A compreensão da dependência espacial das propriedades do solo vem sendo cada vez mais requerida por pesquisadores que objetivam melhorar a interpretação dos resultados de experimentos de campo fornecendo, assim, subsídios para novas pesquisas a custos reduzidos. Em geral, variáveis como, por exemplo, o teor de cálcio no solo, estudado neste trabalho, apresentam grande variabilidade impossibilitando, na maioria das vezes, a detecção de reais diferenças estatísticas entre os efeitos de tratamentos. A consideração de amostras georreferenciadas é uma abordagem importante na análise de dados desta natureza, uma vez que amostras mais próximas são mais similares do que as mais distantes e, assim, cada realização desta variável contém informação de sua vizinhança. Neste trabalho, métodos geoestatísticos que baseiam-se na modelagem da dependência espacial, nas pressuposições Gaussianas e nos estimadores de máxima verossimilhança são utilizados para analisar e interpretar a variabilidade do teor de cálcio no solo, resultado de um experimento realizado na Fazenda Angra localizada no Estado do Rio de Janeiro. A área experimental foi dividida em três regiões em função dos diferentes períodos de adubação realizadas. Neste estudo foram utilizados dados do teor de cálcio obtidos das camadas 0-20cm e 20-40cm do solo, de acordo com as coordenadas norte e leste. Modelos lineares mistos, apropriados para estudar dados com esta característica, e que permitem a utilização de diferentes estruturas de covariâncias e a incorporação da região e tendência linear das coordenadas foram usados. As estruturas de covariâncias utilizadas foram: a exponencial e a Matérn. Para estimar e avaliar a variabilidade dos parâmetros utilizaram-se os métodos de máxima verossimilhança, máxima verossimilhança restrita e o perfil de verossimilhança. A identificação da dependência e a predição foram realizadas por meio de variogramas e mapas de krigagem. Além disso, a seleção do modelo adequado foi feita pelo critério de informação de Akaike e o teste da razão de verossimilhanças. Observou-se, quando utilizado o método de máxima verossimilhança, o melhor modelo foi aquele com a covariável região e, com o método de máxima verossimilhança restrita, o modelo com a covariável região e tendência linear nas coordenadas (modelo 2). Com o teor de cálcio, na camada 0-20cm e considerando-se a estrutura de covariância exponencial foram obtidas as menores variâncias nugget e a maior variância espacial (sill - nugget). Com o método de máxima verossimilhança e com o modelo 2 foram observadas variâncias de predição mais precisas. Por meio do perfil de verossimilhança pode-se observar menor variabilidade dos parâmetros dos variogramas ajustados com o modelo 2. Utilizando-se vários modelos e estruturas de covariâncias, deve-se ser criterioso, pois a precisão das estimativas, depende do tamanho da amostra e da especificação do modelo para a média. Os resultados obtidos foram analisados, com a subrotina geoR desenvolvida por Ribeiro Junior & Diggle (2000), e por meio dela pode-se obter estimativas confiáveis para os parâmetros dos diferentes modelos estimados. / The understanding of the spatial dependence of the properties of the soil becomes more and more required by researchers that attempt to improve the interpretation of the results of field experiments supplying subsidies for new researches at reduced costs. In general, variables as, for example, the calcium content in the soil, studied in this work, present great variability disabling, most of the time, the detection of real statistical differences among the treatment effects. The consideration of georeferenced samples is an important approach in the analysis of data of this nature, because closer samples are more similar than the most distant ones and, thus, each realization of this variable contains information of its neighborhood. In this work, geostatistics methods that are based on the modeling of the spatial dependence, under the Gaussian assumptions and the maximum likelihood estimators, are used to analyze and to interpret the variability of calcium content in the soil, obtained from an experiment carried on at Fazenda Angra, located in Rio de Janeiro, Brazil. The experimental area was divided in three areas depending on the different periods of fertilization. In this study, data of the calcium soil content from the layers 0-20cm and 20-40cm, were used, according to the north and east coordinates. Mixed linear models, ideal to study data with this characteristic, and that allow the use of different covariance structures, and the incorporation of the region and linear tendency of the coordinates, were used. The covariance structures were: the exponential and the Matérn. Maximum likelihood, maximum restricted likelihood and the profile of likelihood methods were used to estimate and to evaluate the variability of the parameters. The identification of the dependence and the prediction were realized using variograms and krigging maps. Besides, the selection of the appropriate model was made through the Akaike information criterion and the likelihood ratio test. It was observed that when maximum likelihood method was used the most appropriate model was that with the region covariate and, with the maximum restricted likelihood method, the best model was the one with the region covariate and linear tendency in the coordinates (model 2). With the calcium content, in the layer 0-20cm and considering the exponential covariance structure, the smallest nugget variances and the largest spatial variance (sill - nugget) were obtained. With the maximum likelihood method and with the model 2 more precise prediction variances were observed. Through the profile of likelihood method, smaller variability of the adjusted variogram parameters can be observed with the model 2. With several models and covariance structures being used, one should be very critical, because the precision of the estimates depends on the size of the sample and on the specification of the model for the average. The obtained results were analyzed, with the subroutine geoR developed by Ribeiro Junior & Diggle (2000), and through this subroutine, reliable estimates for the parameters of the different estimated models can be obtained.

published_or_final_version / Statistics / Master / Master of Philosophy

A compreensão da dependência espacial das propriedades do solo vem sendo cada vez mais requerida por pesquisadores que objetivam melhorar a interpretação dos resultados de experimentos de campo fornecendo, assim, subsídios para novas pesquisas a custos reduzidos. Em geral, variáveis como, por exemplo, o teor de cálcio no solo, estudado neste trabalho, apresentam grande variabilidade impossibilitando, na maioria das vezes, a detecção de reais diferenças estatísticas entre os efeitos de tratamentos. A consideração de amostras georreferenciadas é uma abordagem importante na análise de dados desta natureza, uma vez que amostras mais próximas são mais similares do que as mais distantes e, assim, cada realização desta variável contém informação de sua vizinhança. Neste trabalho, métodos geoestatísticos que baseiam-se na modelagem da dependência espacial, nas pressuposições Gaussianas e nos estimadores de máxima verossimilhança são utilizados para analisar e interpretar a variabilidade do teor de cálcio no solo, resultado de um experimento realizado na Fazenda Angra localizada no Estado do Rio de Janeiro. A área experimental foi dividida em três regiões em função dos diferentes períodos de adubação realizadas. Neste estudo foram utilizados dados do teor de cálcio obtidos das camadas 0-20cm e 20-40cm do solo, de acordo com as coordenadas norte e leste. Modelos lineares mistos, apropriados para estudar dados com esta característica, e que permitem a utilização de diferentes estruturas de covariâncias e a incorporação da região e tendência linear das coordenadas foram usados. As estruturas de covariâncias utilizadas foram: a exponencial e a Matérn. Para estimar e avaliar a variabilidade dos parâmetros utilizaram-se os métodos de máxima verossimilhança, máxima verossimilhança restrita e o perfil de verossimilhança. A identificação da dependência e a predição foram realizadas por meio de variogramas e mapas de krigagem. Além disso, a seleção do modelo adequado foi feita pelo critério de informação de Akaike e o teste da razão de verossimilhanças. Observou-se, quando utilizado o método de máxima verossimilhança, o melhor modelo foi aquele com a covariável região e, com o método de máxima verossimilhança restrita, o modelo com a covariável região e tendência linear nas coordenadas (modelo 2). Com o teor de cálcio, na camada 0-20cm e considerando-se a estrutura de covariância exponencial foram obtidas as menores variâncias nugget e a maior variância espacial (sill – nugget). Com o método de máxima verossimilhança e com o modelo 2 foram observadas variâncias de predição mais precisas. Por meio do perfil de verossimilhança pode-se observar menor variabilidade dos parâmetros dos variogramas ajustados com o modelo 2. Utilizando-se vários modelos e estruturas de covariâncias, deve-se ser criterioso, pois a precisão das estimativas, depende do tamanho da amostra e da especificação do modelo para a média. Os resultados obtidos foram analisados, com a subrotina geoR desenvolvida por Ribeiro Junior & Diggle (2000), e por meio dela pode-se obter estimativas confiáveis para os parâmetros dos diferentes modelos estimados. / The understanding of the spatial dependence of the properties of the soil becomes more and more required by researchers that attempt to improve the interpretation of the results of field experiments supplying subsidies for new researches at reduced costs. In general, variables as, for example, the calcium content in the soil, studied in this work, present great variability disabling, most of the time, the detection of real statistical differences among the treatment effects. The consideration of georeferenced samples is an important approach in the analysis of data of this nature, because closer samples are more similar than the most distant ones and, thus, each realization of this variable contains information of its neighborhood. In this work, geostatistics methods that are based on the modeling of the spatial dependence, under the Gaussian assumptions and the maximum likelihood estimators, are used to analyze and to interpret the variability of calcium content in the soil, obtained from an experiment carried on at Fazenda Angra, located in Rio de Janeiro, Brazil. The experimental area was divided in three areas depending on the different periods of fertilization. In this study, data of the calcium soil content from the layers 0-20cm and 20-40cm, were used, according to the north and east coordinates. Mixed linear models, ideal to study data with this characteristic, and that allow the use of different covariance structures, and the incorporation of the region and linear tendency of the coordinates, were used. The covariance structures were: the exponential and the Matérn. Maximum likelihood, maximum restricted likelihood and the profile of likelihood methods were used to estimate and to evaluate the variability of the parameters. The identification of the dependence and the prediction were realized using variograms and krigging maps. Besides, the selection of the appropriate model was made through the Akaike information criterion and the likelihood ratio test. It was observed that when maximum likelihood method was used the most appropriate model was that with the region covariate and, with the maximum restricted likelihood method, the best model was the one with the region covariate and linear tendency in the coordinates (model 2). With the calcium content, in the layer 0-20cm and considering the exponential covariance structure, the smallest nugget variances and the largest spatial variance (sill - nugget) were obtained. With the maximum likelihood method and with the model 2 more precise prediction variances were observed. Through the profile of likelihood method, smaller variability of the adjusted variogram parameters can be observed with the model 2. With several models and covariance structures being used, one should be very critical, because the precision of the estimates depends on the size of the sample and on the specification of the model for the average. The obtained results were analyzed, with the subroutine geoR developed by Ribeiro Junior & Diggle (2000), and through this subroutine, reliable estimates for the parameters of the different estimated models can be obtained.

In this thesis, we study robust combinatorial problems with interval data. We introduce several new measures of robustness in response to the drawbacks of existing measures of robustness. The idea of these new measures is to ensure that the solutions are satisfactory for the decision maker in all scenarios, including the worst case scenario. Therefore, we have introduced a threshold over the worst case costs, in which above this threshold, solutions are no longer satisfactory for the decision maker. It is, however, important to consider other criteria than just the worst case.<p>Therefore, in each of these new measures, a second criteria is used to evaluate the performance of the solution in other scenarios such as the best case one. <p><p>We also study the robust deviation p-elements problem. In fact, we study when this solution is equal to the optimal solution in the scenario where the cost of each element is the midpoint of its corresponding interval. <p><p>Then, we finally formulate the robust combinatorial problem with interval data as a bicriteria problem. We also integrate the decision maker's preferences over certain types of solutions into the model. We propose a method that uses these preferences to find the set of solutions that are never preferred by any other solution. We call this set the final set. <p><p>We study the properties of the final sets from a coherence point of view and from a robust point of view. From a coherence point of view, we study necessary and sufficient conditions for the final set to be monotonic, for the corresponding preferences to be without cycles, and for the set to be stable.<p>Those that do not satisfy these properties are eliminated since we believe these properties to be essential. We also study other properties such as the transitivity of the preference and indifference relations and more. We note that many of our final sets are included in one another and some are even intersections of other final sets. From a robust point of view, we compare our final sets with different measures of robustness and with the first- and second-degree stochastic dominance. We show which sets contain all of these solutions and which only contain these types of solutions. Therefore, when the decision maker chooses his preferences to find the final set, he knows what types of solutions may or may not be in the set.<p><p>Lastly, we implement this method and apply it to the Robust Shortest Path Problem. We look at how this method performs using different types of randomly generated instances. <p> / Doctorat en sciences, Orientation recherche opérationnelle / info:eu-repo/semantics/nonPublished

This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts.<p><p>The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile log-likelihood function for skewness admits a stationary point whatever the sample under consideration. Since that moment, researchers have tried to determine the subclasses of skew-symmetric distributions who suffer from each of those problems, which has led to the aforementioned two conjectures. This thesis completely solves these two problems.<p><p>The second part of the thesis, namely Chapters 4 and 5, aims at applying and constructing extremely general skewing mechanisms. As such, in Chapter 4, we make use of the univariate mechanism of Ferreira and Steel (2006) to build optimal (in the Le Cam sense) tests for univariate symmetry which are very flexible. Actually, their mechanism allowing to turn a given symmetric distribution into any asymmetric distribution, the alternatives to the null hypothesis of symmetry can take any possible shape. These univariate mechanisms, besides that surjectivity property, enjoy numerous good properties, but cannot be extended to higher dimensions in a satisfactory way. For this reason, we propose in Chapter 5 different general mechanisms, sharing all the nice properties of their competitors in Ferreira and Steel (2006), but which moreover can be extended to any dimension. We formally prove that the surjectivity property holds in dimensions k>1 and we study the principal characteristics of these new multivariate mechanisms.<p><p>Finally, the third part of this thesis, composed of Chapter 6, proposes a test for multivariate central symmetry by having recourse to the concepts of statistical depth and runs. This test extends the celebrated univariate runs test of McWilliams (1990) to higher dimensions. We analyze its asymptotic behavior (especially in dimension k=2) under the null hypothesis and its invariance and robustness properties. We conclude by an overview of possible modifications of these new tests./<p><p>Cette thèse traite de différents aspects statistiques et probabilistes de symétrie et asymétrie univariées et multivariées, et est subdivisée en trois parties distinctes.<p><p>La première partie, qui comprend les chapitres 1, 2 et 3 de la thèse, est destinée à la résolution de deux conjectures associées aux lois skew-symétriques multivariées. Depuis l'introduction en 1985 par Adelchi Azzalini du plus célèbre représentant de cette classe de lois, à savoir la loi skew-normale, il est bien connu qu'en un voisinage de la situation symétrique la matrice d'information de Fisher est singulière et la fonction de vraisemblance profile pour le paramètre d'asymétrie admet un point stationnaire quel que soit l'échantillon considéré. Dès lors, des chercheurs ont essayé de déterminer les sous-classes de lois skew-symétriques qui souffrent de chacune de ces problématiques, ce qui a mené aux deux conjectures précitées. Cette thèse résoud complètement ces deux problèmes.<p><p>La deuxième partie, constituée des chapitres 4 et 5, poursuit le but d'appliquer et de proposer des méchanismes d'asymétrisation très généraux. Ainsi, au chapitre 4, nous utilisons le méchanisme univarié de Ferreira and Steel (2006) pour construire des tests de symétrie univariée optimaux (au sens de Le Cam) qui sont très flexibles. En effet, leur méchanisme permettant de transformer une loi symétrique donnée en n'importe quelle loi asymétrique, les contre-hypothèses à la symétrie peuvent prendre toute forme imaginable. Ces méchanismes univariés, outre cette propriété de surjectivité, possèdent de nombreux autres attraits, mais ne permettent pas une extension satisfaisante aux dimensions supérieures. Pour cette raison, nous proposons au chapitre 5 des méchanismes généraux alternatifs, qui partagent toutes les propriétés de leurs compétiteurs de Ferreira and Steel (2006), mais qui en plus sont généralisables à n'importe quelle dimension. Nous démontrons formellement que la surjectivité tient en dimension k > 1 et étudions les caractéristiques principales de ces nouveaux méchanismes multivariés.<p><p>Finalement, la troisième partie de cette thèse, composée du chapitre 6, propose un test de symétrie centrale multivariée en ayant recours aux concepts de profondeur statistique et de runs. Ce test étend le célèbre test de runs univarié de McWilliams (1990) aux dimensions supérieures. Nous en analysons le comportement asymptotique (surtout en dimension k = 2) sous l'hypothèse nulle et les propriétés d'invariance et de robustesse. Nous concluons par un aperçu sur des modifications possibles de ces nouveaux tests. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Dans ce travail, nous proposons des procédures de test paramétriques et nonparamétrique localement et asymptotiquement optimales au sens de Hajek et Le Cam, pour trois modèles d'asymétrie. <p>La construction de modèles d'asymétrie est un sujet de recherche qui a connu un grand développement ces dernières années, et l'obtention des tests optimaux (pour trois modèles différents) est une étape essentielle en vue de leur mise en application. <p>Notre approche est fondée sur la théorie de Le Cam d'une part, pour obtenir les propriétés de normalité asymptotique, bases de la construction des tests paramétriques optimaux, et la théorie de Hajek d'autre part, qui, via un principe d'invariance permet d'obtenir les procédures non-paramétriques.<p><p>Nous considérons dans ce travail deux classes de distributions univariées asymétriques, l'une fondée sur un développement d'Edgeworth (décrit dans le Chapitre 1), et l'autre construite en utilisant un paramètre d'échelle différent pour les valeurs positives et négatives (le modèle de Fechner, décrit dans le Chapitre 2).<p>Le modèle d'asymétrie elliptique étudié dans le dernier chapitre est une généralisation multivariée du modèle du Chapitre 2.<p>Pour chacun de ces modèles, nous proposons de tester l'hypothèse de symétrie par rapport à un centre fixé, puis par rapport à un centre non spécifié.<p><p>Après avoir décrit le modèle pour lequel nous construisons les procédures optimales, nous obtenons la propriété de normalité locale asymptotique. A partir de ce résultat, nous sommes capable de construire les tests paramétriques localement et asymptotiquement optimaux. Ces tests ne sont toutefois valides que si la densité sous-jacente f est correctement spécifiée. Ils ont donc le mérite de déterminer les bornes d'efficacité paramétrique, mais sont difficilement applicables. <p>Nous adaptons donc ces tests afin de pouvoir tester les hypothèses de symétrie par rapport à un centre fixé ou non, lorsque la densité sous-jacente est considérée comme un paramètre de nuisance. <p>Les tests que nous obtenons restent localement et asymptotiquement optimaux sous f, mais restent valides sous une large classe de densités. <p><p>A partir des propriétés d'invariance du sous-modèle identifié par l'hypothèse nulle, nous obtenons les tests de rangs signés localement et asymptotiquement optimaux sous f, et valide sous une vaste classe de densité. Nous présentons en particulier, les tests fondés sur les scores normaux (ou tests de van der Waerden), qui sont optimaux sous des hypothèses Gaussiennes, tout en étant valides si cette hypothèse n'est pas vérifiée.<p>Afin de comparer les performances des tests paramétriques et non paramétriques présentés, nous calculons les efficacités asymptotiques relatives des tests non paramétriques par rapport aux tests pseudo-Gaussiens, sous une vaste classe de densités non-Gaussiennes, et nous proposons quelques simulations. / Doctorat en sciences, Orientation statistique / info:eu-repo/semantics/nonPublished