The method of least squares has been used in general for regression analysis. It is usually assumed that the errors are confined to the dependent variable, but in many cases both dependent and independent variables are typically measured with some stochastic errors. The statistical method of orthogonal regression has been used when both variables under investigation are subject to stochastic errors. Furthermore, the measurements sometimes may not be exact but have been censored. In this situation doing orthogonal regression with censored data directly between the two variables, it may yield an incorrect estimates of the relationship. In this work we discuss the estimation of orthogonal regression under censored data in one variable and then provide a method of estimation and two criteria on when the method is applicable. When the observations satisfy the criteria provided here, there will not be very large differences between the estimated orthogonal regression line and the theoretical orthogonal regression line.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-30T20:07:56Z No. of bitstreams: 1 robertocarlossoaresnalonpereirasouza.pdf: 1346845 bytes, checksum: e248f967f42f4ef763b613dc39ed0649 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-06-01T11:51:04Z (GMT) No. of bitstreams: 1 robertocarlossoaresnalonpereirasouza.pdf: 1346845 bytes, checksum: e248f967f42f4ef763b613dc39ed0649 (MD5) / Made available in DSpace on 2017-06-01T11:51:04Z (GMT). No. of bitstreams: 1 robertocarlossoaresnalonpereirasouza.pdf: 1346845 bytes, checksum: e248f967f42f4ef763b613dc39ed0649 (MD5) Previous issue date: 2013-02-21 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho apresenta-se uma nova formulação para regressão ortogonal. O problema é definido como a minimização do risco empírico em relação a uma função de perda com tubo desenvolvida para regressão ortogonal, chamada ρ-insensível. Um algoritmo para resolver esse problema é proposto, baseado na abordagem da descida do gradiente estocástica. Quando formulado em variáveis duais o método permite a introdução de funções kernel e flexibilidade do tubo. Até onde se sabe, este é o primeiro método que permite a introdução de kernels, através do chamado “kernel-trick”, para regressão ortogonal. Apresenta-se ainda um algoritmo para regressão clássica que usa a função de perda ε-insensível e segue também a abordagem da descida do gradiente. Para esse algo ritmo apresenta-se uma prova de convergência que garante um número finito de correções. Finalmente, introduz-se uma estratégia incremental que pode ser usada acoplada com ambos os algoritmos para obter soluções esparsas e também uma aproximação para o “tubo mínimo”que contém os dados. Experimentos numéricos são apresentados e os resultados comparados a outros métodos da literatura. / In this work, we introduce a new formulation for orthogonal regression. The problem is defined as minimization of the empirical risk with respect to a tube loss function de veloped for orthogonal regression, named ρ-insensitive. The method is constructed via an stochastic gradient descent approach. The algorithm can be used in primal or in dual variables. The latter formulation allows the introduction of kernels and soft margins. To the best of our knowledge, this is the first method that allows the introduction of kernels via the so-called “kernel-trick” for orthogonal regression. Also, we present an algorithm to solve the classical regression problem using the ε-insensitive loss function. A conver gence proof that guarantees a finite number of updates is presented for this algorithm. In addition, an incremental strategy algorithm is introduced, which can be used to find sparse solutions and also an approximation to the “minimal tube” containing the data. Numerical experiments are shown and the results compared with other methods.

Le contexte de l’étude est celui des mouvements de terrain susceptibles de se produire à la suite d’un affaissement minier caractéristique de Lorraine et de leurs conséquences sur les habitations en maçonnerie traditionnelle. Quand de tels affaissements se produisent, ces habitations subissent en effet des désordres qui résultent des efforts engendrés dans la structure par les mouvements du terrain. La réponse qui caractérise alors l’état global de la structure dépend des caractéristiques géométriques, physiques et mécaniques. Or, la nature discontinue des maçonneries et la complexité des interactions entre blocs dans ces maçonneries rend complexe et difficile la détermination de cette réponse. Il en est de même de l’interaction sol-structure. L’objectif de la recherche consiste donc à étudier, par modélisation numérique avec la méthode des éléments distincts et par la technique des plans d’expérience et des surfaces de réponse, le comportement d’ouvrages en maçonnerie soumis à un affaissement minier caractéristique et à dégager de cette étude des critères permettant d’estimer, à l’échelle d’une ville entière, la vulnérabilité de tous ses bâtiments en maçonnerie. Une première analyse simplifiée expose le principe de la démarche mise en œuvre à l’échelle de la ville de Joeuf, utilisée comme site pilote. Elle repose sur l’analyse de la longueur cumulée des joints ouverts, assimilés à la formation de fissures dans la structure. Ensuite, une analyse typologique permet de distinguer 4 groupes de maisons aux caractéristiques proches. Sur chacun de ces groupes, la démarche est ensuite appliquée de manière systématique. Elle prend en considération des caractéristiques géométriques des façades et aboutit à la formulation de fonctions de vulnérabilité qui font appel à la technique de régression orthogonale / The context of our study concerns ground movements that may occur in Lorraine as a result of mining subsidence events and their impact on traditional masonry houses. When such an event occurs, houses suffer disorders resulting from efforts in the structure caused by the movement of the ground. The response that characterizes the state of the structure depends on the geometrical, physical and mechanical characteristics. However, the discontinuous nature of the masonry and the interactions complexity between masonry blocks makes it difficult to determine that response. The same is true about the soil-structure interaction. The purpose of this research is to study, by numerical modelling with the distinct element method, experimental design planning and response surfaces, the behaviour of masonry structures subjected to a typical mining subsidence event and to define from this study some criteria making possible the estimation of the vulnerability of all the buildings of a city. A first simplified analysis describes the principle of the used methodology which is then applied to the study of all houses of the city of Joeuf, used as a pilot site. This methodology is based on an analysis of the total length of the opened joints, which are considered as similar to cracks in the structure. Then, a typology analysis helps first to distinguish 4 groups (types) of houses which have similar characteristics. On each of these groups, the methodology is applied consistently, based on the geometrical characteristics of the houses facades and then leads to the formulation of vulnerability functions that use the technique of orthogonal regression

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