Advanced Data Analysis from an Elementary Point of View
Advanced Data Analysis from an Elementary Point of View
Cosma Rohilla Shalizi
3
For my parents and in memory of my grandparents
Contents
Introduction
12
Introduction
12
To the Reader
12
Concepts You Should Know
15
Part I Regression and Its Generalizations
17
1 Regression Basics
19
1.1 Statistics, Data Analysis, Regression
19
1.2 Guessing the Value of a Random Variable
20
1.3 The Regression Function
21
1.4 Estimating the Regression Function
25
1.5 Linear Smoothers
30
1.6 Further Reading
41
Exercises
41
2 The Truth about Linear Regression
43
2.1 Optimal Linear Prediction: Multiple Variables
43
2.2 Shifting Distributions, Omitted Variables, and Transformations
48
2.3 Adding Probabilistic Assumptions
57
2.4 Linear Regression Is Not the Philosopher's Stone
60
2.5 Further Reading
61
Exercises
62
3 Model Evaluation
63
3.1 What Are Statistical Models For?
63
3.2 Errors, In and Out of Sample
64
3.3 Over-Fitting and Model Selection
68
3.4 Cross-Validation
72
3.5 Warnings
76
3.6 Further Reading
79
Exercises
80
4 Smoothing in Regression
86
4.1 How Much Should We Smooth?
86
4
15:21 Sunday 21st March, 2021 Copyright c Cosma Rohilla Shalizi; do not distribute without permission updates at
Contents
5
4.2 Adapting to Unknown Roughness
87
4.3 Kernel Regression with Multiple Inputs
94
4.4 Interpreting Smoothers: Plots
96
4.5 Average Predictive Comparisons
97
4.6 Computational Advice: npreg
98
4.7 Further Reading
101
Exercises
102
5 Simulation
115
5.1 What Is a Simulation?
115
5.2 How Do We Simulate Stochastic Models?
116
5.3 Repeating Simulations
120
5.4 Why Simulate?
121
5.5 Further Reading
127
Exercises
127
6 The Bootstrap
128
6.1 Stochastic Models, Uncertainty, Sampling Distributions
128
6.2 The Bootstrap Principle
130
6.3 Resampling
141
6.4 Bootstrapping Regression Models
143
6.5 Bootstrap with Dependent Data
148
6.6 Confidence Bands for Nonparametric Regression
149
6.7 Things Bootstrapping Does Poorly
149
6.8 Which Bootstrap When?
150
6.9 Further Reading
151
Exercises
152
7 Splines
154
7.1 Smoothing by Penalizing Curve Flexibility
154
7.2 Computational Example: Splines for Stock Returns
156
7.3 Basis Functions and Degrees of Freedom
162
7.4 Splines in Multiple Dimensions
164
7.5 Smoothing Splines versus Kernel Regression
165
7.6 Some of the Math Behind Splines
165
7.7 Further Reading
167
Exercises
168
8 Additive Models
170
8.1 Additive Models
170
8.2 Partial Residuals and Back-fitting
171
8.3 The Curse of Dimensionality
174
8.4 Example: California House Prices Revisited
176
8.5 Interaction Terms and Expansions
180
8.6 Closing Modeling Advice
182
8.7 Further Reading
183
Exercises
183
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