Brief Contents

• Chapter 1 Dimensional Analysis
• Chapter 2 Basics of R Programming
• Chapter 3 Linear Models Using Regression and Data
• Chapter 4 Principles of Mathematical Modeling
• Chapter 5 Mathematical Modeling by Linear Algebra
• Chapter 6 Mathematical Modeling by Calculus
• Chapter 7 Probabilistic Models
• Chapter 8 Stochastic Models
• Chapter 9 Visualize Mathematical Models by R
• Chapter 10 Statistical Models and Hypothesis Tests
• Chapter 11 Concept of Big Data Modeling
• Chapter 12 Concepts of Machine Learning
• Chapter 13 Artificial Intelligence Models
• Chapter 14 Network Models
• Chapter 15 Mathematical and Statistical Consulting
• Appendix A Advanced R Graphics
• Appendix B Advanced R Coding

Full Table of Contents

• Foreword vii
• Glossary xi
• 1 Dimensional Analysis 1
  • 1.1 Dimension and Units 1
  • 1.2 Five Fundamental Dimensions: LMTθI 2
  • 1.3 Relationships Between Magnetic and Electric Fields 7
  • 1.4 The General Principle of Dimensional Analysis 7
    • 1.4.1 A Universal Mathematical Model for Nature 7
    • 1.4.2 Dimensional Analysis for a Free-Fall Body using the Universal Model 9
    • 1.4.3 Dimensional Analysis for a Simple Pendulum using the Universal Model 12
  • 1.5 Shock Wave Radius of a Nuclear Explosion 15
  • 1.6 Complications of the Universal Mathematical Model 18
  • Exercises 19


• 2 Basics of R Programming 25
  • 2.1 Download and Install R Software Package 25
  • 2.2 R Tutorial 26
  • 2.2.1 R as a Smart Calculator 26
  • 2.2.2 Write a Function in R 27
  • 2.2.3 Plot with R 27
  • 2.2.4 Symbolic Calculations for Calculus by R 28
  • 2.2.5 Vectors and Matrices 28
  • 2.2.6 Statistics 31
  • 2.3 Directory and Its Path 32
  • 2.4 Online Tutorials 32
  • 2.4.1 Youtube Tutorial: For True Beginners 32
  • 2.4.2 YouTube Tutorial: For Some Basic Statistical Summaries 31
  • 2.4.3 YouTube Tutorial: Input Data by Reading a csv File into R 33
  • 2.4.4 Install Multiple R Packages Using Pacman 34
  • Exercises 34


• 3 Linear Models Using Regression and Data 39
  • 3.1 Introduction to a Linear Model 39
  • 3.1.1 A Linear Model for the Life Expectancy in France 39
  • 3.1.2 Energy Consumption and Heating Degree Data 42
  • 3.2 Formula Derivation and Interpretation for the Trend and Intercept of a Linear Regression 44
  • 3.2.1 Anomaly Data 44
  • 3.2.2 Estimate the Linear Model From the Anomaly Data 46
  • 3.2.3 Derivation of the Linear Model Estimators 46
  • 3.2.4 Percentage of Variance Explained in Terms of R² 48
  • 3.2.5 Geometric Interpretations and Historical Note 48
  • 3.3 An Example of Linear Model and Data Analysis Using R: A Global Warming Dataset 49
  • 3.4 Research Level Exploration for Analyzing the Global Warming Data 55
  • Exercises 57


• 4 Principles of Mathematical Modeling 61
  • 4.1 Principles of Mathematical Modeling and Client Report Template 61
  • 4.2 Zeroing a Rifle: a DAESI Example 62
  • 4.3 Modeling Mortgage Payment 66
  • 4.4 EBM for Modeling the Moon’s Surface Temperature 68
  • 4.4.1 Moon-Earth-Sun Orbit and Lunar Surface 69
  • 4.4.2 Moon’s Surface Temperature 70
  • 4.4.3 EBM Prediction for the Moon Surface Temperature 73
  • 4.5 Zero-Dimensional Energy Balance Model for Earth’s Constant Temperature Climate 77
  • 4.5.1 Earth’s Energy Budget 77
  • 4.5.2 A Uniform Water-Covered Earth 78
  • 4.6 EBM for a Uniform Earth with Nonlinear Albedo Feedback 80
  • 4.7 Template of a Client Report 83
  • 4.8 Term Project #1 84
  • Exercises 85


• 5 Mathematical Modeling by Linear Algebra 89
  • 5.1 Kirchhoff’s Laws and Solution of an Electric Circuit 89
  • 5.2 Mass Balance Models for Chemical Equations 91
  • 5.3 Leontif Production Model: A Balance of the Output and Input 92
  • 5.4 An SVD model to Represent Space-Time Data 95
  • 5.4.1 The Fundamental Idea of SVD: Space-Time-Energy Separation 95
  • 5.4.2 SVD for a 2-Dim Spatial Domain and 1-Dim Temporal Domain 97
  • 5.4.3 An SVD Algorithm and Covariance Matrix 99
  • 5.4.4 SVD Analysis for El Niño Southern Oscillation Data 103
  • 5.4.5 SVD Analysis for the Tropical Pacific’s Precipitation Data 109
  • Exercises 109


• 6 Mathematical Modeling by Calculus 115
  • 6.1 Chemical Mixture Problems in a Natural or Chemical Engineering Process 115
  • 6.2 Optimal Dimensions of Food Cans 117
  • 6.3 A Differential Equation Model for the Vertical Force Balance on a Small Parcel of Atmosphere 120
  • 6.4 Hypsometric Model for Atmosphere: Exponential Decrease of Pressure with Respect to Elevation 123
  • 6.4.1 The General Hypsometric Equation 123
  • 6.4.2 An Application of the Hypsometric Equation: Calculate the Elevation of Mount Mitchell 128
  • 6.4.3 Hypsometric Equation for an Isothermal Layer 129
  • 6.5 Optimal Production Level of Oil 130
  • 6.6 Modeling Blackbody Radiation 132
  • Exercises 137


• 7 Probabilistic Models 141
  • 7.1 The Event-Table Method and Simulation for Two Dice 141
  • 7.2 Geometric Probability Method: Buffon’s Needle Problem 142
  • 7.2.1 Buffon’s Needle Problem 142
  • 7.2.2 The Short Needle Problem: ℓ < d 144
  • 7.2.3 The Long Needle Problem: ℓ ≥ d 148
  • 7.2.4 Computer Simulation of the Buffon’s Needle Problem 151
  • 7.3 Monte Carlo Simulations 152
  • 7.3.1 Use Monte Carlo Simulation to Estimate the Volume of an n-ball 153
  • 7.3.2 Use Monte Carlo Simulation for Numerical Integration 156
  • 7.4 Markov Chains 158
  • 7.4.1 Example 1 158
  • 7.4.2 Example 2: Order of Fish Tanks 162


• 8 Stochastic Models 167
  • 8.1 A Nowhere Differentiable But Everywhere Continuous Model 167
  • 8.2 Brownian Motion 168
  • 8.3 Ito Calculus 170
  • 8.4 Fractal Dimension and Similarity 171
  • 8.4.1 References 171
  • 8.4.2 Dimension of Koch Curve 171
  • 8.4.3 Use R to Calculate the Fractal Dimension 172
  • 8.5 Stochastic Differential Equations 173
  • 8.6 Solving SDE Using R 173


• 9 Visualize Mathematical Models by R 175
  • 9.1 R Graphics Examples 175
  • 9.1.1 Plot Two Different Time Series on the Same Plot 175
  • 9.1.2 Figure Setups: Margins, Fonts, Mathematical Symbols, and More 176
  • 9.1.3 Plot Two or More Panels on the Same Figure 180
  • 9.2 Contour Color Maps 181
  • 9.2.1 Basic Principles for an R Contour Plot 181
  • 9.2.2 Plot Contour Color Maps for Random Values on a Map 181
  • 9.2.3 Plot Contour Maps from Climate Model Data in NetCDF Files 182
  • 9.3 Visualize Regression Models Using R 186
  • 9.4 Animation of a Free Fall Based on Model 186
  • 9.5 Visualize El Niño Models and Data 186
  • 9.5.1 A Sea Level Pressure Model 186
  • 9.5.2 A Surface Temperature Model 186
  • 9.5.3 A Precipitation Model 186


• 10 Statistical Models and Hypothesis Tests 189
  • 10.1 Statistical Indices from the Global Temperature Data from 1880 to 2015 190
  • 10.2 Commonly Used Statistical Plots 194
    • 10.2.1 Histogram of a Set of Data 194
    • 10.2.2 Box Plot 194
    • 10.2.3 Scatter Plot 195
    • 10.2.4 QQ-Plot 198
  • 10.3 Probability Distributions 199
    • 10.3.1 What is a Probability Distribution? 199
    • 10.3.2 Normal Distribution 202
    • 10.3.3 Student’s t-Distribution 204
  • 10.4 Estimate and Its Error 206
    • 10.4.1 Probability of a Sample Inside a Confidence Interval 206
    • 10.4.2 Mean of a Large Sample Size: Approximately Normal Distribution 207
    • 10.4.3 Mean of a Small Sample Size: t-Test 214
  • 10.5 Statistical Inference of a Linear Trend 217
  • 10.6 Free Online Statistics Tutorials 219
  • References 221
  • Exercises 222


• 11 Concept of Big Data Modeling 223
  • 11.1 Big Data: Books for Layman and Techies 223
  • 11.2 A Guide to Propose and Review a Big Data Project 224
  • 11.3 The Concept of Fitting a Model to Data 224
  • 11.4 Why Over Fitting is Bad? 224
  • 11.5 Good Models and Decision-Analytic Thinking 224
  • 11.6 Big Data Practice 224
  • 11.7 Data visualization 225
  • 11.8 Term Project #3 – The Final Project 225
• 12 Concepts of Machine Learning 227
  • 12.1 What is Machine Learning? 228
  • 12.2 K-means Clustering 229
  • 12.3 Logistic Regression 231
  • 12.4 CART Classification 231
  • 12.5 SVM Classification 231
  • References 233
  • Exercises 234


• 13 Artificial Intelligence Models 235
  • 13.1 Introduction 235
  • 13.2 Searching 235
  • 13.3 First-Order Logic 235
  • 13.4 Planning 235
  • 13.5 Knowledge Representation 235
  • 13.6 Uncertainty Quantification 235


• 14 Network Models 237
  • 14.1 Introduction 237
  • 14.2 An Example of Transportation Network 239
  • 14.3 An Example of Communication Network 239
  • 14.4 Matching Algorithms 239
  • 14.5 Flow Maximization 239
  • 14.6 Shortest Path Between Two Nodes in a Network 239
  • 14.7 Neural Network Models and Their Simulations 239


• 15 Mathematical and Statistical Consulting 241
  • 15.1 How to Conduct the First Meeting 241
    • 15.1.1 When to Hold the First Meeting 241
    • 15.1.2 What to Present at the First Meeting as a Consultant 241
    • 15.1.3 What Questions to Ask 241
  • 15.2 How to Write an SOW 241
  • 15.3 Deliver the Consulting Results 241
  • 15.4 Maintain the Conducts 241


• Appendix A Advanced R Graphics 243
  • A.1 Two-Dimensional Line Plots and Setups of Margins and Labels 243
  • A.1.1 Plot Two Different Time Series on the Same Plot 244
  • A.1.2 Figure Setups: Margins, Fonts, Mathematical Symbols, and More 245
  • A.1.3 Plot Two or More Panels on the Same Figure 248
  • A.2 Color Contour Maps 250
    • A.2.1 Basic Principles for an R Contour Plot 181
    • A.2.2 Plot Contour Color Maps for Random Values on a Map 181
    • A.2.3 Plot Contour Maps from Climate Model Data in NetCDF Files 182
  • A.3 Plot Wind Velocity Field on a Map 259
    • A.3.1 Plot a Wind Field Using arrow.plot 259
    • A.3.2 Plot a Surface Wind Field from NetCDF Data 260
  • A.4 ggplot for Data 262
  • A.5 Animation 263
  • References 267
  • Exercises 267


• Appendix B Advanced R Coding 269