Title Page
Abstract
Contents
Chapter 1. Introduction 14
Chapter 2. Review 16
2.1. MCD-based covariance matrix estimation 16
2.1.1. Huang et al. (2006) 18
2.1.2. Other methods 21
2.2. Scaled Lasso 23
2.2.1. Regression method 23
2.2.2. Precision matrix estimation 25
Chapter 3. Modified Cholesky Decomposition based Precision Matrix Estimation via Scaled Lasso 27
3.1. Convergence of SLMCD 30
Chapter 4. Numerical study 35
4.1. The cases of Huang et al. (2006) 35
4.2. Extended study 39
Chapter 5. Data example 51
Chapter 6. Conclusion 58
6.1. Discussion 59
Appendices 60
Appendix A 60
Appendix B 68
Bibliography 72
국문초록 76
Table 4.1. Losses and orders of losses are calculated for cases of Huang et al. (2006) with covariance estimation method of Huang et al. (2006) with the... 38
Table 4.2. Matrix 1-norm (ℓ₁) are calculated for cases of Huang et al. (2006) with some lasso-based precision matrix estimation methods and proposed... 45
Table 4.3. Matrix 2-norm (ℓ₂) are calculated for cases of Huang et al. (2006) with some lasso-based precision matrix estimation methods and proposed... 46
Table 4.4. Frobenius norm (ℓF) are calculated for cases of Huang et al. (2006) with some lasso-based precision matrix estimation methods and proposed...[이미지참조] 47
Table 4.5. Matrix 1-norm (ℓ₁) are calculated for cases of Sun and Zhang (2013) with some lasso-based precision matrix estimation methods and pro-... 48
Table 4.6. Matrix 2-norm (ℓ₂) are calculated for cases of Sun and Zhang (2013) with some lasso-based precision matrix estimation methods and pro-... 49
Table 4.7. Frobenius norm (ℓF) are calculated for cases of Sun and Zhang (2013) with some lasso-based precision matrix estimation methods and pro-...[이미지참조] 50
Table 5.1. Notations used in data analysis for training and testing sets. 53
Table 5.2. Mean values of absolute errors calculated through 7 DOWs compared with various lasso-based precision matrix estimation method and SLMCD. 56
Table 5.3. Standard deviations of absolute errors calculated through 7 DOWs compared with various lasso-based precision matrix estimation method and SLMCD. 57
Table A.1. Running time of R codes are calculated for cases of Huang et al. (2006) and Sun and Zhang (2013) with some lasso-based precision matrix... 60
Table A.2. Ratios of identifying zeros of true zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with n=... 62
Table A.3. Ratios of identifying non-zeros of true zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 62
Table A.4. Ratios of identifying non-zeros of true non-zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 63
Table A.5. Ratios of identifying zeros of true non-zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 63
Table A.6. Ratios of identifying zeros of true zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with n=... 64
Table A.7. Ratios of identifying non-zeros of true zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 64
Table A.8. Ratios of identifying non-zeros of true non-zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 65
Table A.9. Ratios of identifying zeros of true non-zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 65
Table A.10. Ratios of identifying zeros of true zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 66
Table A.11. Ratios of identifying non-zeros of true zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 66
Table A.12. Ratios of identifying non-zeros of true non-zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 67
Table A.13. Ratios of identifying zeros of true non-zeros in Ω with SLMCD to some lasso-based precision matrix estimation methods for comparison with... 67
Figure 5.1. Data preview including the intensity of average electric current with 8mA and above collected by Kyolim soft coorporation used in real data analysis. 52
Figure 5.2. Explaining how data analysis was done by using training and test sets by best linear prediction method by using example of k=2. 54
Figure 5.3. Boxplot of losses calculated through 7 DOWs compared with various lasso-based precision matrix estimation method and SLMCD. 57