This thesis aims to identify new change-point detection methods and/or trend in temporal data. After a brief theoretical introduction on splines, several existing change-point detection already in the literature will be presented. Then, new change-point detection methods using splines and Bayesian statistics will be presented. Moreover, in order to understand the method using Bayesian statistics, an introduction to Bayesian theory will be presented. Using simulations, we will make a comparison of the power of all these methods. Still using simulations, an analysis of the new most effective method will be performed. Then, this method will be applied to real data. A brief conclusion will make a summary of this thesis.

Uplift modelling is a statistical method initially developed in marketing. It has two groups (a control group and a treatment group) that are compared using a binary response variable (the response can be « yes » or « no »). The goal of this model is to detect the treatment e ect on prospects. This e ect can be either negative, null or positive. It depends on characteristics of each individual in each group. The purpose of this master thesis is to compare the Bayesian point of view with the frequentist one on uplift modelling. The uplift models used in this thesis are Lo model (2002) and Lai model (2004). Both of them are originally modeled using the frequentist point of view. Therefore, the Bayesian approach is modeled and compared to the frequentist one. Simulations are done on generated data from logistic regressions. Then regression parameters are estimated with Monte- Carlo simulations for Bayesian approach. They are then compared to parameter estimations from the frequentist approach. Parameter estimations have direct influences on the ability of the modelling to predict treatment e ect on individual. Three priors are considered for the Bayesian estimation of the parameters. These densities are chosen such that they are non-informative. They are the following : transformed beta, Cauchy and normal. In the course of the study, we will notice the Bayesian method has a real positive impact on targeting individual from the small size sample.

Every day, decisions must be made about the amount of hydroelectricity produced in Quebec. These decisions are based on the prediction of water inflow in watersheds based on hydrological models. These models take into account several factors, including the presence or absence of snow. This information is critical during the spring melt to anticipate future flows, since between 30 and 40 % of the flood volume may come from the melting of the snow cover. It is therefore necessary for forecasters to be able to monitor on a daily basis the snow cover to adjust their expectations about the melting phenomenon. Some methods to map snow on the ground are currently used at the Institut de recherche d'Hydro-Québec (IREQ), but they have some shortcomings. This master thesis's main goal is to use remote sensing passive microwave data (the vertically polarized brightness temperature gradient ratio (GTV)) with a statistical approach to produce snow maps and to quantify the classification uncertainty. In order to do this, the GTV has been used to calculate a daily probability of snow via a Gaussian mixture model using Bayesian statistics. Subsequently, these probabilities were modeled using linear regression models on logits and snow cover maps were produced. The models results were validated qualitatively and quantitatively, and their integration at Hydro-Québec was discussed.

Logistic regression is a model of generalized linear regression (GLM) used to explain binary variables. The model seeks to estimate the probability of success of this variable by the linearization of explanatory variables. When the goal is to estimate more accurately the impact of various incentives from a marketing campaign (coefficients of the logistic regression), the identification of the choice of the optimum prior density is sought. In our simulations, using the MCMC method of slice sampling, we compare different prior densities specified by different types of density, location and scale parameters. These comparisons are applied to samples of different sizes generated with different probabilities of success. The maximum likelihood estimate, Gelman’s method and Genkin’s method complement the comparative. Our simulations demonstrate that the MCMC method with a normal prior density centered at 0 with variance of 3,125, the MCMC method with a Student prior density with 3 degrees of freedom centered at 0 with variance of 3,125 and Gelman’s method with a Cauchy density centered at 0 with scale parameter of 2,5 get estimates that are globally the most accurate of the coefficients of the logistic regression.

This master's thesis addresses the problem of the simulation of simultaneous credible intervals in a Bayesian context. First, we will study precipation data and two functions based on these data : the empirical distribution function and the return period, a non-linear function of the empirical distribution. We will review different methods already known to obtain simultaneous confidence intervals of these functions with a polynomial basis and we will present a method to simulate simultaneous credible intervals. Second, we will explore some models of prior distributions and in the more complex one, we will need the Monte-Carlo method to simulate simultaneous posterior credible intervals. Finally, we will use a non-linear basis based on the angular transformation and on monotone splines to obtain valid simultaneous credible intervals for the return period.

For panel data, repeated measures over time can challenge the hypothesis of dependence between subjects. Tests were developped in order to assess if some dependence remains among residuals. The three tests we present in this master thesis are from Pesaran (2004), Friedman (1937) and Frees (1995). These three tests, constructed specifically for linear models, are based on the residuals generated from models (and their correlations). We wish to study in this master thesis the performances of these three tests in the case of generalized linear Poisson models. For that goal, we compare them between each other (level, power, etc.) using two linear models, one with an autoregressive term and the other without. Next, inspired by Hsiao, Pesaran and Pick (2007) who adapt the test from Pesaran (2004), we will study their performances in a generalized Poisson model. All of our comparisons are done with simulations by modifying some variables (number of observations, strength of the dependence). We will observe that when the correlation is always of the same sign, Pesaran’s test is the best in most cases, for the linear models and the generalized linear model. Frees’ test will show good performances when the sign of the correlations alternates.

Cavitation erosion which results from repeated collapse of transient vapor cavities on solid surfaces is a constant problematic in hydraulic turbine runners and continues to enforce costly repair and loss of revenues. A vibratory detection system of cavitation erosion was installed 10 years ago for continuous monitoring of 4 hydropower units. A new hardware version of the system was developed and installed in 2010. This new system configuration is more reliable and allows more accurate evaluation of the cavitation erosion of the runners in kg/10 000 h. The first objective of this study is to investigate cavitation behavior upon one generating unit and to build a statistical model which will allow prediction of instant cavitation related to operating variables, such as gate opening, water flow, headwater level, tailwater levels, etc. The second objective is to develop a methodology for the reproducibility of the studies to other sites. A retrospective study will be conducted and we will mainly focus on data available since the system update in 2010. The preliminary analysis enhanced the complexity of the phenomenon. Indeed, changes in the relationship between cavitation and various operating variables were observed and could be due to a seasonal behavior or different operating conditions. Using hierarchical clustering and regression models, we formalize this heterogeneity by developing a model which includes operating variables such as active power, tailwater level and gate opening.

The main goal of this master's thesis is to find whether the summer convective precipitations simulated by the Canadian Regional Climate Model (CRCM) are stationary over time or not. In order to answer that question, we propose both a frequentist and Bayesian statistical methodology. For the frequentist approach, we used standard quality control and the CUSUM to determine if the mean has increased over the years. For the Bayesian approach, we compared the posterior distributions of the precipitations over time. In order to do the comparison, we used a statistic based on the Hellinger's distance, the J-divergence and the L2 norm. In this master's thesis, we used the ARL (average run length) to calibrate each of our methods. Therefore, a big part of this thesis is about studying the actual property of the ARL. Once our tools are well calibrated, we used the simulation to compare them together. Finally, we studied the data from the CRCM to decide, whether or not, the data are stationary.

Data coming from a fine sampling of a continuous process (random field) can be represented as images. A statistical test aiming at detecting a difference between two images can be seen as a group of tests in which each pixel is compared to the corresponding pixel in the other image. We then use a method to control the type I error over all the tests, such as the Bonferroni correction or the control of the false discovery rate (FDR). Methods of data analysis have been developped in the field of medical imaging, mainly by Keith Worsley, using the geometry of random fields in order to build a global statistical test over the whole image. The expected Euler characteristic of the excursion set of the random field underlying the sample over a given threshold is used in order to determine the probability that the random field exceeds this same threshold under the null hypothesis (topological inference). We present some notions relevant to random fields, in particular isotropy (the covariance function between two given points of a field depends only on the distance between them). We discuss two methods for the analysis of non\-isotropic random fields. The first one consists in deforming the field and then using the intrinsic volumes and the Euler characteristic densities. The second one uses the Lipschitz-Killing curvatures. We then perform a study of sensitivity and power of the topological inference technique comparing it to the Bonferonni correction. Finally, we use topological inference in order to describe the evolution of climate change over Quebec territory between 1991 and 2100 using temperature data simulated and published by the Climate Simulation Team at Ouranos, with the Canadian Regional Climate Model CRCM4.2.

In this master’s thesis, we develop a statistical method to estimate the lifetime expectancy of clients in the insurance domain. The forecasts are personnalized according to the clients’ own features, the most notable being the fact that they can have any combination of automobile and residential insurance products. Three approaches are compared. The first approach is the simple Markov model which assume homogeneity and stationnarity of the transition probabilities. The other model suggested – which is implemented both by direct computation and by simulation – allows for heterogeneity of the transition probabilities, thus providing forecasts which evolve in time along with the characteristics of the clients. The transitions probabilities are estimated using multinomial logistic regressions.

The main interest of this research is the validation of a statistical method in pharmacoepidemiology. Specifically, we will compare the results of a previous study performed with a nested case-control which took into account the average exposure to treatment to : – results obtained in a cohort study, using the time-dependent exposure, with no adjustment for time since exposure ; – results obtained using the cumulative exposure weighted by the recent past ; – results obtained using the Bayesian model averaging. Covariates are estimated by the classical approach and by using a nonparametric Bayesian approach. In the later, the Bayesian model averaging will be used to model the uncertainty in the choice of models. To model the cumulative effect of exposure which varies over time, in the classical approach the function assigning weights according to recency will be estimated using regression splines. In order to compare the results with previous studies, a cohort of people diagnosed with hypertension will be constructed using the databases of the RAMQ and Med-Echo. The Cox model including two variables which vary in time will be used. The time-dependent variables considered in this paper are the dependent variable (first stroke event) and one of the independent variables, namely the exposure.

This thesis presents some estimation methods and algorithms to analyse count data in particular and discrete data in general. It is also part of an NSERC strategic project, named CC-Bio, which aims to assess the impact of climate change on the distribution of plant and animal species in Québec. After a brief introduction to the concepts and definitions of biogeography and those relative to the generalized linear mixed models in chapters 1 and 2 respectively, my thesis will focus on three major and new ideas. First, we introduce in chapter 3 a new form of distribution whose components have marginal distribution Poisson or Skellam. This new specification allows to incorporate relevant information about the nature of the correlations between all the components. In addition, we present some properties of this probability distribution function. Unlike the multivariate Poisson distribution initially introduced, this generalization enables to handle both positive and negative correlations. A simulation study illustrates the estimation in the two-dimensional case. The results obtained by Bayesian methods via Monte Carlo Markov chain (MCMC) suggest a fairly low relative bias of less than 5% for the regression coefficients of the mean. However, those of the covariance term seem a bit more volatile. Later, the chapter 4 presents an extension of the multivariate Poisson regression with random effects having a gamma density. Indeed, aware that the abundance data of species have a high dispersion, which would make misleading estimators and standard deviations, we introduce an approach based on integration by Monte Carlo sampling. The approach remains the same as in the previous chapter. Indeed, the objective is to simulate independent latent variables to transform the multivariate problem estimation in many generalized linear mixed models (GLMM) with conventional gamma random effects density. While the assumption of knowledge a priori dispersion parameters seems too strong and not realistic, a sensitivity analysis based on a measure of goodness of fit is used to demonstrate the robustness of the method. Finally, in the last chapter, we focus on the definition and construction of a measure of concordance or a correlation measure for some zeros augmented count data with Gaussian copula models. In contrast to Kendall's tau whose values lie in an interval whose bounds depend on the frequency of ties observations, this measure has the advantage of taking its values on the interval (-1, 1). Originally introduced to model the correlations between continuous variables, its extension to the discrete case implies certain restrictions and its values are no longer in the entire interval (-1,1) but only on a subset. Indeed, the new measure could be interpreted as the correlation between continuous random variables before being transformed to discrete variables considered as our discrete non negative observations. Two methods of estimation based on integration via Gaussian quadrature and maximum likelihood are presented. Some simulation studies show the robustness and the limits of our approach.

This master’s thesis is about the estimation of bivariate tables which are monotone within the rows and/or the columns, with a special interest in the approximation of life tables. This problem is approached through a nonparametric Bayesian regression model, in particular linear combinations of regression splines. By condensing a life table, our goal is to reduce its storage space without losing the entries’ accuracy. We will also study the reconstruction time of the table with our estimators. The properties of the reference table, specifically its monotonicity, must be preserved in the estimation. We are working with a monotone spline basis since splines are flexible and their derivatives can easily be manipulated. Those properties enable the imposition of constraints of monotonicity on our model. A brief review on univariate approximations of monotone functions is then extended to bivariate estimations. We use hierarchical Bayesian modeling to include the constraints in the prior distributions. We then explain the Markov chain Monte Carlo algorithm to obtain a posterior estimator. Finally, we study the estimator’s behaviour by applying our model on the Standard Normal table and the Student’s t table. We estimate our data of interest, the life table, to establish the improvement in data accessibility.

Zero-inflated models, both discrete and continuous, have a large variety of applications and fairly well-known properties. Some work has been done on zero-deflated and zero-modified discrete models. The usual formulation of continuous zero-inflated models -- a mixture between a continuous density and a Dirac mass at zero -- precludes their extension to cover the zero-deflated case. We introduce an alternative formulation of zero-inflated continuous models, along with a natural extension to the zero-deflated case. Parameter estimation is first studied within the classical frequentist framework. Several methods for obtaining the maximum likelihood estimators are proposed. The problem of point estimation is considered from a Bayesian point of view. Hypothesis testing, aiming at determining whether data are zero-inflated, zero-deflated or not zero-modified, is also considered under both the classical and Bayesian paradigms. The proposed estimation and testing methods are assessed through simulation studies and applied to aggregated rainfall data. The data is shown to be zero-deflated, demonstrating the relevance of the proposed model. We next consider the clustering of samples of zero-deflated data. Such data present strong non-normality. Therefore, the usual methods for determining the number of clusters are expected to perform poorly. We argue that Bayesian clustering based on the marginal distribution of the observations would take into account the particularities of the model and exhibit better performance. Several clustering methods are compared using a simulation study. The proposed method is applied to aggregated rainfall data sampled from 28 measuring stations in British Columbia.

This thesis is about Bayesian functional data analysis in hydrology. The main objective is to model water flow data in a parsimonious fashion while still reproducing the statistical features of the data. Functional data analysis leads us to consider the water flow time series as functions to be modelled with a nonparametric method. First, the functions are registered in order to make them more homogeneous. With a more homogeneous sample of curves, we proceed to model their statistical features by relying on Bayesian regression splines in a fairly broad probabilistic framework. More specifically, we study a family of continuous distributions, which include those of the exponential family, from which the data might have arisen. Furthermore, to have a flexible nonparametric modeling tool, we treat the interior knots, which define the basis elements of the regression splines, as random quantities. We then use MCMC with reversible jumps in order to explore the posterior distribution of the interior knots. In order to simplify the procedure in our general modeling context, we consider some approximations for the marginal distribution of the observations, namely one based on the Schwarz information criterion and another which relies on Laplace's approximation. In addition to modeling the central tendency of a sample of curves, we also propose a methodology to simultaneously model the central tendency and the dispersion of the curves in our general probabilistic framework. Finally, since we study several statistical distributions for the observations, we put forward an approach to determine the most adequate distributions for a given sample of curves.

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