The time it takes an athlete to recover from an injury can be highly influenced by training procedures as well as the medical care and physical therapy received. When an injury occurs to the muscles or tendons of an athlete, it is desirable to cool the muscles and tendons within the body to reduce inflammation, thereby reducing the recovery time. Consequently, finding a method of treatment that is effective in reducing tendon temperatures is beneficial to increasing the speed at which the athlete is able to recover. In this project, Bayesian inference with Gaussian processes will be used to model the effect that different treatments have in reducing tendon temperature within the ankle. Gaussian processes provide a powerful methodology for modeling data that exhibit complex characteristics such as nonlinear behavior while retaining mathematical simplicity.

A main objective of Departments of Transportation is to improve the safety of the roadways over which they have jurisdiction. Safety projects, such as cable barriers and raised medians, are utilized to reduce both crash frequency and crash severity. The efficacy of these projects must be evaluated in order to use resources in the best way possible. Five models are proposed for the evaluation of traffic projects: (1) a Bayesian Poisson regression model; (2) a hierarchical Poisson regression model building on model (1) by adding hyperpriors; (3) a similar model correcting for overdispersion; (4) a dynamic linear model; and (5) a traditional before-after study model. Evaluation of these models is discussed using various metrics including DIC. Using the models selected for analysis, it was determined that cable barriers are quite effective at reducing severe crashes and cross-median crashes on Utah highways. Raised medians are also largely effective at reducing severe crashes. The results of before and after analyses are highly valuable to Departments of Transportation in identifying effective projects and in determining which roadway segments will benefit most from their implementation.

Several passive energy dissipation devices have been implemented in practice as the seismic protective systems to mitigate structural damage caused by earthquakes. The solid viscoelastic dampers are among such passive energy dissipation systems. To examine the response reducing effectiveness of these dampers, it is necessary that engineers are able to conduct response analysis of structures installed with added dampers accurately and efficiently. The main objective of this work, therefore, is to develop formulations that can be effectively used with various models of the viscoelastic dampers to calculate the seismic response of a structure-damper system. To incorporate the mechanical effect from VE dampers in the structural dynamic design, it is important to use a proper force-deformation model to correctly describe the frequency dependence of the damper. The fractional derivative model and the general linear model are capable of capturing the frequency dependence of viscoelastic materials accurately. In our research, therefore, we have focused on the development of systematic procedures for calculating the seismic response for these models. For the fractional derivative model, we use the G1 and L1 algorithms to derive various numerical schemes for solving the fractional differential equations for earthquake motions described by acceleration time histories at discrete time points. For linear systems, we also develop a modal superposition method for this model of the damper. This superposition approach can be implemented to obtain the response time history for seismic input defined by the ground acceleration time history. For random ground motion that is described stochastically by the spectral density function, we derive an expression based on random vibration analysis to compute the mean square response of the system. It is noted that the numerical computations involved with the fractional derivative model can be complicated and cumbersome. To alleviate computation difficulty, we explore the use of a general linear model with Kelvin chain analog as a physical representation of the damper properties. The parameters in the model are determined through a curve fitting optimization process. To simplify the analytical work, a self-adjoint system of state equations are formulated by introducing auxiliary displacements for the internal elements in the Kelvin chain. This self-adjoint system can then be solved by using the modal superposition method, which can be extended to develop a response spectrum approach to calculate the seismic design response for the structural system for seismic inputs defined by design ground response spectra. Numerical studies are carried out to demonstrate the applicability of these formulations. Results show that all the proposed approaches provide accurate response values, and the response reduction effects of the viscoelastic dampers can be evaluated to assess their performance using these models and methods. However, the use of a general linear model of the damper is the most efficient. It can capture frequency dependence of the storage and loss moduli as well as the fractional derivative model. The calculation of the response by direct numerical integration of the equations of motion or through the use of the modal superposition approach is significantly simplified, and response spectrum formulation for the calculation of seismic response of design interest can be conveniently formulated. / Ph. D.

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This dissertation focuses on estimation and selection methods for a simple linear model with two levels of variation. This model provides a foundation for extensions to more levels. We propose new regularization criteria for model selection, subset selection, and variable selection in this context. Regularization is a penalized-estimation approach that shrinks the estimate and selects variables for structured data. This dissertation introduces a procedure (HM-ALASSO) that extends regularized multilevel-model estimation and selection to enforce principles of fixed heredity (e.g., including main effects when their interactions are included) and random heredity (e.g., including fixed effects when their random terms are included). The goals in developing this method were to create a procedure that provided reasonable estimates of all parameters, adhered to fixed and random heredity principles, resulted in a parsimonious model, was theoretically justifiable, and was able to be implemented and used in available software. The HM-ALASSO incorporates heredity-constrained selection directly into the estimation process. HM-ALASSO is shown to enjoy the properties of consistency, sparsity, and asymptotic normality. The ability of HM-ALASSO to produce quality estimates of the underlying parameters while adhering to heredity principles is demonstrated using simulated data. The performance of HM-ALASSO is illustrated using a subset of the High School and Beyond (HS&B) data set that includes math-achievement outcomes modeled via student- and school-level predictors. The HM-ALASSO framework is flexible enough that it can be adapted for various rule sets and parameterizations. / Statistics

Estimations of heritability and variance explained due to environmental exposures and interaction effects help in understanding complex diseases. Current methods to detect such interactions rely on variance component methods. These methods have been neces- sary due to the m » n problem, where the number of predictors (m) vastly outnumbers the number of observations (n). These methods are all computationally intensive, which is further exacerbated when considering gene-environment interactions, as the number of predictors increases from m to 2m+1 in the case of a single environmental exposure. Novel methods are thus needed to enable fast and unbiased calculations of the variance explained (R2) for gene-environment interactions in very large samples on multiple traits. Taking advantage of the large number of participants in contemporary genetic studies, we herein propose a novel method for continuous trait R2 estimates that are up to 20 times faster than current methods. We have devised a novel method, monsterlm, that enables multiple linear regression on large regions encompassing tens of thousands of variants in hundreds of thousands of participants. We tested monsterlm with simulations using real genotypes from the UK Biobank. During simulations we verified the properties of monsterlm to estimate the variance explained by interaction terms. Our preliminary results showcase potential interactions between blood biochemistry biomarkers such as HbA1c, Triglycerides and ApoB with an environmental factor relating to obesity-related lifestyle factor: Waist-hip Ratio (WHR). We further investigate these results to reveal that more than 50% of the interaction variance calculated can be attributed to ∼5% of the single-nucleotide polymorphisms (SNPs) interacting with the environmental trait. Lastly, we showcase the impact of interactions on improving polygenic risk scores. / Thesis / Master of Science (MSc)

Sensitivity analysis quantifies the effect that of perturbations of the model inputs have on the model's outputs. Some of the key insights gained using sensitivity analysis are to understand the robustness of the model with respect to perturbations, and to select the most important parameters for the model. MATLODE is a tool for sensitivity analysis of models described by ordinary differential equations (ODEs). MATLODE implements two distinct approaches for sensitivity analysis: direct (via the tangent linear model) and adjoint. Within each approach, four families of numerical methods are implemented, namely explicit Runge-Kutta, implicit Runge-Kutta, Rosenbrock, and single diagonally implicit Runge-Kutta. Each approach and family has its own strengths and weaknesses when applied to real world problems. MATLODE has a multitude of options that allows users to find the best approach for a wide range of initial value problems. In spite of the great importance of sensitivity analysis for models governed by differential equations, until this work there was no MATLAB ordinary differential equation sensitivity analysis toolbox publicly available. The two most popular sensitivity analysis packages, CVODES [8] and FATODE [10], are geared toward the high performance modeling space; however, no native MATLAB toolbox was available. MATLODE fills this need and offers sensitivity analysis capabilities in MATLAB, one of the most popular programming languages within scientific communities such as chemistry, biology, ecology, and oceanogra- phy. We expect that MATLODE will prove to be a useful tool for these communities to help facilitate their research and fill the gap between theory and practice. / Master of Science

This paper provides an alternative methodology for the analysis of a set of Likert responses measured on a common attitudinal scale when the primary focus of interest is on the relative importance of items in the set. The method makes fewer assumptions about the distribution of the responses than the more usual approaches such as comparisons of means, MANOVA or ordinal data methods. The approach transforms the Likert responses into paired comparison responses between the items. The complete multivariate pattern of responses thus produced can be analysed by an appropriately reformulated paired comparison model. The dependency structure between item responses can also be modelled flexibly. The advantage of this approach is that sets of Likert responses can be analysed simultaneously within the Generalized Linear Model framework, providing standard likelihood based inference for model selection. This method is applied to a recent international survey on the importance of environmental problems. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Les simulations et figures ont été réalisées avec le logiciel R. / Pour des données de panel, les mesures répétées dans le temps peuvent remettre en cause l’hypothèse d’indépendance entre les individus. Des tests ont été développés pour pouvoir vérifier s’il reste de la dépendance entre les résidus d’un modèle. Les trois tests que nous présentons dans ce mémoire sont ceux de Pesaran (2004), Friedman (1937) et Frees (1995). Ces trois tests se basent sur les résidus (et leurs corrélations) et ont été construits pour des modèles linéaires. Nous voulons étudier dans ce mémoire les performances de ces trois tests dans le cadre d’un modèle linéaire généralisé de Poisson. Dans ce but, on compare tout d’abord leurs performances (niveaux et puissances) pour deux modèles linéaires, l’un ayant un terme autorégressif et l’autre non. Par la suite, nous nous intéressons à leurs performances pour un modèle linéaire généralisé de Poisson en s’inspirant de Hsiao, Pesaran et Pick (2007) qui adaptent le test de Pesaran (2004) pour un modèle linéaire généralisé. Toutes nos comparaisons de performances se feront à l’aide de simulations dans lesquelles nous ferons varier un certain nombre de paramètres (nombre d’observations, force de la dépendance, etc.). Nous verrons que lorsque les corrélations sont toutes du même signe, le test de Pesaran donne en général de meilleurs résultats, à la fois dans les cas linéaires et pour le modèle linéaire généralisé. Le test de Frees présentera de bonnes propriétés dans le cas où le signe des corrélations entre les résidus alterne. / 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.