Les travaux présentés dans cette thèse concernent l'analyse statistique de cas particuliers du processus de Cox. Dans une première partie, nous proposons une synthèse des résultats existants sur le processus power-law (processus d'intensité puissance), synthèse qui ne peut être exhaustive étant donné la popularité de ce processus. Nous considérons une approche bayésienne pour l'inférence des paramètres de ce processus qui nous conduit à introduire et à étudier en détails une distribution que nous appelons loi H-B. Cette loi est une loi conjuguée. Nous proposons des stratégies d'élicitation des hyperparamètres et étudions le comportement des estimateurs de Bayes par des simulations. Dans un deuxième temps, nous étendons ces travaux au cas du processus d’intensité exponentielle (exponential-law process). De la même façon, nous définissons et étudions une loi conjuguée pour l'analyse bayésienne de ce dernier. Dans la dernière partie de la thèse, nous considérons un processus auto-excité qui intègre une covariable. Ce travail est motivé, à l'origine, par un problème de fiabilité qui concerne des données de défaillances de matériels exposés à des environnements sévères. Les résultats sont illustrés par des applications sur des données d'activités orageuses collectées dans deux départements français. Enfin, nous donnons quelques directions de travail et perspectives de futurs développements de l'ensemble de nos travaux. / The work presented in this PhD dissertation concerns the statistical analysis of some particular cases of the Cox process. In a first part, we study the power-law process (PLP). Since the literature for the PLP is abundant, we suggest a state-of-art for the process. We consider the classical approach and recall some important properties of the maximum likelihood estimators. Then we investigate a Bayesian approach with noninformative priors and conjugate priors considering different parametrizations and scenarios of prior guesses. That leads us to define a family of distributions that we name H-B distribution as the natural conjugate priors for the PLP. Bayesian analysis with the conjugate priors are conducted via a simulation study and an application on real data. In a second part, we study the exponential-law process (ELP). We review the maximum likelihood techniques. For Bayesian analysis of the ELP, we define conjugate priors: the modified- Gumbel distribution and Gamma-modified-Gumbel distribution. We conduct a simulation study to compare maximum likelihood estimates and Bayesian estimates. In the third part, we investigate self-exciting point processes and we integrate a power-law covariate model to this intensity of this process. A maximum likelihood procedure for the model is proposed and the Bayesian approach is suggested. Lastly, we present an application on thunderstorm data collected in two French regions. We consider a strategy to define a thunderstorm as a temporal process associated with the charges in a particular location. Some selected thunderstorms are analyzed. We propose a reduced maximum likelihood procedure to estimate the parameters of the Hawkes process. Then we fit some thunderstorms to the power-law covariate self-exciting point process taking into account the associated charges. In conclusion, we give some perspectives for further work.

This work is aimed at the understanding of the dynamical behavior of hydrogen in diamond. The investigation was carried out using Transverse Field muon Spin Rotation (TF-SR) and the Longitudinal Field muon Spin Relaxation (LF-SR) techniques. The chemical analogy between hydrogen (p+e-) and muonium (u+e-) enabled the study of the indirect dynamical behavior of hydrogen in diamond. The TF-SR and LF-SR measurements were carried out in an isotopically pure 13C diamond in the temperature ranges 11 mK - 320 K and 10 K - 400 K, respectively. In the TF-SR results, the Prompt Absolute Fraction (PAF) of both diamagnetic and the paramagnetic states are temperature independent. The spin relaxation rate for the state is non-zero and temperature independent, while that of the Mux state is non-zero and temperature dependent. The behavior of indicates that the u+d is immobile, while that of MuT indicates the mobility in diamond. The LF-SR results provide hop rate and associated nuclear hyperfine interaction parameters of the Mut state in diamond. / Physics / M.Sc. (Physics)

L'étude de l'émission intégrée et résolue en temps de quatre configurations d'hétérostructures quantiques de type points-dans-un-fil d'InGaN/GaN nous a permis de déterminer la nature de la localisation et du mécanisme de recombinaison des porteurs de charge dans ces nanofils. Des mesures de comptage de photon unique correlés en temps (TCSPC) étendues sur une plage temporelle allant de 210 à 26000ns ont permis d'observer un comportement fortement non exponentiel de l'émission que nous avons déterminé être une loi de puissance. Nous avons trouvé que le temps de vie de l'émission diminue rapidement avec l'énergie d'émission. Par contre, l'observation d'un effet de la puissance d'excitation sur le temps de vie semble indiquer qu'à une énergie d'émission ne soit pas associée une seule dynamique d'émission à long temps. En utilisant une densité d'excitation laser de seulement quelques dizaines de watt par cm au carré, nous avons pu démontrer, en régime non perturbatif, que le profil des spectres d'émission intégrés en temps ainsi que la dynamique de l'évolution temporelle de l'émission étaient tout à fait compatibles avec une recombinaison radiative centrée sur une distribution de nano-agrégats riches en indium naturellement formés lors de la croissance des nanofils par MBE assistée par plasma. Cette conclusion est supportée par notre incapacité à observer l'effet Stark à confinement quantique, le succès d'un modèle de séparation de charges parfaitement compatible avec l'image des nano-agrégats d'indium et, finalement, par l'observation d'une émission principalement isotrope en polarisation. / We have performed time-integrated and time-resolved photoluminescence measurements on four different configurations of InGaN/GaN dot-in-a-wire heterostructures in order to further our understanding of the localization and radiative recombination mechanism involved in the process of emission. Time correlated single photon counting (TCSPC) measurements from 100 ns to 26000 ns have allowed us to observe a strong non-exponential decay which follows a power law on long time scale. The characteristic exponent of this power law is strongly correlated with the emission energy, causing the life-time of the emission to fall rapidly with increasing of its energies. The observation that the excitation power has an effect on the life-time shows that other factors such as the growth conditions must be involved in the coupling between life-time and energy. Using a low power density of a few tens of watts per cm squared, we have shown, in a non perturbative regime, that the shape of the time-integrated spectra and the dynamics of the time-resolved decay curves were consistent with a radiative recombination process centered on In-rich nanocluster. These nanoclusters naturally occur in the embedded InGaN inclusions during the growth by plasma-assisted MBE. This conclusion is supported by the absence of Quantum confined Stark effect. The success of a charge separation model is perfectly consistent with the emission centered on In-rich nanocluster and the observation of a quasiperfect isotropic emission. / Mesures effectuées dans le laboratoire de caractérisation optique des semi-conducteurs du Prof. Richard Leonelli du département de physique de l'université de Montréal. Les nanofils d'InGaN/GaN ont été fournis par le groupe du Prof. Zetian Mi du département de génie électrique et informatique de l'université McGill.

Mesures effectuées dans le laboratoire de caractérisation optique des semi-conducteurs du Prof. Richard Leonelli du département de physique de l'université de Montréal. Les nanofils d'InGaN/GaN ont été fournis par le groupe du Prof. Zetian Mi du département de génie électrique et informatique de l'université McGill. / L'étude de l'émission intégrée et résolue en temps de quatre configurations d'hétérostructures quantiques de type points-dans-un-fil d'InGaN/GaN nous a permis de déterminer la nature de la localisation et du mécanisme de recombinaison des porteurs de charge dans ces nanofils. Des mesures de comptage de photon unique correlés en temps (TCSPC) étendues sur une plage temporelle allant de 210 à 26000ns ont permis d'observer un comportement fortement non exponentiel de l'émission que nous avons déterminé être une loi de puissance. Nous avons trouvé que le temps de vie de l'émission diminue rapidement avec l'énergie d'émission. Par contre, l'observation d'un effet de la puissance d'excitation sur le temps de vie semble indiquer qu'à une énergie d'émission ne soit pas associée une seule dynamique d'émission à long temps. En utilisant une densité d'excitation laser de seulement quelques dizaines de watt par cm au carré, nous avons pu démontrer, en régime non perturbatif, que le profil des spectres d'émission intégrés en temps ainsi que la dynamique de l'évolution temporelle de l'émission étaient tout à fait compatibles avec une recombinaison radiative centrée sur une distribution de nano-agrégats riches en indium naturellement formés lors de la croissance des nanofils par MBE assistée par plasma. Cette conclusion est supportée par notre incapacité à observer l'effet Stark à confinement quantique, le succès d'un modèle de séparation de charges parfaitement compatible avec l'image des nano-agrégats d'indium et, finalement, par l'observation d'une émission principalement isotrope en polarisation. / We have performed time-integrated and time-resolved photoluminescence measurements on four different configurations of InGaN/GaN dot-in-a-wire heterostructures in order to further our understanding of the localization and radiative recombination mechanism involved in the process of emission. Time correlated single photon counting (TCSPC) measurements from 100 ns to 26000 ns have allowed us to observe a strong non-exponential decay which follows a power law on long time scale. The characteristic exponent of this power law is strongly correlated with the emission energy, causing the life-time of the emission to fall rapidly with increasing of its energies. The observation that the excitation power has an effect on the life-time shows that other factors such as the growth conditions must be involved in the coupling between life-time and energy. Using a low power density of a few tens of watts per cm squared, we have shown, in a non perturbative regime, that the shape of the time-integrated spectra and the dynamics of the time-resolved decay curves were consistent with a radiative recombination process centered on In-rich nanocluster. These nanoclusters naturally occur in the embedded InGaN inclusions during the growth by plasma-assisted MBE. This conclusion is supported by the absence of Quantum confined Stark effect. The success of a charge separation model is perfectly consistent with the emission centered on In-rich nanocluster and the observation of a quasiperfect isotropic emission.

The biomass size spectrum - the power-law scaling relationship between average individual size and total biomass - has often been studied in lake and marine ecosystems, but rarely in lotic systems. The objective of this study was to test for characteristic biomass spectra in small temperate streams. Seasonal fish and macroinvertebrate data, including population abundance and biomass estimates, were collected in three wadeable, southern West Virginia streams from October 2013 to May 2015. Fish abundances were estimated with 3-pass electrofishing (depletion) surveys and individuals were weighed in the field. Macroinvertebrates were collected with a Hess sampler and returned to the lab for identification to the lowest practical level (usually genus). Published length-mass regressions were then used to estimate individual mass. All size spectra relationships (linear regression of log-log data) were highly significant (p<0.001). Size spectra intercepts were variable and may reflect seasonal differences in fish and invertebrate densities. Size spectra slopes were more consistent, with a mean slope of approximately -0.73, suggesting a common scaling relationship between stream consumers at differing trophic levels.

This thesis addresses the observed fluorescence intermittency of single semiconductor nanocrystals, so called Quantum Dots (QDs), which is also referred to as blinking. Despite continuous excitation their fluorescence is randomly interrupted by dark periods that can last over several minutes. Especially the extraction of power law dwell time statistics in bright and dark states indicates very complex underlying processes that are not fully understood to date. Here two approaches are followed to reveal the nature of the blinking mechanism. One addresses the common threshold method for extraction of power law dwell times. Its performance is tested with simulations to a broad range of experimentally determined parameters. Strong deviations are found between input and extracted statistics dependent on input parameters themselves. A comparison with experimental data does not support the assignment of power law statistics for the bright state and indicates the existence of distinct blinking mechanisms. The second approach directly aims at the nature of the dark state, which is mostly attributed to charges in the QD or trap states in its vicinity. A method is developed to detect charging processes on single QDs with their fluorescence. Electrochemistry is combined with confocal microscopy also allowing evaluations of excited state lifetimes and emission spectra. Reduction and oxidation of the QD bands are successfully observed as a quenching of QD fluorescence. Single QD observations identify two independent blinking mechanisms, that are assigned to positive and negative charging. Positive charging is not only observed after hole injection but also the extraction of excited electrons. Three additional quenching mechanisms are identified, two of which are assigned to trap relaxation. Differences between two substrate electrodes demonstrate the importance of the substrate material.

The objective of this study is to compare Bayesian and parametric approaches to determine the best for estimating reliability in complex systems. Determining reliability is particularly important in business and medical contexts. As expected, the Bayesian method showed the best results in assessing the reliability of systems. In the first study, the Bayesian reliability function under the Higgins-Tsokos loss function using Jeffreys as its prior performs similarly as when the Bayesian reliability function is based on the squared-error loss. In addition, the Higgins-Tsokos loss function was found to be as robust as the squared-error loss function and slightly more efficient. In the second study, we illustrated that--through the power law intensity function--Bayesian analysis is applicable in the power law process. The power law intensity function is the key entity of the power law process (also called the Weibull process or the non-homogeneous Poisson process). It gives the rate of change of a system's reliability as a function of time. First, using real data, we demonstrated that one of our two parameters behaves as a random variable. With the generated estimates, we obtained a probability density function that characterizes the behavior of this random variable. Using this information, under the commonly used squared-error loss function and with a proposed adjusted estimate for the second parameter, we obtained a Bayesian reliability estimate of the failure probability distribution that is characterized by the power law process. Then, using a Monte Carlo simulation, we showed the superiority of the Bayesian estimate compared with the maximum likelihood estimate and also the better performance of the proposed estimate compared with its maximum likelihood counterpart. In the next study, a Bayesian sensitivity analysis was performed via Monte Carlo simulation, using the same parameter as in the previous study and under the commonly used squared-error loss function, using mean square error comparison. The analysis was extended to the second parameter as a function of the first, based on the relationship between their maximum likelihood estimates. The simulation procedure demonstrated that the Bayesian estimates are superior to the maximum likelihood estimates and that the selection of the prior distribution was sensitive. Secondly, we found that the proposed adjusted estimate for the second parameter has better performance under a noninformative prior. In the fourth study, a Bayesian approach was applied to real data from breast cancer research. The purpose of the study was to investigate the applicability of a Bayesian analysis to survival time of breast cancer data and to justify the applicability of the Bayesian approach to this domain. The estimation of one parameter, the survival function, and hazard function were analyzed. The simulation analysis showed that the Bayesian estimate of the parameter performed better compared with the estimated value under the Wheeler procedure. The excellent performance of the Bayesian estimate is reflected even for small sample sizes. The Bayesian survival function was also found to be more efficient than its parametric counterpart. In the last study, a Bayesian analysis was carried out to investigate the sensitivity to the choice of the loss function. One of the parameters of the distribution that characterized the survival times for breast cancer data was estimated applying a Bayesian approach and under two different loss functions. Also, the estimates of the survival function were determined under the same setting. The simulation analysis showed that the choice of the squared-error loss function is robust in estimating the parameter and the survival function.

Many software reliability growth models have beenanalyzed for measuring the growth of software reliability. In this dissertation, regression methods are explored to study software reliability models. First, two parametric linear models are proposed and analyzed, the simple linear regression and transformed linearregression corresponding to a power law process. Some software failure data sets do not follow the linear pattern. Analysis of popular real life data showed that these contain outliers andleverage values. Linear regression methods based on least squares are sensitive to outliers and leverage values. Even though the parametric regression methods give good results in terms of error measurement criteria, these results may not be accurate due to violation of the parametric assumptions. To overcome these difficulties, nonparametric regression methods based on ranks are proposed as alternative techniques to build software reliability models. In particular, monotone regre ssion and rank regression methods are used to evaluate the predictive capability of the models. These models are applied to real life data sets from various projects as well as to diverse simulated data sets. Both the monotone and the rank regression methods are robust procedures that are less sensitive to outliers and leverage values. In particular, the regression approach explains predictive properties of the mean time to failure for modeling the patterns of software failure times.In order to decide on model preference and to asses predictive accuracy of the mean time between failure time estimates for the defined data sets, the following error measurements evaluative criteria are used: the mean square error, mean absolute value difference, mean magnitude of relative error, mean magnitude oferror relative to the estimate, median of the absolute residuals, and a measure of dispersion. The methods proposed in this dissertation, when applied to real software failure data, give lesserror in terms of all the measurement criteria compared to other popular methods from literature. Experimental results show that theregression approach offers a very promising technique in software reliability growth modeling and prediction.

Over the last twenty years, as biological, technological, and social net- works have risen in prominence and importance, the study of complex networks has attracted researchers from a wide range of fields. As a result, there is a large and diverse body of literature concerning the properties and development of models for complex networks. However, many of the models that have been previously developed, although quite successful at capturing many observed properties of complex networks, have failed to capture the fundamental semantics of the networks. In this thesis, we propose a robust and general model for complex networks that incorporates at a fundamental level semantic information. We show that for a large range of average degrees and with a suitable choice of parameters, this model exhibits the three hallmark properties of complex networks: small diameter, clustering, and skewed degree distribution. Additionally, we provide a structural interpretation of assortativity and apply this strucutral assortativity to the random dot product graph model. We also extend the results of Chung, Lu, and Vu on the spectral gap of the expected degree sequence model to a general class of random graph models with independent edges. We apply this result to the recently developed Stochastic Kronecker graph model of Leskovec, Chakrabarti, Kleinberg, and Faloutsos.

This work is aimed at the understanding of the dynamical behavior of hydrogen in diamond. The investigation was carried out using Transverse Field muon Spin Rotation (TF-SR) and the Longitudinal Field muon Spin Relaxation (LF-SR) techniques. The chemical analogy between hydrogen (p+e-) and muonium (u+e-) enabled the study of the indirect dynamical behavior of hydrogen in diamond. The TF-SR and LF-SR measurements were carried out in an isotopically pure 13C diamond in the temperature ranges 11 mK - 320 K and 10 K - 400 K, respectively. In the TF-SR results, the Prompt Absolute Fraction (PAF) of both diamagnetic and the paramagnetic states are temperature independent. The spin relaxation rate for the state is non-zero and temperature independent, while that of the Mux state is non-zero and temperature dependent. The behavior of indicates that the u+d is immobile, while that of MuT indicates the mobility in diamond. The LF-SR results provide hop rate and associated nuclear hyperfine interaction parameters of the Mut state in diamond. / Physics / M.Sc. (Physics)