abstract: The Pearson and likelihood ratio statistics are well-known in goodness-of-fit testing and are commonly used for models applied to multinomial count data. When data are from a table formed by the cross-classification of a large number of variables, these goodness-of-fit statistics may have lower power and inaccurate Type I error rate due to sparseness. Pearson's statistic can be decomposed into orthogonal components associated with the marginal distributions of observed variables, and an omnibus fit statistic can be obtained as a sum of these components. When the statistic is a sum of components for lower-order marginals, it has good performance for Type I error rate and statistical power even when applied to a sparse table. In this dissertation, goodness-of-fit statistics using orthogonal components based on second- third- and fourth-order marginals were examined. If lack-of-fit is present in higher-order marginals, then a test that incorporates the higher-order marginals may have a higher power than a test that incorporates only first- and/or second-order marginals. To this end, two new statistics based on the orthogonal components of Pearson's chi-square that incorporate third- and fourth-order marginals were developed, and the Type I error, empirical power, and asymptotic power under different sparseness conditions were investigated. Individual orthogonal components as test statistics to identify lack-of-fit were also studied. The performance of individual orthogonal components to other popular lack-of-fit statistics were also compared. When the number of manifest variables becomes larger than 20, most of the statistics based on marginal distributions have limitations in terms of computer resources and CPU time. Under this problem, when the number manifest variables is larger than or equal to 20, the performance of a bootstrap based method to obtain p-values for Pearson-Fisher statistic, fit to confirmatory dichotomous variable factor analysis model, and the performance of Tollenaar and Mooijaart (2003) statistic were investigated. / Dissertation/Thesis / Doctoral Dissertation Statistics 2018

abstract: It is common in the analysis of data to provide a goodness-of-fit test to assess the performance of a model. In the analysis of contingency tables, goodness-of-fit statistics are frequently employed when modeling social science, educational or psychological data where the interest is often directed at investigating the association among multi-categorical variables. Pearson's chi-squared statistic is well-known in goodness-of-fit testing, but it is sometimes considered to produce an omnibus test as it gives little guidance to the source of poor fit once the null hypothesis is rejected. However, its components can provide powerful directional tests. In this dissertation, orthogonal components are used to develop goodness-of-fit tests for models fit to the counts obtained from the cross-classification of multi-category dependent variables. Ordinal categories are assumed. Orthogonal components defined on marginals are obtained when analyzing multi-dimensional contingency tables through the use of the QR decomposition. A subset of these orthogonal components can be used to construct limited-information tests that allow one to identify the source of lack-of-fit and provide an increase in power compared to Pearson's test. These tests can address the adverse effects presented when data are sparse. The tests rely on the set of first- and second-order marginals jointly, the set of second-order marginals only, and the random forest method, a popular algorithm for modeling large complex data sets. The performance of these tests is compared to the likelihood ratio test as well as to tests based on orthogonal polynomial components. The derived goodness-of-fit tests are evaluated with studies for detecting two- and three-way associations that are not accounted for by a categorical variable factor model with a single latent variable. In addition the tests are used to investigate the case when the model misspecification involves parameter constraints for large and sparse contingency tables. The methodology proposed here is applied to data from the 38th round of the State Survey conducted by the Institute for Public Policy and Michigan State University Social Research (2005) . The results illustrate the use of the proposed techniques in the context of a sparse data set. / Dissertation/Thesis / Ph.D. Mathematics 2011

Os sistemas elétricos de potência modernos apresentam inúmeros desafios em sua operação. Nos sistemas de distribuição de energia elétrica, devido à grande ramificação, presença de extensos ramais monofásicos, à dinâmica das cargas e demais particularidades inerentes, a localização de faltas representa um dos maiores desafios. Das barreiras encontradas, a influência da impedância de falta é uma das maiores, afetando significativamente a aplicação dos métodos tradicionais na localização, visto que a magnitude das correntes de falta é similar à da corrente de carga. Neste sentido, esta tese objetivou desenvolver um sistema inteligente para localização de faltas de alta impedância, o qual foi embasado na aplicação da técnica de decomposição por componentes ortogonais no pré-processamento das variáveis e inferência fuzzy para interpretar as não-linearidades do Sistemas de Distribuição com presença de Geração Distribuída. Os dados para treinamento do sistema inteligente foram obtidos a partir de simulações computacionais de um alimentador real, considerando uma modelagem não-linear da falta de alta impedância. O sistema fuzzy resultante foi capaz de estimar as distâncias de falta com um erro absoluto médio inferior a 500 m e um erro absoluto máximo da ordem de 1,5 km, em um alimentador com cerca de 18 km de extensão. Tais resultados equivalem a um grau de exatidão, para a maior parte das ocorrências, dentro do intervalo de ±10%. / Modern electric power systems present numerous challenges in its operation. Fault location is a major challenge in Power Distribution Systems due to its large branching, presence of single-phase laterals and the dynamic loads. The influence of the fault impedance is one of the largest, significantly affecting the use of traditional methods for its location, since the magnitude of the fault currents is similar to the load current. In this sense, this thesis aimed to develop an intelligent system for location of high impedance faults, which was based on the application of the decomposition technique of orthogonal components in the pre-processing variables and fuzzy inference to interpret the nonlinearities of Power Distribution Systems with the presence of Distributed Generation. The data for training the intelligent system were obtained from computer simulations of an actual feeder, considering a non-linear modeling of the high impedance fault. The resulting fuzzy system was able to estimate distances to fault with an average absolute error of less than 500 m and a maximum absolute error of 1.5 km order, on a feeder about 18 km long. These results are equivalent to a degree of accuracy for the most occurrences within the ± 10% range.

Os sistemas elétricos de potência modernos apresentam inúmeros desafios em sua operação. Nos sistemas de distribuição de energia elétrica, devido à grande ramificação, presença de extensos ramais monofásicos, à dinâmica das cargas e demais particularidades inerentes, a localização de faltas representa um dos maiores desafios. Das barreiras encontradas, a influência da impedância de falta é uma das maiores, afetando significativamente a aplicação dos métodos tradicionais na localização, visto que a magnitude das correntes de falta é similar à da corrente de carga. Neste sentido, esta tese objetivou desenvolver um sistema inteligente para localização de faltas de alta impedância, o qual foi embasado na aplicação da técnica de decomposição por componentes ortogonais no pré-processamento das variáveis e inferência fuzzy para interpretar as não-linearidades do Sistemas de Distribuição com presença de Geração Distribuída. Os dados para treinamento do sistema inteligente foram obtidos a partir de simulações computacionais de um alimentador real, considerando uma modelagem não-linear da falta de alta impedância. O sistema fuzzy resultante foi capaz de estimar as distâncias de falta com um erro absoluto médio inferior a 500 m e um erro absoluto máximo da ordem de 1,5 km, em um alimentador com cerca de 18 km de extensão. Tais resultados equivalem a um grau de exatidão, para a maior parte das ocorrências, dentro do intervalo de ±10%. / Modern electric power systems present numerous challenges in its operation. Fault location is a major challenge in Power Distribution Systems due to its large branching, presence of single-phase laterals and the dynamic loads. The influence of the fault impedance is one of the largest, significantly affecting the use of traditional methods for its location, since the magnitude of the fault currents is similar to the load current. In this sense, this thesis aimed to develop an intelligent system for location of high impedance faults, which was based on the application of the decomposition technique of orthogonal components in the pre-processing variables and fuzzy inference to interpret the nonlinearities of Power Distribution Systems with the presence of Distributed Generation. The data for training the intelligent system were obtained from computer simulations of an actual feeder, considering a non-linear modeling of the high impedance fault. The resulting fuzzy system was able to estimate distances to fault with an average absolute error of less than 500 m and a maximum absolute error of 1.5 km order, on a feeder about 18 km long. These results are equivalent to a degree of accuracy for the most occurrences within the ± 10% range.