Estimation and Detection of Multivariate Gene Regulatory Relationships

The Coefficient of Determination (CoD) plays an important role in Genomics problems, for instance, in the inference of gene regulatory networks from gene- expression data. However, the inference theory about CoD has not been investigated systematically. In this dissertation, we study the inference of discrete CoD from both frequentist and Bayesian perspectives, with its applications to system identification problems in Genomics. From a frequentist viewpoint, we provide a theoretical framework for CoD estimation by introducing nonparametric CoD estimators and parametric maximum-likelihood (ML) CoD estimators based on static and dynamical Boolean models. Inference algorithms are developed to discover gene regulatory relationships, and numerical examples are provided to validate preferable performance of the ML approach with access to sufficient prior knowledge. To make the applications of the CoD independent of user-selectable thresholds, we describe rigorous multiple testing procedures to investigate significant regulatory relation- ships among genes using the discrete CoD, and to discover canalyzing genes using the intrinsically multivariate prediction (IMP) criterion. We develop practical statistic tools that are open to the scientific community. On the other hand, we propose a Bayesian framework for the inference of the CoD across a parametrized family of joint distributions between target and predictors. Examples of applications of the Bayesian approach are provided against those of nonparametric and parametric approaches by using synthetic data. We have found that, with applications to system identification problems in Genomics, both parametric and Bayesian CoD estimation approaches outperform the nonparametric approaches. Hence, we conclude that parametric and Bayesian estimation approaches are preferred when we have partial knowledge about gene regulation. On the other hand, we have shown that the two proposed statistical testing frameworks can detect well-known gene regulation and canalyzing genes like p53 and DUSP1 from real data sets, respectively. This indicates that our methodology could serve as a promising tool for the detection of potential gene regulatory relationships and canalyzing genes. In one word, this dissertation is intended to serve as foundation for a detailed study of applications of CoD estimation in Genomics and related fields.