Table of Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1. Basic Math and Calculus Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Number Theory 2
Order of Operations 3
Variables 5
Functions 6
Summations 11
Exponents 13
Logarithms 16
Euler’s Number and Natural Logarithms 18
Euler’s Number 18
Natural Logarithms 21
Limits 22
Derivatives 24
Partial Derivatives 28
The Chain Rule 31
Integrals 33
Conclusion 39
Exercises 39
2. Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Understanding Probability 42
Probability Versus Statistics 43
Probability Math 44
Joint Probabilities 44
Union Probabilities 45
Conditional Probability and Bayes’ Theorem 47
Joint and Union Conditional Probabilities 49
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Binomial Distribution 51
Beta Distribution 53
Conclusion 60
Exercises 61
3. Descriptive and Inferential Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
What Is Data? 63
Descriptive Versus Inferential Statistics 65
Populations, Samples, and Bias 66
Descriptive Statistics 69
Mean and Weighted Mean 70
Median 71
Mode 73
Variance and Standard Deviation 73
The Normal Distribution 78
The Inverse CDF 85
Z-Scores 87
Inferential Statistics 89
The Central Limit Theorem 89
Confidence Intervals 92
Understanding P-Values 95
Hypothesis Testing 96
The T-Distribution: Dealing with Small Samples 104
Big Data Considerations and the Texas Sharpshooter Fallacy 105
Conclusion 107
Exercises 107
4. Linear Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
What Is a Vector? 110
Adding and Combining Vectors 114
Scaling Vectors 116
Span and Linear Dependence 119
Linear Transformations 121
Basis Vectors 121
Matrix Vector Multiplication 124
Matrix Multiplication 129
Determinants 131
Special Types of Matrices 136
Square Matrix 136
Identity Matrix 136
Inverse Matrix 136
Diagonal Matrix 137
Triangular Matrix 137
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Sparse Matrix 138
Systems of Equations and Inverse Matrices 138
Eigenvectors and Eigenvalues 142
Conclusion 145
Exercises 146
5. Linear Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A Basic Linear Regression 149
Residuals and Squared Errors 153
Finding the Best Fit Line 157
Closed Form Equation 157
Inverse Matrix Techniques 158
Gradient Descent 161
Overfitting and Variance 167
Stochastic Gradient Descent 169
The Correlation Coefficient 171
Statistical Significance 174
Coefficient of Determination 179
Standard Error of the Estimate 180
Prediction Intervals 181
Train/Test Splits 185
Multiple Linear Regression 191
Conclusion 191
Exercises 192
6. Logistic Regression and Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Understanding Logistic Regression 193
Performing a Logistic Regression 196
Logistic Function 196
Fitting the Logistic Curve 198
Multivariable Logistic Regression 204
Understanding the Log-Odds 208
R-Squared 211
P-Values 216
Train/Test Splits 218
Confusion Matrices 219
Bayes’ Theorem and Classification 222
Receiver Operator Characteristics/Area Under Curve 223
Class Imbalance 225
Conclusion 226
Exercises 226
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7. Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
When to Use Neural Networks and Deep Learning 228
A Simple Neural Network 229
Activation Functions 231
Forward Propagation 237
Backpropagation 243
Calculating the Weight and Bias Derivatives 243
Stochastic Gradient Descent 248
Using scikit-learn 251
Limitations of Neural Networks and Deep Learning 253
Conclusion 256
Exercise 256
8. Career Advice and the Path Forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Redefining Data Science 258
A Brief History of Data Science 260
Finding Your Edge 263
SQL Proficiency 263
Programming Proficiency 266
Data Visualization 269
Knowing Your Industry 270
Productive Learning 272
Practitioner Versus Advisor 272
What to Watch Out For in Data Science Jobs 275
Role Definition 275
Organizational Focus and Buy-In 276
Adequate Resources 278
Reasonable Objectives 279
Competing with Existing Systems 280
A Role Is Not What You Expected 282
Does Your Dream Job Not Exist? 283
Where Do I Go Now? 284
Conclusion 285
A. Supplemental Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
B. Exercise Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
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