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Numerical Algorithms for Optimization Problems in Genetical Analysis
Mälardalen University, Department of Mathematics and Physics.
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

The focus of this thesis is on numerical algorithms for efficient solution of QTL analysis problem in genetics.

Firstly, we consider QTL mapping problems where a standard least-squares model is used for computing the model fit. We develop optimization methods for the local problems in a hybrid global-local optimization scheme for determining the optimal set of QTL locations. Here, the local problems have constant bound constraints and may be non-convex and/or flat in one or more directions. We propose an enhanced quasi-Newton method and also implement several schemes for constrained optimization. The algorithms are adopted to the QTL optimization problems. We show that it is possible to use the new schemes to solve problems with up to 6 QTLs efficiently and accurately, and that the work is reduced with up to two orders magnitude compared to using only global optimization.

Secondly, we study numerical methods for QTL mapping where variance component estimation and a REML model is used. This results in a non-linear optimization problem for computing the model fit in each set of QTL locations. Here, we compare different optimization schemes and adopt them for the specifics of the problem. The results show that our version of the active set method is efficient and robust, which is not the case for methods used earlier. We also study the matrix operations performed inside the optimization loop, and develop more efficient algorithms for the REML computations. We develop a scheme for reducing the number of objective function evaluations, and we accelerate the computations of the derivatives of the log-likelihood by introducing an efficient scheme for computing the inverse of the variance-covariance matrix and other components of the derivatives of the log-likelihood.

Place, publisher, year, edition, pages
Västerås: Mälardalens högskola , 2008.
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 59
Keywords [en]
Quantitative Trait Loci (QTL), restricted maximum likelihood (REML), variance components, average information (AI) matrix, Local optimization, Quasi-Newton method, Active Set method, Hessian approximation, BFGS update
National Category
Computational Mathematics
Research subject
Matematik/tillämpad matematik
Identifiers
URN: urn:nbn:se:mdh:diva-650ISBN: 978-91-85485-84-0 (print)OAI: oai:DiVA.org:mdh-650DiVA, id: diva2:121268
Public defence
2008-06-05, Kappa, U, Högskoleplan 1, Västerås, 13:00
Opponent
Supervisors
Available from: 2008-04-28 Created: 2008-04-28
List of papers
1. Efficient Algorithms for Multi-Dimensional Global Optimization in Genetic Mapping of Complex Traits
Open this publication in new window or tab >>Efficient Algorithms for Multi-Dimensional Global Optimization in Genetic Mapping of Complex Traits
2010 (English)In: Advances and Applications in Bioinformatics and Chemistry, ISSN 1178-6949, Vol. 3, no 1, p. 75-88Article in journal (Refereed) Published
Abstract [en]

We present a two-phase strategy for optimizing a multidimensional, nonconvex function arising during genetic mapping of quantitative traits. Such traits are believed to be affected by multiple so called quantitative trait loci (QTL), and searching for d QTL results in a d-dimensional optimization problem with a large number of local optima. We combine the global algorithm DIRECT with a number of local optimization methods that accelerate the final convergence, and adapt the algorithms to problem-specific features. We also improve the evaluation of the QTL mapping objective function to enable exploitation of the smoothness properties of the optimization landscape. Our best two-phase method is demonstrated to be accurate in at least six dimensions and up to ten times faster than currently used QTL mapping algorithms.

National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-4219 (URN)2-s2.0-80051553747 (Scopus ID)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2017-12-14Bibliographically approved
2. Local Optimization for Genetic Mapping of Multiple Quantitative Trait Loci
Open this publication in new window or tab >>Local Optimization for Genetic Mapping of Multiple Quantitative Trait Loci
(English)In: Journal of Optimization and EngineeringArticle in journal (Refereed) Submitted
Identifiers
urn:nbn:se:mdh:diva-4220 (URN)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2015-07-08Bibliographically approved
3. Efficient Implementation of the AI-REML Iteration for Variance Component QTL Analysis
Open this publication in new window or tab >>Efficient Implementation of the AI-REML Iteration for Variance Component QTL Analysis
(English)In: Computational Statistics and Data Analysis, ISSN 0167-9473Article in journal (Refereed) Submitted
Identifiers
urn:nbn:se:mdh:diva-4221 (URN)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2015-07-08Bibliographically approved
4. New Algorithms for Evaluating the Log-likelihood Function Derivatives in the AI-REML Method
Open this publication in new window or tab >>New Algorithms for Evaluating the Log-likelihood Function Derivatives in the AI-REML Method
2009 (English)In: Communications in statistics. Simulation and computation, ISSN 0361-0918, E-ISSN 1532-4141, Vol. 38, no 6, p. 1348-1364Article in journal (Refereed) Published
Abstract [en]

 In this study, we propose several improvements of the Average Information Restricted Maximum Likelihood algorithms for estimating the variance components for genetic mapping of quantitative traits. The improved methods are applicable when two variance components are to be estimated. The improvements are related to the algebraic part of the methods and utilize the properties of the underlying matrix structures. In contrast to previously developed algorithms, the explicit computation of a matrix inverse is replaced by the solution of a linear system of equations with multiple right-hand sides, based on a particular matrix decomposition. The computational costs of the proposed algorithms are analyzed and compared.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:mdh:diva-4222 (URN)10.1080/03610910902912944 (DOI)000270912200008 ()2-s2.0-70449641719 (Scopus ID)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2017-12-14Bibliographically approved
5. Newton-type Methods for REML Estimation in Genetic Analysis of Quantitative Traits
Open this publication in new window or tab >>Newton-type Methods for REML Estimation in Genetic Analysis of Quantitative Traits
2008 (English)In: Journal of Computational Methods in Sciences and Engineering, ISSN 1472-7978, Vol. 8, no 1, p. 53-67Article in journal (Refereed) Published
Abstract [en]

Robust and efficient optimization methods for variance component estimation using Restricted Maximum Likelihood (REML) models for geneticmapping of quantitative traits are considered. We show that the standard Newton-AI scheme may fail when the optimum is located at one of the constraint boundaries, and we introduce different approaches to remedy this by taking the constraints into account. We approximate the Hessian of the objective function using the average information matrix and also by using an inverse BFGS formula. The robustness and efficiency is evaluated for problems derived from two experimental data from the same animal populations.

National Category
Mathematics
Identifiers
urn:nbn:se:mdh:diva-4223 (URN)2-s2.0-58149488846 (Scopus ID)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2013-02-05Bibliographically approved
6. Assessing a Multiple QTL Search Using the Variance Component Model
Open this publication in new window or tab >>Assessing a Multiple QTL Search Using the Variance Component Model
2010 (English)In: Computational biology and chemistry (Print), ISSN 1476-9271, E-ISSN 1476-928X, Vol. 34, no 1, p. 34-41Article in journal (Refereed) Published
Abstract [en]

Development of variance component algorithms in genetics has previously mainly focused on animal breeding models or problems in human genetics with a simple data structure. We study alternative methods for constrained likelihood maximization in quantitative trait loci (QTL) analysis for large complex pedigrees. We apply a forward selection scheme to include several QTL and interaction effects, as well as polygenic effects, with up to five variance components in the model. We show that the implemented active set and primal-dual schemes result in accurate solutions and that they are robust. In terms of computational speed, a comparison of two approaches for approximating the Hessian of the log-likelihood shows that the method using an average information matrix is the method of choice for the five-dimensional problem. The active set method, with the average information method for Hessian computation, exhibits the fastest convergence with an average of 20 iterations per tested position, where the change in variance components <0.0001 was used as convergence criterion.

National Category
Natural Sciences
Identifiers
urn:nbn:se:mdh:diva-4224 (URN)10.1016/j.compbiolchem.2009.12.001 (DOI)000275587800004 ()2-s2.0-75149131993 (Scopus ID)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2017-12-14Bibliographically approved
7. Adapted Downhill Simplex Method for Pricing Convertible Bonds
Open this publication in new window or tab >>Adapted Downhill Simplex Method for Pricing Convertible Bonds
2007 (English)In: Theory of Stochastic Processes, ISSN 0321-3900, Vol. 13, no 4, p. 130-147Article in journal (Refereed) Published
Abstract [en]

The paper is devoted to modeling optimal exercise strategies of thebehavior of investors and issuers working with convertible bonds.This implies solution of the problems of stock price modeling, payoffcomputation and minimax optimization.Stock prices (underlying asset) were modeled under the assumptionof the geometric Brownian motion of their values. The Monte Carlomethod was used for calculating the real payoff which is the objectivefunction. The minimax optimization problem was solved using thederivative-free Downhill Simplex method.The performed numerical experiments allowed to formulate recommendationsfor the choice of appropriate size of the initial simplex inthe Downhill Simplex Method, the number of generated trajectoriesof underlying asset, the size of the problem and initial trajectories ofthe behavior of investors and issuers.

National Category
Computational Mathematics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-4225 (URN)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2013-12-02Bibliographically approved
8. Increasing the Efficiency of Variance Component Quantitative Trait Loci Analysis by Using Reduced-Rank Identity-by-Descent Matrices
Open this publication in new window or tab >>Increasing the Efficiency of Variance Component Quantitative Trait Loci Analysis by Using Reduced-Rank Identity-by-Descent Matrices
2007 (English)In: Genetics, ISSN 0016-6731, Vol. 176, no 3, p. 1935-1938Article in journal (Refereed) Published
Abstract [en]

Recent technological development in genetics has made large-scale marker genotyping fast and practicable, facilitating studies for detection of QTL in large general pedigrees. We developed a method that speeds up restricted maximum-likelihood (REML) algorithms for QTL analysis bysimplifying the inversion of the variance-covariance matrix of the trait vector. The method was tested in an experimental chicken pedigree including 767 phenotyped individuals and 14 genotyped markers on chicken chromosome 1. The computation time in a chromosome scan covering 475 cM was reduced by 43% when the analysis was based on linkage only and by 72% when linkage disequilibrium information was included. The relative advantage of using our method increases with pedigree size, marker density, and linkage disequilibrium, indicating even greater improvements in the future. 

National Category
Genetics
Identifiers
urn:nbn:se:mdh:diva-4226 (URN)10.1534/genetics.107.071977 (DOI)000248416300044 ()2-s2.0-34547132590 (Scopus ID)
Available from: 2008-04-28 Created: 2008-04-28 Last updated: 2015-07-02Bibliographically approved

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