The unimodality of a function seems a simple concept. But in the Euclidean space R^m, m=3,4,..., it is not easy to define. We have an easy tool to find the minimum point of a unimodal function. The goal of this project is to formalize and support distinctive strategies that typically guarantee convergence. Support is given both by analytic arguments and simulation study. Application is envisioned in low-dimensional but non-trivial problems. The convergence of the proposed iterated grid search algorithm is presented along with the results of particular application studies. It has been recognized that the derivative methods, such as the Newton-type method, are not entirely satisfactory, so a variety of other tools are being considered as alternatives. Many other tools have been rejected because of apparent manipulative difficulties. But in our current research, we focus on the simple algorithm and the guaranteed convergence for unimodal function to avoid the possible chaotic behavior of the function. Furthermore, in case the loss function to be optimized is not unimodal, we suggest a weaker condition: almost (noisy) unimodality, under which the iterated grid search finds an estimated optimum point. / Ph. D.

O mapeamento e a estimação de riscos e incidências são ferramentas muito úteis para a Epidemiologia pois, auxiliam na prevenção de agravos da saúde e, também auxiliam no planejamento e avaliação dos serviços de saúde. Este trabalho busca utilizar uma ferramenta estatística que incorpora de forma adequada este tipo de análise ao estudo de outras características que estejam relacionadas a estes agravos. No presente trabalho utiliza-se como aplicação dados do estudo caso-controle espacial com base populacional de acidentes de trabalho com a proposta de estimar a distribuição espacial do risco de sofrer acidente de trabalho na área urbana do município de Piracicaba/SP entre trabalhadores que se encontravam na situação de precarização do trabalho em associação com outras variáveis de interesse através de modelos aditivos generalizados (MAG) e, através disso, mostrar que ao incorporar de forma explícita o espaço no processo de modelagem dos dados ocorre um ganho significativo na explicação da variação do risco. O modelo MAG utilizado tem variável resposta binomial (caso e controle) e multinomial (caso e controle separados pela gravidade do acidente sofrido). Com os modelos ajustados, mapas foram desenhados com indicações de diferentes cores para a intensidade do risco de sofrer acidente de trabalho. Outra abordagem utilizada para os dados espaciais de acidentes de trabalho foi a INLA (INTEGRATED NESTED LAPLACE APPROXIMATIONS), a qual é utilizada como processo de modelagem para a família dos modelos Gaussianos latentes através de novos métodos para esta família de modelos. A intenção foi mostrar como essa nova abordagem lida com dados do tipo espacial e, fazer uma comparação com a abordagem feita pela modelagem GAM / Mapping and estimation of risks and impacts are very useful tools for Epidemiology at the assistance in prevention of injuries and health, also assists in planning and evaluation of health services. This paper seeks to use a statistical tool that adequately incorporates this type of analysis to the study of other characteristics that are related these illnesses. In the present work is used as application data from case-control study space-based population accidents with the proposal to estimate the spatial distribution of risk of suffering an accident at work in the urban area of Piracicaba/SP among workers who were in employed as casual labor in combination with other variables of interest using generalized additive models (GAM) and, thereby, show that by incorporating explicitly space in the process of data modeling is a gain significant in explaining the variation in risk. The GAM model have used binomial response variable (case and control) and multinomial (case and control separated by the severity of the accident suffered). With the adjusted models, maps were drawn with indications of different colors to the intensity of the risk of accident. Another approach used for spatial data on accidents at work was the INLA (INTEGRATED NESTED LAPLACE APPROXIMATIONS), which is used as a modeling process for the family of latent Gaussian models through new methods for this family of models. The intention was to show how this new approach deals with spatial data and a comparison with the approach made by GAM modeling.

O mapeamento e a estimação de riscos e incidências são ferramentas muito úteis para a Epidemiologia pois, auxiliam na prevenção de agravos da saúde e, também auxiliam no planejamento e avaliação dos serviços de saúde. Este trabalho busca utilizar uma ferramenta estatística que incorpora de forma adequada este tipo de análise ao estudo de outras características que estejam relacionadas a estes agravos. No presente trabalho utiliza-se como aplicação dados do estudo caso-controle espacial com base populacional de acidentes de trabalho com a proposta de estimar a distribuição espacial do risco de sofrer acidente de trabalho na área urbana do município de Piracicaba/SP entre trabalhadores que se encontravam na situação de precarização do trabalho em associação com outras variáveis de interesse através de modelos aditivos generalizados (MAG) e, através disso, mostrar que ao incorporar de forma explícita o espaço no processo de modelagem dos dados ocorre um ganho significativo na explicação da variação do risco. O modelo MAG utilizado tem variável resposta binomial (caso e controle) e multinomial (caso e controle separados pela gravidade do acidente sofrido). Com os modelos ajustados, mapas foram desenhados com indicações de diferentes cores para a intensidade do risco de sofrer acidente de trabalho. Outra abordagem utilizada para os dados espaciais de acidentes de trabalho foi a INLA (INTEGRATED NESTED LAPLACE APPROXIMATIONS), a qual é utilizada como processo de modelagem para a família dos modelos Gaussianos latentes através de novos métodos para esta família de modelos. A intenção foi mostrar como essa nova abordagem lida com dados do tipo espacial e, fazer uma comparação com a abordagem feita pela modelagem GAM / Mapping and estimation of risks and impacts are very useful tools for Epidemiology at the assistance in prevention of injuries and health, also assists in planning and evaluation of health services. This paper seeks to use a statistical tool that adequately incorporates this type of analysis to the study of other characteristics that are related these illnesses. In the present work is used as application data from case-control study space-based population accidents with the proposal to estimate the spatial distribution of risk of suffering an accident at work in the urban area of Piracicaba/SP among workers who were in employed as casual labor in combination with other variables of interest using generalized additive models (GAM) and, thereby, show that by incorporating explicitly space in the process of data modeling is a gain significant in explaining the variation in risk. The GAM model have used binomial response variable (case and control) and multinomial (case and control separated by the severity of the accident suffered). With the adjusted models, maps were drawn with indications of different colors to the intensity of the risk of accident. Another approach used for spatial data on accidents at work was the INLA (INTEGRATED NESTED LAPLACE APPROXIMATIONS), which is used as a modeling process for the family of latent Gaussian models through new methods for this family of models. The intention was to show how this new approach deals with spatial data and a comparison with the approach made by GAM modeling.

Statistical process control (SPC) applies the science of statistics to various process control in order to provide higher-quality products and better services. The K chart is one among the many important tools that SPC offers. Creation of the K chart is based on Support Vector Data Description (SVDD), a popular data classifier method inspired by Support Vector Machine (SVM). As any methods associated with SVM, SVDD benefits from a wide variety of choices of kernel, which determines the effectiveness of the whole model. Among the most popular choices is the Euclidean distance-based Gaussian kernel, which enables SVDD to obtain a flexible data description, thus enhances its overall predictive capability. This thesis explores an even more robust approach by incorporating the Mahalanobis distance-based kernel (hereinafter referred to as Mahalanobis kernel) to SVDD and compare it with SVDD using the traditional Gaussian kernel. Method's sensitivity is benchmarked by Average Run Lengths obtained from multiple Monte Carlo simulations. Data of such simulations are generated from multivariate normal, multivariate Student's (t), and multivariate gamma populations using R, a popular software environment for statistical computing. One case study is also discussed using a real data set received from Halberg Chronobiology Center. Compared to Gaussian kernel, Mahalanobis kernel makes SVDD and thus the K chart significantly more sensitive to shifts in mean vector, and also in covariance matrix.