Master of Science / Department of Statistics / Leigh Murray / This case study was motivated by the need for effective statistical analysis for a series of poultry feeding experiments conducted in 2006 by Kansas State University researchers in the department of Animal Science. Some of these experiments involved an automated auger feed line system commonly used in commercial broiler houses and continuous, proportion response data. Two of the feed line experiments are considered in this case study to determine if a statistical model using a non-normal response offers a better fit for this data than a model utilizing a normal approximation. The two experiments involve fixed as well as multiple random effects. In this case study, the data from these experiments is analyzed using a linear mixed model and Generalized Linear Mixed Models (GLMM’s) with the SAS Glimmix procedure. Comparisons are made between a linear mixed model and GLMM’s using the beta and binomial responses. Since the response data is not count data a quasi-binomial approximation to the binomial is used to convert continuous proportions to the ratio of successes over total number of trials, N, for a variety of possible N values. Results from these analyses are compared on the basis of point estimates, confidence intervals and confidence interval widths, as well as p-values for tests of fixed effects. The investigation concludes that a GLMM may offer a better fit than models using a normal approximation for this data when sample sizes are small or response values are close to zero. This investigation discovers that these same instances can cause GLMM’s utilizing the beta response to behave poorly in the Glimmix procedure because lack of convergence issues prevent the obtainment of valid results. In such a case, a GLMM using a quasi-binomial response distribution with a high value of N can offer a reasonable and well behaved alternative to the beta distribution.

Master of Science / Department of Statistics / Susan J. Brown / Haiyan Wang / DNA hybridization microarray technologies have made it possible to gain an unbiased perspective of whole genome transcriptional activity on such a scale that is increasing more and more rapidly by the day. However, due to biologically irrelevant bias introduced by the experimental process and the machinery involved, correction methods are needed to restore the data to its true biologically meaningful state. Therefore, it is important that the algorithms developed to remove any sort of technical biases are accurate and robust. This report explores the concept of background correction in microarrays by using a real data set of five replicates of whole genome tiling arrays hybridized with genetic material from Tribolium castaneum. It reviews the literature surrounding such correction techniques and explores some of the more traditional methods through implementation on the data set. Finally, it introduces an alternative approach, implements it, and compares it to the traditional approaches for the correction of such errors.

The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence. In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will be measured via a direct approach which involves the derivation of representation formulae for solutions to the problem. For a real parameter lambda, several classes of initial data are considered. These include the class of smooth functions with either zero or nonzero mean, a family of piecewise constant functions, and a large class of initial data with a bounded derivative that is, at least, continuous almost everywhere and satisfies Holder-type estimates near particular locations in the domain. Amongst other results, our analysis will indicate that for appropriate values of the parameter, the curvature of the data in a neighborhood of these locations is responsible for an eventual breakdown of solutions, or their persistence for all time. Additionally, we will establish a nontrivial connection between the qualitative properties of L-infinity blow-up in ux, and its Lp regularity. Finally, for smooth and non-smooth initial data, a special emphasis is made on the study of regularity of stagnation point-form solutions to the two and three dimensional incompressible Euler equations subject to periodic or Dirichlet boundary conditions.

This study examined variation in ectoparasite density and grooming behavior of naturally occurring white-tailed deer (Odocoileus virginianus) in southwest Alabama. Stimulus-driven grooming as well as the intraspecific body size and vigilance principles of programmed grooming were tested. During the rut, males had a higher average tick (Ixodidae) density than females and exhibited complete separation of tick parasitism between non-rutting and rutting periods, supporting the vigilance principle. Stimulus-driven grooming was supported, as both fawns and yearlings had significantly higher fly (Hippoboscidae) and combined fly/tick densities than adults, and fawns oral groomed at a significantly higher rate than adults, even in the absence of allogrooming. Programmed and stimulus-driven grooming of deer examined in this study were not mutually exclusive but ectoparasite dependent.

"This project utilized generalized estimating equations and general linear modeling to model learning curves for sonographer performance in emergency ultrasonography. Performance was measured in two ways: image quality (interpretable vs. possible hindrance in interpretation) and agreement of findings between the sonographer and an expert reviewing sonographer. Records from 109 sonographers were split into two data sets-- training (n=50) and testing (n=59)--to conduct exploratory analysis and fit the final models for analysis, respectively. We determined that the number of scans of a particular exam type required for a sonographer to obtain quality images on that exam type with a predicted probability of 0.9 is highly dependent upon the person conducting the review, the indication of the scan (educational or medical), and the outcome of the scan (whether there is a pathology positive finding). Constructing family-wise 95% confidence intervals for each exam type demonstrated a large amount of variation for the number of scans required both between exam types and within exam types. It was determined that a sonographer's experience with a particular exam type is not a significant predictor of future agreement on that exam type and thus no estimates were made based on the agreement learning curves. In addition, we concluded based on a type III analysis that when already considering exam type related experience, the consideration of experience on other exam types does not significantly impact the learning curve for quality. However, the learning curve for agreement is significantly impacted by the additional consideration of experience on other exam types."

A modelagem de processos em física e engenharia frequentemente resulta em problemas inversos. Em geral, esses problemas apresentam difícil resolução, pois são classificados como mal-postos. Resolvê-los, tratando-os como problemas de otimização, requer a minimização de uma função objetivo, que mede a discrepância entre os dados experimentais e os obtidos pelo modelo teórico, somada a uma função de regularização. Na maioria dos problemas práticos, essa função objetivo é não-convexa e requer o uso de métodos de otimização estocásticos. Dentre eles, tem-se o algoritmo de recozimento simulado (Simulated Annealing), que é baseado em três pilares: i) distribuição de visitação no espaço de soluções; ii) critério de aceitação; e iii) controle da estocasticidade do processo. Aqui, propomos uma nova generalização do algoritmo de recozimento simulado e da função de regularização. No algoritmo de otimização, generalizamos o cronograma de resfriamento, que usualmente são considerados algébricos ou logarítmicos, e o critério de Metropolis. Com relação à função de regularização, unificamos as versões mais utilizadas, em uma única fórmula. O parâmetro de controle dessa generalização permite transitar continuamente entre as regularizações de Tikhonov e entrópica. Por meio de experimentos numéricos, aplicamos nosso algoritmo na resolução de dois importantes problemas inversos na área de Física Médica: a determinação do espectro de um feixe de raios X, a partir de sua curva de atenuação, e a reconstrução da imagem na tomografia de impedância elétrica. Os resultados mostram que o algoritmo de otimização proposto é eficiente e apresenta um regime ótimo de parâmetros, relacionados à divergência do segundo momento da distribuição de visitação. / Modeling of processes in Physics and Engineering frequently yields inverse problems. These problems are normally difficult to be solved since they are classified as ill-posed. Solving them as optimization problems require the minimization of an objective function which measures the difference between experimental and theoretical data, added to a regularization function. For most of practical inverse problems, this objective function is non-convex and needs a stochastic optimization method. Among them, we have Simulated Annealing algorithm, which is based on three fundamentals: i) visitation distribution in the search space; ii) acceptance criterium; and iii) control of process stochasticity. Here, we propose a new generalization of simulated annealing algorithm and of the regularization function. On the optimization algorithm, we have generalized both the cooling schedule, which usually is algebric or logarithmic, and the Metropolis acceptance criterium. Regarding to regularization function, we have unified the most used versions in an unique equation. The generalization control parameter allows exchange continuously between the Tikhonov and entropic regularization. Through numerical experiments, we applied our algorithm to solve two important inverse problems in Medical Physics: determination of a beam X-rays spectrum from its attenuation curve and the image reconstruction of electrical impedance tomography. Results show that the proposed algorithm is efficient and presents an optimal arrangement of parameters, associated to the divergence of the visitation distribution.

Dados composicionais são vetores, chamados de composições, cujos componentes são todos positivos, satisfazem a soma igual a 1 e possuem um espaço amostral próprio chamado Simplex. A restrição da soma induz a correlação entre os componentes. Isso exige que os métodos estatísticos para análise desses conjuntos de dados considerem esse fato. A teoria para dados composicionais foi desenvolvida inicialmente por Aitchison na década de 80. Desde então, várias técnicas e métodos têm sido desenvolvidos para a modelagem dos dados composicionais. Este trabalho apresenta as principais abordagens para a análise estatística de dados composicionais independentes. Sendo, regressão Dirichlet (distribuição natural aos dados composicionais) ou o uso de transformações em razões logarítmicas que saem do espaço simplex para o espaço real. Também descreve os métodos para os casos em que a suposição de independência não pode ser atendida. Por exemplo, dados composionais com dependência espacial. Para esses casos, há na literatura métodos baseados nas teorias desenvolvidas para análise geoestatística de dados univariados; ou, no uso de transformações em razões logarítmicas com a inclusão da dependência espacial. Além de revisitar os métodos já difundidos, propõe-se o uso do método de Equações de Estimação Generalizadas (EEG) como alternativa para a análise de dados composicionais independentes e com dependência espacial. A principal vantagem é que as equações de estimação necessitam apenas da especificação de funções que descrevam a média e a estrutura de covariância. Assim, não é necessário atribuir uma distribuição de probabilidade aos dados ou fazer o uso de transformações. A aplicação do método EEG para dados composicionais independentes apresentou resultados tão eficientes quanto a regressão Dirichlet ou transformação em razões logarítmicas. Para os dados composicionais com dependência espacial, o método baseado em verossimilhança foi o que apresentou valores preditos mais próximos aos valores reais. O método EEG foi mais eficaz do que a abordagem geoestatística dos componentes individuais, porém, comparado com os demais métodos, foi o que apresentou maior valor residual. / C ompositional data are vectors, called compositions, whose components are all positive, it satisfies the sum equal one and has a Simplex space. The sum constraint induces the correlation between the components and this requires that the statistical methods for the analysis of datasets consider this fact. The theory for compositional data was developed mainly by Aitchison in the 1980s, and since then, several techniques and methods have been developed for compositional data modelling. This work presents the main approaches for the statistical analysis of independent compositional data, such as Dirichlet regression (natural distribution to compositional data) or the use of transformations log-ratios that aim to leave the simplex space for to Euclidean space. Also describes the methods for cases where the assumption of independence cannot be satisfied, for example, spatial dependence compositional data. For these cases, there are in the literature methods of analysis based on the theories developed for univariate geostatistics analysis or use of logratios transformations with the inclusion of the spatial dependence generated by the distance between the points. In addition, to revisiting the already diffused methods, this work propose the use of the Generalized Estimation Equation (GEE) method as an alternative for the analysis of independent compositional data and with spatial dependence. The GEE only requires the specification of functions that describe the mean and correlation matrix (covariance structure, therefore, it is not necessary to assign a probability distribution to the data or transformations. The application of the GEE method for independent compositional data presented results as efficient as Dirichlet regression or log-ratios transformation. Compositional data with spatial dependence, log-ratios transformations presented predicted values close to the real values. GEE method was more effective than the traditional geostatistical approach, however, compared with the other methods, It was the one that presented the high residual values.

A distribuição espectral de raios X de megavoltagem utilizados em departamentos de radioterapia é uma grandeza fundamental a partir da qual, em princípio, todas as informações requeridas relevantes para tratamentos de radioterapia podem ser determinadas. A medição direta é difícil de realizar clinicamente, e a análise da transmissão é um método indireto clinicamente viável para determinar espectros de fótons de aceleradores lineares clínicos. Neste método, os sinais de transmissão são adquiridos após o feixe passar através de diferentes espessuras de atenuadores. O objetivo deste trabalho foi o estabelecimento e a aplicação de um método indireto que utilizou um modelo espectral baseado no algoritmo de recozimento simulado generalizado para determinar o espectro de fótons de aceleradores lineares clínicos com base na curva de transmissão. A análise dos espectros obtidos foi feita por determinação analítica de grandezas dosimétricas e parâmetros relacionados. / The spectral distribution of megavoltage X-rays used in radiotherapy departments is a fundamental quantity from which, in principle, all relevant information required for radiotherapy treatments can be determined. The direct measurement is difficult to achieve clinically and analyzing the transmission is a clinically viable indirect method for determining clinical linear accelerators photon spectra. In this method, transmission signals are acquired after the beam passes through different thicknesses of attenuators. The objective of this work was the establishment and application of an indirect method that used a spectral model based on generalized simulated annealing algorithm to determine the spectrum of clinical linear accelerators photons based on the transmission curve. Analysis of the spectra was made by analytical determination of dosimetric quantities and related parameters.

A integração de diferentes modalidades de imagens médicas possibilita uma análise mais detalhada de seu conteúdo, visando-se um diagnóstico mais preciso da patologia presente. Este processo, conhecido como corregistro, busca o alinhamento das imagens através da transformação rígida (ou não rígida) das mesmas, por algoritmos matemáticos de distorção, translação, rotação e ajuste de escala. A amplitude de cada transformação é determinada por uma medida de similaridade das imagens. Quanto menor a similaridade, maior será a transformação aplicada. Neste sentido, a métrica de similaridade é uma peça chave do processo de corregistro. No presente trabalho, inicialmente são propostas novas definições para o cálculo dos erros de alinhamento nas transformações de translação, rotação e escala, com o objetivo de se avaliar o desempenho do corregistro. Em seguida, cinco experimentos são realizados. No primeiro, a Informação Mútua Generalizada é avaliada como medida de similaridade para corregistro em imagens fractais e cerebrais. Neste caso, os resultados sugerem a viabilidade do emprego desta métrica, pois em geral conduz a erros de alinhamento muito pequenos, mas sem vantagens aparentes em relação à formulação de Shannon. No segundo experimento, um estudo comparativo entre a Informação Mútua e as métricas clássicas (Coeficiente de Correlação, Média dos Quadrados, Diferença de Gradiente e Cardinalidade) é então realizado. Para as imagens binárias analisadas, as métricas com menores valores de erro de alinhamento para os corregistros de translação e rotação foram a Informação Mútua e a Diferença de Gradiente. Para o corregistro de escala, todas as métricas conduziram a erros de alinhamento próximos de zero. No terceiro experimento, o processo de alinhamento é investigado em termos do número de iterações do algoritmo de corregistro. Considerando-se ambas as variáveis erro de alinhamento e número de iterações, conclui-se que o uso da Informação Mútua Generalizada com q = 1.0 é adequado ao corregistro. No quarto experimento, a influência da dimensão fractal no corregistro de imagens fractais binárias foi estudada. Para algumas métricas, a tendência geral observada é a de uma diminuição do erro de alinhamento em resposta ao aumento da dimensão fractal. Finalmente, no quinto experimento, constatou-se a existência de correlação linear entre os erros de alinhamento de imagens em tons de cinza do córtex cerebral e de fractais do conjunto Julia. / The integration of different modalities of medical images provides a detailed analysis of its contents, aiming at a more accurate diagnosis of the pathology. This process, known as coregistration, seeks to align the images through rigid (or non-rigid) transformations, by mathematical algorithms of distortion, translation, rotation and scaling. The amplitude of each transformation is determined by a similarity measure of the images. The lower the similarity, the greater the transformation applied. In this sense, the similarity metric is the key for the coregistration process. In this work, new definitions are proposed for the calculation of alignment errors in the transformations of translation, rotation and scale, with the objective of evaluating the performance of coregistration. Then, five experiments are performed. In the first one, the Generalized Mutual Information is evaluated as a similarity measure for coregistration of brain images and fractals. In this case, the results suggest the feasibility of using this measure, since it leads to very small alignment errors, although no advantages in relation to Shannon formulation are evident. In the second experiment, a comparative study between Mutual Information and the classical metrics (Correlation Coefficient, Mean Squares, Gradient Difference and Cardinality) is performed. For the binary images analyzed, the metrics with lower alignment errors for translation and rotation are the Mutual Information and Gradient Difference. For scaling transformation, all the metrics lead to alignment errors close to zero. In the third experiment, the alignment process is investigated in terms of number of iterations of the coregistration algorithm. Considering both variables alignment error and number of iterations, it is concluded that the use of Generalized Mutual Information with q =1 is appropriate for coregistration. In the fourth experiment, it is studied the influence of fractal dimension in coregistration of binary fractal images. For some metrics, as a general trend, one observes the decay of the alignment error in response to the increase of the fractal dimension. Finally, in the fifth experiment, the results indicate the existence of a linear correlation between the alignment errors of grayscale images of the cerebral cortex and Julia set fractals.

A demanda por abordagens eficientes para o problema de reconhecer a estrutura de cada gene numa sequência genômica motivou a implementação de um grande número de programas preditores de genes. Fizemos uma análise dos programas de sucesso com abordagem probabilística e reconhecemos semelhanças na implementação dos mesmos. A maior parte desses programas utiliza a cadeia oculta generalizada de Markov (GHMM - generalized hiddenMarkov model) como um modelo de gene. Percebemos que muitos preditores têm a arquitetura da GHMM fixada no código-fonte, dificultando a investigação de novas abordagens. Devido a essa dificuldade e pelas semelhanças entre os programas atuais, implementamos o sistema MYOP (Make Your Own Predictor) que tem como objetivo fornecer um ambiente flexível o qual permite avaliar rapidamente cada modelo de gene. Mostramos a utilidade da ferramenta através da implementação e avaliação de 96 modelos de genes em que cada modelo é formado por um conjunto de estados e cada estado tem uma distribuição de duração e um outro modelo probabilístico. Verificamos que nem sempre um modelo probabilísticomais sofisticado fornece um preditor melhor, mostrando a relevância das experimentações e a importância de um sistema como o MYOP. / The demand for efficient approaches for the gene structure prediction has motivated the implementation of different programs. In this work, we have analyzed successful programs that apply the probabilistic approach. We have observed similarities between different implementations, the same mathematical framework called generalized hidden Markov chain (GHMM) is applied. One problem with these implementations is that they maintain fixed GHMM architectures that are hard-coded. Due to this problem and similarities between the programs, we have implemented the MYOP framework (Make Your Own Predictor) with the objective of providing a flexible environment that allows the rapid evaluation of each gene model. We have demonstrated the utility of this tool through the implementation and evaluation of 96 gene models in which each model has a set of states and each state has a duration distribution and a probabilistic model. We have shown that a sophisticated probabilisticmodel is not sufficient to obtain better predictor, showing the experimentation relevance and the importance of a system as MYOP.