Social Exclusion Orderings

We consider the problem of measuring social exclusion using qualitative data. We suggest a class of social exclusion indicators deriving the partial orderings associated with dominace for these indicators. We characterize the set of transformations on the distribution of individual deprivation scores underlying the dominace conditions proposed.Social esclusion, dominance, measures.

A Deep Study of Fuzzy Implications

This thesis contributes a deep study on the extensions of the IMPLY operator in classical binary logic to fuzzy logic, which are called fuzzy implications. After the introduction in Chapter 1 and basic notations about the fuzzy logic operators In Chapter 2 we first characterize In Chapter 3 S- and R- implications and then extensively investigate under which conditions QL-implications satisfy the thirteen fuzzy implication axioms. In Chapter 4 we develop the complete interrelationships between the eight supplementary axioms FI6-FI13 for fuzzy implications satisfying the five basic axioms FI1-FI15. We prove all the dependencies between the eight fuzzy implication axioms, and provide for each independent case a counter-example. The counter-examples provided in this chapter can be used in the applications that need different fuzzy implications satisfying different fuzzy implication axioms. In Chapter 5 we study proper S-, R- and QL-implications for an iterative boolean-like scheme of reasoning from classical binary logic in the frame of fuzzy logic. Namely, repeating antecedents $n$ times, the reasoning result will remain the same. To determine the proper S-, R- and QL-implications we get a full solution of the functional equation $I(x,y)=I(x,I(x,y))$, for all $x$, $y\in[0,1]$. In Chapter 6 we study for the most important t-norms, t-conorms and S-implications their robustness against different perturbations in a fuzzy rule-based system. We define and compare for these fuzzy logical operators the robustness measures against bounded unknown and uniform distributed perturbations respectively. In Chapter 7 we use a fuzzy implication $I$ to define a fuzzy $I$-adjunction in $\mathcal{F}(\mathbb{R}^{n})$. And then we study the conditions under which a fuzzy dilation which is defined from a conjunction $\mathcal{C}$ on the unit interval and a fuzzy erosion which is defined from a fuzzy implication $I^{'}$ to form a fuzzy $I$-adjunction. These conditions are essential in order that the fuzzification of the morphological operations of dilation, erosion, opening and closing obey similar properties as their algebraic counterparts. We find out that the adjointness between the conjunction $\mathcal{C}$ on the unit interval and the implication $I$ or the implication $I^{'}$ play important roles in such conditions

Fitting aggregation operators to data

Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /

Comparison of random variables from a game-theoretic perspective

This work consists of four related parts, divided into eight chapters. A ¯rst part introduces the framework of cycle-transitivity, developed by De Baets et al. It is shown that this framework is ideally suited for describing and compar- ing forms of transitivity of probabilistic relations. Not only does it encompass most already known concepts of transitivity, it is also ideally suited to describe new types of transitivity that are encountered in this work (such as isostochas- tic transitivity and dice-transitivity). The author made many non-trivial and sometimes vital contributions to the development of this framework. A second part consists of the development and study of a new method to compare random variables. This method, which bears the name generalized dice model, was developed by De Meyer et al. and De Schuymer et al., and can be seen as a graded alternative to the well-known concept of ¯rst degree stochastic dominance. A third part involves the determination of the optimal strategies of three game variants that are closely related to the developed comparison scheme. The de¯nitions of these variants di®er from each other solely by the copula that is used to de¯ne the payo® matrix. It turns out however that the characterization of the optimal strategies, done by De Schuymer et al., is completely di®erent for each variant. A last part includes the study of some combinatorial problems that orig- inated from the investigation of the transitivity of probabilistic relations ob- tained by utilizing the developed method to compare random variables. The study, done by De Schuymer et al., includes the introduction of some new and interesting concepts in partition theory and combinatorics. A more thorough discussion, in which each section of this work is taken into account, can be found in the overview at the beginning of this manuscript. Although this work is oriented towards a mathematical audience, the intro- duced concepts are immediately applicable in practical situations. Firstly, the framework of cycle-transitivity provides an easy means to represent and compare obtained probabilistic relations. Secondly, the generalized dice model delivers a useful alternative to the concept of stochastic dominance for comparing random variables. Thirdly, the considered dice games can be viewed in an economical context in which competitors have the same resources and alternatives, and must choose how to distribute these resources over their alternatives. Finally, it must be noted that this work still leaves opportunities for future research. As immediate candidates we see, ¯rstly the investigation of the tran- sitivity of generalized dice models in which the random variables are pairwisely coupled by a di®erent copula. Secondly, the characterization of the transitivity of higher-dimensional dice models, starting with dimension 4. Thirdly, the study of the applicability of the introduced comparison schemes in areas such as mar- ket e±ciency, portfolio selection, risk estimation, capital budgeting, discounted cash °ow analysis, etc

Game Theory (Open Access textbook with 165 solved exercises)

This is an Open Access textbook on non-cooperative Game Theory with 165 solved exercises.Comment: 578 pages, 163 figure

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