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We are entering an exciting era, rich in the availability of data via sources such as the Internet, satellites, particle colliders, telecommunication networks, computer simulations, and the like. The confluence of increasing computational resources, volumes of data, and variety of statistical procedures has brought us to a modern enlightenment. Within the next century, these tools will combine to reveal unforeseeable insights into the social and natural sciences. Perhaps the largest headwind we now face is our collectively slow-moving imagination. Like a car on an open road, learning is limited by its own rate. Historically, slow information dissemination and the unavailability of experimental resources limited our learning. To that point, any methodological contribution that helps in the conversion of data into knowledge will accelerate us along this open road. Furthermore, if that contribution is accessible to others, the speedup in knowledge discovery scales exponentially. Markov chain Monte Carlo (MCMC) is a broad class of powerful algorithms, typically used for Bayesian inference. Despite their variety and versatility, these algorithms rarely become mainstream workhorses because they can be difficult to implement. The humble goal of this work is to bring to the table a few more highly versatile and robust, yet easily-tuned algorithms. Specifically, we introduce weighted particle tempering, a parallelizable MCMC procedure that is adaptable to large computational resources. We also explore and develop a highly practical implementation of reversible jump, the most generalized form of MetropolisHastings. Finally, we combine these two algorithms into reversible jump weighted particle tempering, and apply it on a model and dataset that was partially collected by the author and his collaborators, halfway around the world. It is our hope that by introducing, developing, and exhibiting these algorithms, we can make a reasonable contribution to the ever-growing body of MCMC research. / Ph. D.

Muitas são as características quantitativas que são, significativamente, influenciadas por fatores genéticos, em geral, existem vários genes que colaboram para a variação de uma ou mais características quantitativas. As informações ausentes a respeito dos genótipos nos marcadores moleculares é um problema comum em estudo de mapeamento genético e, por conseguinte, no mapeamento dos locus que controlam estas características fenotípicas (QTL). Os dados que não foram observados ocorrem, principalmente, devido a erros de genotipagem e de marcadores não informativos. Para solucionar este problema foi utilizado o método do modelo oculto de Markov para inferir estes dados. Os métodos de acurácias evidenciaram o sucesso da aplicação desta técnica de imputa- ção. Uma vez imputado, na inferência bayesiana estes dados não serão mais tratados como uma variável aleatória resultando assim, numa redução no espaço paramétrico do modelo. Outra grande dificuldade no mapeamento de QTL se deve ao fato de que não se conhece ao certo a quantidade destes que influenciam uma dada característica, fazendo com que surjam diversos problemas, um deles é a dimensão do espaço paramétrico e, consequentemente, a obtenção da amostra a posteriori. Assim, com o objetivo de contornar este problema foi proposta a utilização do método Monte Carlo via cadeia de Markov com Saltos Reversíveis, uma vez que este permite flutuar, entre cada iteração, modelos com diferentes quantidades de parâmetros. A utilização da abordagem bayesiana permitiu detectar cinco QTL para a característica estudada. Todas as análises foram implementadas no programa estatístico R. / There are many quantitative characteristics which are significantly influenced by genetic factors, in general, there are several genes that contribute to the variation of one or more quantitative trait. The missing information about the genotypes in molecular markers is a common problem in studying genetic mapping and therefore the mapping of loci that control these phenotypic traits (QTL). The data were not observed occur mainly due to errors in genotyping and uninformative markers. To solve this problem the method of occult Markov model to infer this information was used. Techniques accuracies demonstrated the successful application of this technique of imputation. Once allocated, in the Bayesian inference this data will no longer be treated as a random variable thus resulting in a reduction in the parameter space of the model. Another great difficulty in mapping QTL is due to the fact that no one knows exactly the amount of these which influence a given characteristic, so that several problems arise, one of them is dimension of the parameter space and, consequently, obtaining the sample a posterior. Thus, in order to solve this problem using the method via Monte Carlo Markov chain Reversible Jump was proposed, since this allows fluctuate between each iteration, models with different numbers of parameters. The use of the Bayesian approach allowed five QTL detected for the studied trait. All analyzes were implemented in the statistical software R.

Bayesian surface smoothing using splines usually proceeds by choosing the smoothness parameter through the use of data driven methods like generalized cross validation. In this methodology, knots of the splines are assumed to lie at the data locations. When anisotropy is present in the data, modeling is done via parametric functions. In the present thesis, we have proposed a non-parametric approach to Bayesian surface smoothing in the presence of anisotropy. We use eigenfunctions generated by thin-plate splines as our basis functions. Using eigenfunctions does away with having to place knots arbitrarily, as is done customarily. The smoothing parameter, the anisotropy matrix, and other parameters are simultaneously updated by a Reversible Jump Markov Chain Monte Carlo (RJMCMC) sampler. Unique in our implementation is model selection, which is again done concurrently with the parameter updates. Since the posterior distribution of the coefficients of the basis functions for any given model order is available in closed form, we are able to simplify the sampling algorithm in the model selection step. This also helps us in isolating the parameters which influence the model selection step. We investigate the relationship between the number of basis functions used in the model and the smoothness parameter and find that there is a delicate balance which exists between the two. Higher values of the smoothness parameter correspond to more number of basis functions being selected. Use of a non-parametric approach to Bayesian surface smoothing provides for more modeling flexibility. We are not constrained by the shape defined by a parametric shape of the covariance as used by earlier methods. A Bayesian approach also allows us to include the results obtained from previous analysis of the same data, if any, as prior information. It also allows us to evaluate pointwise estimates of variability of the fitted surface. We believe that our research also poses many questions for future research.

The main goal of this research is to estimate the model parameters and to detect multiple change points in the long memory parameter of Gaussian ARFIMA(p, d, q) processes. Our approach is Bayesian and inference is done on wavelet domain. Long memory processes have been widely used in many scientific fields such as economics, finance and computer science. Wavelets have a strong connection with these processes. The ability of wavelets to simultaneously localize a process in time and scale domain results in representing many dense variance-covariance matrices of the process in a sparse form. A wavelet-based Bayesian estimation procedure for the parameters of Gaussian ARFIMA(p, d, q) process is proposed. This entails calculating the exact variance-covariance matrix of given ARFIMA(p, d, q) process and transforming them into wavelet domains using two dimensional discrete wavelet transform (DWT2). Metropolis algorithm is used for sampling the model parameters from the posterior distributions. Simulations with different values of the parameters and of the sample size are performed. A real data application to the U.S. GNP data is also reported. Detection and estimation of multiple change points in the long memory parameter is also investigated. The reversible jump MCMC is used for posterior inference. Performances are evaluated on simulated data and on the Nile River dataset.

The main goal of this research is to estimate the model parameters and to detect multiple change points in the long memory parameter of Gaussian ARFIMA(p, d, q) processes. Our approach is Bayesian and inference is done on wavelet domain. Long memory processes have been widely used in many scientific fields such as economics, finance and computer science. Wavelets have a strong connection with these processes. The ability of wavelets to simultaneously localize a process in time and scale domain results in representing many dense variance-covariance matrices of the process in a sparse form. A wavelet-based Bayesian estimation procedure for the parameters of Gaussian ARFIMA(p, d, q) process is proposed. This entails calculating the exact variance-covariance matrix of given ARFIMA(p, d, q) process and transforming them into wavelet domains using two dimensional discrete wavelet transform (DWT2). Metropolis algorithm is used for sampling the model parameters from the posterior distributions. Simulations with different values of the parameters and of the sample size are performed. A real data application to the U.S. GNP data is also reported. Detection and estimation of multiple change points in the long memory parameter is also investigated. The reversible jump MCMC is used for posterior inference. Performances are evaluated on simulated data and on the Nile River dataset.

Muitas são as características quantitativas que são, significativamente, influenciadas por fatores genéticos, em geral, existem vários genes que colaboram para a variação de uma ou mais características quantitativas. As informações ausentes a respeito dos genótipos nos marcadores moleculares é um problema comum em estudo de mapeamento genético e, por conseguinte, no mapeamento dos locus que controlam estas características fenotípicas (QTL). Os dados que não foram observados ocorrem, principalmente, devido a erros de genotipagem e de marcadores não informativos. Para solucionar este problema foi utilizado o método do modelo oculto de Markov para inferir estes dados. Os métodos de acurácias evidenciaram o sucesso da aplicação desta técnica de imputa- ção. Uma vez imputado, na inferência bayesiana estes dados não serão mais tratados como uma variável aleatória resultando assim, numa redução no espaço paramétrico do modelo. Outra grande dificuldade no mapeamento de QTL se deve ao fato de que não se conhece ao certo a quantidade destes que influenciam uma dada característica, fazendo com que surjam diversos problemas, um deles é a dimensão do espaço paramétrico e, consequentemente, a obtenção da amostra a posteriori. Assim, com o objetivo de contornar este problema foi proposta a utilização do método Monte Carlo via cadeia de Markov com Saltos Reversíveis, uma vez que este permite flutuar, entre cada iteração, modelos com diferentes quantidades de parâmetros. A utilização da abordagem bayesiana permitiu detectar cinco QTL para a característica estudada. Todas as análises foram implementadas no programa estatístico R. / There are many quantitative characteristics which are significantly influenced by genetic factors, in general, there are several genes that contribute to the variation of one or more quantitative trait. The missing information about the genotypes in molecular markers is a common problem in studying genetic mapping and therefore the mapping of loci that control these phenotypic traits (QTL). The data were not observed occur mainly due to errors in genotyping and uninformative markers. To solve this problem the method of occult Markov model to infer this information was used. Techniques accuracies demonstrated the successful application of this technique of imputation. Once allocated, in the Bayesian inference this data will no longer be treated as a random variable thus resulting in a reduction in the parameter space of the model. Another great difficulty in mapping QTL is due to the fact that no one knows exactly the amount of these which influence a given characteristic, so that several problems arise, one of them is dimension of the parameter space and, consequently, obtaining the sample a posterior. Thus, in order to solve this problem using the method via Monte Carlo Markov chain Reversible Jump was proposed, since this allows fluctuate between each iteration, models with different numbers of parameters. The use of the Bayesian approach allowed five QTL detected for the studied trait. All analyzes were implemented in the statistical software R.

This paper focuses on how to improve strategic asset allocation in practice. Strategic asset allocation is perhaps the most fundamental issue in portfolio management and it has been thoroughly discussed in previous research. We take our starting point in the traditional work of Markowitz within portfolio optimization. We provide a new solution of how to perform portfolio optimization in practice, or more specifically how to estimate the covariance matrix, which is needed to perform conventional portfolio optimization. Many researchers within this field have noted that the return distribution of financial assets seems to vary over time, so called regime switching, which makes it dicult to estimate the covariance matrix. We solve this problem by using a Bayesian approach for developing a Markov chain Monte Carlo algorithm that detects break points in the return distribution of financial assets, thus enabling us to improve the estimation of the covariance matrix. We find that there are two break points during the time period studied and that the main difference between the periods are that the volatility was substantially higher for all assets during the period that corresponds to the financial crisis, whereas correlations were less affected. By evaluating the performance of the algorithm we find that the algorithm can increase the Sharpe ratio of a portfolio, thus that our algorithm can improve strategic asset allocation over time. / Detta examensarbete fokuserar på hur man kan förbättra tillämpningen av strategisk tillgångsslagsallokering i praktiken. Hur man allokerar kapital mellan tillgångsslag är kanske de mest fundamentala beslutet inom kapitalförvaltning och ämnet har diskuterats grundligt i litteraturen. Vårt arbete utgår från Markowitz traditionella teorier inom portföljoptimering och utifrån dessa tar vi fram ett nytt angreppssätt för att genomföra portföljoptimering i praktiken. Mer specifikt utvecklar vi ett nytt sätt att uppskatta kovar-iansmatrisen för avkastningsfördelningen för finansiella tillgångar, något som är essentiellt för att kunna beräkna de optimala portföljvikterna enligt Markowitz. Det påstås ofta att avkastningens fördelning förändras över tid; att det sker så kallade regimskiften, vilket försvårar uppskattningen av kovariansmatrisen. Vi löser detta problem genom att använda ett Bayesiansk angreppssätt där vi utvecklar en Markov chain Monte Carlo-algoritm som upptäcker brytpunkter i avkastningsfördelningen, vilket gör att uppskattningen av kovar-iansmatrisen kan förbättras. Vi finner två brytpunkter i fördelningen under den studerade tidsperioden och den huvudsakliga skillnaden mellan de olika tidsperioderna är att volatiliten var betydligt högre för samtliga tillgångar under den tidsperiod som motsvaras av finanskrisen, medan korrelationerna mellan tillgångsslagen inte påverkades lika mycket. Genom att utvärdera hur algoritmen presterar finner vi att den ökar en portföljs Sharpe ratio och således att den kan förbättra den strategiska allokeringen mellan tillgångsslagen över tid.

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