Statistical methods and applications to animal breeding

This thesis comprises a collection of 39 research papers divided into three groups. The first group discusses the development of statistical methods, especially novel methods of variance component estimation, with general application. The second group examines the potential use of statistical methods in animal breeding studies, ranging from the construction of new experimental designs to the analysis of non-normal data. The third group reports on studies on animal breeding data in beef and dairy cattle.Group I is entitled "Statistical methods, including variance component estimation, with general application". The major theme of this group is the estimation of variance components. Some previous work based on methods for balanced data, gave rise to methods that were neither unique nor efficient and other methods gave results that are inconsistent with the analysis of variance for balanced data. A method was introduced, now known as REML (Residual Maximum Likelihood) that unifies the area. The method was introduced for the analysis of incomplete block designs with unequal block size but was found to have important applications in the analysis of groups of similar trials, time-series and animal breeding. Papers investigating REML estimation for multivariate data, time-series and detecting outliers are included. The relationship of REML to other methods is elucidated, especially for balanced and partially balanced designs. Computational strategies are discussed.The last two papers in the group illustrate a method of analysis of dial lei crosses that involves using multiple copies of the data. This idea of using multiple copies was shown also to be useful in the analysis of rectangular lattice designs and in the interpretation of some recently introduced neighbour analyses of field trials.The next group of papers, Group II, report on "Application of statistical methods to animal breeding studies". The work on variance components has some application in animal breeding and I have built on these links. Four papers consider efficient designs for estimation of genetic parameters, including designs for estimating heritability from data on two generations of data, for estimating maternal genetic variances, for estimating parent-offspri ng regression and for estimating multivariate genetic parameters. These designs can lead to substantial reductions in the variances of the estimates of the parameters, compared with classical designs, halving variances in some cases. Other papers have shown how to efficiently estimate heritability from unbalanced data, both from two generations of data and from more than two generations.Often in animal breeding experiments animals used as parents are not selected at random, but selected on phenotypic measurements, perhaps of relatives. This can cause bias in some methods of estimation. On the other hand REML estimates can take account of the selection process. Selection experiments and the estimation of realised heritability are discussed.REML estimation has found widespread acceptance by animal breeders, partly because some quantities arising in the methods were terms that animal breeders use in evaluating animals. It was shown how to improve one method of evaluation and methods of evaluating sires were reviewedSome work is included on multivariate evaluation. It is shown how the complex multivariate calculations can be reduced to simpler univariate calculations using a canonical transformation, how results on selection indices can be used to interpret multivariate predictions. A simple interpretation of quadratic selection indices is given.Other work considered some parallel problems with non-normal data. In particular for binary data, estimation of heritability, optimal designs for estimation of heritability and prediction of breeding values. It was shown how to estimate genotype frequencies using generalised linear model methods and > h? suggested how to evaluate animals worth and estimate genetic parameters when the data fits a generalised linear model.The last group, Group III, is entitled "Experimental studies". These include reports on a long term study of evaluation of breeds and cross-breeding in beef cattle in Zambia. The section also examines the genetic relationship between meat and milk production in British Friesian cattle. The validity of models used in dairy sire evaluation are investigated including the heterogeneity of heritability of milk yield at different levels of production and the use of a novel model for taking account of environmental variation within herds.GROUP I: STATISTICAL METHODS INCLUDING VARIANCE COMPONENT ESTIMATE WITH GENERAL APPLICATION01. R. THOMPSON. 1969. Iterative estimation of variance components for non-orthogonal data. Biometrics 25, 767-773. || 02. H.D. PATTERSON and R. THOMPSON. 1971. Recovery of inter-block information when block sizes are unequal. Biometrika 58, 545-554. || 03. H.D. PATTERSON and R. THOMPSON. 1975. Maximum likelihood estimation of components of variance. Proceedings of the 8th International Biometric Conference. Ed. L.C.A. Corsten and T. Postelnicu, 199-207. || 04. R. THOMPSON. 1980. Maximum likelihood estimation of variance components. Math. Operationsforsh. Statist. ljU 545-561. || 05. R. THOMPSON. 1978. The estimation of variance and covariance components with an application when records are subject to culling. Biometrics 29, 527-550. || 06. L.R. SCHAEFFER, J.W. WILTON and R. THOMPSON. 1978. Simultaneous estimation of variance and covariance components from multitrait mixed model equations. Biometrics 34, 199-208. || 07. D.M. COOPER and R. THOMPSON. 1977 . A note on the estimation of the parameters of the autoregressive-moving average process. Biometrika 64, 625-628. || 08. R. THOMPSON. 1985. A note on restricted maximum likelihood estimation with an alternative outlier model. J.R. Statist. Soc. B 47, 53-55. || 09. R. THOMPSON. 1975. A note on the W transformation. Technometrics J7, 511-512. || 10. R. THOMPSON and K. MEYER. 1986. Estimation of variance components : what is missing in the EM algorithm? J. Statist. Comput. Simul. 24 215-230. || 11. D.L. ROBINSON, R. THOMPSON and P.G.N. DIGBY. REML. 1982. A program for the analysis of non-orthogonal data by restricted maximum likelihood. COMPSTAT 1982, II. Eds. H. Cassinus, P. Ettinger and J.R. Mattieu. Physica-Verlag, Wien 231-232. || 12. R. THOMPSON. 1983. Dial lei crosses, partially balanced incomplete block designs with triangular association schemes and rectangular lattices. GENSTAT newsletter JJ3, 16-32. || 13. R. THOMPSON. 1984. The use of multiple copies of data in forming and interpreting analysis of variance. Experimental design, Statistical Methods and Genetic Statistics. Ed. K. Hinkelmann. Marcel Dekker, New York, 155-174.GROUP II: APPLICATION OF STATISTICAL METHODS TO ANIMAL BREEDING STUDIES14. R. THOMPSON. 1976. The estimation of maternal genetic variances. Biometrics 32 903-917. || 15. R. THOMPSON. 1976. Design of experiments to estimate heritability when observations are available on parents and offspring. Biometrics 32 283-304. || 16. W.G. HILL and R. THOMPSON. 1977. Design of experiments to estimate parent-offspring regression using selected parents. Anim. Prod. 24, 163-168. || 17. N.D. CAMERON and R. THOMPSON. 1986. Design of multivariate selection experiments to estimate genetic parameters. Theor. Appl. Genet. 72, 466-476. || 18. R. THOMPSON. 1977. The estimation of heritability with unbalanced data. I. Observations available on parents and offspring. Biometrics 33, 485-495. || 19. R. THOMPSON. 1977. The estimation of heritability with unbalanced data. II. Data available on more than two generations. Biometrics 33, 495-504. || 20. R. THOMPSON. 1977. The estimation of heritability with unbalanced data. III. Unpublished Appendices, 1-17. || 21. R. THOMPSON. 1976. Estimation of quantitative genetic parameters. Proceedings of the International Conference on Quantitative Genetics. Ed. 0. Kempthorne, E. Pollak and T. Bailey. Iowa State University press, Ames, Iowa, 639-657. (vii) || 22. W.G. HILL and R. THOMPSON. 1978. Probabilities of non-positive definite between group or genetic covariance matrices. Biometrics 34, 429-439. || 23. K. MEYER and R. THOMPSON. 1984. Bias in variance and covariance component estimators due to selection on a correlated trait. Z. Tierzucht. Zuchtungsbiol. 101, 33-50. || 24. R. THOMPSON. 1976. Relationship between the cumulative different and best linear unbiased predictor methods of evaluating bulls. Anim. Prod. 23^, 15-24. || 25. R. THOMPSON. 1979. Sire Evaluation. Biometrics 35, 339-353. || 26. R. THOMPSON. 1986. Estimation of realised heritability in a selected population using mixed model methods. Genet. Sel. Evol . 475-484. || 27. R. THOMPSON. 1972. The maximum likelihood approach to the estimate of liability. Anim. Hum. Genet. 36, 221-231. || 28. R. THOMPSON, B.J. McGUIRK and A.R. GILMOUR. 1985. Estimating the heritability of all-or-none and categorical traits by offspring-parent regression. Z. Tierzucht. Zuchtungsbiol. 102, 342-354. || 29. J.L. FOULLEY, D. GIANOLA and R. THOMPSON. 1983. Prediction of genetic merit from data on binary and quantitative variates with an application to calving difficulty, birth weight and pelvic opening. Genet. Sel. Evol. 15, 401-424. || 30. R. THOMPSON and R.J. BAKER. 1981. Composite link functions in generalised linear models. J.R. Statist. Soc. B. 30, 125-131. || 31. R. THOMPSON. 1980. A note on the estimation of economic values for selection indices. Anim. Prod. 31, 115-117.GROUP III: EXPERIMENTAL STUDIES32. W. THORPE, D.K.R. CRUICKSHANK and R. THOMPSON. 1980. Genetic and evironmental influences on beef cattle production in Zambia. Factors affecting weaner production from Angoni, Barotse and Boran dams. Anim. Prod. 30, 217-234. || 33. W. THORPE, D.K.R. CRUICKSHANK and R. THOMPSON. 1980. Genetic and environmental influences on beef cattle production in Zambia. 2. Sire weights for age of purebred and reciprocally crossbred progeny. Anim. Prod. 30, 235-243. || 34. W. THORPE, D.K.R. CRUICKSHANK and R. THOMPSON. 1980. Genetic and environmental influences on beef cattle production in Zambia. 3. Carcass characteristics of purebred and reciprocally crossbred progeny. Anim. Prod. 30, 245-252. || 35. W. THORPE, D.C.K. CRUICKSHANK and R. THOMPSON. 1982. Genetic and environmental influences on beef cattle in Zambia. 4. Weaner production from purebred and reciprocally crossbred progeny. Anim. Prod. 33, 165-177. || 36. W. THORPE, D.K.R. CRUICKSHANK and R. THOMPSON. 1979. The growth and carcass character!- sti cs of crosses of Hereford and Friesian with Angoni, Barotse and Boran cattle in Zambia. J. Agric. Sci., Camb. 93, 423-430. || 37. I.L. MASON, V.E. VIAL and R. THOMPSON. 1972. Genetic parameters of beef characteristics and the genetic relationship between meat and milk production in British Friesian cattle. Anim. Prod. 135-148. || 38. W.G. HILL, M.R. EDWARDS, M-K A. AHMED and R. THOMPSON. 1983. Heritability of milk yield and composition at different levels and variability of production. Anim. Prod. 36, 59-68. || 39. V.P.S. CHAUHAN and R. THOMPSON. 1986. Dairy sire evaluation using a "rolling months" model. Z. Tierzucht. Zuchtungsbiol 103, 321-333

Modelling techniques for biological systems

The objective of this investigation has been to develop and evaluate techniques which are appropriate to the modelling and simulation of biological reaction system behaviour. The model used as the basis for analysis of modelling and simulation techniques is a reduced version of the biological model proposed by the IAWPRC Task Group for mathematical modell ing in wastewater treatment design. This limited model has the advantage of being easily manageable in terms of analysis and presentation of the simulation techniQues whilst at the same time incorporating a range of features encountered with biological growth applications in general. Because a model may incorporate a number of different components and large number of biological conversion processes, a convenient method of presentation was found to be a matrix format. The matrix representation ensures clarity as to what compounds, processes and react ion terms are to be incorporated and allows easy comparison of different models. In addition, it facilitates transforming the model into a computer program. Simulation of the system response first involves specifying the reactor configuration and flow patterns. With this information fixed, mass balances for each compound in each reactor can be completed. These mass balances constitute a set of simultaneous non-linear differential and algebraic eQuations which, when solved, characterise the system behaviour

Convection in Corrugated Slots

This thesis consists of two parts. The first part deals with the development of proper methodology, i.e. a spectrally accurate algorithm suitable for analysis of convection problems in corrugated slots. The second part is devoted to the study of natural convection in corrugated slots. The algorithm uses the immersed boundary conditions (IBC) concept to deal with the irregular form of the solution domain associated with the presence of corrugated plates. The field equations are discretized on a regular domain surrounding the flow domain using Fourier expansions in the horizontal direction and Chebyshev expansions in the vertical direction. The boundary conditions are expressed in the form of constraints and the spectrally accurate discretization of these constraints has been proposed. The buoyancy forces associated with the temperature difference between isothermal plates drive the natural convection. This temperature difference is expressed in terms of the Rayleigh number Ra and the analysis is limited to its subcritical values where no secondary motion takes place in the absence of corrugation. Corrugations have a sinusoidal form characterized by the wave number a, the upper and lower amplitudes and the phase difference between the upper and lower corrugation systems. They create horizontal temperature gradients which lead to the formation of vertical and horizontal pressure gradients which drive the motion regardless of the intensity of the heating. Presence of corrugations affects the conductive heat flow and creates the convective heat flow. The increase of the heat flow induced by the corrugations has been determined. The convection is qualitatively similar for all Prandtl numbers with the intensity of convection increasing for smaller Pr’s and with the heat transfer augmentation increasing for larger Pr’s

Gradient and relaxation nonlinear techniques for the analysis of cable supported structures

The purpose of this work is to investigate the efficiency of numerical nonlinear solution procedures when applied to the static analysis of cable supported structures. Gradient and relaxation methods are developed and compared with existing nonlinear solution techniques. In order to obtain a more general picture of the performances of the above methods, stiffness methods with Newton Raphson iterative schemes have also been included in the comparative study. Chapter 1 examines the behaviour and characteristics of cable supported structures and investigates the analytical requirements for static analysis. A state of the art of numerical solution techniques used to analyse these structures is presented. An extensive review of published work in relation to the analysis of single unstiffened cables, dual cables and cable networks is also presented. Chapter 2 approaches the solution of the structural problem through total energy formulations. Three basic energy formulations are discussed with particular emphasis given to the total potential energy formulation. The principles of the unconstrained minimization method are considered and different search techniques for approximating the minimum are discussed. Expressions for the gradient vector of the total potential energy are obtained and the tangent stiffness matrix is evaluated as the matrix of the second partial derivatives of the total potential energy formulation. Different scaling techniques are reviewed and the effects of the termination criterion used, for different methods of analysis, on the final accuracy of the methods is also discussed. In Chapter 3 there is an extensive theoretical treatment of gradient methods for the nonlinear solution of structural problems. Particular emphasis is given to the conjugate gradient algorithm and the modifications proposed by various investigators since it first appeared in 1952. A number of one dimensional linear searches are studied which approximate the minimum along the p direction and determine the scalar parameter a for the next iteration. And extensions of the conjugate gradient algorithm for the evaluation of the scalar parameter e, as proposed by Sorenson and Polak and Ribiere are discussed, Finally, the memory gradient method which employs a two dimensional linear search for a simultaneously evaluation of a and e is also presented. Chapter 4 examines the efficiency of the methods discussed in Chapter 3 when applied to the nonlinear solution of a number of test problems. The problems are selected to have varying numbers of degrees of freedom and the respective stiffness matrices to have differing condition numbers in order to study the response of the methods for different structural characteristics. The Fletcher and Reeves method with Davidon's linear search with a cubic equation to approximate the minimum, Stanton's algorithm for bracketing the solution and the regular falsi-bisection algorithm to approximate the minimum, a combined algorithm of Davidon and Stanton's techniques, Buchholdt's method, Polak and Ribiere's algorithm, Sorenson's version, the memory gradient method and a number of linearized conjugate gradient algorithms are developed and their convergence characteristics are compared. The effects of scaling and reinitialization are also studied. In Chapter 5 there is a theoretical investigation of relaxation methods and in particular the dynamic relaxation and the successive overelaxation methods. A rigorous examination of the characteristic properties of dynamic relaxation is carried out. The method is treated as a standard eigenvalue problem for error vectors and expressions for the iteration parameters are developed with respect to the minimum and maximum eigenvalue of the current stiffness matrix. A theoretical comparison of a number of pure iterative methods is performed and relationships between the iteration or scalar parameters of the conjugate gradient method, the dynamic relaxation method, the jacobi semi-iterative method, and the Tchebycheff methods, are established. This suggests that all these methods in fact belong to the same family of methods called "three term recursion formulae". A combined conjugate gradient and Tchebycheff type method is also studied. A method for the automatic evaluation of the dynamic relaxation parameters is developed by the author which can guarantee convergence for almost any arbitrary initial estimate of the minimum and maximum eigenvalues of the current stiffness matrix. The concept of using kinetic energy damping instead of viscous damping in the dynamic relaxation iterative process is also examined. Finally, the successive overelaxation method is modified to be applicable to the nonlinear analysis of structural problems, and two ongoing processes for automatic evaluation of the optimum overelaxation parameter w , proposed by Carre and Hageman, are also examined. Chapter 6 is devoted to a theoretical and numerical investigation of the problem of finding the minimum and maximum eigenvalues of a symmetric matrix. The power method, the steepest descent method, the conjugate gradient method, and the coordinate relaxation method, are among the techniques examined and compared in this Chapter. Several other modifications to the initial conjugate gradient algorithm are also studied, including the modification proposed by Fried for the evaluation of the scalar parameters and the one proposed by Geradin. An orthogonalization process is also applied to alleviate the dependency of the convergence of the method on the initial approximation for the final eigenvector. In Chapter 7 numerical studies of the relaxation methods discussed in Chapter 5 are performed. Alternative forms of the dynamic relaxation methods with an "a priori" evaluation of the iteration parameters (using one of the methods discussed in Chapter 6), with automatic adjustment of the relaxation parameters based on the method developed in Chapter 5, and with the incorporation of kinetic damping, are applied for different test problems. Techniques to avoid the occurrence of instability of the method, when the current maximum eigenvalue of the iteration matrix becomes greater than the estimated maximum eigenvalue, are also developed and compared. Finally, the efficiency of the successive overelaxation method, with both constant and adjustable relaxation parameters is examined and compared with the efficiency of the dynamic relaxation method. In Chapter 8 a review of methods operating through the formulation of the overall stiffness matrix is carried out. The efficiency of these methods is dependent on both the method employed to perform the linear solution when this is necessary and the nonlinear technique used to approximate the nonlinear equilibrium position in each iteration. A compact store elimination scheme, proposed by Jennings, is studied in conjunction with the Gaussian elimination procedure. Three different classes of nonlinear techniques are discussed together with the area in which each one has proved to be more suitable. Chapter 9 performs a general comparative study of the convergence characteristics of the best methods from each classification {gradient, relaxation and stiffness methods}, and examines the advantages and disadvantages involved in the application of the methods to the nonlinear elastic analysis of cable supported structures with members being allowed to slacken. The computer time required to obtain a certain degree of accuracy, the storage requirements and the cost involved are all examined and compared in an effort to select the most suitable method for each particular class of problem. In Chapter 10 the ultimate load carrying capacity of cable structures is studied, with members being allowed to slacken and with the inclusion of nonlinear stress-strain relationships. Two different solution procedures are employed: the stiffness method with or without the compact store elimination scheme in conjunction with Newton Raphson iteration, and Stanton's conjugate gradient algorithm. The convergence of the methods are tested for different values of the termination parameters and load increments. A continuous stress-strain curve as proposed by Jonatowski is used and provision for the cable members to reload following a different path is also included. Finally, Chapter 11 reviews the general conclusions resulting from the experience gained from the theoretical and numerical treatment of the methods discussed in this work, together with suggestions for further research

Gratings: Theory and Numeric Applications, Second Revisited Edition

International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11

Mathematical software for gas transmission networks

This thesis is concerned with the development of numerical software for the simulation of gas transmission networks. This involves developing software for the solution of a large system of stiff differential/algebraic equations (DAE) containing frequent severe disturbances. The disturbances arise due to the varying consumer demands and the operation of network controlling devices such as the compressors. Special strategies are developed to solve the DAE system efficiently using a variable-step integrator. Two sets of strategies are devised; one for the implicit methods such as the semi-implicit Runge-Kutta method, and the other for the linearly implicit Rosenbrock-type method. Four integrators, based on different numerical methods, have been implemented and the performance of each one is compared with the British Gas network analysis program PAN, using a number of large, realistic transmission networks. The results demonstrate that the variable-step integrators are reliable and efficient. An efficient sparse matrix decomposition scheme is developed to solve the large, sparse system of equations that arise during the integration of the DAE system. The decomposition scheme fully exploits the special structure of the coefficient matrix. Lastly, for certain networks, the existing simulation programs fail to compute a feasible solution because of the interactions of the controlling devices in the network. To overcome this difficulty, the problem is formulated as a variational inequality model and solved numerically using an optimization routine from the NAG library (NAGFLIB(l982)). The reliability of the model is illustrated using three test networks

Fast, High-Order Accurate Integral Equation Methods and Application to PDE-Constrained Optimization

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part. Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension. BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169695/1/hszhu_1.pd