Effective Statistical Learning Methods for Actuaries II: Tree-Based Methods and Extensions

Preface
Contents
1 Introduction
	1.1 The Risk Classification Problem
		1.1.1 Insurance Risk Diversification
		1.1.2 Why Classifying Risks?
		1.1.3 The Need for Regression Models
		1.1.4 Observable Versus Hidden Risk Factors
		1.1.5 Insurance Ratemaking Versus Loss Prediction
	1.2 Insurance Data
		1.2.1 Claim Data
		1.2.2 Frequency-Severity Decomposition
		1.2.3 Observational Data
		1.2.4 Format of the Data
		1.2.5 Data Quality Issues
	1.3 Exponential Dispersion (ED) Distributions
		1.3.1 Frequency and Severity Distributions
		1.3.2 From Normal to ED Distributions
		1.3.3 Some ED Distributions
		1.3.4 Mean and Variance
		1.3.5 Weights
		1.3.6 Exposure-to-Risk
	1.4 Maximum Likelihood Estimation
		1.4.1 Likelihood-Based Statistical Inference
		1.4.2 Maximum-Likelihood Estimator
		1.4.3 Derivation of the Maximum-Likelihood Estimate
		1.4.4 Properties of the Maximum-Likelihood Estimators
		1.4.5 Examples
	1.5 Deviance
	1.6 Actuarial Pricing and Tree-Based Methods
	1.7 Bibliographic Notes and Further Reading
	References
2 Performance Evaluation
	2.1 Introduction
	2.2 Generalization Error
		2.2.1 Definition
		2.2.2 Loss Function
		2.2.3 Estimates
		2.2.4 Decomposition
	2.3 Expected Generalization Error
		2.3.1 Squared Error Loss
		2.3.2 Poisson Deviance Loss
		2.3.3 Gamma Deviance Loss
		2.3.4 Bias and Variance
	2.4 (Expected) Generalization Error for Randomized Training Procedures
	2.5 Bibliographic Notes and Further Reading
	References
3 Regression Trees
	3.1 Introduction
	3.2 Binary Regression Trees
		3.2.1 Selection of the Splits
		3.2.2 The Prediction in Each Terminal Node
		3.2.3 The Rule to Determine When a Node Is Terminal
		3.2.4 Examples
	3.3 Right Sized Trees
		3.3.1 Minimal Cost-Complexity Pruning
		3.3.2 Choice of the Best Pruned Tree
	3.4 Measure of Performance
	3.5 Relative Importance of Features
		3.5.1 Example 1
		3.5.2 Example 2
		3.5.3 Effect of Correlated Features
	3.6 Interactions
	3.7 Limitations of Trees
		3.7.1 Model Instability
		3.7.2 Lack of Smoothness
	3.8 Bibliographic Notes and Further Reading
	References
4 Bagging Trees and Random Forests
	4.1 Introduction
	4.2 Bootstrap
	4.3 Bagging Trees
		4.3.1 Bias
		4.3.2 Variance
		4.3.3 Expected Generalization Error
	4.4 Random Forests
	4.5 Out-of-Bag Estimate
	4.6 Interpretability
		4.6.1 Relative Importances
		4.6.2 Partial Dependence Plots
	4.7 Example
	4.8 Bibliographic Notes and Further Reading
	References
5 Boosting Trees
	5.1 Introduction
	5.2 Forward Stagewise Additive Modeling
	5.3 Boosting Trees
		5.3.1 Algorithm
		5.3.2 Particular Cases
		5.3.3 Size of the Trees
	5.4 Gradient Boosting Trees
		5.4.1 Numerical Optimization
		5.4.2 Steepest Descent
		5.4.3 Algorithm
		5.4.4 Particular Cases
	5.5 Boosting Versus Gradient Boosting
	5.6 Regularization and Randomness
		5.6.1 Shrinkage
		5.6.2 Randomness
	5.7 Interpretability
		5.7.1 Relative Importances
		5.7.2 Partial Dependence Plots
		5.7.3 Friedman\'s H-Statistics
	5.8 Example
	5.9 Bibliographic Notes and Further Reading
	References
6 Other Measures for Model Comparison
	6.1 Introduction
	6.2 Measures of Association
		6.2.1 Context
		6.2.2 Probability of Concordance
		6.2.3 Kendall\'s Tau
		6.2.4 Spearman\'s Rho
		6.2.5 Numerical Example
	6.3 Measuring Lift
		6.3.1 Motivation
		6.3.2 Predictors Characteristics
		6.3.3 Convex Order
		6.3.4 Concentration Curve
		6.3.5 Assessing the Performances of a Given Predictor
		6.3.6 Comparison of the Performances of Two Predictors
		6.3.7 Ordered Lorenz Curve
		6.3.8 Numerical Illustration
		6.3.9 Case Study
	6.4 Bibliographic Notes and Further Reading
	References