This thesis consists of three parts. In the first part, we recall some background theory that will be used throughout the thesis. In the second part, we studied the existence and uniqueness of solutions of the stochastic heat equations with Markovian switching. In the third part, we investigate the properties of solutions, such as Feller property, strong Feller property and stability.

Operational risk losses are heavy tailed and are likely to be asymmetric and extremely dependent among business lines/event types. The analysis of dependence via copula models has been focussed on the bivariate case mainly. In the vast majority of instances symmetric elliptical copulas are employed to model dependence for severities. This thesis proposes a new methodology to assess, in a multivariate way, the asymmetry and extreme dependence between severities, and to calculate the capital for operational risk. This methodology simultaneously uses (i) several parametric distributions and an alternative mixture distribution (the Lognormal for the body of losses and the generalised Pareto Distribution for the tail) using a technique from extreme value theory, (ii) the multivariate skew t-copula applied for the first time across severities and (iii) Bayesian theory. The former to model severities, I test simultaneously several parametric distributions and the mixture distribution for each business line. This procedure enables me to achieve multiple combinations of the severity distribution and to find which fits most closely. The second to effectively model asymmetry and extreme dependence in high dimensions. The third to estimate the copula model, given the high multivariate component (i.e. eight business lines and seven event types) and the incorporation of mixture distributions it is highly difficult to implement maximum likelihood. Therefore, I use a Bayesian inference framework and Markov chain Monte Carlo simulation to evaluate the posterior distribution to estimate and make inferences of the parameters of the skew t-copula model. The research analyses an updated operational loss data set, SAS® Operational Risk Global Data (SAS OpRisk Global Data), to model operational risk at international financial institutions. I then evaluate the impact of this multivariate, asymmetric and extreme dependence on estimating the total regulatory capital, among other established multivariate copulas. My empirical findings are consistent with other studies reporting thin and medium-tailed loss distributions. My approach substantially outperforms symmetric elliptical copulas, demonstrating that modelling dependence via the skew t-copula provides a more efficient allocation of capital charges of up to 56% smaller than that indicated by the standard Basel model.

O procedimento de controle on-line de processos por atributos, proposto por Taguchi et al. (1989), consiste em amostrar um item a cada m produzidos e decidir, a cada inspeção, se houve ou não aumento na fração de itens não-conformes produzidos. Caso o item inspecionado seja não-conforme, pára-se o processo para ajuste supondo-se que tenha ocorrido uma mudança para a condição fora de controle. Como i) o sistema de inspeção pode estar sujeito a erros de classificação e o item inspecionado ser submetido à classificações repetidas; ii) a fração de itens não-conformes no estado fora de controle pode variar em função do número de itens produzidos (x) segundo uma função y (x); iii) e a decisão de parar o processo pode ser tomada com base no resultado das últimas h inspeções, desenvolve-se um modelo que engloba estes pontos. Utilizando-se as propriedades de uma cadeia de Markov ergódica, obtém-se a expressão do custo médio do sistema de controle, que é minimizada por parâmetros que vão além do intervalo de inspeção m: o número de classificações repetidas r; o número mínimo de classificações conformes para declarar um item como conforme s, o comprimento do histórico de inspeções considerado h e o critério de parada para ajuste u. Os resultados obtidos mostram que: o uso de classificações repetidas pode ser uma alternativa econômica quando apenas um item é considerado na decisão sobre o ajuste do processo; uma cadeia da Markov finita pode ser utilizada para representar o sistema de controle na presença de uma função y (x) não-constante; tomar a decisão de ajuste com base na observação de uma seqüência de itens inspecionados é a alternativa de maior impacto sobre o custo de controle do processo. / The quality control procedure for attributes, proposed by Taguchi et al. (1989), consists in inspecting a single item at every m produced items and, based on the result of each inspection, deciding weather the non-conforming fraction has increased or not. If an inspected item is declared non-conforming, the process is stopped and adjusted, assuming that it has changed to out-of-control condition. Once: i) the inspection system is subject to misclassification and it is possible to carry out repetitive classifications in the inspected item; ii) the non-conforming fraction, when the process is out-of-control, can be described by y(x); iii) the decision about stopping the process can be based on last h inspections, a model which considers those points is developed. Using properties of ergodic Markov chains, the average cost expression is calculated and can be minimized by parameters beyond m: number of repetitive classifications (r); minimum number of classifications as conforming to declare an item as conforming (s); number of inspections taken into account (h) and stopping criteria (u). The results obtained show that: repetitive classifications of the inspected item can be a viable option if only one item is used to decide about the process condition; a finite Markov chain can be used to represent the control procedure in presence of a function y(x); deciding about the process condition based on last h inspections has a significant impact on the average cost.

Mapas genéticos são arranjos lineares que indicam a ordem e distância entre locos nos cromossomos de uma determinada espécie. Recentemente, a grande disponibilidade de marcadores moleculares tem tornado estes mapas cada vez mais saturados, sendo necessários métodos eficientes para sua construção. Uma das etapas que merece mais atenção na construção de mapas de ligação é a ordenação dos marcadores genéticos dentro de cada grupo de ligação. Tal ordenação é considerada um caso especial do clássico problema do caixeiro viajante (TSP), que consiste em escolher a melhor ordem entre todas as possíveis. Entretanto, a estratégia de busca exaustiva torna-se inviável quando o número de marcadores é grande. Nesses casos, para que esses mapas possam ser construídos uma alternativa viável é a utilização de algoritmos que forneçam soluções aproximadas. O objetivo desse trabalho foi avaliar a eficiência dos algoritmos Try (TRY), Seriation (SER), Rapid Chain Delineation (RCD), Recombination Counting and Ordering (RECORD) e Unidirectional Growth (UG), além dos critérios PARF (produto mínimo das frações de recombinação adjacentes), SARF (soma mínima das frações de recombinação adjacentes), SALOD (soma máxima dos LOD scores adjacentes) e LMHC (verossimilhança via cadeias de Markov ocultas), usados juntamente com o algoritmo de verificação de erros RIPPLE, para a construção de mapas genéticos. Para tanto, foi simulado um mapa de ligação de uma espécie vegetal hipotética, diplóide e monóica, contendo 21 marcadores com distância fixa entre eles de 3 centimorgans. Usando o método Monte Carlo, foram obtidas aleatoriamente 550 populações F2 com 100 e 400 indivíduos, além de diferentes combinações de marcadores dominantes e codominantes. Foi ainda simulada perda de 10% e 20% dos dados. Os resultados mostraram que os algoritmos TRY e SER tiveram bons resultados em todas as situações simuladas, mesmo com presença de elevado número de dados perdidos e marcadores dominantes ligados em repulsão, podendo ser então recomendado em situações práticas. Os algoritmos RECORD e UG apresentaram bons resultados na ausência de marcadores dominantes ligados em repulsão, podendo então ser recomendados em situações com poucos marcadores dominantes. Dentre todos os algoritmos, o RCD foi o que se mostrou menos eficiente. O critério LHMC, aplicado com o algoritmo RIPPLE, foi o que apresentou melhores resultados quando se deseja fazer verificações de erros na ordenação. / Genetic linkage maps are linear arrangements showing the order and distance between loci in chromosomes of a particular species. Recently, the availability of molecular markers has made such maps more saturated and efficient methods are needed for their construction. One of the steps that deserves more attention in the construction of genetic linkage maps is the ordering of genetic markers within each linkage group. This ordering is considered a special case of the classic traveling salesman problem (TSP), which consists in choosing the best order among all possible ones. However, the strategy of exhaustive search becomes unfeasible when the number of markers is large. One possible alternative to construct such maps is to use algorithms that provide approximate solutions. Thus, the aim of this work was to evaluate the efficiency of algorithms Try (TRY), Seriation (SER), Rapid Chain Delineation (RCD), Recombination Counting and Ordering (RECORD) and Unidirectional Growth (UG), as well as the criteria PARF (product of adjacent recombination fractions), SARF (sum of adjacent recombination fractions), SALOD (sum of adjacent lod scores) and LMHC (likelihood via hidden Markov chains), used with the RIPPLE algorithm for error verification, in the construction of genetic linkage maps. For doing so, a linkage map of a hypothetical diploid and monoecious plant species was simulated, containing 21 markers with fixed distance of 3 centimorgans between them. Using Monte Carlo methods, 550 F2 populations were randomly simulated with 100 and 400 individuals, together with different combinations of dominant and codominant markers. 10 % and 20 % of missing data was also included. Results showed that the algorithms TRY and SER gave good results in all situations, even with presence of a large number of missing data and dominant markers linked in repulsion phase. Thus, these can be recommended for analyzing real data. The algorithms RECORD and UG gave good results in the absence of dominant markers linked in repulsion phase and can be used in this case. Among all algorithms, RCD was the least efficient. The criterion LHMC, applied with the RIPPLE algorithm, showed the best results when the goal is to check ordering errors.

Estudamos o comportamento da familia aresta-triangulo de grafos aleatorios exponenciais (ERG) usando metodos de Monte Carlo baseados em Cadeias de Markov. Comparamos contagens de subgrafos e correlacoes entre arestas de ergs as de Grafos Aleatorios Binomiais (BRG, tambem chamados de Erdos-Renyi). E um resultado teorico conhecido que para algumas parametrizacoes os limites das contagens de subgrafos de ERGs convergem para os de BRGs, assintoticamente no numero de vertices [BBS11, CD11]. Observamos esse fenomeno em grafos com poucos (20) vertices em nossas simulacoes. / We study the behavior of the edge-triangle family of exponential random graphs (ERG) using the Markov Chain Monte Carlo method. We compare ERG subgraph counts and edge correlations to those of the classic Binomial Random Graph (BRG, also called Erdos-Renyi model). It is a known theoretical result that for some parameterizations the limit ERG subgraph counts converge to those of BRGs, as the number of vertices grows [BBS11, CD11]. We observe this phenomenon on graphs with few (20) vertices in our simulations.

abstract: In this work, I present a Bayesian inference computational framework for the analysis of widefield microscopy data that addresses three challenges: (1) counting and localizing stationary fluorescent molecules; (2) inferring a spatially-dependent effective fluorescence profile that describes the spatially-varying rate at which fluorescent molecules emit subsequently-detected photons (due to different illumination intensities or different local environments); and (3) inferring the camera gain. My general theoretical framework utilizes the Bayesian nonparametric Gaussian and beta-Bernoulli processes with a Markov chain Monte Carlo sampling scheme, which I further specify and implement for Total Internal Reflection Fluorescence (TIRF) microscopy data, benchmarking the method on synthetic data. These three frameworks are self-contained, and can be used concurrently so that the fluorescence profile and emitter locations are both considered unknown and, under some conditions, learned simultaneously. The framework I present is flexible and may be adapted to accommodate the inference of other parameters, such as emission photophysical kinetics and the trajectories of moving molecules. My TIRF-specific implementation may find use in the study of structures on cell membranes, or in studying local sample properties that affect fluorescent molecule photon emission rates. / Dissertation/Thesis / Masters Thesis Applied Mathematics 2019

<div>Multi-model inference covers a wide range of modern statistical applications such as variable selection, model confidence set, model averaging and variable importance.</div><div>The performance of multi-model inference depends on the availability of candidate models, whose quality has been rarely studied in literature. In this dissertation, we study genetic algorithm (GA) in order to obtain high-quality candidate models. Inspired by the process of natural selection, GA performs genetic operations such as selection, crossover and mutation iteratively to update a collection of potential solutions (models) until convergence. The convergence properties are studied based on the Markov chain theory and used to design an adaptive termination criterion that vastly reduces the computational cost. In addition, a new schema theory is established to characterize how the current model set is improved through evolutionary process. Extensive numerical experiments are carried out to verify our theory and demonstrate the empirical power of GA, and new findings are obtained for two real data examples. </div>

Dementia is a clinical syndrome caused by neurodegeneration or cerebrovascular injury. Patients with dementia suffer from deterioration in memory, thinking, behavior and the ability to perform everyday activities. Since there are no cures or disease-modifying therapies for dementia, there is much interest in identifying modifiable risk factors that may help prevent or slow the progression of cognitive decline. Medications are a common focus of this type of research. Importantly, according to a report from the Centers for Disease Control and Prevention (CDC), 19.1% of the population aged 60 and over report taking antidepressants during 2011-2014, and this number tends to increase. However, antidepressant use among the elderly may be concerning because of the potentially harmful effects on cognition. To assess the impacts of antidepressants on the risk of dementia, we conducted three consecutive projects. In the first project, a retrospective cohort study using Marginal Structural Cox Proportional Hazards regression model with Inverse Probability Weighting (IPW) was conducted to evaluate the average causal effects of different classes of antidepressant on the risk of dementia. Potential causal effects of selective serotonin reuptake inhibitors (SSRIs), serotonin and norepinephrine reuptake inhibitors (SNRIs), atypical anti-depressants (AAs) and tri-cyclic antidepressants (TCAs) on the risk of dementia were observed at the 0.05 significance level. Multiple sensitivity analyses supported these findings. Unmeasured confounding is a threat to the validity of causal inference methods. In evaluating the effects of antidepressants, it is important to consider how common comorbidities of depression, such as sleep disorders, may affect both the exposure to anti-depressants and the onset of cognitive impairment. In this dissertation, sleep apnea and rapid-eye-movement behavior disorder (RBD) were unmeasured and thus uncontrolled confounders for the association between antidepressant use and the risk of dementia. In the second project, a bias factor formula for two binary unmeasured confounders was derived in order to account for these variables. Monte Carlo analysis was implemented to estimate the distribution of the bias factor for each class of antidepressant. The effects of antidepressants on the risk of dementia adjusted for both measured and unmeasured confounders were estimated. Sleep apnea and RBD attenuated the effect estimates for SSRI, SNRI and AA on the risk of dementia. In the third project, to account for potential time-varying confounding and observed time-varying treatment, a multi-state Markov chain with three transient states (normal cognition, mild cognitive impairment (MCI), and impaired but not MCI) and two absorbing states (dementia and death) was performed to estimate the probabilities of moving between finite and mutually exclusive cognitive state. This analysis also allowed participants to recover from mild impairments (i.e., mild cognitive impairment, impaired but not MCI) to normal cognition, and accounted for the competing risk of death prior to dementia. These findings supported the results of the main analysis in the first project.

Queuing systems are important parts of our daily lives and to keep their operations at an efficient level they need to be monitored by using queuing Performance Metrics, such as average queue lengths and average waiting times. On the other hand queue lengths and waiting times are generally random variables and their distributions depend on different properties like arrival rates, service times, number of servers. We focused on detecting the change in service rates in this report. Therefore, we monitored queues by using Cumulative Sum(CUSUM) charts based on likelihood ratios and compared the Average Run Length values of different service rates.

Statistical analyses and modeling have contributed greatly to our understanding of the pathogenesis of HIV-1 infection; they also provide guidance for the treatment of AIDS patients and evaluation of antiretroviral (ARV) therapies. Various statistical methods, nonlinear mixed-effects models in particular, have been applied to model the CD4 and viral load trajectories. A common assumption in these methods is all patients come from a homogeneous population following one mean trajectories. This assumption unfortunately obscures important characteristic difference between subgroups of patients whose response to treatment and whose disease trajectories are biologically different. It also may lack the robustness against population heterogeneity resulting misleading or biased inference. Finite mixture models, also known as latent class models, are commonly used to model nonpredetermined heterogeneity in a population; they provide an empirical representation of heterogeneity by grouping the population into a finite number of latent classes and modeling the population through a mixture distribution. For each latent class, a finite mixture model allows individuals in each class to vary around their own mean trajectory, instead of a common one shared by all classes. Furthermore, a mixture model has ability to cluster and estimate class membership probabilities at both population and individual levels. This important feature may help physicians to better understand a particular patient disease progression and refine the therapeutical strategy in advance. In this research, we developed mixture dynamic model and related Bayesian inferences via Markov chain Monte Carlo (MCMC). One real data set from HIV/AIDS clinical management and another from clinical trial were used to illustrate the proposed models and methods. This dissertation explored three topics. First, we modeled the CD4 trajectories using a finite mixture model with four distinct components of which the mean functions are designed based on Michaelis-Menten function. Relevant covariates both baseline and time-varying were considered and model comparison and selection were based on such-criteria as Deviance Information Criteria (DIC). Class membership model was allowed to depend on covariates for prediction. Second, we explored disease status prediction HIV/AIDS using the latent class membership model. Third, we modeled viral load trajectories using a finite mixture model with three components of which the mean functions are designed based on published HIV dynamic systems. Although this research is motivated by HIV/AIDS studies, the basic concepts and methods developed here have much broader applications in management of other chronic diseases; they can also be applied to dynamic systems in other fields. Implementation of our methods using the publicly- vailable WinBUGS package suggest that our approach can be made quite accessible to practicing statisticians and data analysts.