Low-rank and sparse decomposition in multivariate regional quantile regression

다국어 초록 (Multilingual Abstract)

We propose multiple response regional quantile regression by imposing low-rank plus sparse structure assumption on the underlying coefficient matrix. This work is motivated by the analysis of cancer cell line encyclopedia (CCLE), which consists of resistance responses to multiple drugs and gene expression of cancer cell line. In the CCLE data analyses, we assume that only a few genes are relevant to the effect of drug resistance and some genes could have similar effects on multiple responses. To estimate the drug resistance response from gene information and to identify the genes responsible for the sensitivity of the resistance response to each drug, we propose a penalized multivariate quantile regression by decomposing the quantile coefficient function into the low-rank and sparse matrices. Low-rank part is a constant function of quantile levels, which represents the global pattern of the coefficient function, whereas the sparse matrix can be smoothly varying by quantile levels, which represents a local and specific pattern of the coefficient function. We prove low-rank and sparse consistency under regularity conditions. We compute the proposed penalized method via alternating direction method of multipliers (ADMM) algorithm. We also propose the novel tuning parameter selection using Generalized information criterion (GIC) to select parsimonious model with good prediction ability. In our numerical analysis using simulated data, the proposed method better predicts drug responses compared with the other methods. Real data application via CCLE reveals the usefulness of the proposed method.

국문 초록 (Abstract)

CCLE (암 세포주 백과사전) 데이터는 세포주들에 대한 24개 약물에 대한 저항반응과 유전자 정보로 이루어져 있다. 이러한 CCLE 데이터를 분석하기 위하여 다중 반응의 분위수 구간 회귀 계수 행렬에 구조적 조건을 준 모형을 제안한다. CCLE 데이터를 분석함에 있어서 회귀 계수 행렬이 분위수와 상관 없이 약물들끼리 전반적으로 비슷한 영향을 주는 하위 랭크와 약물별로 분위수마다 다른 영향을 미치는 성김 행렬의 합으로 표현될 수 있다고 가정한다. 하위 랭크 행렬과 성김 행렬은 각각 핵 노름(Nuclear Norm)과 L1 노름 제한자를 이용한 벌점화된 다중 반응 분위수 구간 회귀로 추정되었으며 목적 함수는 Alternative Direction Method of Multipliers (ADMM) 방법을 이용하여 최적화되었다. 시뮬레이션 데이터와 CCLE 데이터를 통해 제안된 방법이 다른 방법들에 비해 저항 반응과 유전자 정보의 상관 관계를 더 잘 규명하는 것을 확인할 수 있다.

목차 (Table of Contents)

• 1 Introduction 1
• 2 Preliminaries 5
• 2.1 Notations 5
• 2.2 Quantile regression 5
• 2.3 Regional quantile regression 7
• 2.4 CCLE 7
• 2.5 Matrix decomposition into low-rank and sparse matrices 8
• 2.6 Matrix decomposition in regional quantile regression 10
• 3 The Model and Method 12
• 3.1 Multiple responses quantile regression 12
• 3.2 Method 14
• 4 ADMM Algorithm 16
• 4.1 Updating H0 and H 19
• 4.2 Updating Theta0 and Theta 19
• 4.3 Updating Z 20
• 4.4 Updating E 21
• 4.5 Stopping criteria for ADMM 22
• 5 Tuning Parameters Selection 23
• 5.1 Generalized Information Criterion 23
• 6 Numerical Example 24
• 6.1 Simulation setting 24
• 6.2 Simulation results 26
• 6.2.1 Finite sample performances 28
• 6.2.2 When n increases 29
• 7 Application to CCLE Data 35
• 7.1 Screening data 35
• 7.2 Prediction error 35
• 7.3 When ∆ = [0.1, 0.3] 38
• 7.3.1 Low-rank pattern 39
• 7.3.2 Sparse pattern 42
• 7.4 When ∆ = [0.4, 0.6] 45
• 7.4.1 Low-rank pattern 45
• 7.4.2 Sparse pattern 46
• 8 Conclusion 51
• References 52
• A Graphic Materials 59