Efficient algorithms for detecting genetic interactions in genome-wide association study

Genome-wide association study (GWAS) aims to find genetic factors underlying complex phenotypic traits, for which epistasis or gene-gene interaction detection is often preferred over a single-locus approach. However, the computational burden has been a major hurdle to apply epistasis test at the genome-wide scale due to the large number of single nucleotide polymorphism (SNP) pairs to be tested. We have developed and implemented a series of efficient algorithms, i.e., FastANOVA, FastChi, COE, and TEAM, which support epistasis tests in a wide range of problem settings. These algorithms utilize a permutation test for proper error control. Unlike heuristic approaches, they guarantee to find the optimal solutions. It has been shown theoretically and experimentally that these algorithms significantly speed up the process of epistasis detection

Large-scale linear regression: Development of high-performance routines

In statistics, series of ordinary least squares problems (OLS) are used to study the linear correlation among sets of variables of interest; in many studies, the number of such variables is at least in the millions, and the corresponding datasets occupy terabytes of disk space. As the availability of large-scale datasets increases regularly, so does the challenge in dealing with them. Indeed, traditional solvers---which rely on the use of black-box" routines optimized for one single OLS---are highly inefficient and fail to provide a viable solution for big-data analyses. As a case study, in this paper we consider a linear regression consisting of two-dimensional grids of related OLS problems that arise in the context of genome-wide association analyses, and give a careful walkthrough for the development of {\sc ols-grid}, a high-performance routine for shared-memory architectures; analogous steps are relevant for tailoring OLS solvers to other applications. In particular, we first illustrate the design of efficient algorithms that exploit the structure of the OLS problems and eliminate redundant computations; then, we show how to effectively deal with datasets that do not fit in main memory; finally, we discuss how to cast the computation in terms of efficient kernels and how to achieve scalability. Importantly, each design decision along the way is justified by simple performance models. {\sc ols-grid} enables the solution of $10^{11}$ correlated OLS problems operating on terabytes of data in a matter of hours

Efficient algorithms in analyzing genomic data

With the development of high-throughput and low-cost genotyping technologies, immense data can be cheaply and efficiently produced for various genetic studies. A typical dataset may contain hundreds of samples with millions of genotypes/haplotypes. In order to prevent data analysis from becoming a bottleneck, there is an evident need for fast and efficient analysis methods. My thesis focuses on two interesting and important genetic analyzing problems. Genome-wide Association mapping. The goal of genome wide association mapping is to identify genes or narrow regions in the genome which have significant statistical correlations to the given phenotypes. The discovery of these genes offers the potential for increased understanding of biological processes affecting phenotypes such as body weight and blood pressure. Sample selection for maximal Genetic Diversity. Given a large set of samples, it is usually more efficient to first conduct experiments on a small subset. Then the following question arises: What subset to use? There are many experimental scenarios where the ultimate objective is to maintain, or at least maximize, the genetic diversity within relatively small breeding populations. In my thesis, I developed the following efficient and effective algorithms to address these problems. Phylogeny-based Genom-wide association mapping: TreeQA: The algorithm uses local perfect phylogeny tree in genome wide analysis for genotype/phenotype association mapping. Samples are partitioned according to the sub-trees they belong to. The association between a tree and the phenotype is measured by some statistic tests. TreeQA+: TreeQA+ inherits all the advantages of TreeQA. Moreover, it improves TreeQA by incorporating sample correlations into the association study. Sample selection for maximal genetic diversity: Sample Selection in biallelic SNP Data: Samples are selected based on their genetic diversity among a set of SNPs. Given a set of samples, the algorithms search for the minimum subset that retains all diversity (or a high percentage of diversity). Representative Sample Selection in Non-Biallelic Data: For more general data (non-biallelic), information-theoretic measurements such as entropy and mutual information are used to measure the diversity of a sample subset. Samples are selected to maximize the original information retained

Computing Petaflops over Terabytes of Data: The Case of Genome-Wide Association Studies

In many scientific and engineering applications, one has to solve not one but a sequence of instances of the same problem. Often times, the problems in the sequence are linked in a way that allows intermediate results to be reused. A characteristic example for this class of applications is given by the Genome-Wide Association Studies (GWAS), a widely spread tool in computational biology. GWAS entails the solution of up to trillions ($10^{12}$) of correlated generalized least-squares problems, posing a daunting challenge: the performance of petaflops ($10^{15}$ floating-point operations) over terabytes of data. In this paper, we design an algorithm for performing GWAS on multi-core architectures. This is accomplished in three steps. First, we show how to exploit the relation among successive problems, thus reducing the overall computational complexity. Then, through an analysis of the required data transfers, we identify how to eliminate any overhead due to input/output operations. Finally, we study how to decompose computation into tasks to be distributed among the available cores, to attain high performance and scalability. With our algorithm, a GWAS that currently requires the use of a supercomputer may now be performed in matter of hours on a single multi-core node. The discussion centers around the methodology to develop the algorithm rather than the specific application. We believe the paper contributes valuable guidelines of general applicability for computational scientists on how to develop and optimize numerical algorithms

Strong rules for nonconvex penalties and their implications for efficient algorithms in high-dimensional regression

We consider approaches for improving the efficiency of algorithms for fitting nonconvex penalized regression models such as SCAD and MCP in high dimensions. In particular, we develop rules for discarding variables during cyclic coordinate descent. This dimension reduction leads to a substantial improvement in the speed of these algorithms for high-dimensional problems. The rules we propose here eliminate a substantial fraction of the variables from the coordinate descent algorithm. Violations are quite rare, especially in the locally convex region of the solution path, and furthermore, may be easily detected and corrected by checking the Karush-Kuhn-Tucker conditions. We extend these rules to generalized linear models, as well as to other nonconvex penalties such as the $\ell_2$-stabilized Mnet penalty, group MCP, and group SCAD. We explore three variants of the coordinate decent algorithm that incorporate these rules and study the efficiency of these algorithms in fitting models to both simulated data and on real data from a genome-wide association study

Algorithms for Large-scale Whole Genome Association Analysis

In order to associate complex traits with genetic polymorphisms, genome-wide association studies process huge datasets involving tens of thousands of individuals genotyped for millions of polymorphisms. When handling these datasets, which exceed the main memory of contemporary computers, one faces two distinct challenges: 1) Millions of polymorphisms come at the cost of hundreds of Gigabytes of genotype data, which can only be kept in secondary storage; 2) the relatedness of the test population is represented by a covariance matrix, which, for large populations, can only fit in the combined main memory of a distributed architecture. In this paper, we present solutions for both challenges: The genotype data is streamed from and to secondary storage using a double buffering technique, while the covariance matrix is kept across the main memory of a distributed memory system. We show that these methods sustain high-performance and allow the analysis of enormous datase