization　65
3.3.1.　Necessary and sufficient conditions of local minimizers　65
33.2.　Convexity and global minimizers　69
3.3.3.　Gradient descent　74
3.4.　Logistic regression　76
3.5.　K-means　79
3.6.　Support vector machine　80
3.7.　Neural networks　83
3.7.1.　Mathematical formulation　83
3.7.2.　Activation functions　85
3.7.3.　Cost function　87
3.7.4.　Backpropagation　87
3.7.5.　Backpropagation algorithm　88
4. Network analysis　91
4.1.　Introduction　91
4.2.　Graph modeling　92
4.3.　Spectral graph bipartitioning　99
4.4.　Network embedding　105
4.5.　Network based influenza prediction　106
4.5.1.　I ntrod uction　107
4.5.2.　Data analysis with spatial networks　111
4.5.3.　ANN method for prediction　118
5. Ordinary differential equations　129
5.1.　Introduction　129
5.2.　Basic differential equation models　130
5.2.1.　Logistic differential equations　130
5.2.2.　Epidemical model　131
5.3.　Prediction of daily PM2.5 concentration　134
5.3.1.　Introduction　134
5.3.2.　Genetic programming for ODE　136
5.3.3.　Experimental results and prediction analysis　143
5.4.　Analysis of COVID-19　148
5.4.1.　Introduction　148
5.4.2.　Modeling and parameter estimation　152
5.4.3.　Model simulations　160
5.4.4.　Conclusion and perspective　165
5.5.　Analysis of COVID-19 in Arizona　166
5.5.1.　Introduction　166
5.5.2.　Data sources and collection　167
5.5.3.　Model simulations　168
5.5.4.　Remarks　170
6.　Partial differential equations　173
6.1.　Introduction　173
6.2.　Formulation of partial differential equation　models　174
6.3.　Bitcoin price prediction　179
6.3.1.　Network analysis for bitcoin　181
6.3.2.　PDE modeling　184
6.3.3.　Bitcoin price prediction　186
6.3.4.　Remarks　190
6.4.　Prediction of PM2.5 in China　192
6.4.1.　Introduction　192
6.4.2.　PDE model for PM2.5　194
6.4.3.　Data collection and clustering　195
6.4.4.　PM2.5 prediction　197
6.4.5.　Remarks　199
6.5.　Prediction of COVID-19 in Arizona　202
6.5.1.　Introduction　202
6.5.2.　Arizona COVID data　204
6.5.3.　PDE modeling of Arizona COVID-19　206
6.5.4.　Model prediction　209
6.5.5.　Remarks　210
6.6.　Compliance with COVID-19 mitigation policies　in the US　213
6.6.1.　I ntrod uction　213
6.6.2.　Data set sources and collection　216
6.6.3.　PDE model for quantifying compliance with COVID-19 policies　217
6.6.4.　Model prediction　221
6.6.5.　Analysis of compliance with the US COVID-19 mitigation policy　224
6.6.6.　Remarks　226
Bibliography　229
Index　241