This thesis addresses the recurrent threshold crossing behavior of long-time correlated noise. The behavior of long-time correlated noise like f / 1 , 5 . 1 / 1 f , and 2 / 1 f can be associated with the behavior of many phenomena in nature, so it is of interest to study the behavior of this noise. Our method of modeling their recurring behavior relies on setting a particular threshold level for a particular level of noise and observing how frequently the noise crosses the threshold level. We also add a periodic drive to the noise which enables it to cross the threshold level easily when it is at peak, and vice versa. This technique provides a model for the changing seasons that occur during every year. We also compare the recurrence behavior of threshold crossings from our computer simulations with theoretical results from the Rice formula. We have related the recurrence of these threshold crossings with the recurrence of natural disasters. Therefore we are providing a model to predict the recurrence of a natural disaster once that disaster has previously occurred. From our results, we conclude that once a natural disaster has occurred, there is a high probability of its recurrence in a short time, and this probability gradually decreases with time.

This paper proposes a mixture of experts recurrent events model. This general model accommodates an unobservable frailty variable, intervention effect, influence of accumulating event occurrences, and covariate effects. A latent class variable is utilized to deal with a heterogeneous population and associated covariates. A homogeneous nonparametric baseline hazard and heterogeneous parametric covariate effects are assumed. Maximum likelihood principle is employed to obtain parameter estimates. Since the frailty variable and latent classes are unobserved, an estimation procedure is derived through the EM algorithm. A simulated data set is generated to illustrate the data structure of recurrent events for a heterogeneous population.

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Processsos de renovação são um caso especial de processos pontuais envolvendo eventos recorrentes nos quais um item ou unidade, após a ocorrência de uma falha, é recolocado na mesma condição de novo. Devido a essa propriedade os tempos entre ocorrências para um processo de renovação são independentes e a sua função intensidade é dada pela função de risco. Fatores que interferem nos tempos de recorrência de unidades distintas, ou indivíduos, e que não são observados, podem ser modelados com a inclusão de um termo de fragilidade no modelo. Neste trabalho é apresentado o desenvolvimento de um modelo de regressão para um processo de renovação com tempos entre ocorrências com distribuição de Weibull. Na modelagem foi considerada, ainda, a presença de censuras e a inclusão de um termo de fragilidade para explicar a relação existente entre os tempos de recorrências de uma unidade. A metodologia é desenvolvida para o caso em que várias unidades são acometidas por eventos recorrentes. Nas simulações realizadas foram analisadas as probabilidades de cobertura empíricas do intervalo de confiança normal assintótico e também o comportamento das variâncias dos estimadores. A presença de censuras na amostra inflacionou as variâncias dos estimadores de máxima verossimilhança além de produzir estimativas viciadas para um dos parâmetros da regressão, sendo que o vício do estimador foi corrigido por meio de um processo "bootstrap". Na modelagem sem termo de fragilidade, os resultados das análises das probabilidades de cobertura empírica dos intervalos de confiança assintóticos mostraram uma boa aproximação com os valores esperados, mas com certos cuidados a serem tomados, especialmente nos procedimentos baseados na simetria das distribuições empíricas. A inclusão de um termo de fragilidade na modelagem, por sua vez, causou uma perturbação na estimação máxima verossimilhança com um aumento nas variâncias dos estimadores diretamente associados à variabilidade do termo de fragilidade. Além disso, as coberturas empíricas dos intervalos de confiança assintóticos foram, na grande maioria superestimadas, com resultados satisfatórios apenas para o parâmetro de forma da distribuição Weibull. / Renewal Processes are a special case of point processes involving recurrent events in which a unit, after a failure, is restored to the like new condition. Due to that property the times between occurrences for a renewal process are independent and its intensity function is given by the hazard function. Random factors not observed, that afects the recurrence times of the units, can be explained by a frailty term added in the model. In this work a regression model is presented for a renewal process with Weibull distribution for the times between occurrences. The modeling considers censored times and a frailty variable to explain the relationship among the recurrence times of a unit. The methodology was developed for the situation where several units are submitted by recurrent events. The empirical probabilities of coverage of the asymptotic normal confidence interval and the behavior of the variances of the estimators were analyzed in the simulations performed. The presence of censures in the sample inflated the variances of the maximum likelihood estimators besides to produce biased estimates for the regression parameters. The bias of the estimator was corrected by "bootstrap" procedure. The analysis of the probability of empirical coverage of the asymptotic confidence intervals, without frailty, presented a good approximation to the nominal values, but some observations about procedures have to be made on the symmetry of the empirical distributions. The frailty term incorporated at the modeling disturbed the maximum likelihood estimation increasing estimators' variability, directly associated to the variance of the fragility term. In the most of the cases, the empirical coverages of the asymptotic confidence intervals were overestimated, with satisfactory results just for the shape parameter of the Weibull distribution.

L'analyse du risque de survenue d'événements récurrents est une motivation majeure dans de nombreuses études de recherche clinique ou épidémiologique. En cancérologie, certaines stratégies thérapeutiques doivent être évaluées au cours d'essais randomisés où l'efficacité est mesurée à partir de la survenue d'événements successifs marquant la progression de la maladie. L'état de santé de patients infectés par le VIH évolue en plusieurs étapes qui ont pu être définies par la survenue d'événements cliniques successifs.Ce travail de thèse porte sur les modèles de régression du risque pour l'analyse de la survenue d'événements successifs. En pratique, la présence de corrélations entre les temps d'attente séparant les événements successifs est une hypothèse qui peut rarement être écartée d'emblée. L'objectif de la thèse porte sur le développement de modèles de régression permettant d'évaluer une telle corrélation. Dans ce cadre, la méthode le plus souvent utilisée suppose que la corrélation entre les délais successifs a pour origine une hétérogénéité aléatoire, non observée, entre sujets. Le modèle correspondant définit le risque instantané individuel en fonction d'un terme aléatoire, ou « fragilité », de distribution gamma et dont la variance quantifie l'hétérogénéité entre sujets et donc la corrélation entre délais d'un même sujet. Cependant, l'utilisation de ce modèle pour évaluer l'ampleur des corrélations présente l'inconvénient de conduire à une estimation biaisée de la variance de la fragilité.Une première approche a été définie pour deux événements successifs dans une échelle de temps « par intervalles », c'est-à-dire où le risque est exprimé en fonction du temps écoulé depuis l'événement précédent. L'approche mise au point a été obtenue à partir d'une approximation du risque de second événement conditionnellement au premier délai dans un modèle à fragilité pour plusieurs distributions de fragilité. Une seconde approche a été définie en échelle de temps « calendaire », où le risque est exprimé en fonction du temps écoulé depuis le début du suivi du sujet. L'approche retenue a été obtenue à partir d'une approximation de l'intensité conditionnelle au passé dans un modèle à fragilité. Dans les deux échelles de temps, l'approche mise au point consiste à introduire une covariable interne, calculée sur le passé du processus, qui correspond à la différence entre le nombre d'événements observés pour le sujet sur la période passée, et le nombre attendu d'événements pour ce sujet sur la même période compte tenu de ses covariables externes. Une revue de la littérature des études de simulations a montré que le cas d'une hétérogénéité dans la population face au risque d'événement était souvent envisagé par les auteurs. En revanche, dans beaucoup d'études de simulations, le cas d'un risque dépendant du temps, ou d'une dépendance entre événements, n'étaient pas considérés. Des études de simulations ont permis de montrer dans les deux échelles de temps considérées un gain de puissance du test mis au point par rapport au test d'homogénéité correspondant au modèle à fragilité gamma. Ce gain est plus marqué en échelle de temps par intervalles. Par ailleurs, dans cette échelle de temps, le modèle proposé permet une amélioration de l'estimation de la variance de la fragilité dans le cas d'une hétérogénéité faible ou modérée, plus particulièrement pour de petits échantillons.L'approche développée en échelle de temps par intervalles a été utilisée pour analyser les données d'une cohorte de patients infectés par le VIH, montrant une corrélation négative entre le délai entre infection et première manifestation mineure d'immunodéficience et le délai entre première manifestation mineure d'immunodéficience et stade SIDA déclaré. / The risk analysis for the occurrence of recurrent events is a major concern in many clinical research studies or epidemiological studies. In the field of oncology, therapeutic strategies are evaluated in randomised clinical trials in which efficacy is assessed through the occurrence of sequential events that define the progression of the disease. In HIV-infected patients, the infection evolves in several stages that have been defined by the occurrence of successive clinical events. The frame of this work is the regression models for the risk of multiple successive events. In practice, the hypothesis of existing correlations between the inter-event times cannot be a priori discarded. The aim of this work is to develop a regression model that would assess such correlations. In this setting, the most common method is to assume that correlations between inter-event times are induced by a random, unobserved heterogeneity across individuals. The corresponding model defines the individual hazard as a function of a random variable, or " frailty ", assumed to be gamma-distributed with a variance that quantifies the heterogeneity across individuals and incidentally the correlations between inter-event times. However, the use of this model when evaluating the correlations has the drawback that it tends to underestimate the variance of the frailty.A first approach was proposed for two sequential events in a "gap-timescale", in which the risk is defined as a function of the time elapsed since the previous event. The proposed method was derived from an approximation of the risk of second event given the first time-to-event in a frailty model for various frailty distributions. Another approach was defined in "calendar-time", in which the risk is expressed as a function of the time elapsed since the beginning of the subject's follow-up. The proposed method was derived from an approximation of the intensity conditional on the past in a frailty model. In both timescales, the method that was developed consists in including in the model an internal covariate, that is calculated on the history of the process, and that corresponds to the difference between the observed number of events and the expected number of events in the past period given the individual's other covariates.A review of the literature involving simulation studies showed that when defining the generation processes, most authors considered the case of heterogeneity in the population. However, in many simulation studies, only constant hazards are considered, and no event-dependence is introduced. Simulations studies showed that in both timescales, the test of the effect of the internal covariate in the proposed model proved more powerful that the usual test of homogeneity in the gamma frailty model. This gain of power is more noticeable in gap-time. Additionally, in this timescale, the proposed model provides a better estimation of the variance of the frailty when heterogeneity is low or moderate, more particularly in small samples.The method developed in gap-time was used to analyse data from a cohort of HIV-infected patients. It showed a negative correlation between the time from infection to first minor manifestation of immunodeficiency and the time from first minor manifestation of immunodeficiency to AIDS. The method developed in calendar-time was used to study the occurrence of repeated progressions and severe toxicities in a clinical trial for patients with advanced colorectal cancer. In this example, the method corroborated the results obtained with a gamma frailty model which showed a significant heterogeneity.

Processsos de renovação são um caso especial de processos pontuais envolvendo eventos recorrentes nos quais um item ou unidade, após a ocorrência de uma falha, é recolocado na mesma condição de novo. Devido a essa propriedade os tempos entre ocorrências para um processo de renovação são independentes e a sua função intensidade é dada pela função de risco. Fatores que interferem nos tempos de recorrência de unidades distintas, ou indivíduos, e que não são observados, podem ser modelados com a inclusão de um termo de fragilidade no modelo. Neste trabalho é apresentado o desenvolvimento de um modelo de regressão para um processo de renovação com tempos entre ocorrências com distribuição de Weibull. Na modelagem foi considerada, ainda, a presença de censuras e a inclusão de um termo de fragilidade para explicar a relação existente entre os tempos de recorrências de uma unidade. A metodologia é desenvolvida para o caso em que várias unidades são acometidas por eventos recorrentes. Nas simulações realizadas foram analisadas as probabilidades de cobertura empíricas do intervalo de confiança normal assintótico e também o comportamento das variâncias dos estimadores. A presença de censuras na amostra inflacionou as variâncias dos estimadores de máxima verossimilhança além de produzir estimativas viciadas para um dos parâmetros da regressão, sendo que o vício do estimador foi corrigido por meio de um processo "bootstrap". Na modelagem sem termo de fragilidade, os resultados das análises das probabilidades de cobertura empírica dos intervalos de confiança assintóticos mostraram uma boa aproximação com os valores esperados, mas com certos cuidados a serem tomados, especialmente nos procedimentos baseados na simetria das distribuições empíricas. A inclusão de um termo de fragilidade na modelagem, por sua vez, causou uma perturbação na estimação máxima verossimilhança com um aumento nas variâncias dos estimadores diretamente associados à variabilidade do termo de fragilidade. Além disso, as coberturas empíricas dos intervalos de confiança assintóticos foram, na grande maioria superestimadas, com resultados satisfatórios apenas para o parâmetro de forma da distribuição Weibull. / Renewal Processes are a special case of point processes involving recurrent events in which a unit, after a failure, is restored to the like new condition. Due to that property the times between occurrences for a renewal process are independent and its intensity function is given by the hazard function. Random factors not observed, that afects the recurrence times of the units, can be explained by a frailty term added in the model. In this work a regression model is presented for a renewal process with Weibull distribution for the times between occurrences. The modeling considers censored times and a frailty variable to explain the relationship among the recurrence times of a unit. The methodology was developed for the situation where several units are submitted by recurrent events. The empirical probabilities of coverage of the asymptotic normal confidence interval and the behavior of the variances of the estimators were analyzed in the simulations performed. The presence of censures in the sample inflated the variances of the maximum likelihood estimators besides to produce biased estimates for the regression parameters. The bias of the estimator was corrected by "bootstrap" procedure. The analysis of the probability of empirical coverage of the asymptotic confidence intervals, without frailty, presented a good approximation to the nominal values, but some observations about procedures have to be made on the symmetry of the empirical distributions. The frailty term incorporated at the modeling disturbed the maximum likelihood estimation increasing estimators' variability, directly associated to the variance of the fragility term. In the most of the cases, the empirical coverages of the asymptotic confidence intervals were overestimated, with satisfactory results just for the shape parameter of the Weibull distribution.

Recurrent data are widely encountered in many applications. This thesis work focuses on how the recurrent hospital admissions relate to the air pollutants. In particular, we consider the data for two major cities in Saskatchewan. The study period ranges from January 1, 2005 to December 30, 2011 and involves 20,284 patients aged 40 years and older. The hospital admission data is from the Canadian Institute for Health Information (CIHI). The air pollutants data is from the National Air Pollution Surveillance Program (NAPS) from Environment Canada. The data set has been approved by the Biomedical Research Ethics Board, University of Saskatchewan. The gaseous pollutants included in this study are carbon monoxide (CO), nitrogen dioxide (NO2), sulfur dioxide (SO2), ozone (O3), as well as particulate matter PM2:5 (tiny particles in the air that are 2:5 microns in width). In the data analysis, we applied three different existing models to all respiratory diseases and asthma, respectively. The three models are the Poisson process model (also called Andersen-Gill model), the Poisson process model with the number of previous events as a covariate and the Poisson process model with shared gamma distributed frailties (random effects). For all respiratory diseases, the Poisson process model with random effects provides the best t in comparison to the other two models. The model output suggests that the increased risk of hospital readmission is significantly associated with increased CO and O3. For asthma, the Poisson process model provides the best t in comparison to the other two models. We found that only CO and O3 have significant effects on recurrent hospital admissions due to asthma. We concluded this thesis with the discussion on the current and potential future work.

We present a theoretical approach to the statistical analysis of the dependence of the gap time length between consecutive recurrent events, on a set of explanatory random variables and in the presence of right censoring. The dependence is expressed through regression-like and overdispersion parameters, estimated via estimating functions and equations. The mean and variance of the length of each gap time, conditioned on the observed history of prior events and other covariates, are known functions of parameters and covariates, and are part of the estimating functions. Under certain conditions on censoring, we construct normalized estimating functions that are asymptotically unbiased and contain only observed data. We then use modern mathematical techniques to prove the existence, consistency and asymptotic normality of a sequence of estimators of the parameters. Simulations support our theoretical results.

Ce travail a eu pour objectif de proposer des modèles conjoints d'intensités de processus d'événements récurrents et d'un événement terminal dépendant. Nous montrons que l'analyse séparée de ces événements conduit à des biais d'estimation importants. C'est pourquoi il est nécessaire de prendre en compte les dépendances entre les différents événements d'intérêt. Nous avons choisi de modéliser ces dépendances en introduisant des effets aléatoires (ou fragilités) et de travailler sur la structure de dépendance. Ces effets aléatoires prennent en compte les dépendances entre événements, les dépendances inter-récurrences et l'hétérogénéité non-observée. Nous avons, en premier lieu, développé un modèle conjoint à fragilités pour un type d'événement récurrent et un événement terminal dépendant en introduisant deux effets aléatoires indépendants pour prendre en compte et distinguer la dépendance inter-récurrences et celle entre les risques d'événements récurrents et terminal. Ce modèle a été ajusté pour des données de patients atteints de lymphome folliculaire où les événements d'intérêt sont les rechutes et le décès. Le second modèle développé permet de modéliser conjointement deux types d'événements récurrents et un événement terminal dépendant en introduisant deux effets aléatoires corrélés et deux paramètres de flexibilités. Ce modèle s'avère adapté pour l'analyse des risques de récidives locorégionales, de récidives métastatiques et de décès chez des patientes atteintes de cancer du sein. Nous confirmons ainsi que le décès est lié aux récidives métastatiques mais pas aux récidives locorégionales tandis que les deux types de récidives sont liés. Cependant ces approches font l'hypothèse de proportionnalité des intensités conditionnellement aux fragilités, que nous allons tenter d'assouplir. Dans un troisième travail, nous proposons de modéliser un effet potentiellement dépendant du temps des covariables en utilisant des fonctions B-Splines. / This work aimed to propose joint models for recurrent events and a dependent terminal event. We show how separate analyses of these events could lead to important biases. That is why it seems necessary to take into account the dependencies between events of interest. We choose to model these dependencies through random effects (or frailties) and work on the dependence structure. These random effects account for dependencies between events, inter-dependence recurrences and unobserved heterogeneity. We first have developed a joint frailty model for one type of recurrent events and a dependent terminal event with two independent random effects to take into account and distinguish the inter-recurrence dependence and between recurrent events and terminal event. This model was applied to follicular lymphoma patient’s data where events of interest are relapses and death. The second proposed model is used to model jointly two types of recurrent events and a dependent terminal event by introducing two correlated random effects and two flexible parameters. This model is suitable for analysis of locoregional recurrences, metastatic recurrences and death for breast cancer patients. It confirms that the death is related to metastatic recurrence but not locoregional recurrence while both types of recurrences are related. However, these approaches do the assumption of proportional intensities conditionally on frailties, which we want to relax. In a third study, we propose to model potentially time-dependent regression coefficient using B-splines functions.

Doctor of Philosophy / With the fast development of artificial intelligence (AI) technology, the reliability of AI needs to be investigated for confidently using AI products in our daily lives. This dissertation includes three projects introducing the statistical models and model estimation methods that can be used in the reliability analysis of AI systems. The first project analyzes the recurrent events data from autonomous vehicles (AVs). A nonparametric model is proposed to study the reliability of AI systems in AVs, and a statistical framework is introduced to evaluate the adequacy of using traditional parametric models in the analysis. The proposed model and framework are then applied to analyze AV data from four manufacturers that participated in an AV driving testing program overseen by the California Department of Motor Vehicles. The second project develops a survival model to investigate the failure times of graphics processing units (GPUs) used in supercomputers. The model considers several covariates, the spatial correlation, and the correlation among multiple types of failures. In addition, unique spatial correlation functions and a special distance function are introduced to quantify the spatial correlation inside supercomputers. The model is applied to explore the GPU failure times in the Titan supercomputer. The third project proposes a new Markov chain Monte Carlo sampler that can be used in the estimation and inference of spatial survival models. The sampler can generate a reasonable amount of samples within a shorter computing time compared with existing popular samplers. Important factors that can influence the performance of the proposed sampler are explored, and the sampler is used to analyze the Titan GPU failures to illustrate its usefulness in solving real-world problems.