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Part I: Motivation and the Basics
Chapter 1: Introduction
1.1 Least Squares and Regularized Regression
1.2 Lasso: Survival of The Bigger
1.3 Thresholding The Sample Covariance Matrix
1.4 Sparse PCA and Regression
1.5 Graphical Models: Nodewise Regression
1.6 Cholesky Decomposition and Regression
1.7 The Bigger Picture: Latent Factor Models
1.8 Further Reading
Chapter 2: Data, Sparsity, and Regularization
2.1 Data Matrix: Examples
2.2 Shrinking The Sample Covariance Matrix
2.3 Distribution of The Sample Eigenvalues
2.4 Regularizing Covariances Like a Mean
2.5 The Lasso Regression
2.6 Lasso: Variable Selection and Prediction
2.7 Lasso: Degrees of Freedom and Bic
2.8 Some Alternatives to The Lasso Penalty
Chapter 3: Covariance Matrices
3.1 Definition and Basic Properties
3.2 The Spectral Decomposition
3.3 Structured Covariance Matrices
3.4 Functions of a Covariance Matrix
3.5 PCA: The Maximum Variance Property
3.6 Modified Cholesky Decomposition
3.7 Latent Factor Models
3.8 GLM for Covariance Matrices
3.9 GLM via the Cholesky Decomposition
3.10 GLM for Incomplete Longitudinal Data
3.11 A Data Example: Fruit Fly Mortality Rate
3.12 Simulating Random Correlation Matrices
3.13 Bayesian Analysis of Covariance Matrices
Part II: Covariance Estimation: Regularization
Chapter 4: Regularizing the Eigenstructure
4.1 Shrinking The Eigenvalues
4.2 Regularizing The Eigenvectors
4.3 A Duality Between PCA and SVD
4.4 Implementing Sparse PCA: A Data Example
4.5 Sparse Singular Value Decomposition (SSVD)
4.6 Consistency of PCA
4.7 Principal Subspace Estimation
4.8 Further Reading
Chapter 5: Sparse Gaussian Graphical Models
5.1 Covariance Selection Models: Two Examples
5.2 Regression Interpretation of Entries of Σ−1
5.3 Penalized Likelihood and Graphical Lasso
5.4 Penalized Quasi-Likelihood Formulation
5.5 Penalizing The Cholesky Factor
5.6 Consistency and Sparsistency
5.7 Joint Graphical Models
5.8 Further Reading
Chapter 6: Banding, Tapering, and Thresholding
6.1 Banding The Sample Covariance Matrix
6.2 Tapering The Sample Covariance Matrix
6.3 Thresholding The Sample Covariance Matrix
6.4 Low-Rank Plus Sparse Covariance Matrices
6.5 Further Reading
Chapter 7: Multivariate Regression: Accounting for
Correlation
7.1 Multivariate Regression and LS Estimators
7.2 Reduced Rank Regressions (RRR)
7.3 Regularized Estimation of B
7.4 Joint Regularization of (B, Ω)
7.5 Implementing MRCE: Data Examples
7.6 Further Reading
Bibliography
Index
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