Dissertations / Theses: 'Methods of association and difference'

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Bouaziz, Matthieu. "Statistical methods to account for different sources of bias in Genome-Wide association studies." Thesis, Evry-Val d'Essonne, 2012. http://www.theses.fr/2012EVRY0023/document.

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Les études d'association à grande échelle sont devenus un outil très performant pour détecter les variants génétiques associés aux maladies. Ce manuscrit de doctorat s'intéresse à plusieurs des aspects clés des nouvelles problématiques informatiques et statistiques qui ont émergé grâce à de telles recherches. Les résultats des études d'association à grande échelle sont critiqués, en partie, à cause du biais induit par la stratification des populations. Nous proposons une étude de comparaison des stratégies qui existent pour prendre en compte ce problème. Leurs avantages et limites sont discutés en s'appuyant sur divers scénarios de structure des populations dans le but de proposer des conseils et indications pratiques. Nous nous intéressons ensuite à l'interférence de la structure des populations dans la recherche génétique. Nous avons développé au cours de cette thèse un nouvel algorithme appelé SHIPS (Spectral Hierarchical clustering for the Inference of Population Structure). Cet algorithme a été appliqué à un ensemble de jeux de données simulés et réels, ainsi que de nombreux autres algorithmes utilisés en pratique à titre de comparaison. Enfin, la question du test multiple dans ces études d'association est abordée à plusieurs niveaux. Nous proposons une présentation générale des méthodes de tests multiples et discutons leur validité pour différents designs d'études. Nous nous concertons ensuite sur l'obtention de résultats interprétables aux niveaux de gènes, ce qui correspond à une problématique de tests multiples avec des tests dépendants. Nous discutons et analysons les différentes approches dédiées à cette fin
Genome-Wide association studies have become powerful tools to detect genetic variants associated with diseases. This PhD thesis focuses on several key aspects of the new computational and methodological problematics that have arisen with such research. The results of Genome-Wide association studies have been questioned, in part because of the bias induced by population stratification. Many stratégies are available to account for population stratification scenarios are highlighted in order to propose pratical guidelines to account for population stratification. We then focus on the inference of population structure that has many applications for genetic research. We have developed and present in this manuscript a new clustering algoritm called Spectral Hierarchical clustering for the Inference of Population Structure (SHIPS). This algorithm in the field to propose a comparison of their performances. Finally, the issue of multiple-testing in Genome-Wide association studies is discussed on several levels. We propose a review of the multiple-testing corrections and discuss their validity for different study settings. We then focus on deriving gene-wise interpretation of the findings that corresponds to multiple-stategy to obtain valid gene-disease association measures

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Gerdsen, Willy. "Investigating suitable pitch sizes for young football players in New Zealand." Click here to access this resource online, 2008. http://hdl.handle.net/10292/399.

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Whilst smaller pitches have become the norm in junior football, they may still be too large for certain ages and levels of skill. To date there has been no research into the relationships between size of the pitch and the technical kicking ability (distance and accuracy) of young players (8 - 14 years of age). The purpose of this investigation was to examine variables that influence actual kicking distance and accuracy and also to measure what differences in play behaviour (passing and dribbling) emerge from self-selected changes in pitch size for different age groups (9 and 10 years of age). Data were collected on 120 (N=120) junior football players: Playing experience (M=2.85 years, SD=2.56), Height (M=1.44 m, SD=1.08), Weight (M=37.8 kg, SD=7.69), Lengths of lower limbs (knee/ankle: M=35.54 cm, SD=4.27; hip/knee: M=35.54 cm, SD=5.25), Step lengths (M=37.66 cm, SD=8.693), Estimated kicking distance (M=31.13 m, SD=16.63). Participants performed a series of three kicks along the ground, using the inside of the foot, and aiming for a target (25 m away). The distance (M=18.04 m; SD=6.56) and accuracy (M=8.32 m; SD=4.38) of each kick were measured. The children (9 and 10 years of age) were then assigned to teams and asked to construct a small-sided game on two different pitch sizes. The first pitch size used was the recommended regulation size. The second pitch size was self-selected by the players. Any changes to the playing dimensions (e.g., width of the pitch) and playing behaviours (e.g., total number of passes, dribbling) were measured and analysed. Kicking distance is best predicted by the player’s height (20.0%, P < 0.000), or a combination of the player’s height and estimated kicking distance (30.0%, P = 0.002). Kicking accuracy can be attributed to the influence of the player’s step lengths (8.1%, P = 0.016) and both their step lengths and estimated kicking distance (15.1%, P = 0.020). Furthermore, our findings demonstrate that the increase in pitch size (18.5% and 25%) resulted in a greater amount of dribbling (63% and 33%) and passing (12%). In general, our results support the idea that young children in New Zealand should be playing on a pitch and at a skill level which matches their football abilities. Grouping young players on a pitch according to their physical (e.g., height) and technical kicking ability (e.g., distance, accuracy) instead of their chronological age, seems to be the key factor to any other set of proposals. The findings of this thesis have important messages that could enhance the effectiveness of coaching, competitive game-play (pitch sizes) and consequently performance at all junior levels of football in New Zealand. Further research should manipulate the number of players per team to see if this factor affects competitive game-play in junior football. Additionally, the mean distances maintained between players in the same team (team-mates) during game play needs to be considered (with regard to the ‘beehive effect’). Further studies should examine different age groups of equivalent skill level and assess their performance in relation to their technical kicking ability on different pitches.

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Garcia, Neto Baltasar Fernandes. "Power of QTL mapping of different genome-wide association methods for traits under different genetic structures : a simulation study /." Universidade Estadual Paulista (UNESP), 2018. http://hdl.handle.net/11449/152982.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
A complexidade das características que podem apresentar diferentes estruturas de ação gênica como, por exemplo, poligênicas ou afetadas por genes de efeito maior, aliado a diferentes herdabilidades, entre outros fatores, tornam a detecção de QTLs desafiadora. Diversos métodos têm sido empregados com o intuito de realizar estudos de associação ampla do genoma (GWAS), objetivando o mapeamento de QTL. A metodologia weighted single-step GBLUP (wssGBLUP), por exemplo, é uma alternativa para a realização de GWAS, que permite o uso simultâneo de informações genotípicas, de pedigree e fenotípicas, mesmo de animais não genotipados. Métodos Bayesianos também são utilizados para a realização de GWAS, partindo da premissa básica de que a variância observada pode variar em cada locus em uma distribuição a priori específica. O objetivo do presente estudo foi avaliar, por meio de simulações, quais métodos, dentre os avaliados, mais auxiliaria na identificação de QTLs para características poligênicas e afetadas por genes de efeito maior, apresentando diferentes herdabilidades. Utilizamos os métodos: wssGBLUP, com a inclusão ou não de informação adicional fenotípica de animais não genotipados e dois distintos ponderadores para os marcadores, onde w1 representou a mesma ponderação (w1=1) e w2 a ponderação calculada de acordo com o processo de iteração anterior (w1) ; Bayes C, assumindo dois valores para π (π=0.99 and π=0.999), onde π é a proporção de SNPs não incluída no modelo, além do LASSO Bayesiano. Os resultados mostraram que para cenários poligênicos o poder de detecção é menor e o uso adicional de fenótipos de animais não genotipados pode ajudar na detecção, ainda que com pouca intensidade. Para cenários com característica sob efeito maior, houve maior poder na detecção de QTL pelos diferentes métodos em comparação aos cenários poligênicos com destaque para a leve vantagem do método Bayes C. A inclusão de informação fenotípica adicional, entretanto, causou viés nas estimativas e atrapalhou o desempenho do wssGBLUP na presença de QTL com efeito maior. O aumento da v herdabilidade para ambas as estruturas melhorou o desempenho dos métodos e o poder de mapeamento. O método mais adequado para a detecção de QTL depende da estrutura genética e da herdabilidade da característica, não existindo um método que seja superior para todos os cenários.
The complexity of the traits that can present different genetic structures, such as polygenic or affected by genes of major effect, in addition to different heritabilities, among other factors, make the detection of QTLs challenging. Several methods have been employed with the purpose of performing genome wide association studies (GWAS), aiming the mapping of QTL. The single-step weighted GBLUP (wssGBLUP) method, for example, is an alternative to GWAS, which allows the simultaneous use of genotypic, pedigree and phenotypic information, even from non-genotyped animals. Bayesian methods are also used to perform GWAS, starting from the basic premise that the observed variance can vary at each locus with a specific priori distribution. The objective of the present study was to evaluate, through simulation, which methods, among the evaluated ones, more assist in the identification of QTLs for polygenic and major gene affected traits, presenting different heritabilities. We used the following methods: wssGBLUP, with or without additional phenotypic information from non-genotyped animals and two different weights for markers, where w1 represented the same weight (w1=1) and w2 the weight calculated according to the previous iteration process (w1); Bayes C, assuming two values for π (π = 0.99 and π = 0.999), where π is the proportion of SNPs not included in the model, and Bayesian LASSO. The results showed that for polygenic scenarios the detection power is lower and the additional use of phenotypes from non-genotyped animals may help in the detection, yet with low intensity. For scenarios with major effect, there was greater power in the detection of QTL by all different methods with slighter superior performance for the Bayes C method. However, the inclusion of additional phenotypic information caused bias in the estimates and harmed the performance of the wssGBLUP in the presence of major QTL. The increase in heritability for both structures improved the performance of the methods and the power of mapping. The most suitable method for the iii detection of QTL is dependent on the genetic structure and the heritability of the trait, and there is not a superior method for all scenarios.

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Salazar-Lazaro, Carlos Harold Wilson R. M. Wilson R. M. "Association schemes, codes, and difference sets /." Diss., Pasadena, Calif. : California Institute of Technology, 2007. http://resolver.caltech.edu/CaltechETD:etd-05222007-003651.

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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.

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Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.

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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.

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Hayman, Kenneth John. "Finite-difference methods for the diffusion equation." Title page, table of contents and summary only, 1988. http://web4.library.adelaide.edu.au/theses/09PH/09phh422.pdf.

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Persson, Jonas. "Accurate Finite Difference Methods for Option Pricing." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7097.

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Steinle, Peter John. "Finite difference methods for the advection equation /." Title page, table of contents and abstract only, 1993. http://web4.library.adelaide.edu.au/theses/09PH/09phs8224.pdf.

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Filipovic, Zlatko. "Finite difference methods for pricing financial derivatives." Thesis, Imperial College London, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.420931.

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Demir, Ismail. "Seismic wave modelling using finite difference methods." Thesis, University of South Wales, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.284896.

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REN, XIAOHUI. "COMPARING QUANTITATIVE ASSOCIATION RULE METHODS." University of Cincinnati / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1089133333.

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ZHANG, GE. "STATISTICAL METHODS IN GENETIC ASSOCIATION." University of Cincinnati / OhioLINK, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1196099744.

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Svärd, Magnus. "Stable High-Order Finite Difference Methods for Aerodynamics." Doctoral thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621.

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In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates. The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version. With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes. We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.

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Svärd, Magnus. "Stable high-order finite difference methods for aerodynamics /." Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621.

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Kama, Phumezile. "Non-standard finite difference methods in dynamical systems." Thesis, Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-07132009-163422.

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Saleemi, Asima Parveen. "Finite Difference Methods for the Black-Scholes Equation." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48660.

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Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

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Nehrbass, John Wayne. "Advances in finite-difference methods for electromagnetic modeling /." The Ohio State University, 1996. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487942182326148.

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Ackerman, Hans Christian. "Disease association mapping : methods and markers." Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.249649.

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Alshahrani, Mohammed Nasser D. "Statistical methods for rare variant association." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/22436/.

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Deoxyribonucleic acid (DNA) sequencing allows researchers to conduct more complete assessments of low-frequency and rare genetic variants. In anticipation of the availability of next-generation sequencing data, there is increasing interest in investigating associations between complex traits and rare variants (RVs). In contrast to association studies of common variants (CVs), due to the low frequencies of RVs, common wisdom suggests that existing statistical tests for CVs might not work, motivating the recent development of several new tests that analyze RVs, most of which are based on the idea of pooling/collapsing RVs. Genome-wide association studies (GWAS) based on common SNPs gained more attention in the last few years and have been regularly used to examine complex genetic compositions of diseases and quantitative traits. GWASs have not discovered everything associated with diseases and genetic variations. However, recent empirical evidence has demonstrated that low-frequency and rare variants are, in fact, connected to complex diseases. This thesis will focus on the study of rare variant association. Aggregation tests, where multiple rare variants are analyzed jointly, have incorporated weighting schemes on variants. However, their power is very much dependent on the weighting scheme. I will address three topics in this thesis: the definition of rare variants and their call file (VCF) and a description of the methods that have been used in rare variant analysis. Finally, I will illustrate challenges involved in the analysis of rare variants and propose different weighting schemes for them. Therefore, since the efficiency of rare variant studies might be considerably improved by the application of an appropriate weighting scheme, choosing the proper weighting scheme is the topic of the thesis. In the following chapters, I will propose different weighting schemes, where weights are applied at the level of the variant, the individual or the cell (i.e. the individual genotype call), as well as a weighting scheme that can incorporate quality measures for variants (i.e., a quality score for variant calls) and cells (i.e., genotype quality).

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Camlica, Sedat. "Recursive Passive Localization Methods Using Time Difference Of Arrival." Master's thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/2/12611032/index.pdf.

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In this thesis, the passive localization problem is studied. Robust and recursive solutions are presented by the use of Time Difference of Arrival (TDOA). The TDOA measurements are assumed to be gathered by moving sensors which makes the number of the sensors increase synthetically. First of all, a location estimator should be capable of processing the new measurements without omitting the past data. This task can be accomplished by updating the estimate recursively whenever new measurements are available. Convenient forms of the recursive filters, such as the Kalman filter, the Extended Kalman filter etc., can be applied. Recursive filter can be divided to two major groups: (a) The first type of recursive estimators process the TDOA measurements directly, and (b) the second type of the recursive estimators is the post processing estimators which process the TDOA indirectly, instead they fuse or smooth available location estimates. In this sense, recursive passive localization methods are presented for both types. In practice, issues like being spatially distant from each other and/or a radar with a rotating narrow beam may prevent the sensors to receive the same pulse. In such a case, the sensors can not construct common TDOA measurements which means that they can not accomplish the location estimation procedure. Additionally, there may be more than one sensor group making TDOA measurements. An estimator should be capable of fusing the measurements from different sensor groups. A sensor group consists of sensors which are able to receive the same pulse. In this work, solutions of these tasks are also given. Performances of the presented methods are compared by simulation studies. The method having the best performance, which is based on the Kalman Filter, is also capable of estimating the track of a moving emitter by directly processing the TDOA measurements.

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Kress, Wendy. "High Order Finite Difference Methods in Space and Time." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3559.

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Grilli, Roberto. "Methods for Trace Gas Detection Using Difference Frequency Generation." Thesis, University of Bristol, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.520211.

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Pant, Dibakar Raj. "Line element and variational methods for color difference metrics." Phd thesis, Université Jean Monnet - Saint-Etienne, 2012. http://tel.archives-ouvertes.fr/tel-00981484.

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Visual sensitivity to small color difference is an important factor for precision color matching. Small color differences can be measured by the line element theory in terms of color distances between a color point and neighborhoods of points in a color space. This theory gives a smooth positive definite symmetric metric tensor which describes threshold of color differences by ellipsoids in three dimensions and ellipses in two dimensions. The metric tensor is also known as the Riemannian metric tensor. In regard to the color differences, there are many color difference formulas and color spaces to predict visual difference between two colors but, it is still challenging due to the nonexistence of a perfect uniform color space. In such case, the Riemannian metric tensor can be used as a tool to study the performance of various color spaces and color difference metrics for measuring the perceptual color differences. It also computes the shortest length or the distance between any two points in a color space. The shortest length is called a geodesic. According to Schrödinger's hypothesis geodesics starting from the neutral point of a surface of constant brightness correspond to the curves of constant hue. The chroma contours are closed curves at constant intervals from the origin measured as the distance along the constant hue geodesics. This hypothesis can be utilized to test the performance of color difference formulas to predict perceptual attributes (hue and chroma) and distribution of color stimulus in any color space. In this research work, a method to formulate line element models of color difference formulas the ΔE*ab, the ΔE*uv, the OSA-UCS ΔEE and infinitesimal approximation of CIEDE2000 (ΔE00) is presented. The Jacobian method is employed to transfer their Riemannian metric tensors in other color spaces. The coefficients of such metric tensors are used to compute ellipses in two dimensions. The performance of these four color difference formulas is evaluated by comparing computed ellipses with experimentally observed ellipses in different chromaticity diagrams. A method is also developed for comparing the similarity between a pair of ellipses. The technique works by calculating the ratio of the area of intersection and the area of union of a pair of ellipses. Similarly, at a fixed value of lightness L*, hue geodesics originating from the achromatic point and their corresponding chroma contours of the above four formulas in the CIELAB color space are computed by solving the Euler-Lagrange equations in association with their Riemannian metrics. They are compared with with the Munsell chromas and hue circles at the Munsell values 3, 5 and 7. The result shows that neither formulas are fully perfect for matching visual color difference data sets. However, Riemannized ΔE00 and the ΔEE formulas measure the visual color differences better than the ΔE*ab and the ΔE*uv formulas at local level. It is interesting to note that the latest color difference formulas like the OSA-UCS ΔEE and the Riemannized ΔE00 do not show better performance to predict hue geodesics and chroma contours than the conventional CIELAB and CIELUV color difference formulas and none of these formulas fit the Munsell data accurately

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Bowen, Matthew K. "High-order finite difference methods for partial differential equations." Thesis, Loughborough University, 2005. https://dspace.lboro.ac.uk/2134/13492.

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General n-point formulae for difference operators and their errors are derived in terms of elementary symmetric functions. These are used to derive high-order, compact and parallelisable finite difference schemes for the decay-advection-diffusion and linear damped Korteweg-de Vnes equations. Stability calculations are presented and the speed and accuracy of the schemes is compared to that of other finite difference methods in common use. Appendices contain useful tables of difference operators and errors and present a stability proof for quadratic inequalities. For completeness, the appendices conclude with the standard Thomas method for solving tri-diagonal systems.

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Cheema, Tasleem Akhter. "Higher-order finite-difference methods for partial differential equations." Thesis, Brunel University, 1997. http://bura.brunel.ac.uk/handle/2438/7131.

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This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.

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Seibold, Benjamin [Verfasser]. "M-Matrices in Meshless Finite Difference Methods / Benjamin Seibold." Aachen : Shaker, 2006. http://d-nb.info/1166514307/34.

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Lin, Yuan. "High-order finite difference methods for solving heat equations /." Available to subscribers only, 2008. http://proquest.umi.com/pqdweb?did=1559848541&sid=1&Fmt=2&clientId=1509&RQT=309&VName=PQD.

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Thesis (Ph. D.)--Southern Illinois University Carbondale, 2008.
"Department of Mathematics." Keywords: High-order finite difference, Heat equations Includes bibliographical references (p. 64-68). Also available online.

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Demirayak, Murat Neslitürk Ali İhsan. "Analysis Of Finite Difference Methods For Convection-Diffusion Problem/." [s.l.]: [s.n.], 2004. http://library.iyte.edu.tr/tezler/master/matematik/T000481.pdf.

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Göransson, Albin. "Stability and accuracy for difference methods using asynchronous processors." Thesis, Linköpings universitet, Matematiska institutionen, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-146045.

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We solve initial boundary value problems with information unavailable at random time-steps. The randomly unavailable information represents asynchrony between processing elements. To approximate the initial boundary value problem, finite difference operators with summation-by-parts proper-ties and weak boundary procedures are used. Utilizing the energy method, we derive energy estimates for synchronous and asynchronous problems. The simulations show that the solutions may remain accurate and stable, even in the asynchronous case.

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Wang, Siyang. "Finite Difference and Discontinuous Galerkin Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320614.

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Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.

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Mossberg, Eva. "Higher order finite difference methods for wave propagation problems." Licentiate thesis, Uppsala universitet, Avdelningen för teknisk databehandling, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-86003.

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Wave propagation is described by the wave equation, or in the time-periodic case, by the Helmholtz equation. For problems with small wavelengths, high order discretizations must be used to resolve the solution. Two different techniques for finding compact finite difference schemes of high order are studied and compared. The first approach is Numerov's idea of using the equation to transfer higher derivatives to lower order ones for the Helmholtz equation, or, for the wave equation, from time to space. The second principle is the method of deferred correction, where a lower order approximation is used for error correction. For the time-independent Helmholtz problem, sharp estimates for the error are derived, in order to compare the arithmetic complexity for both approaches with a non-compact scheme. The characteristics of the errors for fourth order as well as sixth order accuracy are demonstrated and the advantages and disadvantages of the methods are discussed. A time compact, Numerov-type, fourth order method and a fourth order method using deferred correction in time are studied for the wave equation. Schemes are derived for both the second order formulation of the equation, and for the system in first order form. Stability properties are analyzed and numerical experiments have been performed, for both constant and variable coefficients in the equations. For the first order formulation, a staggered grid is used.

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Sullivan, Erin T. "A Communication Approach to Mimesis and Gender Performance: What Difference does Difference make in the NBA?" Fogler Library, University of Maine, 2004. http://www.library.umaine.edu/theses/pdf/SullivanET2004.pdf.

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Koh, Hyunwook. "Adaptive Statistical Methods for Microbiome Association Studies." Thesis, New York University, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10750033.

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The human microbiome studies have been accelerated by the advances in next-generation sequencing technologies. There has also been increasing interest in discovering microbial taxa that are associated with diverse host phenotypes, environmental factors or clinical interventions. Here, I first describe unique features of microbiome data and the resulting demand for adaptive association analysis which robustly suits different association patterns, while providing valid statistical inferences. Then, I introduce two adaptive microbiome association tests as follows.

My first method, namely, optimal microbiome-based association test (OMiAT), relates microbial composition with continuous (e.g., body mass index) or binary (e.g., disease status) traits. OMiAT is a data-driven adaptive testing method which approximates to the most powerful performance among different candidate tests from the sum of powered score tests (SPU) and microbiome regression-based kernel association test (MiRKAT). I illustrate that OMiAT robustly discovers underlying association signals arising from highly imbalanced microbial abundances and phylogenetic tree structure, while correctly controlling type I error rates. I also propose a way to apply it to fine association mapping of diverse higher-level taxa at different taxonomic levels within a newly introduced microbial taxa discovery framework, microbiome comprehensive association mapping (MiCAM).

My second method, namely, optimal microbiome-based survival analysis (OMiSA), relates microbial composition with survival (i.e., time to event) traits. OMiSA approximates to the most powerful association test within two test domains, 1) microbiome-based survival analysis using linear and non-linear bases of OTUs (MiSALN) and 2) microbiome-based kernel association test for survival traits (MiRKAT-S). I illustrate that OMiSA powerfully discovers underlying associated lineages whether they are rare or abundant and phylogenetically related or not, while correctly controlling type I error rates.

OMiAT and OMiSA are attractive in practice due to the high complexity of microbiome data and the unknown true nature of the state. MiCAM also provides a hierarchical microbiome association map through a breadth of taxonomic levels, which can be used as a guideline for further investigation on the roles of discovered taxa in human health or disease.

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Porter, Heather Frances. "Multi-trait methods for genetic association testing." Thesis, King's College London (University of London), 2018. https://kclpure.kcl.ac.uk/portal/en/theses/multitrait-methods-for-genetic-association-testing(d040066c-8b0f-4f1b-bd0c-7be3f144b7e2).html.

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The early stages of the genome-wide association study (GWAS) era were dominated by studies focusing on single phenotypes, while in recent years there has been growing interest in multi-trait GWAS. A wide variety of multi-trait GWAS methods have been developed, but publications introducing new methods are highly inconsistent in their evaluation of method performance, obscuring their relative merit. Facilitated by burgeoning national biobank resources, multi-trait analyses are set to become more routinely applied, making understanding their relative performance increasingly important. We develop a simulation framework to model the complex networks underlying multivariate genetic epidemiology. We exploit our simulation framework to perform a comprehensive comparison study of the leading multi-trait GWAS methods, providing a web application and open-source software program implementing our simulation framework for further benchmarking of multi-trait GWAS methods. Motivated by our comparison results, we develop novel methodology and present a series of multi-trait analyses. We perform multi-trait genome-wide analyses on publicly available GWAS summary statistics on 19 traits – metabolic, anthropometric and psychiatric. We develop and apply two summary statistic methods: one that has increased power to detect pleiotropic effects on multiple traits, and one that is more powerful for detecting heterogeneous genetic effects. Polygenic risk scores (PRS) are now a commonly used tool for performing phenotype prediction from genetics, assessing the genetic aetiology underlying diseases, and testing for shared genetic aetiology among traits. Using UK Biobank data, we explore the predictive ability of PRS computed across multiple traits for Major Depressive Disorder (MDD). The MDD PRS itself has so far offered modest prediction of MDD case/control status; we explore the use of PRS built on traits correlated with MDD to improve predictive ability. We build main effect and interaction models, using both AIC and BIC stepwise variable selection, and cross-validation, to establish the most predictive models.

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Liu, Jin. "Penalized methods in genome-wide association studies." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1242.

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Penalized regression methods are becoming increasingly popular in genome-wide association studies (GWAS) for identifying genetic markers associated with disease. However, standard penalized methods such as the LASSO do not take into account the possible linkage disequilibrium between adjacent markers. We propose a novel penalized approach for GWAS using a dense set of single nucleotide polymorphisms (SNPs). The proposed method uses the minimax concave penalty (MCP) for marker selection and incorporates linkage disequilibrium (LD) information by penalizing the difference of the genetic effects at adjacent SNPs with high correlation. A coordinate descent algorithm is derived to implement the proposed method. This algorithm is efficient and stable in dealing with a large number of SNPs. A multi-split method is used to calculate the p-values of the selected SNPs for assessing their significance. We refer to the proposed penalty function as the smoothed MCP (SMCP) and the proposed approach as the SMCP method. Performance of the proposed SMCP method and its comparison with a LASSO approach are evaluated through simulation studies, which demonstrate that the proposed method is more accurate in selecting associated SNPs. Its applicability to real data is illustrated using data from a GWAS on rheumatoid arthritis. Based on the idea of SMCP, we propose a new penalized method for group variable selection in GWAS with respect to the correlation between adjacent groups. The proposed method uses the group LASSO for encouraging group sparsity and a quadratic difference for adjacent group smoothing. We call it smoothed group LASSO, or SGL for short. Canonical correlations between two adjacent groups of SNPS are used as the weights in the quadratic difference penalty. Principal components are used to reduced dimensionality locally within groups. We derive a group coordinate descent algorithm for computing the solution path of the SGL. Simulation studies are used to evaluate the finite sample performance of the SGL and group LASSO. We also demonstrate its applicability on rheumatoid arthritis data.

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Chiang, Weng Cheng Venus. "High-order finite difference methods for solving convection diffusion equations." Thesis, University of Macau, 2008. http://umaclib3.umac.mo/record=b1807119.

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Arakelyan, Avetik. "The Finite Difference Methods for Multi-phase Free Boundary Problems." Doctoral thesis, KTH, Matematik (Avd.), 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-33543.

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This thesis consist of an introduction and four research papers concerning numerical analysis for a certain class of free boundary problems. Paper I is devoted to the numerical analysis of the so-called two-phase membrane problem. Projected Gauss-Seidel method is constructed. We prove general convergence of the algorithm as well as obtain the error estimate for the finite difference scheme. In Paper II we have improved known results on the error estimates for a Classical Obstacle (One-Phase) Problem with a finite difference scheme. Paper III deals with the parabolic version of the two-phase obstacle-like problem. We introduce a certain variational form, which allows us to definea notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. In the last paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support. The proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions.
QC 20110510

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Batty, Christopher. "Simulating viscous incompressible fluids with embedded boundary finite difference methods." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/28642.

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The behaviour of liquids and gases ranks among the most familiar and yet complex physical phenomena commonly encountered in daily life. To create a seamless approximation of the real world, it is clear that we must be able to accurately simulate fluids. However, a crucial element of what makes fluid behaviour so complex and compelling is its interactions with its surroundings. To simulate the motion of a fluid we cannot consider the Navier-Stokes equations in isolation; we must also examine the boundary conditions at the point where the fluid meets the world. Enforcing these boundary conditions has traditionally been a source of tremendous difficulty. Cartesian grid-based methods typically approximate the world in an unrealistic, axis-aligned block representation, while conforming mesh methods frequently suffer from poor mesh quality and expensive mesh generation. This thesis examines the use of embedded boundary finite difference methods to alleviate these shortcomings by providing a degree of sub-grid information that enables more efficient, flexible, accurate, and realistic simulations. The first key contribution of the thesis is the use of a variational approach to derive novel embedded boundary finite difference methods for fluids, by exploiting the concept of natural boundary conditions. This idea is applied first to animate the interaction between incompressible fluids and irregularly shaped dynamic rigid bodies. I then apply a similar technique to properly handle viscous free surfaces, enabling realistic buckling and coiling in viscous flows. Lastly, I unify these ideas to simulate Stokes flows in the presence of both free surface and solid boundaries, and demonstrate the method's convergence on a range of examples. The second main contribution is a study of embedded boundary methods for pressure projection in the context of unstructured Delaunay and Voronoi meshes. By eliminating the need for boundary-conforming meshes, this work enables efficient high-quality adaptive mesh generation and improves simulation accuracy. Furthermore, it demonstrates that by placing simulation samples at Voronoi sites, and choosing these sites intelligently with respect to liquid geometry, one can eliminate surface noise, improve the realism and stability of surface tension, and plausibly simulate nearly arbitrarily thin sheets and droplets.

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Bø, Ruben Kristoffer Thomasse. "On Mimetic Finite Difference Methods for Grids with Curved Faces." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2012. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-19356.

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In this thesis the mimetic finite difference method for grids with curved faces is presented, implemented and tested with an emphasis on applications in reservoir simulation. The thesis gives a brief introduction to reservoir modeling and introduce the mimetic method for flat and for curved faces. Then the continuity condition for the curved mimetic method is discussed. It is shown that the suggested continuity condition is not valid for cases with a difference in permeability between two cells separated by a curved face. An alternative continuity condition is discussed and implemented. Numerical examples confirm that the original continuity condition is incorrect for general examples with heterogeneous permeability. Numerical examples for the alternative continuity condition shows that it is correct for simple cases, and that it gives no gain in accuracy compared to the mimetic method. In conclusion the curved mimetic method is primarily of academic interest.

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Shen, Jonah Wai. "Finite difference methods applied to biot theory in porous medium." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1995. http://handle.dtic.mil/100.2/ADA306214.

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Thesis (M.S. in Engineering Acoustics) Naval Postgraduate School, September 1995.
Thesis advisor(s):, Clyde Scandrett,Anthony A. Atchley. "September 1995." Bibliography: p. 43. Also available online.

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Mattsson, Ken. "Summation-by-Parts Operators for High Order Finite Difference Methods." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3434.

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Gao, Liping. "Splitting finite difference methods for the time-dependent Maxwell equations." Thesis, Coventry University, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.429698.

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Pindza, Edson. "Spectral difference methods for solving equations of the KdV hierarchy." Thesis, Stellenbosch : Stellenbosch University, 2008. http://hdl.handle.net/10019.1/2168.

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Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008.
The Korteweg-de Vries (KdV) hierarchy is an important class of nonlinear evolution equa- tions with various applications in the physical sciences and in engineering. In this thesis analytical solution methods were used to ¯nd exact solutions of the third and ¯fth order KdV equations, and numerical methods were used to compute numerical solutions of these equations. Analytical methods used include the Fan sub-equation method for constructing exact trav- eling wave solutions, and the simpli¯ed Hirota method for constructing exact N-soliton solutions. Some well known cases were considered. The Fourier spectral method and the ¯nite di®erence method with Runge-Kutta time dis- cretisation were employed to solve the third and the ¯fth order KdV equations with periodic boundary conditions. The one soliton and the two soliton solutions were used as initial conditions. The numerical solutions are obtained and compared with the exact solutions. The propagation of a single soliton as well as the interaction of double soliton solutions is modeled well by both numerical methods, although the Fourier spectral method performs better. The stability, consistency and convergence of these numerical methods were investigated. Error propagation is studied. The theoretically predicted quadratic convergence of the ¯nite di®erence method as well as the exponential convergence of the Fourier spectral method is con¯rmed in numerical experiments.

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Virta, Kristoffer. "Difference methods with boundary and interface treatment for wave equations." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-209139.

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Wave motion in acoustic and elastic media is highly influenced by the presence of outer boundaries and media interfaces. The solutions to the equations governing the wave motion at any point in the domain as a function of time can be sought either through analytical or numerical techniques. This thesis proposes provably stable finite difference schemes to accurately investigate wave interaction with boundaries and interfaces. Schemes for the acoustic wave equation in three spatial coordinates, general domains and heterogeneous media and the elastic wave equation in two spatial dimensions and layered media are presented. A study of the Rayleigh surface wave in almost incompressible media is carried through. Extensive numerical experiments designed to verify stability and accuracy as well as applicability to nontrivial boundary and interface phenomena are given.

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Yung, Godwin Yuen Han. "Statistical methods for analyzing genetic sequencing association studies." Thesis, Harvard University, 2016. http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493313.

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Case-control genetic sequencing studies are increasingly being conducted to identify rare variants associated with complex diseases. Oftentimes, these studies collect a variety of secondary traits--quantitative and qualitative traits besides the case-control disease status. Reusing the data and studying the association between rare variants and secondary phenotypes provide an attractive and cost effective approach that can lead to discovery of new genetic associations. In Chapter 1, we carry out an extensive investigation of the validity of ad hoc methods, which are simple, computationally efficient methods frequently applied in practice to study the association between secondary phenotypes and single common genetic variants. Though other researchers have investigated the same problem, we make two key contributions to existing literature. First, we show that in taking an ad hoc approach, it may be desirable to adjust for covariates that affect the primary disease in the secondary phenotype model, even though these covariates are not necessarily associated with the secondary phenotype in the population. Second, we show that when the disease is rare, ad hoc methods can lead to severely biased estimation and inference if the true disease model follows a non-logistic model such as the probit model. Spurious associations can be avoided by including interaction terms in the fitted regression model. Our results are justified theoretically and via simulations, and illustrated by a genome-wide association study of smoking using a lung cancer case-control study. In Chapter 2, we consider the problem of testing associations between secondary phenotypes and sets of rare genetic variants. We show that popular region-based methods such as the burden test and the sequence kernel association test (SKAT) can only be applied under the same conditions as those applicable to ad hoc methods (Chapter 1). For a more robust alternative, we propose an inverse-probability-weighted version of the optimal SKAT (SKAT-O) to account for unequal sampling of cases and controls. As an extension of SKAT-O, our approach is data adaptive and includes the weighted burden test and weighted SKAT as special cases. In addition to weighting individuals to account for the biased sampling, we can also consider weighting the variants in SKAT-O. Decreasing the weight of non-causal variants and increasing the weight of causal variants can improve power. However, since researchers do not know which variants are actually causal, it is common practice to weight genetic variants as a function of their minor allele frequencies. This is motivated by the belief that rarer variants are more likely to have larger effects. In Chapter 3, we propose a new unsupervised statistical framework for predicting the functional status of genetic variants. Compared to existing methods, the proposed algorithm integrates a diverse set of annotations---which are partitioned beforehand into multiple groups by the user---and predicts the functional status for each group, taking into account within- and between-group correlations. We demonstrate the advantages of the algorithm through application to real annotation data and conclude with future directions.
Biostatistics

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Porretta'S, Luciano. "MODELS AND METHODS IN GENOME WIDE ASSOCIATION STUDIES." Doctoral thesis, Universite Libre de Bruxelles, 2018. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/265314.

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The interdisciplinary field of systems biology has evolved rapidly over the last few years. Different disciplines have contributed to the development of both its experimental and theoretical branches.Although computational biology has been an increasing activity in computer science for more than a two decades, it has been only in the past few years that optimization models have been increasingly developed and analyzed by researchers whose primary background is Operations Research(OR). This dissertation aims at contributing to the field of computational biology by applying mathematical programming to certain problems in molecular biology.Specifically, we address three problems in the domain of Genome Wide Association Studies}:(i) the Pure Parsimony Haplotyping Under uncertatind Data Problem that consists in finding the minimum number of haplotypes necessary to explain a given set of genotypes containing possible reading errors; (ii) the Parsimonious Loss Of Heterozygosity Problem that consists of partitioning suspected polymorphisms from a set of individuals into a minimum number of deletion areas; (iii) and the Multiple Individuals Polymorphic Alu Insertion Recognition Problem that consists of finding the set of locations in the genome where ALU sequences are inserted in some individual(s).All three problems are NP-hard combinatorial optimization problems. Therefore, we analyse their combinatorial structure and we propose an exact approach to solution for each of them. The proposed models are efficient, accurate, compact, polynomial-sized and usable in all those cases for which the parsimony criterion is well suited for estimation.
Option Informatique du Doctorat en Sciences
info:eu-repo/semantics/nonPublished

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48

Hamilton, Brian. "Finite difference and finite volume methods for wave-based modelling of room acoustics." Thesis, University of Edinburgh, 2016. http://hdl.handle.net/1842/22940.

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Wave-based models of sound propagation can be used to predict and synthesize sounds as they would be heard naturally in room acoustic environments. The numerical simulation of such models with traditional time-stepping grid-based methods can be an expensive process, due to the sheer size of listening environments (e.g., auditoriums and concert halls) and due to the temporal resolution required by audio rates that resolve frequencies up to the limit of human hearing. Finite difference methods comprise a simple starting point for such simulations, but they are known to suffer from approximation errors that may necessitate expensive grid refinements in order to achieve sufficient levels of accuracy. As such, a significant amount of research has gone into designing finite difference methods that are highly accurate while remaining computationally efficient. The problem of designing and using accurate finite difference schemes is compounded by the fact that room acoustics models require complex boundary conditions to model frequency-dependent wall impedances over non-trivial geometries. The implementation of such boundary conditions in a numerically stable manner has been a challenge for some time. Stable boundary conditions for finite difference room acoustics simulations have been formulated in the past, but generally they have only been useful in modelling trivial geometries (e.g., idealised shoebox halls). Finite volume methods have recently been shown to be a viable solution to the problem of complex boundary conditions over non-trivial geometries, and they also allow for the use of energy methods for numerical stability analyses. Finite volume methods lend themselves naturally to fully unstructured grids and they can simplify to the types of grids typically used in finite difference methods. This allows for room acoustics simulation models that balance the simplicity of finite difference methods for wave propagation in air with the detail of finite volume methods for the modelling of complex boundaries. This thesis is an exploration of these two distinct, yet related, approaches to wave-based room acoustic simulations. The overarching theme in this investigation is the balance between accuracy, computational efficiency, and numerical stability. Higher-order and optimised schemes in two and three spatial dimensions are derived and compared, towards the goal of finding accurate and efficient finite difference schemes. Numerical stability is analysed using frequency-domain analyses, as well as energy techniques whenever possible, allowing for stable and frequency-dependent boundary conditions appropriate for room acoustics modelling. Along the way, the use of non-Cartesian grids is investigated, geometric relationships between certain finite difference and finite volume schemes are explored, and some problems associated to staircasing effects at boundaries are considered. Also, models of sound absorption in air are incorporated into these numerical schemes, using physical parameters that are appropriate for room acoustic scenarios.

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McGuckin, Conor. "School bullying in Northern Ireland : prevalence, measurement, and association with individual difference variables." Thesis, University of Ulster, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403944.

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Duru, Kenneth. "Perfectly Matched Layers and High Order Difference Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-173009.

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The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete. We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials. We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML. Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.

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Dissertations / Theses on the topic 'Methods of association and difference'

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