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This paper is concerned with a study of approximation order and construction of locally supported elements for the space S41 (Δ) of Cl quartic pp (piecewise polynomial) functions on a triangulation Δ of a connected polygonal domain Ω in R2. It is well known that, when Δ is a three-directional mesh Δ(1), the order of approximation of S41(Δ(1)) is only 4, not 5. Though a local Clough-Tocher refinement procedure of an arbitrary triangulation A yields the optimal (fifth) order of approximation from the space S41(Δ) (see [1]), it needs more data points in addition to the vertex set of the triangulation A. In this paper, we will introduce a particular mixed three-directional mesh Δ(3)) and construct so-called mixed three-directional elements. We prove that the space S41(Δ(3)) achieves its optimal-order of approximation by constructing an interpolation scheme using mixed three-directional elements.

Étalonner une caméra numérique consiste à calculer ses paramètres intrinsèques et extrinsèques qui décrivent la transformation du système de coordonnées de la scène vers celui de l'image. Plusieurs méthodes existent et permettent le calcul de ses éléments en se basant sur la minimisation d'un critère linéaire ou non. Le but de ce mémoire est de proposer un modèle simple et précis permettant, à partir d'un site d'étalonnage au laboratoire de métrologie du Département des sciences géomatiques de l'Université Laval, d'extraire les paramètres d'une caméra numérique. Nous avons modélisé la distorsion et nous avons corrigé la déformation des droites dans l'image afin de se placer dans un modèle projectif. Nous avons utilisé la spline cubique composée qui a permis de calculer les deux types de distorsions radiale et tangentielle et a montré que l'erreur d'interpolation est négligeable devant la fonction de distorsion. En conclusion, nous présentons un algorithme permettant d'étalonner une caméra en deux étapes. La première étape consiste à corriger l'image afin de maintenir la notion de la projection perspective et de déduire la fonction de la distorsion. La deuxième étape permet de calculer les paramètres qui identifient la position et l'orientation de la caméra par rapport à la scène ainsi que ses caractéristiques.

Les courbes de Bezier et les courbes splines ont trouve un cadre de présentation simple et naturel avec la notion de floraison d'une fonction polynomiale, telle qu'elle a été présentée dans les travaux de Lyle Ramshaw (1987). Notre but a consiste a étendre cette présentation au cas des surfaces et aussi des variétés de dimension supérieure. Les splines simpliciales sont une généralisation naturelle des b-splines au cas de plusieurs variables. Nous présentons leurs principales propriétés ainsi qu'une définition de différences divisées pour fonctions de plusieurs variables. Un algorithme d'évaluation d'une spline simpliciale est propose et teste. Floraisons et splines simpliciales sont les éléments essentiels d'un nouveau schéma de b-splines introduit par Dahmen, Micchelli et Seidel (1992). Ce schéma est étudié et ses principales propriétés sont présentées. Une grande similarité avec l'étude des courbes est retrouvée

Studien är utförd i form av ett examensarbete vid JTH (Jönköpings Tekniska Högskola) i samarbete med komponentgruppen på Husqvarna AB. Syftet med arbetet var att genom en produktutvecklingsprocess hitta en annan metod för att fästa centrifugalkopplingen på motorsågen 560 XP. På Husqvarnas motorsågar drivs sågkedjan och oljepumpen med hjälp av en centrifugalkoppling som i dagsläget fästs med ett skruvförband på vevaxeln. För att dagens koppling inte skall lossa under drift rekommenderas ett åtdragningsmoment på 20–25 Nm. Till nästkommande produktgeneration har detta höjts till 30–40 Nm vilket försämrar produktens servicebarhet samt att risken för att ha sönder kopplingen under demontering ökar. Därav uppkom behovet av ett nytt sätt att fästa kopplingen på. Lösningen borde klara de ökade lasterna samt förbättra servicebarhet genom underlättad montering och demontering. Genom en förstudie bestående av genomgång av befintlig produkt och intervjuer med insatt personal på Husqvarna samlades information in om problemet. Det användes för att sammanställa en konstruktionskriterielista med de krav som borde och önskades uppfyllas. De viktigaste kraven visade sig vara att kunna överföra vridmoment, förhindra att förbandet lossar under drift, minimera ett eventuellt åtdragningsmoment samt att erbjuda montering och demontering med verktyg som medföljer motorsågen vid inköp. Vidare utfördes en konceptstudie för att hitta en potentiell lösning till problemet. Genom en brainstormingprocess producerades drygt 20 koncept och idéer som sedan sållades och utvärderades med metoderna genomförbarhetsbedömning, Go/No-Go och Pughs matris. Koncepten bedömdes utefter kraven i konstruktionskriterielistan där ett koncept valdes att vidareutvecklas. Det valda konceptet var ett splinesförband som förhindrade axiell rörelse med hjälp av ett E-clip och fjäderbricka. Styrkorna hos detta koncept låg speciellt i att åtdragningsmomentet kunde elimineras, vilket underlättade montering och demontering. E-clipet hanterades enkelt med motorsågens medföljande verktyg. Beräkningar och grov konstruktion av konceptet utfördes därefter. Det visade sig vara svårt att dimensionera splinesförbandet inom de designbegränsningar studien ställt upp för att minimera påverkan av övriga komponenter i systemet. Skjuvspänningarna i vevaxel och splineständer blev höga på grund av säkerhetsfaktorer för utmattning och användningsområde. Höga nog att överstiga vevaxelmaterialets maximalt tillåtna skjuvspänning flera gånger om. För att minska dessa spänningar behöver troligtvis designbegränsningarna överskridas vilket leder till stora omkonstruktioner. Dock rekommenderas Husqvarna att fortsätta studera konceptet mer detaljerat och ingående innan det utesluts. Vid jämförelse med befintlig lösning har slutsatsen kunnat dras att skillnaden i skjuvspänningarna mellan båda lösningarna inte borde vara så stora som teorin säger. Detta öppnade upp flertalet förslag på fortsatt arbete. Bland annat att göra datorstödda simuleringar på en detaljkonstruktion av konceptet för att kunna göra mer noggranna hållfasthetsbedömningar. En riskbedömning borde även göras för att öka medvetenheten om de risker och konsekvenser konceptet för med sig. / This study is performed as a final thesis project at School of Engineering, Jönköping University in collaboration with the Global Components Group at Husqvarna AB. The purpose of the work was to find another way to attach the centrifugal clutch to the chainsaw model 560 XP. On Husqvarna's chainsaws, the cutting chain and oil pump are driven by a centrifugal clutch that currently is fastened with a screw joint on the crank shaft. To make sure the clutch won't come loose during operation, a tightening torque of 20–25 Nm is recommended. For the next product generation, the tightening torque has been raised to 30–40 Nm which will worsen the serviceability. Furthermore, the risk of breaking the clutch during disassembly will increase. This led to the need of a new way of fastening the clutch. The solution should withstand the increased loads as well as improve the serviceability through easier assembly and disassembly. Information about the problem were collected through a pilot study, including a review of the current product and interviews with involved personnel at Husqvarna. This was used to compile a list of necessary and desirable requirements for the solution. The most important requirements proved to be the ability to transfer torque, stop the joint from getting loose during operation, minimize the tightening torque as well as offering assembly and disassembly with the including tool upon purchase. Furthermore, a conceptual study was carried out to be able to find a solution to the problem. A brainstorming process generated just over 20 concepts and ideas that later were screened and evaluated with various well proven methods, such as Go/No-Go and Pugh's matrix. The concepts were evaluated against the requirements specified in the pilot study and one single concept was chosen for further development. The chosen concept was a spline joint that prevented axial movement with the help of an E-clip and a spring washer. The strength of this solution was mainly the elimination of the tightening torque, which led to easier assembly and disassembly. The E-clip was easily handled with the tool provided in the purchase. Calculations and a rough design were then performed. It turned out that it was hard to dimension the spline joint within the design limits stated to minimize the impact on the other components in the system. The shear stresses in the crankshaft and the spline teeth ended up high because of safety factors for fatigue life and type of application. High enough to exceed the maximum allowable shear stress of the shaft material several times. To lower these stresses, the design limitations will most likely have to be exceeded which leads to extensive design changes. However, Husqvarna are recommended to continue the study on this concept in more detail before it is to be precluded. Comparison with the current solution led to the conclusion that the differences in shear stresses between the both solutions shouldn't be as big as the theory claims. This opened a series of suggestions for future work. For instance, computer aided simulations of a detailed design of the concept would more accurately evaluate the strength of the concept. Furthermore, a risk assessment should be performed to address all the risks and consequences the concept brings with it.

MSc (Statistics) / Department of Statistics / The baseline parameter vector in a discrete-time survival model is determined by the number of time points. The larger the number of the time points, the higher the dimension of the baseline parameter vector which often leads to biased maximum likelihood estimates. One of the ways to overcome this problem is to use a simpler parametrization that contains fewer parameters. A simulation approach was used to compare the accuracy of three variants of penalised regression spline methods in smoothing the baseline hazard function. Root mean squared error (RMSE) analysis suggests that generally all the smoothing methods performed better than the model with a discrete baseline hazard function. No single smoothing method outperformed the other smoothing methods. These methods were also applied to data on age at rst alcohol intake in Thohoyandou. The results from real data application suggest that there were no signi cant di erences amongst the estimated models. Consumption of other drugs, having a parent who drinks, being a male and having been abused in life are associated with high chances of drinking alcohol very early in life. / NRF

Cette thèse porte sur l'analyse bayésienne de données fonctionnelles dans un contexte hydrologique. L'objectif principal est de modéliser des données d'écoulements d'eau d'une manière parcimonieuse tout en reproduisant adéquatement les caractéristiques statistiques de celles-ci. L'analyse de données fonctionnelles nous amène à considérer les séries chronologiques d'écoulements d'eau comme des fonctions à modéliser avec une méthode non paramétrique. Dans un premier temps, les fonctions sont rendues plus homogènes en les synchronisant. Ensuite, disposant d'un échantillon de courbes homogènes, nous procédons à la modélisation de leurs caractéristiques statistiques en faisant appel aux splines de régression bayésiennes dans un cadre probabiliste assez général. Plus spécifiquement, nous étudions une famille de distributions continues, qui inclut celles de la famille exponentielle, de laquelle les observations peuvent provenir. De plus, afin d'avoir un outil de modélisation non paramétrique flexible, nous traitons les noeuds intérieurs, qui définissent les éléments de la base des splines de régression, comme des quantités aléatoires. Nous utilisons alors le MCMC avec sauts réversibles afin d'explorer la distribution a posteriori des noeuds intérieurs. Afin de simplifier cette procédure dans notre contexte général de modélisation, nous considérons des approximations de la distribution marginale des observations, nommément une approximation basée sur le critère d'information de Schwarz et une autre qui fait appel à l'approximation de Laplace. En plus de modéliser la tendance centrale d'un échantillon de courbes, nous proposons aussi une méthodologie pour modéliser simultanément la tendance centrale et la dispersion de ces courbes, et ce dans notre cadre probabiliste général. Finalement, puisque nous étudions une diversité de distributions statistiques au niveau des observations, nous mettons de l'avant une approche afin de déterminer les distributions les plus adéquates pour un échantillon de courbes donné. / This thesis is about Bayesian functional data analysis in hydrology. The main objective is to model water flow data in a parsimonious fashion while still reproducing the statistical features of the data. Functional data analysis leads us to consider the water flow time series as functions to be modelled with a nonparametric method. First, the functions are registered in order to make them more homogeneous. With a more homogeneous sample of curves, we proceed to model their statistical features by relying on Bayesian regression splines in a fairly broad probabilistic framework. More specifically, we study a family of continuous distributions, which include those of the exponential family, from which the data might have arisen. Furthermore, to have a flexible nonparametric modeling tool, we treat the interior knots, which define the basis elements of the regression splines, as random quantities. We then use MCMC with reversible jumps in order to explore the posterior distribution of the interior knots. In order to simplify the procedure in our general modeling context, we consider some approximations for the marginal distribution of the observations, namely one based on the Schwarz information criterion and another which relies on Laplace's approximation. In addition to modeling the central tendency of a sample of curves, we also propose a methodology to simultaneously model the central tendency and the dispersion of the curves in our general probabilistic framework. Finally, since we study several statistical distributions for the observations, we put forward an approach to determine the most adequate distributions for a given sample of curves.

Nesta dissertação apresentamos métodos de diagnóstico em modelos semiparamétricos sob erros normais, em especial os modelos semiparamétricos com uma variável explicativa não paramétrica, conhecidos como modelos lineares parciais. São utilizados splines cúbicos para o ajuste da variável resposta e são aplicadas funções de verossimilhança penalizadas para a obtenção dos estimadores de máxima verossimilhança com os respectivos erros padrão aproximados. São derivadas também as propriedades da matriz hat para esse tipo de modelo, com o objetivo de utilizá-la como ferramenta na análise de diagnóstico. Gráficos normais de probabilidade com envelope gerado também foram adaptados para avaliar a adequabilidade do modelo. Finalmente, são apresentados dois exemplos ilustrativos em que os ajustes são comparados com modelos lineares normais usuais, tanto no contexto do modelo aditivo normal simples como no contexto do modelo linear parcial. / In this master dissertation we present diagnostic methods in semiparametric models under normal errors, specially in semiparametric models with one nonparametric explanatory variable, also known as partial linear model. We use cubic splines for the nonparametric fitting, and penalized likelihood functions are applied for obtaining maximum likelihood estimators with their respective approximate standard errors. The properties of the hat matrix are also derived for this kind of model, aiming to use it as a tool for diagnostic analysis. Normal probability plots with simulated envelope graphs were also adapted to evaluate the model suitability. Finally, two illustrative examples are presented, in which the fits are compared with usual normal linear models, such as simple normal additive and partially linear models.

O método de elementos analíticos simula escoamentos subterrâneos por meio da superposição de soluções conceituais. No contexto do método, inomogeneidade é uma região bem definida de condutividade hidráulica constante. A diferença de condutividade hidráulica entre a inomogeneidade e o meio em que está inserida causa uma descontinuidade (salto) no potencial de descarga. Tradicionalmente este salto é simulado usando polinômios de primeiro ou segundo grau. O presente trabalho usa polinômios splines quadráticos para interpolar os saltos ocorridos no potencial de descarga ao longo das bordas de inomogeneidades. Paralelamente, a formulação tradicional de interpolação dos saltos no potencial de descarga é estendida para qualquer grau. Os principais elementos que compõe o método são descritos e implementados. O programa computacional resultante (AEM) foi desenvolvido integrado a um sistema de informações geográficas de código-aberto (JUMP). O programa permite a integração com outros sistemas de informações geográficas baseados em JAVA, guardando independência do SIG residente. O projeto do programa AEM/JUMP é baseado na programação orientada a objetos e apresentou grande afinidade com o método de elementos analíticos, havendo identificação entre os conceitos de elemento (usado pelo método) e de objeto (usado pela programação). Conceitos de padrões de projeto são utilizados objetivando ampliar as facilidades de leitura, entendimento, otimização e modificação do código fonte, já disponibilizadas pela programação orientada a objetos. Problemas conceituais são abordados usando as formulações propostas. A interpolação por splines quadráticas mostrou-se eficiente e precisa. Considerando as soluções exatas, o erro médio sobre a área de estudo foi inferior a 0,12%. O AEM/JUMP foi aplicado à região da Lagoa do Bonfim - RN com o objetivo de determinar as isolinhas de cargas hidráulicas. Os resultados foram comparados com estudo anterior, onde obteve resultados compatíveis, comprovando a aplicação do método e de sua implementação. Foram incorporadas ao problema da Lagoa do Bonfim características geométricas do contorno do oceano e de aluviões existentes no entorno da lagoa, demonstrando a utilidade do programa para gerar diferentes cenários de simulação. / The analytical elements method simulates underground draining through the superposition of conceptual solutions. In the method\'s context, inhomogeneity in defined as a clearly set region of constant hydraulic conductivity. Inhomogeneity hydraulic conductivity differences and the environment in which they are inserted cause a discontinuity (jump) in the discharge potential. Traditionally, this jump is simulated using first or second degree polynomials.The present work presents a formulation that uses quadratic spline polynomials to interpolate jumps occurred in the discharge potential through inhomogeneity borders. At the same time, the traditional formulation of discharge potential jump interpolation is extended to any degree. The main elements that compose the method are described and implemented. The resulting computational program (AEM) was developed integrated to an open code geographic information system (JUMP). The program permits the integration with other geographic information systems based on JAVA, keeping its independence from resident SIG. The architecture project program AEM/JUMP is based on object-oriented programming and presented great affinity with the analytical element method, showing identification among element concepts (used by the method) and the object (used by the program). Standard project concepts are used, seeking to widen source code reading possibilities, understanding, optimization and modifications already available through the object-oriented programming. Conceptual problems are approached with proposed formulations. Quadratic spline interpolation proved to be efficient and precise. Considering exact solutions, average mistake on study area was lower than 0.12%. AEM/JUMP was applied to the Lagoa do Bonfim (RN) lake region with the aim of establishing hydraulic charge isolines. Results were compared with the previous study, where compatible results had been obtained, thus proving method feasibility and implementation. Geometric features of surrounding areas and alluvion regions present around the lake area were incorporated to the original problem, demonstrating the usefulness of the program to generate different simulation scenarios.

O objetivo principal deste trabalho é discutir estimação e diagnóstico em modelos semiparamétricos com resposta binomial negativa, mais especificamente, modelos de regressão com resposta binomial negativa em que uma das variáveis explicativas contínuas é modelada de forma não paramétrica. Iniciamos o trabalho com um exemplo ilustrativo e fazemos uma breve revisão dos modelos paramétricos com resposta binomial negativa. Em seguida, introduzimos os modelos semiparamétricos com resposta binomial negativa e discutimos alguns aspectos de estimação, inferência e seleção de modelos. Dedicamos um capítulo a procedimentos de diagnóstico, tais como desenvolvimento de medidas de alavanca e de influência sob os aspectos de deleção de pontos e influência local, além de abordar a análise de resíduos. Reanalizamos o exemplo ilustrativo sob o enfoque semiparamétrico e apresentamos algumas conclusões. / The aim of this work is to discuss some aspects on estimation and diagnostics in negative binomial regression models which an explanatory continuous variable is modeled nonparametrically. First, an illustrative example is presented and analyzed under parametric negative binomial regression models. The proposed models are then introduced and some aspects on estimations, inference and model selection are presented. Particular emphasis is given on the development of diagnostic procedures, such as leverage measures, Cook distances, local influence approach and residuals. The motivated example is reanalyzed under the semiparametric viewpoint and some conclusions are given.