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<p>When researchers work on time series or sequence, certain fundamental questions will naturally arise. One of them will be whether the series or sequence exhibits a gradual trend over time. In this thesis, we propose a test statistic based on moving order statistics and establish an exact procedure to test for the presence of monotone trends. We show that the test statistic under the null hypothesis that there is no trend follows the closed skew normal distribution. An efficient algorithm is then developed to generate realizations from this null distribution. A simulation study is conducted to evaluate the proposed test under the alternative hypotheses with linear, logarithmic and quadratic trend functions. Finally, a practical example is provided to illustrate the proposed test procedure.</p> / Master of Science (MSc)

El tema principal de esta tesis es el desarrollo de técnicas de actualización de precondicionadores para resolver sistemas lineales de gran tamaño y dispersos Ax=b mediante el uso de métodos iterativos de Krylov. Se consideran dos tipos interesantes de problemas. En el primero se estudia la solución iterativa de sistemas lineales no singulares y antisimétricos, donde la matriz de coeficientes A tiene parte antisimétrica de rango bajo o puede aproximarse bien con una matriz antisimétrica de rango bajo. Sistemas como este surgen de la discretización de PDEs con ciertas condiciones de frontera de Neumann, la discretización de ecuaciones integrales y métodos de puntos interiores, por ejemplo, el problema de Bratu y la ecuación integral de Love. El segundo tipo de sistemas lineales considerados son problemas de mínimos cuadrados (LS) que se resuelven considerando la solución del sistema equivalente de ecuaciones normales. Concretamente, consideramos la solución de problemas LS modificados y de rango incompleto. Por problema LS modificado se entiende que el conjunto de ecuaciones lineales se actualiza con alguna información nueva, se agrega una nueva variable o, por el contrario, se elimina alguna información o variable del conjunto. En los problemas LS de rango deficiente, la matriz de coeficientes no tiene rango completo, lo que dificulta el cálculo de una factorización incompleta de las ecuaciones normales. Los problemas LS surgen en muchas aplicaciones a gran escala de la ciencia y la ingeniería como, por ejemplo, redes neuronales, programación lineal, sismología de exploración o procesamiento de imágenes. Los precondicionadores directos para métodos iterativos usados habitualmente son las factorizaciones incompletas LU, o de Cholesky cuando la matriz es simétrica definida positiva. La principal contribución de esta tesis es el desarrollo de técnicas de actualización de precondicionadores. Básicamente, el método consiste en el cálculo de una descomposición incompleta para un sistema lineal aumentado equivalente, que se utiliza como precondicionador para el problema original. El estudio teórico y los resultados numéricos presentados en esta tesis muestran el rendimiento de la técnica de precondicionamiento propuesta y su competitividad en comparación con otros métodos disponibles en la literatura para calcular precondicionadores para los problemas estudiados. / The main topic of this thesis is updating preconditioners for solving large sparse linear systems Ax=b by using Krylov iterative methods. Two interesting types of problems are considered. In the first one is studied the iterative solution of non-singular, non-symmetric linear systems where the coefficient matrix A has a skew-symmetric part of low-rank or can be well approximated with a skew-symmetric low-rank matrix. Systems like this arise from the discretization of PDEs with certain Neumann boundary conditions, the discretization of integral equations as well as path following methods, for example, the Bratu problem and the Love's integral equation. The second type of linear systems considered are least squares (LS) problems that are solved by considering the solution of the equivalent normal equations system. More precisely, we consider the solution of modified and rank deficient LS problems. By modified LS problem, it is understood that the set of linear relations is updated with some new information, a new variable is added or, contrarily, some information or variable is removed from the set. Rank deficient LS problems are characterized by a coefficient matrix that has not full rank, which makes difficult the computation of an incomplete factorization of the normal equations. LS problems arise in many large-scale applications of the science and engineering as for instance neural networks, linear programming, exploration seismology or image processing. Usually, incomplete LU or incomplete Cholesky factorization are used as preconditioners for iterative methods. The main contribution of this thesis is the development of a technique for updating preconditioners by bordering. It consists in the computation of an approximate decomposition for an equivalent augmented linear system, that is used as preconditioner for the original problem. The theoretical study and the results of the numerical experiments presented in this thesis show the performance of the preconditioner technique proposed and its competitiveness compared with other methods available in the literature for computing preconditioners for the problems studied. / El tema principal d'esta tesi és actualitzar precondicionadors per a resoldre sistemes lineals grans i buits Ax=b per mitjà de l'ús de mètodes iteratius de Krylov. Es consideren dos tipus interessants de problemes. En el primer s'estudia la solució iterativa de sistemes lineals no singulars i antisimètrics, on la matriu de coeficients A té una part antisimètrica de baix rang, o bé pot aproximar-se amb una matriu antisimètrica de baix rang. Sistemes com este sorgixen de la discretització de PDEs amb certes condicions de frontera de Neumann, la discretització d'equacions integrals i mètodes de punts interiors, per exemple, el problema de Bratu i l'equació integral de Love. El segon tipus de sistemes lineals considerats, són problemes de mínims quadrats (LS) que es resolen considerant la solució del sistema equivalent d'equacions normals. Concretament, considerem la solució de problemes de LS modificats i de rang incomplet. Per problema LS modificat, s'entén que el conjunt d'equacions lineals s'actualitza amb alguna informació nova, s'agrega una nova variable o, al contrari, s'elimina alguna informació o variable del conjunt. En els problemes LS de rang deficient, la matriu de coeficients no té rang complet, la qual cosa dificultata el calcul d'una factorització incompleta de les equacions normals. Els problemes LS sorgixen en moltes aplicacions a gran escala de la ciència i l'enginyeria com, per exemple, xarxes neuronals, programació lineal, sismologia d'exploració o processament d'imatges. Els precondicionadors directes per a mètodes iteratius utilitzats més a sovint són les factoritzacions incompletes tipus ILU, o la factorització incompleta de Cholesky quan la matriu és simètrica definida positiva. La principal contribució d'esta tesi és el desenvolupament de tècniques d'actualització de precondicionadors. Bàsicament, el mètode consistix en el càlcul d'una descomposició incompleta per a un sistema lineal augmentat equivalent, que s'utilitza com a precondicionador pel problema original. L'estudi teòric i els resultats numèrics presentats en esta tesi mostren el rendiment de la tècnica de precondicionament proposta i la seua competitivitat en comparació amb altres mètodes disponibles en la literatura per a calcular precondicionadors per als problemes considerats. / Guerrero Flores, DJ. (2018). On Updating Preconditioners for the Iterative Solution of Linear Systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/104923

This thesis has discussed both properties and geometry of concrete slabs used in bridges. It gave understanding on behavior of concrete in both tension and compression zones and how crack propagates in specimens by presenting both theory of fracture and performing concrete tests like tension splitting, uniaxial compression and uniaxial tension tests. Furthermore, it supported experimental tests with finite elements modelling for each test, and illustrated both boundary conditions and loads. The thesis has used ARAMIS cameras to observe crack propagations in all experimental tests, and its first study at LNU that emphasized on Brazilian test, because of importance of this test to describe both crushing and cracking behavior of concrete under loading. It’s an excellent opportunity to understand how concrete and steel behave individually and in combination with each other, and to understand fracture process zone, and this has been discussed in theory chapter. The geometry change that could affect stresses distributions has also described in literature and modelled to give good idea on how to model slabs in different angles in the methodology chapter. Thus, thesis will use finite elements program (Abaqus) to model both experimental specimens and concrete slabs without reinforcement to emphasize on concrete behavior and skewness effect. This means studying both properties of concrete and geometry of concrete slabs. This thesis has expanded experimental tests and chose bridges as an application.

Thesis (MScEng)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Different types of live loading due to traffic may act on bridges. The focus of this study is on normal traffic loading according to the South African specification of TMH7. Heavy vehicles are not included in normal traffic loading. TMH7 represents the code of practice for the design of highway bridges and culverts in South Africa. The aim of the study is to provide an insight into the flexural analysis of skew bridges, under the effects of normal traffic loading. The need for the study arose since the specification of TMH7 does not explicitly specify application patterns for normal traffic loading. Only the intensity of normal traffic loading is specified and it should be applied to yield the most adverse effects. For these reasons, a set of so-called standard application patterns are investigated and developed through the course of this study. The envelope of the values from the standard application patterns are compared to the most adverse application pattern for flexural effects in certain design regions of the bridge deck. Flexure, as in the context of this study, translates into the bending and twisting of the bridge deck under loads. A number of numerical experiments are performed for typical single span and multi-span continuous carriageways, where the standard application patterns are compared to the most adverse application patterns. The results from the numerical experiments are documented and compared as the angle of skew of the bridge deck increases in plan-view. For this purpose, the development of effective and specialized software was necessary. It was found that the set of standard application patterns can be used as a preliminary approximation for the most adverse effects of normal traffic loading, for specific flexural resultants in certain design regions of a bridge deck. However, for a large number of secondary flexural effects, the set of standard application patterns did not represent a good approximation for the most adverse values. / AFRIKAANSE OPSOMMING: Verskillende tipes lewendige belasting, as gevolg van verkeer, kan op brûe inwerk. Die fokus van die studie is op normale verkeers-belasting volgens die Suid-Afrikaanse spesifikasie van TMH7. Swaar-voertuie word nie ingesluit by normale verkeers-belasting nie. TMH7 verteenwoordig die kode vir die ontwerp van padbrûe en duikers in Suid-Afrika. Die doel van die studie is om insig te verskaf in die buig-analise van skewe brûe, as gevolg van die werking van normale verkeers-belasting. Die rede vir hierdie studie ontstaan aangesien die spesifikasie van TMH7 nie eksplisiet aanwendingspatrone vir normale verkeers-belasting voorskryf nie. Slegs die intensiteit van normale verkeersbelasting word voorgeskryf en dit moet aangewend word om die negatiefste effekte te verkry. Vir hierdie redes word 'n versameling van sogenaamde standaard aanwendings-patrone deur die loop van die studie ondersoek en ontwikkel. Die omhullings-kurwe van die waardes wat deur die standaard patrone gelewer word, word vergelyk met die waarde van die aanwendings-patroon wat die negatiefste buig-effek in sekere ontwerp-areas van die brugdek veroorsaak. Buig-effekte, soos van toepassing op hierdie studie, verwys na buig en wring van die brugdek as gevolg van belastings. 'n Aantal numeriese eksperimente, vir enkel-span sowel as multi-span deurlopende brugdekke, word uitgevoer en die standaard aanwendings-patrone word vergelyk met die aanwendings-patrone wat die negatiefste waardes lewer. Die resultate van die numeriese eksperimente word gedokumenteer en vergelyk soos die hoek van skeefheid van die brugdek in plan-aansig toeneem. Vir hierdie doel is die ontwikkeling van effektiewe en gespesialiseerde sagteware dus nodig. Daar is gevind dat die standaard aanwendings-patrone, vir spesifieke buig-resultante in sekere ontwerp-areas van die brugdek, as 'n voorlopige benadering vir die negatiefste effekte van normale verkeers-belasting gebruik kan word. Dit was egter verder gevind dat vir 'n groot aantal sekondêre buig-effkte, die versameling standaard aanwendings-patrone nie as 'n goeie benadering vir die negatiefste waardes dien nie.

本文以台灣股價加權指數，使用 AR(3)-GJR-GRACH(1,1) 模型，白噪音假設為 Normal 、 Skew-Normal 、 Student t 、 skew-t 、 EPD 、 SEPD 、與 AEPD 等七種分配。著重於兩個部份，(一) Student t 分配一族與 EPD 分配一族在模型配適與風險值估計的比較；(二) 預測風險值區分為低震盪與高震盪兩個區間，比較不同分配在兩區間預測風險值的差異。 實證分析顯示， t 分配一族與 EPD 分配一族配適的結果，無論是只考慮峰態 ( t 分配與 EPD 分配) ，或者加入影響偏態的參數 ( skew-t 分配與 SEPD 分配) ， t 分配一族的配適程度都較 EPD 分配一族為佳。更進一步考慮分配兩尾厚度不同的 AEPD 分配，配適結果為七種分配中最佳。 風險值的估計在低震盪的區間，常態分配與其他厚尾分配皆能通過回溯測試，採用厚尾分配效果不大；在高震盪的區間，左尾風險值回溯測試結果，常態分配與其他厚尾分配皆無法全數通過，但仍以 AEPD 分配為最佳。最後比較損失函數，左尾風險值估計以 AEPD 分配為最佳，右尾風險值則無一致的結果。因此我們認為 AEPD 分配可作為風險管理有用的工具。

In this thesis, we study the existence of random periodic solutions of random dynamical systems (RDS) by geometric and topological approach. We employed an extension of ergodic theory to random setting to prove that a random invariant set with some kind of dissipative structure, can be expressed as union of random periodic curves. We extensively characterize the dissipative structure by random invariant measures and Lyapunov exponents. For stochastic flows induced by stochastic differential equations (SDEs), we studied the dissipative structure by two point motion of the SDE and prove the existence exponential stable random periodic solutions.

In this thesis we introduce, implement and compare several multivariate goodness-of-fit tests. First of all, we will focus on universal mul- tivariate tests that do not place any assumptions on parametric families of null distributions. Thereafter, we will be concerned with testing of multi- variate normality and, by using Monte Carlo simulations, we will compare power of five different tests of bivariate normality against several alternati- ves. Then we describe multivariate skew-normal distribution and propose a new test of multivariate skew-normality based on empirical moment genera- ting functions. In the final analysis, we compare its power with other tests of multivariate skew-normality. 1

In this dissertation, we approach three different contributions in measurement error model (MEM). Initially, we carry out maximum penalized likelihood inference in MEMs under the normality assumption. The methodology is based on the method proposed by Firth (1993), which can be used to improve some asymptotic properties of the maximum likelihood estimators. In the second contribution, we develop two new estimation methods based on generalized fiducial inference for the precision parameters and the variability product under the Grubbs model considering the two-instrument case. One method is based on a fiducial generalized pivotal quantity and the other one is built on the method of the generalized fiducial distribution. Comparisons with two existing approaches are reported. Finally, we propose to study inference in a heteroscedastic MEM with known error variances. Instead of the normal distribution for the random components, we develop a model that assumes a skew-t distribution for the true covariate and a centered Students t distribution for the error terms. The proposed model enables to accommodate skewness and heavy-tailedness in the data, while the degrees of freedom of the distributions can be different. We use the maximum likelihood method to estimate the model parameters and compute them via an EM-type algorithm. All proposed methodologies are assessed numerically through simulation studies and illustrated with real datasets extracted from the literature. / Neste trabalho abordamos três contribuições diferentes em modelos com erros de medição (MEM). Inicialmente estudamos inferência pelo método de máxima verossimilhança penalizada em MEM sob a suposição de normalidade. A metodologia baseia-se no método proposto por Firth (1993), o qual pode ser usado para melhorar algumas propriedades assintóticas de os estimadores de máxima verossimilhança. Em seguida, propomos construir dois novos métodos de estimação baseados na inferência fiducial generalizada para os parâmetros de precisão e a variabilidade produto no modelo de Grubbs para o caso de dois instrumentos. O primeiro método é baseado em uma quantidade pivotal generalizada fiducial e o outro é baseado no método da distribuição fiducial generalizada. Comparações com duas abordagens existentes são reportadas. Finalmente, propomos estudar inferência em um MEM heterocedástico em que as variâncias dos erros são consideradas conhecidas. Nós desenvolvemos um modelo que assume uma distribuição t-assimétrica para a covariável verdadeira e uma distribuição t de Student centrada para os termos dos erros. O modelo proposto permite acomodar assimetria e cauda pesada nos dados, enquanto os graus de liberdade das distribuições podem ser diferentes. Usamos o método de máxima verossimilhança para estimar os parâmetros do modelo e calculá-los através de um algoritmo tipo EM. Todas as metodologias propostas são avaliadas numericamente em estudos de simulação e são ilustradas com conjuntos de dados reais extraídos da literatura

Apresentamos algumas aplicações de ferramentas estatísticas que são comumente utilizadas na melhoria da qualidade de processos industriais. Inicialmente, desenvolveu-se procedimentos para testar a competência de laboratórios que participam de programas de ensaios de proficiência. Em situações onde os laboratórios medem várias vezes no mesmo ponto, utilizou-se o modelo de erros de medição, proposto por Jaech [39](1985). Além disso, a inferência sobre os parâmetros de tendência aditiva foi generalizada para a classe de distribuições elípticas. A competência dos laboratórios é avaliada pelo teste da razão de verossimilhança generalizada, do qual, obtemos a distribuição exata para a estatística proposta. Em situações onde os laboratórios medem várias vezes em vários pontos e a variável em análise apresenta variações naturais, utilizou-se o modelo com erro nas variáveis. Diante disso, vamos estender o modelo estrutural definido em Barnett [13] (1969) para o modelo ultra-estrutural com réplicas. Neste caso, vamos avaliar não somente a tendência aditiva, mas também, a tendência multiplicativa, ou seja, avaliar a linearidade das medições. As estimativas dos parâmetros foram obtidas via procedimento do algorítmo EM, com isso, desenvolvemos os teste de Wald, razão de verossimilhança e escore para avaliar a competência dos laboratórios. Nos dois modelos propostos, generalizamos o erro normalizado (En) sugerido pelo Guia 43 [37] para testar a competência dos laboratórios participantes de programas de ensaio de proficiência. Apresentamos também, um procedimento para calcular índices de performance para processos univariados e multivariados. Nestes casos, consideramos que a distribuição dos dados segue uma distribuição Normal assimétrica. Além disso, apresentamos uma análise de simulação onde concluímos que a presença de assimetria nos dados pode causar interpretações erradas sobre o processo, quando a distribuição assumida para os dados é a Normal / We present some applications of statistical tools that are used in the improvement of the quality of industrial processes. Initially, we develop procedures to test the ability of laboratories that participate of programs of proficiency test. In situations where the laboratories measure several times in the same point, we use the model of errors of measurement, considered for Jaech [39](1985). Moreover, the inference on the parameters additive bias was generalized for the class of elliptical distributions. The ability of the laboratories is evaluated by the generalized likelihood ratio test, of which, we get the accurate distribution for the statistics proposal. In situations where the laboratories measure some times in some points and the variable in analysis presents natural variations, uses the model with error in the variable. With this, we go to extend the model structural defined in Barnett [13] (1969) for the ultrastructural model with replicate. In this case, we go to not only evaluate the bias additive, but also, the bias multiplicative, that is, to evaluate the linearity of the measurements. The estimates of the parameters had been gotten by the procedure of the EM algorithm, with this, develop of Wald, likelihood ratio and score test to evaluate the ability of the laboratories. In the two considered models, we generalize the normalized error (En) suggested for Guide 43 [37] to test the ability of the participant laboratories of programs of proficiency test. We also present, a procedure to calculate index of performance for univariate and multivariate processes. In these cases, we consider that the distribution of the data follows a skew Normal distribution. Moreover, we present a simulation analysis where we conclude that the presence of asymmetry in the data can cause interpretations missed on the process, when the distribution assumed for the data is the Normal