臺灣博碩士論文加值系統

Researchers in biological sciences nowadays often encounter the curse of
high-dimensionality. A serious consequence is that many traditional statistical
methods fail to fit for high-dimensional models. The problem becomes even
more severe when the interest is in interactions between variables, as there will
be p(p−1)/2 interaction terms with p variables. To improve the performance,
in this thesis we model the interaction effects utilizing its matrix form with
a low-rank structure. A low-rank model for symmetric matrix then greatly
reduces the number of parameters required, and hence, increases the stability
and quality of statistical analysis. Individual hypothesis tests are then carried
out on each interaction effect to wash out insignificant interactions. A low-
rank matrix, however, is not necessarily sparse. We thus impose a sparsity
constraint in the second stage to select interactions.
Due to the extremely high-dimensionality for gene×gene interactions, a
single-stage method is not adequately flexible enough for variable selection.
Our sparse low-rank approach for interactions is a modification of a multi-
stage screen-and-clean procedure byWasserman and Roeder (2009) andWu et
al. (2010). We replace their mere sparsity constraint by combining a low-rank
structure and a sparsity constraint to the interactions. In simulation studies,
we show that the proposed low-rank approximation-aided screen and clean
procedure often can achieve higher power and higher selection-consistency
probability.

Contents
Acknowledgements i
Abstract (in Chinese) ii
Abstract (in English) iii
Contests iv
Figures v
Tables vi
1 Introduction and model specification 1
2 Estimation procedure for sparse low-rank interaction model 4
2.1 Estimation for low-rank model with 2-norm penalty . . . . . . . . . . 5
2.1.1 Rank-2r model implementation . . . . . . . . . . . . . . . . . 5
2.1.2 Rank-1 model implementation . . . . . . . . . . . . . . . . . . 7
2.2 Estimation for sparse model with 1-norm penalty . . . . . . . . . . . 8
2.3 Why a sparse low-rank model, why not a direct sparse model? . . . . 8
3 Low-rank screening by hypothesis testing 10
3.1 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Asymptotic testing procedure . . . . . . . . . . . . . . . . . . . . . . 11
4 Multistage variable selection for detecting
gene×gene interactions 13
4.1 Review of screen-and-clean method . . . . . . . . . . . . . . . . . . . 13
4.2 Low-rank aided screen-and-clean method . . . . . . . . . . . . . . . . 14
5 Simulation studies 16
5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Conclusion discussion 33
Appendix 33
References 37

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