Determining risk to human populations due to natural disasters has been a topic of interest in the STEM fields for centuries. Earthquakes and the tsunamis they cause are of particular interest due to their repetition cycles. These cycles can last hundreds of years but we have only had modern measuring instruments for the last century or so which makes analysis difficult. In this document, we explore ways to improve upon an existing method for reconstructing earthquakes from historical accounts of tsunamis. This method was designed and implemented by Jared P Whitehead's research group over the last 5 years. The issue of this method that we address is the relatively slow convergence. One reason for this slow convergence is caused by the random walk proposal step in the Markov Chain Monte Carlo (MCMC) sampling. We explore ways of constructing an approximate gradient of the model in order to apply a more robust MCMC Method called MALA that uses a gradient combined with some randomness to propose new samples. The types of approximate gradients we explore were a heuristic gradient, a data driven gradient and a gradient of a surrogate model. We chose to use the gradient of a simplified tsunami formula for our implementation. Our MALA algorithm under performed the previous random walk method which we believe implies that the simplified tsunami model didn't give sufficient information to guide the proposed samples in the optimal direction. Further experimentation would be needed to confirm this and we are confident that there are other ways we can improve our convergence as specified in the future work section. Our method is built into the existing Python package tsunamibayes. It is available, open-source, on GitHub: https://github.com/jwp37/tsunamibayes.

Year after years, the amount of data that we continuously generate is increasing. When this situation started the main challenge was to find a way to store the huge quantity of information. Nowadays, with the increasing availability of storage facilities, this problem is solved but it gives us a new issue to deal with: find tools that allow us to learn from this large data sets. In this thesis, a framework for Bayesian learning with the ability to scale to large data sets is studied. We present the Stochastic Gradient Langevin Dynamics (SGLD) framework and show that in some cases its approximation of the posterior distribution is quite poor. A reason for this can be that SGLD estimates the gradient of the log-likelihood with a high variability due to naïve sampling. Our approach combines accurate proxies for the gradient of the log-likelihood with SGLD. We show that it produces better results in terms of convergence to the correct posterior distribution than the standard SGLD, since accurate proxies dramatically reduce the variance of the gradient estimator. Moreover, we demonstrate that this approach is more efficient than the standard Markov Chain Monte Carlo (MCMC) method and that it exceeds other techniques of variance reduction proposed in the literature such as SAGA-LD algorithm. This approach also uses control variates to improve SGLD so that it is straightforward the comparison with our approach. We apply the method to the Logistic Regression model.

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Le sujet de notre mémoire est l'intersection entre deux domaines : La théorie des valeurs extrêmes (TVE) et les copules. L'objet de la TVE est de trouver la loi limite du maximum d'un échantillon. Grâce aux résultats de la TVE, on peut modéliser les phénomènes extrêmes. Par aillleurs, il existe une variante bivariée de la TVE. La variante bivariée de la TVE utilise une famille de copules appelées copules de valeurs extrêmes pour tenir compte de la liaison entre les deux phénomènes extrêmes. En dimension 2, toute copule de valeurs extrêmes dépend d'une fonction de Pickands. L'objet de notre mémoire est d'estimer la fonction de Pickands à partir de données. Nous avons trouvé un moyen de construire une fonction de Pickands grâce à des splines cubiques. À partir de cette construction, on obtient une famille élargie de fonctions de Pickands dans laquelle nous effectuons notre inférence statistique. Nous avons choisit l'approche bayésienne pour construire l'estimateur et les méthodes de MCMC pour les évaluations numériques. La méthode a été appliquée sur des données simulées et réelles. / The subject of our thesis is intersection between two fields: The Extreme Value Theory (EVT) and copulas. The object of EVT is to find the limit law of the maximum of a sample. Due to the results of EVT, we can model extreme phenomena. In addition, there is a bivariate variant of EVT. The bivariate variant of EVT uses a family of copulas called extreme value copulas to account for the connection between the two extreme events. Any copula with extreme values depends on a Pickands function. The object of our thesis is to estimate the Pickands function from data. We have found a way to build a Pickands function using cubic splines. From this construction, we obtain an extended family of Pickands functions in which we perform our statistical inference. We chose the Bayesian approach to build the estimator and the MCMC methods for the estimates. The method was applied on simulated and real data.

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No description available.

Clinical trials are major and costly undertakings for researchers. Planning a clinical trial involves careful selection of the primary and secondary efficacy endpoints. The 2010 draft FDA guidance on adaptive designs acknowledges possible study design modifications, such as selection and/or order of secondary endpoints, in addition to sample size re-estimation. It is essential for the integrity of a double-blind clinical trial that individual treatment allocation of patients remains unknown. Methods have been proposed for re-estimating the sample size of clinical trials, without unblinding treatment arms, for both categorical and continuous outcomes. Procedures that allow a blinded estimation of the treatment effect, using knowledge of trial operational characteristics, have been suggested in the literature. Clinical trials are designed to evaluate effects of one or more treatments on multiple primary and secondary endpoints. The multiplicity issues when there is more than one endpoint require careful consideration for controlling the Type I error rate. A wide variety of multiplicity approaches are available to ensure that the probability of making a Type I error is controlled within acceptable pre-specified bounds. The widely used fixed sequence gate-keeping procedures require prospective ordering of null hypotheses for secondary endpoints. This prospective ordering is often based on a number of untested assumptions about expected treatment differences, the assumed population variance, and estimated dropout rates. We wish to update the ordering of the null hypotheses based on estimating standardized treatment effects. We show how to do so while the study is ongoing, without unblinding the treatments, without losing the validity of the testing procedure, and with maintaining the integrity of the trial. Our simulations show that we can reliably order the standardized treatment effect also known as signal-to-noise ratio, even though we are unable to estimate the unstandardized treatment effect. In order to estimate treatment difference in a blinded setting, we must define a latent variable substituting for the unknown treatment assignment. Approaches that employ the EM algorithm to estimate treatment differences in blinded settings do not provide reliable conclusions about ordering the null hypotheses. We developed Bayesian approaches that enable us to order secondary null hypotheses. These approaches are based on posterior estimation of signal-to-noise ratios. We demonstrate with simulation studies that our Bayesian algorithms perform better than existing EM algorithm counterparts for ordering effect sizes. Introducing informative priors for the latent variables, in settings where the EM algorithm has been used, typically improves the accuracy of parameter estimation in effect size ordering. We illustrate our method with a secondary analysis of a longitudinal study of depression. / Statistics

Mountain meadows in the western USA are experiencing increased rates of conifer encroachment due to climate change and land management practices. Past research has focused on conifer removal as a meadow restoration strategy, but there has been limited work on conifer transpiration in a pre-restoration state. Meadow restoration by conifer removal has the primary goal of recovering sufficient growing season soil moisture necessary for endemic, herbaceous meadow vegetation. Therefore, conifer water use represents an important hydrologic output toward evaluating the efficacy of this active management approach. This study quantified and evaluated transpiration of encroached conifers in a mountain meadow using sap flow prior to restoration by tree removal. We report results of lodgepole pine transpiration estimates for an approximate 1-year period and an evaluation of key environmental variables influencing water use during a dry growing season. The study was conducted at Rock Creek Meadow (RCM) in the southern Cascade Range near Chester, CA, USA. Sap flow data were collected in a sample of lodgepole pine and scaled on a per-plot basis to the larger meadow using tree survey data within a stratified random sampling design (simple scaling). These estimates were compared to a MODIS evapotranspiration (ET) estimate for the meadow. The 1-year period for transpiration estimates overlapped each of the 2019 and 2020 growing seasons partially. The response of lodgepole pine transpiration to solar radiation, air temperature, vapor pressure deficit, and volumetric soil water content was investigated by calibrating a modified Jarvis-Stewart (MJS) model to hourly sap flow data collected during the 2020 growing season, which experienced below average antecedent winter precipitation. The model was validated using spatially different sap flow data in the meadow from the 2021 growing season, also part of a dry year. Calibration and validation were completed using a MCMC approach via the DREAM(ZS) algorithm and a generalized likelihood (GL) function, enabling model parameter and total uncertainty assessment. We also used the model to inform transpiration scaling for the calibration period in select plots in the meadow, which allowed comparison with simple scaling transpiration estimates. Average total lodgepole pine transpiration at RCM was estimated between 220.57 ± 25.28 and 393.39 ± 45.65 mm for the entire campaign (mid-July 2019 to mid-August 2020) and between 100.22 ± 11.49 and 178.75 ± 20.74 mm for the 2020 partial growing season (April to mid-August). The magnitude and seasonal timing were similar to MODIS ET. The model showed good agreement between observed and predicted sap velocity for the 2020 partial growing season (RMSE = 1.25 cm h-1), with meteorological variables modulating early growing season sap flow and volumetric soil water content decline imposing transpiration decrease in the late growing season. The model validation performed similarly to calibration in terms of performance metrics and the influence of meteorological variables. The consistency of the declining volumetric soil water content effect during the late growing season between periods could not be evaluated due to an abridged validation period. Overall, the implementation GL-DREAM(ZS) showed promise for future use in MJS models. Lastly, the model derived transpiration estimates for the 2020 partial growing season showed some of the potential utility in using the MJS model to scale sap flow at the study locale. It also highlights some of the key limitations of this approach as it is executed in the present study.

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No description available.