고려대학교

목차

Contents = 4
Acknowledgements = 7
Abstract in Korean = 8
Abstract = 9
List of Tables = 10
List of Figures = 12
Chapter 1. Introduction = 14
Chapter 2. Overview of Compositional Data = 18
2.1 Definition of Compositional Data = 18
2.2 Difficulties in the Analysis of Compositional data = 19
2.2.1 Negative Biasedness = 19
2.2.2 Null Correlation = 20
2.2.3 Statistical Modeling = 21
2.2.4 Geometric Representation and Transformation = 22
Chapter 3. Box-Cox Contrast Transformation on Compositional Data = 25
3.1 Definition and Motivation = 25
3.2 Covariance Structure and Its properties = 29
3.2.1 Specification of Two Types of Covariance Matrix = 30
3.2.2 Permutation Invariance Property Involving Covariance Matrix = 32
3.2.3 Covariance Structure Related to the Null Correlation = 36
3.3 Other Transformations = 39
3.3.1 Additive Logratio Transformation: Aitchison(1982) = 40
3.3.2 Box-Cox Power Transformation on the Ratio: Rayens and Srinivasan(1991a, 1991b) = 41
Chapter 4. Estimation of Parameters in the Box-Cox Contrast Transformation = 45
4.1 Estimation of Parameters = 45
4.2 Algorithm of Nelder-Mead Simplex Method = 48
4.3 Numerical Illustration = 50
Chapter 5. Box-Cox Multiplicative Transformation on Compositional Data = 56
5.1 Introduction = 56
5.2 Box-Cox Multiplicative Transformation = 59
5.3 Test of Compositional Independence = 63
5.3.1 Neutrality at Fixed Level = 63
5.3.2 Neutrality up to Level c < p = 66
5.3.3 Complete Neutrality = 67
5.4 Numerical Illustration = 68
Chapter 6. Application to Principal Component Analysis = 74
6.1 Introduction: Principal Component Analysis on Compositional Data = 74
6.2 Box-Cox Contrast Principal Component Analysis = 76
6.3 Numerical Illustration = 78
Chapter 7. Conclusions = 96
References = 99