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Este modelo contempla o cenário em que a sequência de frações não-conformes no decorrer de um ciclo do processo de produção aumenta gradativamente (situação comum, por exemplo, quando o desgaste de um equipamento é gradual), diferentemente dos modelos de Taguchi, Nayebpour e Woodall e Nandi e Sreehari (1997), que acomodam sequências de frações não-conformes assumindo no máximo três valores, e de Nandi e Sreehari (1999) e Trindade, Ho e Quinino (2007) que contemplam funções de degradação mais simples. O desenvolvimento é baseado nos trabalhos de Ferguson e Antoniak para o cálculo da distribuição a posteriori de uma medida P desconhecida, associada a uma função de distribuição F desconhecida que representa a sequência de frações não-conformes ao longo de um ciclo, supondo, a priori, mistura de Processos Dirichlet. A aplicação consiste na estimação da função de distribuição F e as estimativas de Bayes são analisadas através de alguns casos particulares / In this work, we propose an alternative model for Taguchi´s On-Line Quality-Monitoring Procedure for Attributes under a Bayesian nonparametric framework. This model may be applied to production processes the sequences of defective fractions during a cycle of which increase gradually (for example, when an equipment deteriorates little by little), differently from either Taguchi\'s, Nayebpour and Woodall\'s and Nandi and Sreehari\'s models that allow at most three values for the defective fraction or Nandi and Sreehari\'s and Trindade, Ho and Quinino\'s which take into account simple deterioration functions. The development is based on Ferguson\'s and Antoniak\'s papers to obtain a posteriori distribution for an unknown measure P, associated with an unknown distribution function F that represents the sequence of defective fractions, considering a prior mixture of Dirichlet Processes. The results are applied to the estimation of the distribution function F and the Bayes estimates are analised through some particular cases.

Este modelo contempla o cenário em que a sequência de frações não-conformes no decorrer de um ciclo do processo de produção aumenta gradativamente (situação comum, por exemplo, quando o desgaste de um equipamento é gradual), diferentemente dos modelos de Taguchi, Nayebpour e Woodall e Nandi e Sreehari (1997), que acomodam sequências de frações não-conformes assumindo no máximo três valores, e de Nandi e Sreehari (1999) e Trindade, Ho e Quinino (2007) que contemplam funções de degradação mais simples. O desenvolvimento é baseado nos trabalhos de Ferguson e Antoniak para o cálculo da distribuição a posteriori de uma medida P desconhecida, associada a uma função de distribuição F desconhecida que representa a sequência de frações não-conformes ao longo de um ciclo, supondo, a priori, mistura de Processos Dirichlet. A aplicação consiste na estimação da função de distribuição F e as estimativas de Bayes são analisadas através de alguns casos particulares / In this work, we propose an alternative model for Taguchi´s On-Line Quality-Monitoring Procedure for Attributes under a Bayesian nonparametric framework. This model may be applied to production processes the sequences of defective fractions during a cycle of which increase gradually (for example, when an equipment deteriorates little by little), differently from either Taguchi\'s, Nayebpour and Woodall\'s and Nandi and Sreehari\'s models that allow at most three values for the defective fraction or Nandi and Sreehari\'s and Trindade, Ho and Quinino\'s which take into account simple deterioration functions. The development is based on Ferguson\'s and Antoniak\'s papers to obtain a posteriori distribution for an unknown measure P, associated with an unknown distribution function F that represents the sequence of defective fractions, considering a prior mixture of Dirichlet Processes. The results are applied to the estimation of the distribution function F and the Bayes estimates are analised through some particular cases.

Selection of signi?cant genes via expression patterns is important in a microarray problem. Owing to small sample size and large number of variables (genes), the selection process can be unstable. This research proposes a hierarchical Bayesian model for gene (variable) selection. We employ latent variables in a regression setting and use a Bayesian mixture prior to perform the variable selection. Due to the binary nature of the data, the posterior distributions of the parameters are not in explicit form, and we need to use a combination of truncated sampling and Markov Chain Monte Carlo (MCMC) based computation techniques to simulate the posterior distributions. The Bayesian model is ?exible enough to identify the signi?cant genes as well as to perform future predictions. The method is applied to cancer classi?cation via cDNA microarrays. In particular, the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the method is used to identify the set of signi?cant genes to classify BRCA1 and others. Microarray data can also be applied to survival models. We address the issue of how to reduce the dimension in building model by selecting signi?cant genes as well as assessing the estimated survival curves. Additionally, we consider the wellknown Weibull regression and semiparametric proportional hazards (PH) models for survival analysis. With microarray data, we need to consider the case where the number of covariates p exceeds the number of samples n. Speci?cally, for a given vector of response values, which are times to event (death or censored times) and p gene expressions (covariates), we address the issue of how to reduce the dimension by selecting the responsible genes, which are controlling the survival time. This approach enables us to estimate the survival curve when n << p. In our approach, rather than ?xing the number of selected genes, we will assign a prior distribution to this number. The approach creates additional ?exibility by allowing the imposition of constraints, such as bounding the dimension via a prior, which in e?ect works as a penalty. To implement our methodology, we use a Markov Chain Monte Carlo (MCMC) method. We demonstrate the use of the methodology with (a) di?use large B??cell lymphoma (DLBCL) complementary DNA (cDNA) data and (b) Breast Carcinoma data. Lastly, we propose a mixture of Dirichlet process models using discrete wavelet transform for a curve clustering. In order to characterize these time??course gene expresssions, we consider them as trajectory functions of time and gene??speci?c parameters and obtain their wavelet coe?cients by a discrete wavelet transform. We then build cluster curves using a mixture of Dirichlet process priors.

Selection of signi?cant genes via expression patterns is important in a microarray problem. Owing to small sample size and large number of variables (genes), the selection process can be unstable. This research proposes a hierarchical Bayesian model for gene (variable) selection. We employ latent variables in a regression setting and use a Bayesian mixture prior to perform the variable selection. Due to the binary nature of the data, the posterior distributions of the parameters are not in explicit form, and we need to use a combination of truncated sampling and Markov Chain Monte Carlo (MCMC) based computation techniques to simulate the posterior distributions. The Bayesian model is ?exible enough to identify the signi?cant genes as well as to perform future predictions. The method is applied to cancer classi?cation via cDNA microarrays. In particular, the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the method is used to identify the set of signi?cant genes to classify BRCA1 and others. Microarray data can also be applied to survival models. We address the issue of how to reduce the dimension in building model by selecting signi?cant genes as well as assessing the estimated survival curves. Additionally, we consider the wellknown Weibull regression and semiparametric proportional hazards (PH) models for survival analysis. With microarray data, we need to consider the case where the number of covariates p exceeds the number of samples n. Speci?cally, for a given vector of response values, which are times to event (death or censored times) and p gene expressions (covariates), we address the issue of how to reduce the dimension by selecting the responsible genes, which are controlling the survival time. This approach enables us to estimate the survival curve when n << p. In our approach, rather than ?xing the number of selected genes, we will assign a prior distribution to this number. The approach creates additional ?exibility by allowing the imposition of constraints, such as bounding the dimension via a prior, which in e?ect works as a penalty. To implement our methodology, we use a Markov Chain Monte Carlo (MCMC) method. We demonstrate the use of the methodology with (a) di?use large B??cell lymphoma (DLBCL) complementary DNA (cDNA) data and (b) Breast Carcinoma data. Lastly, we propose a mixture of Dirichlet process models using discrete wavelet transform for a curve clustering. In order to characterize these time??course gene expresssions, we consider them as trajectory functions of time and gene??speci?c parameters and obtain their wavelet coe?cients by a discrete wavelet transform. We then build cluster curves using a mixture of Dirichlet process priors.