Die 2^(k – 1) Typen der ungeordneten Zahlpartitionen mit k Parts (k-Partitionen) werden hier mit Hilfe der geordneten Partitionen von k definiert. Für jeden Typ gibt es eine erzeugende Funktion der geschlossenen Form mit eindeutiger Nummerierung. Die bekannte erzeugende Funktion der k-Partitionen ist die Summe dieser 2^(k – 1) typspezifischen erzeugenden Funktionen. Die Expansion dieser typspezifischen erzeugenden Funktionen in (unendlich lange) Potenzreihen ist rekursiv möglich. Untersucht werden Zerlegungen von erzeugenden Funktionen der einfachen Typen in erzeugende Funktionen anderer Typen. Damit lassen sich Bijektionen zwischen den Partitionen verschiedener Typen aufspüren. Die typspezifischen Betrachtungen werden auf die geordneten Partitionen und auf ihre erzeugenden Funktionen ausgeweitet.

Made available in DSpace on 2016-06-02T20:06:01Z (GMT). No. of bitstreams: 1 1988.pdf: 463519 bytes, checksum: ca45424706e2fdb08c40f42f1f560364 (MD5) Previous issue date: 2008-06-13 / Financiadora de Estudos e Projetos / In survival analysis some studies show a meaningful cure rate after treatment followup, so considering standard survival models can not be appropriate. In this work is extended the long-term survival model proposed by Chen, Ibrahim and Sinha (1999) via generating function of a real sequence introduced by Feller (1967). This new formulation is the uni_cation of the long-term survival models proposed by Rodrigues el al. (2008). Also, as in Rodrigues el al. (2008) it is shown that the long-term survival generating function satis_es the proportional hazard property if only if the number of competing causes related to the occurence of a event of interest follows a Poisson distribution. A real data set is considered to illustrate this approach. / Em análise de sobrevivência, determinados estudos caracterizam-se por apresentar uma fração significativa de sobreviventes, ou seja, pacientes em tratamento que não apresentaram o evento de interesse, mesmo após um longo período de acompanhamento. Assim considerar modelos de sobrevivência usuais, que assumem que a função de sobrevivência converge para zero quando a variável tempo tende a infinito, pode não ser adequado. Nesse trabalho é apresentado uma extensão do modelo proposto por Chen, Ibrahim e Sinha (1999), usando a função geradora de uma sequência de números reais introduzida por Feller (1967). Essa extensão possibilitou o desenvolvimento de uma teoria unificada para os modelos de sobrevivência de longa-duração, Rodrigues et al. (2008). Mostra-se que modelos já existentes na literatura são considerados casos particulares da teoria unificada, por exemplo, o modelo de Berkson & Gage (1952). Também tem-se em Rodrigues et al. (2008), que a função geradora de longa-duração satisfaz a propriedade de risco proporcional se, e somente se, o número de causas competitivas relacionadas a ocorrência do evento de interesse segue uma distribuição de Poisson. Como ilutração utiliza-se um conjunto de dados reais.

Nous étudions, dans la première partie de notre thèse, les polysimplexes d’un espace euclidien de dimension quelconque, c’est-à-dire les objets consistant en une juxtaposition de simplexes réguliers (de tétraèdres si la dimension est 3) accolés le long de leurs faces. Nous étudions principalement le groupe des symétries de ces polysimplexes. Nous présentons une façon de représenter un polysimplexe à l’aide d’un diagramme. Ceci fournit une classification complète des polysimplexes à similitude près. De plus, le groupe des symétries se déduit du groupe des automorphismes du diagramme. Il découle en particulier de notre étude qu’en dimension supérieure à 2, une telle structure ne possède jamais deux faces parallèles et ne contient jamais de circuit fermé de simplexes.<p><p>Dans la seconde partie de notre thèse, nous abordons un problème classique de combinatoire :l’énumération des pavages d’un rectangle mxn à l’aide de dominos. Klarner et Pollack ont montré qu’en fixant m la suite obtenue vérifie une relation de récurrence linéaire à coefficients constants. Nous établissons une nouvelle méthode nous permettant d’obtenir la fonction génératrice correspondante et la calculons pour m <= 16, alors qu’elle n’était connue que pour m <= 10.<p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In this study we propose three new distributions and a study with longitudinal data. The first was the Odd log-logistic normal distribution: theory and applications in analysis of experiments, the second was Odd log-logistic t Student: theory and applications, the third was the Odd log-logistic skew normal: the new distribution skew-bimodal with applications in analysis of experiments and the fourth regression model with random effect of the Odd log-logistic skew normal distribution: an application in longitudinal data. Some have been demonstrated such as symmetry, quantile function, some expansions, ordinary incomplete moments, mean deviation and the moment generating function. The estimation of the model parameters were approached by the method of maximum likelihood. In applications were used regression models to data from a completely randomized design (CRD) or designs completely randomized in blocks (DBC). Thus, the models can be used in practical situations for as a completely randomized designs or completely randomized blocks designs, mainly, with evidence of asymmetry, kurtosis and bimodality. / A distribuição normal é uma das mais importantes na área de estatística. Porém, não é adequada para ajustar dados que apresentam características de assimetria ou de bimodalidade, uma vez que tal distribuição possui apenas os dois primeiros momentos, diferentes de zero, ou seja, a média e o desvio-padrão. Por isso, muitos estudos são realizados com a finalidade de criar novas famílias de distribuições que possam modelar ou a assimetria ou a curtose ou a bimodalidade dos dados. Neste sentido, é importante que estas novas distribuições tenham boas propriedades matemáticas e, também, a distribuição normal como um submodelo. Porém, ainda, são poucas as classes de distribuições que incluem a distribuição normal como um modelo encaixado. Dentre essas propostas destacam-se: a skew-normal, a beta-normal, a Kumarassuamy-normal e a gama-normal. Em 2013 foi proposta a nova família X de distribuições Odd log-logística-G com o objetivo de criar novas distribuições de probabildade. Assim, utilizando as distribuições normal e a skew-normal como função base foram propostas três novas distribuições e um quarto estudo com dados longitudinais. A primeira, foi a distribuição Odd log-logística normal: teoria e aplicações em dados de ensaios experimentais; a segunda foi a distribuição Odd log-logística t Student: teoria e aplicações; a terceira foi a distribuição Odd log-logística skew-bimodal com aplicações em dados de ensaios experimentais e o quarto estudo foi o modelo de regressão com efeito aleatório para a distribuição distribuição Odd log-logística skew-bimodal: uma aplicação em dados longitudinais. Estas distribuições apresentam boas propriedades tais como: assimetria, curtose e bimodalidade. Algumas delas foram demonstradas como: simetria, função quantílica, algumas expansões, os momentos incompletos ordinários, desvios médios e a função geradora de momentos. A flexibilidade das novas distrições foram comparada com os modelos: skew-normal, beta-normal, Kumarassuamy-normal e gama-normal. A estimativas dos parâmetros dos modelos foram obtidas pelo método da máxima verossimilhança. Nas aplicações foram utilizados modelos de regressão para dados provenientes de delineamentos inteiramente casualizados (DIC) ou delineamentos casualizados em blocos (DBC). Além disso, para os novos modelos, foram realizados estudos de simulação para verificar as propriedades assintóticas das estimativas de parâmetros. Para verificar a presença de valores extremos e a qualidade dos ajustes foram propostos os resíduos quantílicos e a análise de sensibilidade. Portanto, os novos modelos estão fundamentados em propriedades matemáticas, estudos de simulação computacional e com aplicações para dados de delineamentos experimentais. Podem ser utilizados em ensaios inteiramente casualizados ou em blocos casualizados, principalmente, com dados que apresentem evidências de assimetria, curtose e bimodalidade.

Die 2^(k – 1) Typen der ungeordneten Zahlpartitionen mit k Parts (k-Partitionen) werden hier mit Hilfe der geordneten Partitionen von k definiert. Für jeden Typ gibt es eine erzeugende Funktion der geschlossenen Form mit eindeutiger Nummerierung. Die bekannte erzeugende Funktion der k-Partitionen ist die Summe dieser 2^(k – 1) typspezifischen erzeugenden Funktionen. Die Expansion dieser typspezifischen erzeugenden Funktionen in (unendlich lange) Potenzreihen ist rekursiv möglich. Untersucht werden Zerlegungen von erzeugenden Funktionen der einfachen Typen in erzeugende Funktionen anderer Typen. Damit lassen sich Bijektionen zwischen den Partitionen verschiedener Typen aufspüren. Die typspezifischen Betrachtungen werden auf die geordneten Partitionen und auf ihre erzeugenden Funktionen ausgeweitet.:1. Kurze Vorbetrachtung 2. Die Typen der ungeordneten k-Partitionen 3. Konstruktion einer typspezifischen GF (generating function) 4. Nummerierung und Symbolik für typspezifische GF’s 5. Die Summe aller typspezifischen GF’s 6. Multiplizieren elementarer Potenzreihen, Erzeugungsformeln 7. Rekursives Expandieren typspezifischer GF’s 8. Zahlen, die in k-Partitionen aller 2^(k – 1) Typen zerlegbar sind 9. Die Konjugierten der typspezifischen k-Partitionen 10. GF-Zerlegungen 10.1 Zerlegung der GF des Typs r = 2 10.2 Zerlegung der GF des Typs r = 3 11. Die typspezifischen GF’s der geordneten Partitionen 12. Literaturverzeichnis 13. Nachwort

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L'objet de cette thèse concerne la proposition de modèles paramétriques des processus de branchement uni et multi-types. Nous mettons en valeur l’intérêt de la théorie des processus de branchement et du développement nécessaire des différents outils et de concepts propres à plusieurs domaines. Pour cela, nous commençons par rappeler quelques définitions et résultats de la théorie des processus de branchement uni et multi-types, et ce en temps discret comme en temps continu. On se consacre par la suite au développement méthodologique de ces modèles.Dans la deuxième partie de ce mémoire, nous étudions seulement l'évolution d’une seule population en temps continu, et présentons quelques familles de lois paramétriques, associées à des processus de branchement homogènes particuliers. Des méthodes récursives de calcul, ainsi que des propriétés pertinentes, concernant ces distributions de probabilité, sont dérivées des fonctions génératrices satisfaisant certaines équations aux dérivées partielles linéaires précisés. Les familles proposées seront utiles à la modélisation de systèmes plus cohérents en dynamique de populations, puisqu'on y montre que les hypothèses usuelles de distributions de Poisson ne peuvent être argumentées.Dans la troisième partie, nous étudions le comportement de l'évolution de plusieurs populations en interactions. Nous y présentons aussi des modèles paramétriques de lois, associés à des processus de branchement multi-types en temps continu et homogènes en temps. Nous considérons ensuite un modèle particulier, où une population ``mère donneuse" autonome alimente en individus K populations filles, qui sont, elles, en interaction. Ce modèle est bien adapté à l'étude des systèmes dynamiques des populations en interaction qui reste à la fois simple, mais riche en variétés de comportement. L'étude du système multi-types se fait via l'évolution des fonctions génératrices de la loi multidimensionnelles des effectifs. Pour cela, utilisant les équations différentielles ordinaires et aux dérivées partielles, nous établissons les équations implicites des distributions temporelles et multidimensionnelles, et discutons des méthodes analytiques ou numériques de leur résolution. Nous développons ensuite des exemples de modèles et en particulier celui concernant 3 et 4 populations.En conclusion, nous argumentons la pertinence de cette approche, et l’interprétation des paramètres, qui sont d'un grand intérêt pour le développement de méthodes d'inférence statistique, pour de nombreux domaines d'applications. / This thesis aims to propose parametric models for single and multi-type branching processes. The importance of the theory of branching processes is pointed out. Hence, developing various tools and specific concepts in several domains is important for applications. For those purpose, we recall some definitions and results of the single-and-multi-type branching processes theory in discrete and continuous case. Afterward, we focus on the methodological development of those models.In the second part, the evolution of a single population in the continuous case has been studied. Then, some parametric distribution families associated to particular branching mechanisms are explored. Recursive computational procedure and relevant properties concerning the associted probability distributions are derived from generating functions that satisfy specified linear partial differential equations. The suggested families are useful for the modeling of systems that are more coherent with population dynamics, contrarily to the usual hypothesis of Poisson distributions, that cannot be argued.In the third part, the evolution of different populations with interaction is explored. Similarly, some parametric models of homogeneous multi-type branching processes in continuous time are proposed. Afterwards, we consider a particular model where an autonomous donor parent population feeds in individuals, K types progeny populations that interacts. This model is well adapted to the study of dynamical systems of populations in interaction. This simple model, but has a rich variety of behaviors.The study of such systems is also done regarding the evolution of generating functions of multidimensional ndividual countrings. To achievea such study, ordinary and partial differential equations are used to establish the implicit equations of temporal and multidimensional distributions. Analytical and numerical methods for equation resolution are then discussed, and examples of particular models are developed.In conclusion, the relevancy of this approach is argumed, censidering parameters interpretation in the development of inference methods for the various applied domains.

How to quantify security is a classic question in the security community that until today has had no plausible answer. Unfortunately, current security evaluation models are often either quantitative but too specific (i.e., applicability is limited), or comprehensive (i.e., system-level) but qualitative. The importance of quantifying security cannot be overstated, but doing so is difficult and complex, for many reason: the “physics” of the amount of security is ambiguous; the operational state is defined by two confronting parties; protecting and breaking systems is a cross-disciplinary mechanism; security is achieved by comparable security strength and breakable by the weakest link; and the human factor is unavoidable, among others. Thus, security engineers face great challenges in defending the principles of information security and privacy. This thesis addresses model-based system-level security quantification and argues that properly addressing the quantification problem of security first requires a paradigm shift in security modeling, addressing the problem at the abstraction level of what defines a computing system and failure model, before any system-level analysis can be established. Consequently, we present a candidate computing systems abstraction and failure model, then propose two failure-centric model-based quantification approaches, each including a bounding system model, performance measures, and evaluation techniques. The first approach addresses the problem considering the set of controls. To bound and build the logical network of a security system, we extend our original work on the Information Security Maturity Model (ISMM) with Reliability Block Diagrams (RBDs), state vectors, and structure functions from reliability engineering. We then present two different groups of evaluation methods. The first mainly addresses binary systems, by extending minimal path sets, minimal cut sets, and reliability analysis based on both random events and random variables. The second group addresses multi-state security systems with multiple performance measures, by extending Multi-state Systems (MSSs) representation and the Universal Generating Function (UGF) method. The second approach addresses the quantification problem when the two sets of a computing system, i.e., assets and controls, are considered. We adopt a graph-theoretic approach using Bayesian Networks (BNs) to build an asset-control graph as the candidate bounding system model, then demonstrate its application in a novel risk assessment method with various diagnosis and prediction inferences. This work, however, is multidisciplinary, involving foundations from many fields, including security engineering; maturity models; dependability theory, particularly reliability engineering; graph theory, particularly BNs; and probability and stochastic models.