Functional magnetic resonance imaging (fMRI) is a prime example of multi-disciplinary research. Without the beautiful physics of MRI, there wouldnot be any images to look at in the first place. To obtain images of goodquality, it is necessary to fully understand the concepts of the frequencydomain. The analysis of fMRI data requires understanding of signal pro-cessing, statistics and knowledge about the anatomy and function of thehuman brain. The resulting brain activity maps are used by physicians,neurologists, psychologists and behaviourists, in order to plan surgery andto increase their understanding of how the brain works. This thesis presents methods for real-time fMRI and non-parametric fMRIanalysis. Real-time fMRI places high demands on the signal processing,as all the calculations have to be made in real-time in complex situations.Real-time fMRI can, for example, be used for interactive brain mapping.Another possibility is to change the stimulus that is given to the subject, inreal-time, such that the brain and the computer can work together to solvea given task, yielding a brain computer interface (BCI). Non-parametricfMRI analysis, for example, concerns the problem of calculating signifi-cance thresholds and p-values for test statistics without a parametric nulldistribution. Two BCIs are presented in this thesis. In the first BCI, the subject wasable to balance a virtual inverted pendulum by thinking of activating theleft or right hand or resting. In the second BCI, the subject in the MRscanner was able to communicate with a person outside the MR scanner,through a virtual keyboard. A graphics processing unit (GPU) implementation of a random permuta-tion test for single subject fMRI analysis is also presented. The randompermutation test is used to calculate significance thresholds and p-values forfMRI analysis by canonical correlation analysis (CCA), and to investigatethe correctness of standard parametric approaches. The random permuta-tion test was verified by using 10 000 noise datasets and 1484 resting statefMRI datasets. The random permutation test is also used for a non-localCCA approach to fMRI analysis.

Previous work by Flaxman (2004) and Biers-Ariel et al. (2018) focused on the number of distinct words embedded in a string of words of length n. In this thesis, we will extend this work to permutations, focusing on the maximum number of distinct permutations contained in a permutation on [n] = {1,2,...,n} and on the expected number of distinct permutations contained in a random permutation on [n]. We further considered the problem where repetition of subsequences are as a result of the occurrence of (Type A and/or Type B) replications. Our method of enumerating the Type A replications causes double counting and as a result causes the count of the number of distinct sequences to go down.

In its theoretical part, this thesis sums up the basic knowledge concerning permutations. Besides the representation of permutations and determination of their fundamental characteristics, the theoretical part is, first of all, aimed at results concerning the decomposition of permutations into disjoint cycles and at finding the number of permutations with a certain characteristic. We introduce the fundamental bijection that is useful for solving many problems concerning the permutations. Further on, we focus on the number of permutations without a fixed point, Eulerian numbers expressing the number of permutations with a given number of descents, and the number of permutations with a given number of excedances, Stirling numbers of the first kind expressing the number of permutations with a given number of cycles, and Catalan numbers representing the number of permutations avoiding a chosen pattern of length three. Attention is also paid to the Gilbreath permutations and their characteristics. The practical part consists of 14 solved problems. The solutions rely on the results presented in the theoretical part, and there are deduced some further interesting results concerning random permutations.

Estudamos o problema de obtenção de estimadores/preditores ótimos para combinações lineares de respostas coletadas de uma população finita por meio de amostragem aleatória simples. Nesse contexto, estendemos o modelo misto para populações finitas proposto por Stanek, Singer & Lencina (2004, Journal of Statistical Planning and Inference) para casos em que se incluem erros de medida (endógenos e exógenos) e informação auxiliar. Admitindo que as variâncias são conhecidas, mostramos que os estimadores/preditores propostos têm erro quadrático médio menor dentro da classe dos estimadores lineares não viciados. Por meio de estudos de simulação, comparamos o desempenho desses estimadores/preditores empíricos, i.e., obtidos com a substituição das componentes de variância por estimativas, com aquele de competidores tradicionais. Também, estendemos esses modelos para análise de estudos com estrutura do tipo pré-teste/pós-teste. Também por intermédio de simulação, comparamos o desempenho dos estimadores empíricos com o desempenho do estimador obtido por meio de técnicas clássicas de análise de medidas repetidas e com o desempenho do estimador obtido via análise de covariância por meio de mínimos quadrados, concluindo que os estimadores/ preditores empíricos apresentaram um menor erro quadrático médio e menor vício. Em geral, sugerimos o emprego dos estimadores/preditores empíricos propostos para dados com distribuição assimétrica ou amostras pequenas. / We consider optimal estimation of finite population parameters with data obtained via simple random samples. In this context, we extend a finite population mixed model proposed by Stanek, Singer & Lencina (2004, Journal of Statistical Planning and Inference) by including measurement errors (endogenous or exogenous) and auxiliary information. Assuming that variance components are known, we show that the proposed estimators/predictors have the smallest mean squared error in the class of unbiased estimators. Using simulation studies, we compare the performance of the empirical estimators/predictors obtained by replacing variance components with estimates with the performance of a traditional estimator. We also extend the finite population mixed model to data obtained via pretest-posttest designs. Through simulation studies, we compare the performance of the empirical estimator of the difference in gain between groups with the performance of the usual repeated measures estimator and with the performance of the usual analysis of covariance estimator obtained via ordinary least squares. The empirical estimator has smaller mean squared error and bias than the alternative estimators under consideration. In general, we recommend the use of the proposed estimators/ predictors for either asymmetric response distributions or small samples.

Estudamos o problema de obtenção de estimadores/preditores ótimos para combinações lineares de respostas coletadas de uma população finita por meio de amostragem aleatória simples. Nesse contexto, estendemos o modelo misto para populações finitas proposto por Stanek, Singer & Lencina (2004, Journal of Statistical Planning and Inference) para casos em que se incluem erros de medida (endógenos e exógenos) e informação auxiliar. Admitindo que as variâncias são conhecidas, mostramos que os estimadores/preditores propostos têm erro quadrático médio menor dentro da classe dos estimadores lineares não viciados. Por meio de estudos de simulação, comparamos o desempenho desses estimadores/preditores empíricos, i.e., obtidos com a substituição das componentes de variância por estimativas, com aquele de competidores tradicionais. Também, estendemos esses modelos para análise de estudos com estrutura do tipo pré-teste/pós-teste. Também por intermédio de simulação, comparamos o desempenho dos estimadores empíricos com o desempenho do estimador obtido por meio de técnicas clássicas de análise de medidas repetidas e com o desempenho do estimador obtido via análise de covariância por meio de mínimos quadrados, concluindo que os estimadores/ preditores empíricos apresentaram um menor erro quadrático médio e menor vício. Em geral, sugerimos o emprego dos estimadores/preditores empíricos propostos para dados com distribuição assimétrica ou amostras pequenas. / We consider optimal estimation of finite population parameters with data obtained via simple random samples. In this context, we extend a finite population mixed model proposed by Stanek, Singer & Lencina (2004, Journal of Statistical Planning and Inference) by including measurement errors (endogenous or exogenous) and auxiliary information. Assuming that variance components are known, we show that the proposed estimators/predictors have the smallest mean squared error in the class of unbiased estimators. Using simulation studies, we compare the performance of the empirical estimators/predictors obtained by replacing variance components with estimates with the performance of a traditional estimator. We also extend the finite population mixed model to data obtained via pretest-posttest designs. Through simulation studies, we compare the performance of the empirical estimator of the difference in gain between groups with the performance of the usual repeated measures estimator and with the performance of the usual analysis of covariance estimator obtained via ordinary least squares. The empirical estimator has smaller mean squared error and bias than the alternative estimators under consideration. In general, we recommend the use of the proposed estimators/ predictors for either asymmetric response distributions or small samples.