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Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Various properties of integer partitions are studied in this work, in particular the number of summands, the number of ascents and the multiplicities of parts. We work on random partitions, where all partitions from a certain family are equally likely, and determine moments and limiting distributions of the different parameters. The thesis focuses on three main problems: the first of these problems is concerned with the length of prime partitions (i.e., partitions whose parts are all prime numbers), in particular restricted partitions (i.e., partitions where all parts are distinct). We prove a central limit theorem for this parameter and obtain very precise asymptotic formulas for the mean and variance. The second main focus is on the distribution of the number of parts of a given multiplicity, where we obtain a very interesting phase transition from a Gaussian distribution to a Poisson distribution and further to a degenerate distribution, not only in the classical case, but in the more general context of ⋋-partitions: partitions where all the summands have to be elements of a given sequence ⋋ of integers. Finally, we look into another phase transition from restricted to unrestricted partitions (and from Gaussian to Gumbel-distribution) as we study the number of summands in partitions with bounded multiplicities. / AFRIKAANSE OPSOMMING: Verskillende eienskappe van heelgetal-partisies word in hierdie tesis bestudeer, in die besonder die aantal terme, die aantal stygings en die veelvoudighede van terme. Ons werk met stogastiese partisies, waar al die partisies in ’n sekere familie ewekansig is, en ons bepaal momente en limietverdelings van die verskillende parameters. Die teses fokusseer op drie hoofprobleme: die eerste van hierdie probleme gaan oor die lengte van priemgetal-partisies (d.w.s., partisies waar al die terme priemgetalle is), in die besonder beperkte partisies (d.w.s., partisies waar al die terme verskillend is). Ons bewys ’n sentrale limietstelling vir hierdie parameter en verkry baie presiese asimptotiese formules vir die gemiddelde en die variansie. Die tweede hooffokus is op die verdeling van die aantal terme van ’n gegewe veelvoudigheid, waar ons ’n baie interessante fase-oorgang van ’n normaalverdeling na ’n Poisson-verdeling en verder na ’n ontaarde verdeling verkry, nie net in die klassieke geval nie, maar ook in die meer algemene konteks van sogenaamde ⋋-partities: partisies waar al die terme elemente van ’n gegewe ry ⋋ van heelgetalle moet wees.

Neste trabalho, alguns resultados, tais como, função geradora de momentos, relações de recorrência para os momentos e alguns teoremas da classe de distribuições em séries de potencias modificadas (MPSD) proposta por Gupta (1974) e da classe de distribuições em séries de potências modificadas inflacionadas (IMPSD) tanto em um ponto diferente de zero como no ponto zero são apresentados. Uma aplicação do Modelo Poisson padrão, do modelo binomial negativo padrão e dos modelos inflacionados de zeros para dados de contagem, ZIP e ZINB, utilizando-se as técnicas dos MLGs, foi realizada para dois conjuntos de dados reais juntamente com o gráfico normal de probabilidade com envelopes simulados. Também foi proposta a distribuição Weibull binomial negativa (WNB) que é bastante flexível em análise de dados positivos e foram estudadas algumas de suas propriedades matemáticas. Esta é uma importante alternativa para os modelos Weibull e Weibull geométrica, sub-modelos da WNB. A demostração de que a densidade da distribuição Weibull binomial negativa pode ser expressa como uma mistura de densidades Weibull é apresentada. Fornecem-se, também, seus momentos, função geradora de momentos, gráficos da assimetria e curtose, expressoes expl´citas para os desvios médios, curvas de Bonferroni e Lorenz, função quantílica, confiabilidade e entropia, a densidade da estat´stica de ordem e expressões explícita para os momentos da estatística de ordem. O método de máxima verossimilhança é usado para estimar os parametros do modelo. A matriz de informação esperada ´e derivada. A utilidade da distribuição WNB está ilustrada na an´alise de dois conjuntos de dados reais. / In this paper, some result such as moments generating function, recurrence relations for moments and some theorems of the class of modified power series distributions (MPSD) proposed by Gupta (1974) and of the class of inflated modified power series distributions (IMPSD) both at a different point of zero as the zero point are presented. The standard Poisson model, the standard negative binomial model and zero inflated models for count data, ZIP and ZINB, using the techniques of the GLMs, were used to analyse two real data sets together with the normal plot with simulated envelopes. The new distribution Weibull negative binomial (WNB) was proposed. Some mathematical properties of the WNB distribution which is quite flexible in analyzing positive data were studied. It is an important alternative model to the Weibull, and Weibull geometric distributions as they are sub-models of WNB. We demonstrate that the WNB density can be expressed as a mixture of Weibull densities. We provide their moments, moment generating function, plots of the skewness and kurtosis, explicit expressions for the mean deviations, Bonferroni and Lorenz curves, quantile function, reliability and entropy, the density of order statistics and explicit expressions for the moments of order statistics. The method of maximum likelihood is used for estimating the model parameters. The expected information matrix is derived. The usefulness of the new distribution is illustrated in two analysis of real data sets.

The environmental conditions driving the microbial community dynamics in crop soils remain unclear. Here, we focused on the spatial and temporal dynamics of microbial communities in soils cultivated with sugarcane under different soil managements, during two years. Our work was divided into three essential parts, where i) we discuss ecological models and theories for the microbial exploration in crop soils, arguing that those ecological models, which partitioned the microbial communities, may increase the resolution of the environmental and the microbial interactions; ii) we developed a probabilistic model based on the occurrence frequency of microorganisms across systems identifying the core microbial community. The model is based on the Poisson distribution, and it was tested in four datasets available in the Earth Microbiome Project; iii) we identified the core bacterial and fungal communities across soils cultivated with sugarcane, verifying which abiotic components could drive the composition of groups. We increased the resolution of the environmental and the microbial interactions, showing that the core and the variable microbial communities are driven by distinct abiotic components. We also observed that the core and variable microbial communities harbor distinct potential functionality, as nitrogen fixation being more predicted to the core bacterial commmunity, and nitrification process for the variable bacterial community. Our finds increase the knowledge of microbial dynamics and functionality, helping to reveal and explore the crop system microbiome. / As condições ambientais que podem modular a dinâmica da comunidade microbiana em solos de culturas são pouco conhecidas. O presente trabalho foi dividido em três partes essenciais, onde i) discutiu-se modelos e teorias ecológicas para a exploração microbiana em solo agrícolas, argumentando-se que os modelos ecológicos que particionam as comunidades microbianas, poderiam aumentar a resolução entre interações microbianas e o ambiente, ii) desenvolveu-se um modelo probabilístico baseado na freqüência de ocorrência de microorganismos através de sistema identificando a comunidade microbiana \"core\". O modelo baseou-se na distribuição de Poisson, sendo este testado em quatro conjuntos de dados disponíveis no Projeto \"Earth Microbiome\", e iii) identificou-se as comunidades bacterianas e fúngicas core em solos cultivados com cana-de-açúcar, verificando-se quais componentes abióticos poderiam modular a composição dos grupos. Com isso, elevou-se a resolução das interações ambiental e microbiana, indicando que o core microbiano e as comunidades microbianas variáveis são moduladas por componentes abióticos distintos. Observou-se também que as comunidades core e variável possuem funcionalidade potencial distinta, como fixação de nitrogênio mais predita para o core bacteriano e processo de nitrificação para a comunidade variável de bactérias. Os resultados do presente trabalho elevam o conhecimento da dinâmica e funcionalidade microbiana, ajudando a revelar e explorar o microbioma do sistema de cultivo.

by Kwok-Fai Mo. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 74-79). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Statistical Modeling --- p.5 / Chapter 2.1 --- Poisson Regression --- p.5 / Chapter 2.2 --- Overdispersion and Autorrelation --- p.7 / Chapter 2.3 --- Generalized Estimating Equation --- p.9 / Chapter 2.4 --- Zeger's Mehthod --- p.12 / Chapter 2.5 --- Multicollinearity --- p.18 / Chapter 2.5.1 --- The Modified Generalized Estimating Equation --- p.18 / Chapter 2.6 --- Bootstrapping Method --- p.21 / Chapter 2.7 --- The Bootstrap Choice of Ridge Parameter --- p.23 / Chapter 3 --- The Robustness of Zeger's Approach to the Specification of ηt - Simulation Study --- p.26 / Chapter 3.1 --- Introduction --- p.26 / Chapter 3.2 --- Zeger's Algorithm with Varoious Time Series Data --- p.27 / Chapter 3.2.1 --- Data without Multicollinearity --- p.27 / Chapter 3.2.1 --- Data with Multicollinearity --- p.34 / Chapter 3.3 --- Modified Generalized Estimating Equation Approach --- p.40 / Chapter 3.3 --- The Choice of Ridge Paramter in Bootstrap --- p.42 / Chapter 4 --- Real Example --- p.46 / Chapter 4.1 --- Data Structure --- p.46 / Chapter 4.2 --- Model Building --- p.49 / Chapter 4.3 --- Single Pollutant Model --- p.57 / Chapter 4.4 --- Multiple Pollutant Model --- p.62 / Chapter 5 --- Conclusion and Discussion --- p.64 / Appendix --- p.69 / References --- p.74

Este trabalho apresenta uma análise estatística de contagens multivariadas proveniente de várias populações através de modelos de regressão. Foram considerados casos onde os vetores respostas obedeçam às distribuições Poisson multivariada e Poisson log-normal multivariada. Esta distribuição admite correlação de ambos sinais entre componentes do vetor resposta, enquanto que as distribuições mais usuais para dados de contagens (como a Poisson multivariada) admitem apenas correlação positiva entre as componentes do vetor resposta. São discutidos métodos de estimação e testes de hipóteses sobre os parâmetros do modelo para o caso bivariado. Estes modelos de regressão foram aplicados a um conjunto de dados referentes a contagens de dois tipos de defeitos em 100 gramas de fibras têxteis de quatro máquinas craqueadeiras, sendo duas de um fabricante e as outras de um segundo fabricante. Os resultados obtidos nos diferentes modelos de regressão foram comparados. Para estudar o comportamento das estimativas dos parâmetros de uma distribuição Poisson Log-Normal, amostras foram simuladas segundo esta distribuição. / Regression models are presented to analyse multivariate counts from many populations. Due to the random vector characteristic, we consider two classes of probability models: Multivariate Poisson distribution and Multivariate Poisson Log-Normal distribution. The last distribution admits negative and positive correlations between two components of a random vector under study, while other distributions (as Multivariate Poisson) admit only positive correlation. Estimation methods and test of hypothese on the parameters in bivariate case are discussed. The proposed techniques are illustrated by numerical examples, considering counts of two types of defects in 100g of textile fibers produced by four machines, two from one manufacturer and the other two from another one. The results from different regression models are compared. The empirical distribution of Poisson Log-Normal parameter estimations are studied by simulated samples.

The environmental conditions driving the microbial community dynamics in crop soils remain unclear. Here, we focused on the spatial and temporal dynamics of microbial communities in soils cultivated with sugarcane under different soil managements, during two years. Our work was divided into three essential parts, where i) we discuss ecological models and theories for the microbial exploration in crop soils, arguing that those ecological models, which partitioned the microbial communities, may increase the resolution of the environmental and the microbial interactions; ii) we developed a probabilistic model based on the occurrence frequency of microorganisms across systems identifying the core microbial community. The model is based on the Poisson distribution, and it was tested in four datasets available in the Earth Microbiome Project; iii) we identified the core bacterial and fungal communities across soils cultivated with sugarcane, verifying which abiotic components could drive the composition of groups. We increased the resolution of the environmental and the microbial interactions, showing that the core and the variable microbial communities are driven by distinct abiotic components. We also observed that the core and variable microbial communities harbor distinct potential functionality, as nitrogen fixation being more predicted to the core bacterial commmunity, and nitrification process for the variable bacterial community. Our finds increase the knowledge of microbial dynamics and functionality, helping to reveal and explore the crop system microbiome. / As condições ambientais que podem modular a dinâmica da comunidade microbiana em solos de culturas são pouco conhecidas. O presente trabalho foi dividido em três partes essenciais, onde i) discutiu-se modelos e teorias ecológicas para a exploração microbiana em solo agrícolas, argumentando-se que os modelos ecológicos que particionam as comunidades microbianas, poderiam aumentar a resolução entre interações microbianas e o ambiente, ii) desenvolveu-se um modelo probabilístico baseado na freqüência de ocorrência de microorganismos através de sistema identificando a comunidade microbiana \"core\". O modelo baseou-se na distribuição de Poisson, sendo este testado em quatro conjuntos de dados disponíveis no Projeto \"Earth Microbiome\", e iii) identificou-se as comunidades bacterianas e fúngicas core em solos cultivados com cana-de-açúcar, verificando-se quais componentes abióticos poderiam modular a composição dos grupos. Com isso, elevou-se a resolução das interações ambiental e microbiana, indicando que o core microbiano e as comunidades microbianas variáveis são moduladas por componentes abióticos distintos. Observou-se também que as comunidades core e variável possuem funcionalidade potencial distinta, como fixação de nitrogênio mais predita para o core bacteriano e processo de nitrificação para a comunidade variável de bactérias. Os resultados do presente trabalho elevam o conhecimento da dinâmica e funcionalidade microbiana, ajudando a revelar e explorar o microbioma do sistema de cultivo.

In the development of any product to perform a specific function the first concern of the engineer is to design for satisfactory operation. Engineers originally approached the reliability problem by using excessive safety factors to be assured the structure or material would withstand the calculated loads and stresses. The engineer also learned from operating or testing the equipment until failures occurred and then redesigning as mistakes became apparent. These methods were time consuming and often resulted in bulky over designed products. These approaches became impractical with the advent of new technological advancements. The accelerated industrial development of aircraft, missiles, and modern electronics coupled with a need for a drastic reduction in weight and size magnified the problem. As products became more complex the problem of building a reliable product was intensified. An appreciation for the increase in complexity can be gained from considering that in a period of fifteen years the requirements for electronic tubes on a U.S. Navy destroyer changed from sixty to thirty-six hundred (14). During World War II new equipment was developed that had to be operational for extended period of time if the military mission was to be accomplished. The addition of a time requirement added to the already difficult problem caused by the increasing complexity of equipment. It soon became obvious that new techniques had to be developed that would assist the manufacturer in designing a reliable product.

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion process with discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuous model that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (All Ordinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P. Morgan???s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

Multivariate count data are found in a variety of fields. For modeling such data, one may consider the multivariate Poisson distribution. Overdispersion is a problem when modeling the data with the multivariate Poisson distribution. Therefore, in this thesis we propose a new multivariate Poisson hidden Markov model based on the extension of independent multivariate Poisson finite mixture models, as a solution to this problem. This model, which can take into account the spatial nature of weed counts, is applied to weed species counts in an agricultural field. The distribution of counts depends on the underlying sequence of states, which are unobserved or hidden. These hidden states represent the regions where weed counts are relatively homogeneous. Analysis of these data involves the estimation of the number of hidden states, Poisson means and covariances. Parameter estimation is done using a modified EM algorithm for maximum likelihood estimation. <p>We extend the univariate Markov-dependent Poisson finite mixture model to the multivariate Poisson case (bivariate and trivariate) to model counts of two or three species. Also, we contribute to the hidden Markov model research area by developing Splus/R codes for the analysis of the multivariate Poisson hidden Markov model. Splus/R codes are written for the estimation of multivariate Poisson hidden Markov model using the EM algorithm and the forward-backward procedure and the bootstrap estimation of standard errors. The estimated parameters are used to calculate the goodness of fit measures of the models.<p>Results suggest that the multivariate Poisson hidden Markov model, with five states and an independent covariance structure, gives a reasonable fit to this dataset. Since this model deals with overdispersion and spatial information, it will help to get an insight about weed distribution for herbicide applications. This model may lead researchers to find other factors such as soil moisture, fertilizer level, etc., to determine the states, which govern the distribution of the weed counts.