STATISTICAL METHODS IN GENETIC STUDIES

This dissertation includes three Chapters. A brief description of each chapter is organized as follows. In Chapter 1, we proposed a new method, called MF-TOWmuT, for genome-wide association studies with multiple genetic variants and multiple phenotypes using family samples. MF-TOWmuT uses kinship matrix to account for sample relatedness. It is worth mentioning that in simulations, we considered hidden polygenic effects and varied the proportion of variance contributed by it to generate phenotypes. Simulation studies show that MF-TOWmuT can preserve the type I error rates and is more powerful than several existing methods in different simulation scenarios, MFTOWmuT is also quite robust to the proportion of variance explained by invisible polygenic effects and to the direction of effects of genetic variants. In Chapter 2, we proposed a fast and efficient low rank penalized regression with the Elastic Net penalty for the eQTL mapping, called LORSEN. By considering the Elastic Net penalty instead of the L1 penalty, our method can overcome two crucial drawbacks of the L1 penalty, and outperforms two commonly used methods for the eQTL mapping, LORS and FastLORS, in many simulation scenarios in terms of average Area Under the Curve (AUC). In Chapter 3, we proposed a bipartite network-based penalized regression model for the eQTL mapping, called BiNetPeR. This method takes into account the SNPgene marginal association evidence to construct the SNP-gene bipartite network, then uses such a bipartite network to obtain the projected SNP network. Based on the normalized Laplacian matrix of the projected SNP network, we then formulate the eQTL mapping into a penalized regression model. Our simulation results show that our proposed method can maintain the appropriate false positive rate and outperforms two competing methods for the eQTL mapping, FastLORS and mtLasso2G

A note on irreducible maps with several boundaries

We derive a formula for the generating function of d-irreducible bipartite planar maps with several boundaries, i.e. having several marked faces of controlled degrees. It extends a formula due to Collet and Fusy for the case of arbitrary (non necessarily irreducible) bipartite planar maps, which we recover by taking d=0. As an application, we obtain an expression for the number of d-irreducible bipartite planar maps with a prescribed number of faces of each allowed degree. Very explicit expressions are given in the case of maps without multiple edges (d=2), 4-irreducible maps and maps of girth at least 6 (d=4). Our derivation is based on a tree interpretation of the various encountered generating functions.Comment: 18 pages, 8 figure

Arithmetic level raising on triple product of Shimura curves and Gross-Schoen diagonal cycles II: Bipartite Euler system

In this article, we study the Gross-Schoen diagonal cycle on the triple product of Shimura curves at a place of good reduction. We prove the unramified arithmetic level raising theorem for triple product of Shimura curves and we deduce from it the second reciprocity law which relates the image of the diagonal cycle under the Abel-Jacobi map to certain period integral of Gross-Kudla type. Along with the first reciprocity law we proved in a previous work, we show that the Gross-Schoen diagonal cycles form a Bipartite Euler system for the symmetric cube motive of modular forms. As an application we provide some evidence for the rank $1$ case of the Bloch-Kato conjecture for the symmetric cube motive of a modular form

Generic canonical form of pairs of matrices with zeros

We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all of these pairs reduce to the same pair (C, D) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (C D) are constructed by a combinatorial method based on properties of a certain graph.Comment: 13 page