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Causal models have been used in different areas of knowledge in order to comprehend the causal associations between variables. Over the past decades, the amount of studies using these models have been growing a lot, especially those related to biological systems where studying and learning causal relationships among traits are essential for predicting the consequences of interventions in such system. Graph analysis (GA) and structural equation modeling (SEM) are tools used to explore such associations. While GA allows searching causal structures that express qualitatively how variables are causally connected, fitting SEM with a known causal structure allows to infer the magnitude of causal effects. Also SEM can be viewed as multiple regression models in which response variables can be explanatory variables for others. In quantitative genetics studies, SEM aimed to study the direct and indirect genetic effects associated to individuals through information related to them, beyond the observed characteristics, such as the kinship relations. In those studies typically the assumptions of linear relationships among traits are made. However, in some scenarios, nonlinear relationships can be observed, which make unsuitable the mentioned assumptions. To overcome this limitation, this paper proposes to use a mixed effects polynomial structural equation model, second or superior degree, to model those nonlinear relationships. Two studies were developed, a simulation and an application to real data. The first study involved simulation of 50 data sets, with a fully recursive causal structure involving three characteristics in which linear and nonlinear causal relations between them were allowed. The second study involved the analysis of traits related to dairy cows of the Holstein breed. Phenotypic relationships between traits were calving difficulty, gestation length and also the proportion of perionatal death. We compare the model of multiple traits and polynomials structural equations models, under different polynomials degrees in order to assess the benefits of the SEM polynomial of second or higher degree. For some situations the inappropriate assumption of linearity results in poor predictions of the direct, indirect and total of the genetic variances and covariance, either overestimating, underestimating, or even assign opposite signs to covariances. Therefore, we conclude that the inclusion of a polynomial degree increases the SEM expressive power. / Modelos causais têm sido muitos utilizados em estudos em diferentes áreas de conhecimento, a fim de compreender as associações ou relações causais entre variáveis. Durante as últimas décadas, o uso desses modelos têm crescido muito, especialmente estudos relacionados à sistemas biológicos, uma vez que compreender as relações entre características são essenciais para prever quais são as consequências de intervenções em tais sistemas. Análise do grafo (AG) e os modelos de equações estruturais (MEE) são utilizados como ferramentas para explorar essas relações. Enquanto AG nos permite buscar por estruturas causais, que representam qualitativamente como as variáveis são causalmente conectadas, ajustando o MEE com uma estrutura causal conhecida nos permite inferir a magnitude dos efeitos causais. Os MEE também podem ser vistos como modelos de regressão múltipla em que uma variável resposta pode ser vista como explanatória para uma outra característica. Estudos utilizando MEE em genética quantitativa visam estudar os efeitos genéticos diretos e indiretos associados aos indivíduos por meio de informações realcionadas aos indivíduas, além das característcas observadas, como por exemplo o parentesco entre eles. Neste contexto, é tipicamente adotada a suposição que as características observadas são relacionadas linearmente. No entanto, para alguns cenários, relações não lineares são observadas, o que torna as suposições mencionadas inadequadas. Para superar essa limitação, este trabalho propõe o uso de modelos de equações estruturais de efeitos polinomiais mistos, de segundo grau ou seperior, para modelar relações não lineares. Neste trabalho foram desenvolvidos dois estudos, um de simulação e uma aplicação a dados reais. O primeiro estudo envolveu a simulação de 50 conjuntos de dados, com uma estrutura causal completamente recursiva, envolvendo 3 características, em que foram permitidas relações causais lineares e não lineares entre as mesmas. O segundo estudo envolveu a análise de características relacionadas ao gado leiteiro da raça Holandesa, foram utilizadas relações entre os seguintes fenótipos: dificuldade de parto, duração da gestação e a proporção de morte perionatal. Nós comparamos o modelo misto de múltiplas características com os modelos de equações estruturais polinomiais, com diferentes graus polinomiais, a fim de verificar os benefícios do MEE polinomial de segundo grau ou superior. Para algumas situações a suposição inapropriada de linearidade resulta em previsões pobres das variâncias e covariâncias genéticas diretas, indiretas e totais, seja por superestimar, subestimar, ou mesmo atribuir sinais opostos as covariâncias. Portanto, verificamos que a inclusão de um grau de polinômio aumenta o poder de expressão do MEE.

針對金融時序資料變異數不齊一的性質，隨機波動模型除了提供於ARCH族外的另一選擇；且由於其設定隱含波動本身亦為一個隨機波動函數，藉由設定隨時間改變且自我相關的條件變異數，使得隨機波動模型較ARCH族來得有彈性且符合實際。傳統上處理隨機波動模型的參數估計往往需要面對到複雜的多維積分，此問題可藉由貝氏分析裡的馬可夫鏈蒙地卡羅法解決。本文主要的探討標的，即在於利用馬可夫鏈蒙地卡羅法估計美元/新台幣匯率隨機波動模型參數。除原始模型之外，模型的擴充分為三部分：其一為隱含波動的二階自我回歸模型；其二則為藉由基本模型的修改，檢測匯率市場上的槓桿效果；最後，我們嘗試藉由加入scale mixture的方式以驗證金融時序資料中常見的厚尾分配。

設二個隨機變數X1 和X2，所可能的發生值分別為{1,…,I}和{1,…,J}。條件機率分配模型為二個I × J 的矩陣A 和B，分別代表在X2 給定的條件下X1的條件機率分配和在X1 給定的條件下X2的條件機率分配。若存在一個聯合機率分配，而且它的二個條件機率分配剛好就是A 和B，則稱A和B相容。我們透過圖形表示法，提出新的二個離散條件機率分配會相容的充分必要條件。另外，我們證明，二個相容的條件機率分配會有唯一的聯合機率分配的充分必要條件為:所對應的圖形是相連的。我們也討論馬可夫鏈與相容性的關係。 / For two discrete random variables X1 and X2 taking values in {1,…,I} and {1,…,J}, respectively, a putative conditional model for the joint distribution of X1 and X2 consists of two I × J matrices representing the conditional distributions of X1 given X2 and of X2 given X1. We say that two conditional distributions (matrices) A and B are compatible if there exists a joint distribution of X1 and X2 whose two conditional distributions are exactly A and B. We present new versions of necessary and sufficient conditions for compatibility of discrete conditional distributions via a graphical representation. Moreover, we show that there is a unique joint distribution for two given compatible conditional distributions if and only if the corresponding graph is connected. Markov chain characterizations are also presented.

Causal models have been used in different areas of knowledge in order to comprehend the causal associations between variables. Over the past decades, the amount of studies using these models have been growing a lot, especially those related to biological systems where studying and learning causal relationships among traits are essential for predicting the consequences of interventions in such system. Graph analysis (GA) and structural equation modeling (SEM) are tools used to explore such associations. While GA allows searching causal structures that express qualitatively how variables are causally connected, fitting SEM with a known causal structure allows to infer the magnitude of causal effects. Also SEM can be viewed as multiple regression models in which response variables can be explanatory variables for others. In quantitative genetics studies, SEM aimed to study the direct and indirect genetic effects associated to individuals through information related to them, beyond the observed characteristics, such as the kinship relations. In those studies typically the assumptions of linear relationships among traits are made. However, in some scenarios, nonlinear relationships can be observed, which make unsuitable the mentioned assumptions. To overcome this limitation, this paper proposes to use a mixed effects polynomial structural equation model, second or superior degree, to model those nonlinear relationships. Two studies were developed, a simulation and an application to real data. The first study involved simulation of 50 data sets, with a fully recursive causal structure involving three characteristics in which linear and nonlinear causal relations between them were allowed. The second study involved the analysis of traits related to dairy cows of the Holstein breed. Phenotypic relationships between traits were calving difficulty, gestation length and also the proportion of perionatal death. We compare the model of multiple traits and polynomials structural equations models, under different polynomials degrees in order to assess the benefits of the SEM polynomial of second or higher degree. For some situations the inappropriate assumption of linearity results in poor predictions of the direct, indirect and total of the genetic variances and covariance, either overestimating, underestimating, or even assign opposite signs to covariances. Therefore, we conclude that the inclusion of a polynomial degree increases the SEM expressive power. / Modelos causais têm sido muitos utilizados em estudos em diferentes áreas de conhecimento, a fim de compreender as associações ou relações causais entre variáveis. Durante as últimas décadas, o uso desses modelos têm crescido muito, especialmente estudos relacionados à sistemas biológicos, uma vez que compreender as relações entre características são essenciais para prever quais são as consequências de intervenções em tais sistemas. Análise do grafo (AG) e os modelos de equações estruturais (MEE) são utilizados como ferramentas para explorar essas relações. Enquanto AG nos permite buscar por estruturas causais, que representam qualitativamente como as variáveis são causalmente conectadas, ajustando o MEE com uma estrutura causal conhecida nos permite inferir a magnitude dos efeitos causais. Os MEE também podem ser vistos como modelos de regressão múltipla em que uma variável resposta pode ser vista como explanatória para uma outra característica. Estudos utilizando MEE em genética quantitativa visam estudar os efeitos genéticos diretos e indiretos associados aos indivíduos por meio de informações realcionadas aos indivíduas, além das característcas observadas, como por exemplo o parentesco entre eles. Neste contexto, é tipicamente adotada a suposição que as características observadas são relacionadas linearmente. No entanto, para alguns cenários, relações não lineares são observadas, o que torna as suposições mencionadas inadequadas. Para superar essa limitação, este trabalho propõe o uso de modelos de equações estruturais de efeitos polinomiais mistos, de segundo grau ou seperior, para modelar relações não lineares. Neste trabalho foram desenvolvidos dois estudos, um de simulação e uma aplicação a dados reais. O primeiro estudo envolveu a simulação de 50 conjuntos de dados, com uma estrutura causal completamente recursiva, envolvendo 3 características, em que foram permitidas relações causais lineares e não lineares entre as mesmas. O segundo estudo envolveu a análise de características relacionadas ao gado leiteiro da raça Holandesa, foram utilizadas relações entre os seguintes fenótipos: dificuldade de parto, duração da gestação e a proporção de morte perionatal. Nós comparamos o modelo misto de múltiplas características com os modelos de equações estruturais polinomiais, com diferentes graus polinomiais, a fim de verificar os benefícios do MEE polinomial de segundo grau ou superior. Para algumas situações a suposição inapropriada de linearidade resulta em previsões pobres das variâncias e covariâncias genéticas diretas, indiretas e totais, seja por superestimar, subestimar, ou mesmo atribuir sinais opostos as covariâncias. Portanto, verificamos que a inclusão de um grau de polinômio aumenta o poder de expressão do MEE.

Genome-wide association studies have greatly improved our understanding of the contribution of common variants to the genetic architecture of complex traits. However, two major limitations have been highlighted. First, common variant associations typically do not identify the causal variant and/or the gene that it is exerting its effect on to influence a trait. Second, common variant associations usually consist of variants with small effects. As a consequence, it is more challenging to harness their translational impact. Association studies of rare variants and complex traits may be able to help address these limitations. Empirical population genetic data shows that deleterious variants are rare. More specifically, there is a very strong depletion of common protein truncating variants (PTVs, commonly referred to as loss-of-function variants) in the genome, a group of variants that have been shown to have large effect on gene function, are enriched for severe disease-causing mutations, but in other instances may actually be protective against disease. This thesis is divided into three parts dedicated to the study of protein truncating variants, their medical relevance, and their functional consequences. First, I present statistical, bioinformatic, and computational methods developed for the study of protein truncating variants and their association to complex traits, and their functional consequences. Second, I present application of the methods to a number of case-control and quantitative trait studies discovering new variants and genes associated to breast and ovarian cancer, type 1 diabetes, lipids, and metabolic traits measured with NMR spectroscopy. Third, I present work on improving annotation of protein truncating variants by studying their functional consequences. Taken together, these results highlight the utility of interrogating protein truncating variants in medical and functional genomic studies.

We consider a class of periodic preventive maintenance (PM) optimization problems, for a single piece of equipment that deteriorates with time or use, and can be repaired upon failure, through corrective maintenance (CM). We develop analytical and simulation-based optimization models that seek an optimal periodic PM policy, which minimizes the sum of the expected total cost of PMs and the risk-averse cost of CMs, over a finite planning horizon. In the simulation-based models, we assume that both types of maintenance actions are imperfect, whereas our analytical models consider imperfect PMs with minimal CMs. The effectiveness of maintenance actions is modeled using age reduction factors. For a repairable unit of equipment, its virtual age, and not its calendar age, determines the associated failure rate. Therefore, two sets of parameters, one describing the effectiveness of maintenance actions, and the other that defines the underlying failure rate of a piece of equipment, are critical to our models. Under a given maintenance policy, the two sets of parameters and a virtual-age-based age-reduction model, completely define the failure process of a piece of equipment. In practice, the true failure rate, and exact quality of the maintenance actions, cannot be determined, and are often estimated from the equipment failure history. We use a Bayesian approach to parameter estimation, under which a random-walk-based Gibbs sampler provides posterior estimates for the parameters of interest. Our posterior estimates for a few datasets from the literature, are consistent with published results. Furthermore, our computational results successfully demonstrate that our Gibbs sampler is arguably the obvious choice over a general rejection sampling-based parameter estimation method, for this class of problems. We present a general simulation-based periodic PM optimization model, which uses the posterior estimates to simulate the number of operational equipment failures, under a given periodic PM policy. Optimal periodic PM policies, under the classical maximum likelihood (ML) and Bayesian estimates are obtained for a few datasets. Limitations of the ML approach are revealed for a dataset from the literature, in which the use of ML estimates of the parameters, in the maintenance optimization model, fails to capture a trivial optimal PM policy. Finally, we introduce a single-stage and a two-stage formulation of the risk-averse periodic PM optimization model, with imperfect PMs and minimal CMs. Such models apply to a class of complex equipment with many parts, operational failures of which are addressed by replacing or repairing a few parts, thereby not affecting the failure rate of the equipment under consideration. For general values of PM age reduction factors, we provide sufficient conditions to establish the convexity of the first and second moments of the number of failures, and the risk-averse expected total maintenance cost, over a finite planning horizon. For increasing Weibull rates and a general class of increasing and convex failure rates, we show that these convexity results are independent of the PM age reduction factors. In general, the optimal periodic PM policy under the single-stage model is no better than the optimal two-stage policy. But if PMs are assumed perfect, then we establish that the single-stage and the two-stage optimization models are equivalent. / text