Essential Math for Data Science: Take Control of Your Data with Fundamental Linear Algebra, Probability, and Statistics

Preface....6 Conventions Used in This Book....9 Using Code Examples....10 O’Reilly Online Learning....11 How to Contact Us....11 Acknowledgments....12 1. Basic Math and Calculus Review....14 Number Theory....14 Order of Operations....17 Variables....19 Functions....20 Summations....30 Exponents....33 Logarithms....37 Euler’s Number and Natural Logarithms....41 Euler’s Number....41 Natural Logarithms....46 Limits....46 Derivatives....49 Partial Derivatives....55 The Chain Rule....59 Integrals....62 Conclusion....71 Exercises....71 2. Probability....73 Understanding Probability....73 Probability Versus Statistics....75 Probability Math....76 Joint Probabilities....76 Union Probabilities....77 Conditional Probability and Bayes’ Theorem....79 Joint and Union Conditional Probabilities....82 Binomial Distribution....84 Beta Distribution....87 Conclusion....100 Exercises....100 3. Descriptive and Inferential Statistics....102 What Is Data?....102 Descriptive Versus Inferential Statistics....105 Populations, Samples, and Bias....106 Descriptive Statistics....111 Mean and Weighted Mean....111 Median....113 Mode....114 Variance and Standard Deviation....115 The Normal Distribution....122 The Inverse CDF....135 Z-Scores....137 Inferential Statistics....140 The Central Limit Theorem....140 Confidence Intervals....144 Understanding P-Values....149 Hypothesis Testing....150 The T-Distribution: Dealing with Small Samples....163 Big Data Considerations and the Texas Sharpshooter Fallacy....166 Conclusion....167 Exercises....168 4. Linear Algebra....169 What Is a Vector?....169 Adding and Combining Vectors....177 Scaling Vectors....181 Span and Linear Dependence....186 Linear Transformations....191 Basis Vectors....192 Matrix Vector Multiplication....200 Matrix Multiplication....208 Determinants....211 Special Types of Matrices....217 Square Matrix....217 Identity Matrix....217 Inverse Matrix....218 Diagonal Matrix....219 Triangular Matrix....219 Sparse Matrix....219 Systems of Equations and Inverse Matrices....220 Eigenvectors and Eigenvalues....225 Conclusion....229 Exercises....230 5. Linear Regression....232 A Basic Linear Regression....234 Residuals and Squared Errors....243 Finding the Best Fit Line....248 Closed Form Equation....249 Inverse Matrix Techniques....251 Gradient Descent....254 Overfitting and Variance....262 Stochastic Gradient Descent....266 The Correlation Coefficient....268 Statistical Significance....273 Coefficient of Determination....282 Standard Error of the Estimate....283 Prediction Intervals....285 Train/Test Splits....290 Multiple Linear Regression....300 Conclusion....301 Exercises....302 6. Logistic Regression and Classification....303 Understanding Logistic Regression....303 Performing a Logistic Regression....307 Logistic Function....307 Fitting the Logistic Curve....309 Multivariable Logistic Regression....315 Understanding the Log-Odds....320 R-Squared....324 P-Values....329 Train/Test Splits....330 Confusion Matrices....332 Bayes’ Theorem and Classification....336 Receiver Operator Characteristics/Area Under Curve....337 Class Imbalance....340 Conclusion....340 Exercises....341 7. Neural Networks....342 When to Use Neural Networks and Deep Learning....342 A Simple Neural Network....343 Activation Functions....347 Forward Propagation....356 Backpropagation....363 Calculating the Weight and Bias Derivatives....363 Stochastic Gradient Descent....368 Using scikit-learn....371 Limitations of Neural Networks and Deep Learning....372 Conclusion....375 Exercise....376 8. Career Advice and the Path Forward....377 Redefining Data Science....378 A Brief History of Data Science....381 Finding Your Edge....385 SQL Proficiency....385 Programming Proficiency....388 Data Visualization....393 Knowing Your Industry....395 Productive Learning....396 Practitioner Versus Advisor....397 What to Watch Out For in Data Science Jobs....400 Role Definition....401 Organizational Focus and Buy-In....402 Adequate Resources....404 Reasonable Objectives....405 Competing with Existing Systems....407 A Role Is Not What You Expected....409 Does Your Dream Job Not Exist?....412 Where Do I Go Now?....412 Conclusion....414 A. Supplemental Topics....416 Using LaTeX Rendering with SymPy....416 Binomial Distribution from Scratch....418 Beta Distribution from Scratch....419 Deriving Bayes’ Theorem....421 CDF and Inverse CDF from Scratch....423 Use e to Predict Event Probability Over Time....425 Hill Climbing and Linear Regression....427 Hill Climbing and Logistic Regression....430 A Brief Intro to Linear Programming....431 MNIST Classifier Using scikit-learn....439 B. Exercise Answers....441 Chapter 1....441 Chapter 2....444 Chapter 3....446 Chapter 4....449 Chapter 5....453 Chapter 6....458 Chapter 7....462 Index....465 About the Author....510