Title Page
Abstract
Contents
Chapter 1. Introduction 11
1.1. Precision matrix estimation 11
1.2. Constrained ℓ₁ Minimization 14
1.3. Outline 16
Chapter 2. Path algorithm for adaptive constrained ℓ₁ minimization for inverse matrix estimation 18
2.1. Introduction 18
2.2. Preliminaries of the solution path algorithm for CLIME 22
2.3. Modification of FASTCLIME algorithm 34
2.4. Solution path algorithm for adaptive CLIME 43
2.5. Numerical Study 45
2.6. Concluding remark 53
Chapter 3. Adaptive constrained ℓ₁ minimization for inverse covariance matrix estimation via scaled Lasso 54
3.1. Introduction 54
3.2. Preliminaries 55
3.2.1. Adaptive CLIME and its properties 55
3.2.2. Scaled Lasso and its properties 58
3.3. Adaptive CLIME using the diagonal estimates from the scaled Lasso 61
3.4. Numerical Study 62
3.5. Concluding remark 69
Appendix 69
Chapter 4. Conclusion 73
Bibliography 75
초록 79
TABLE 2.1. Comparison of computation times in seconds for CLIME, FastCLIME, mFastCLIME, ACLIME-Diag, ACLIME-Path, and ACLIME-Total. The numbers in... 48
TABLE 3.1. Comparison of estimation errors with matrix ℓ₂ and Frobenius norms for each structure of Ω. The numbers in parentheses denote the standard errors. 67
TABLE 3.2. Comparison of measures for each structure of Ω. The numbers in parentheses denote the standard errors (n = 100, p = 200). 67
TABLE 3.3. Comparison of measures for each structure of Ω. The numbers in parentheses denote the standard errors (n = 200, p = 200). 68
TABLE 3.4. Simulation setting 1 (N = 50, P = 200) 70
TABLE 3.5. Simulation setting 2 (N = 100, P = 200) 71
TABLE 3.6. Simulation setting 3 (N = 200, P = 200) 72
FIGURE 1.1. Solution paths for CLIME and FastCLIME estimators 17
FIGURE 2.1. Solution paths for CLIME and fastCLIME estimators of (ω₁₂,...,ω₂₄) 41
FIGURE 2.2. Solution paths for CLIME and fastCLIME estimators of (ω₂₅,...,ω₄₅) 42
FIGURE 2.3. Solution paths of the CLIME, FastCLIME, and mFastCLIME estimators of (ω₁₂,...,ω₂₄) 49
FIGURE 2.4. Solution paths of the CLIME, FastCLIME, and mFastCLIME estimators of (ω₂₅,...,ω₄₅) 50
FIGURE 2.5. Solution paths of the mFastCLIME and ACLIME of (ω₁₂,...,ω₂₄) 51
FIGURE 2.6. Solution paths of the mFastCLIME and ACLIME of (ω₂₅,...,ω₄₅) 52
FIGURE 3.1. The structure of true precision matrix 64