As of now, not many pharmaceutical companies are producing new categories of antibiotics to fight bacterial infections. Therefore, bacteria are building up a resistance to the medications commonly used. Often, antibiotic resistance begins within a hospital. To combat resistance, researchers completed several studies using cycling of the medications that are already in place, but they found either no improvement or the resistance increased with this type of setting. In addition, although preventative infection control measures have been shown to decrease antibiotic resistance for some antibiotics, the level of antibiotic resistance found in hospitals is still extremely high. This motivates the main goal of this thesis: to quantify how much the overall resistance can be lowered by simply adding one new drug to the regimen. The process of adding a new antibiotic can be quantified using mathematical models that show the flow of patients colonized with various types of bacteria into, out of, and within the hospital. Deterministic models can be used to model the spread of resistant bacteria in hospitals with a relatively large number of beds. However, not all hospitals are large enough to accurately determine the effects using a deterministic model; thus, we must use stochastic models, where mathematical formulations include probability in ways that describe intrinsic random fluctuations, typical of infection processes at smaller scales. In examining the addition of a new antibiotic within a hospital, we consider different administration protocols, either assuming that physicians are equally likely to prescribe the new antibiotic as they are to prescribe existing antibiotics or that physicians prescribe the new antibiotic to only a targeted population of patients. We will examine the variation in the expected level of overall resistance in a hospital depending on the administration procedure as well as the whether the hospital is large (deterministic model) or small (stochastic model). We will conclude with initial results for fitting these models to simulated data using common inverse problem methodology.

Experiments to learn about the effect of light, sex, and diet on the activity of flies generate great quantities of data that is necessary to analyze. Since different researches and students participate in the analysis of those experiments, it is convenient to have a methodology to analyze the experimental data using software so that the data can be analyzed in a uniform way. Being a double major in mathematics and biology, I am interested in:Deciding which statistical procedure to use to analyze the data so that the research questions of the researchers in biology are answered.To recommend how to implement those procedures using software in an efficient way.To write a prototype for the interpretation of the results.Those are the objectives of this work. In the thesis, we first applied two-way ANOVA to analyze the effect of two selected factors, sex (female and male) and diet (liver and non-liver), on the fly activity under dark condition and under light condition, respectively. Next, we employed the repeated measures to capture how fly activity changes over time (day in this case) and to relate the changes to the selected factors, sex and diet, also under dark condition and under light condition, respectively. Finally, we did a little research on the analysis of circadian rhythms and compared the results with that obtained from honey bee activity experiments carried out before.

We introduce a novel stochastic Petri net formalism where discrete and continuous phase-type firing delays can appear in the same model. By capturing deterministic and generally random behavior in discrete or continuous time, as appropriate, the formalism affords higher modeling fidelity and efficiencies to use in practice. We formally specify the underlying stochastic process as a general state space Markov chain and show that it is regenerative, thus amenable to renewal theory techniques to obtain steady-state solutions. We present two steady-state analysis methods depending on the class of problem: one using exact numerical techniques, the other using simulation. Although regenerative structures that ease steady-state analysis exist in general, a noteworthy problem class arises when discrete-time transitions are synchronized. In this case, the underlying process is semi-regenerative and we can employ Markov renewal theory to formulate exact and efficient numerical solutions for the stationary distribution. We propose a solution method that shows promise in terms of time and space efficiency. Also noteworthy are the computational tradeoffs when analyzing the "embedded" versus the "subordinate" Markov chains that are hidden within the original process. In the absence of simplifying assumptions, we propose an efficient regenerative simulation method that identifies hidden regenerative structures within continuous state spaces. The new formalism and solution methods are demonstrated with two applications.

Let a probability driven switch be defined as a switch of three input paths and three output paths. The status of the input paths defines a probability for each output path (as to whether it will generate a signal or not.) One output path is linked to one input path, so the results of the switch at time t can affect the switch at time t+1. A switch so constructed can be defined (by the probabilities) to take on the function of the standard logic gates (AND, OR, …) A net constructed of these switches can be “taught” by “reward” and “punish” algorithms to recognize input patterns. A simulation model showed that a repetitive learning algorithm coupled with a base knowledge (where new patterns are learned while continually checking past learned patterns) gives best results as a function of time. A good measure for the level of stability in response is to notice how many probabilities have converged to one or zero. The larger the number, the more stable the net.

This work presents an enhancement to the classification tree algorithm which forms the basis for Random Forests. Differently from the classical tree-based methods that focus on one variable at a time to separate the observations, the new algorithm performs the search for the best split in two-dimensional space using a linear combination of variables. Besides the classification, the method can be used to determine variables interaction and perform feature extraction. Theoretical investigations and numerical simulations were used to analyze the properties and performance of the new approach. Comparison with other popular classification methods was performed using simulated and real data examples. The algorithm was implemented as an extension package for the statistical computing environment R and is available for free download under the GNU General Public License.

This work presents an enhancement to the classification tree algorithm which forms the basis for Random Forests. Differently from the classical tree-based methods that focus on one variable at a time to separate the observations, the new algorithm performs the search for the best split in two-dimensional space using a linear combination of variables. Besides the classification, the method can be used to determine variables interaction and perform feature extraction. Theoretical investigations and numerical simulations were used to analyze the properties and performance of the new approach. Comparison with other popular classification methods was performed using simulated and real data examples. The algorithm was implemented as an extension package for the statistical computing environment R and is available for free download under the GNU General Public License.

Basketball is a sport that has become increasingly popular world-wide. At the professional level it is a game in which each of the five positions has a specific responsibility that requires unique skills. It seems likely that it would be valuable for coaches to know which skills for each position are most conducive to winning. Knowing which skills to develop for each position could help coaches optimize each player's ability by customizing practice to contain drills that develop the most important skills for each position that would in turn improve the team's overall ability. Through the use of Bayesian hierarchical modeling and NBA box-score performance categories, this project will determine how each position needs to perform in order for their team to be successful.

The International Normalizing Ratio (INR) measures the speed at which blood clots. Healthy people have an INR of about one. Some people are at greater risk of blood clots and their physician prescribes a target INR range, generally 2-3. The farther a patient is above or below their prescribed range, the more dangerous their situation. A variety of point-of-care (POC) devices has been developed to monitor patients. The purpose of this research was to develop innovative graphics to help describe a highly unbalanced dataset and to carry out Bayesian analyses to determine which of five devices best manages patients. An initial Bayesian analysis compared a machine-identical beta-binomial model to a machine-specific beta-binomial model. The response variable was number of in-range visits. A second Bayesian analysis compared a machine-identical lognormal model, a machine-specific lognormal model, and a machine-specific lognormal model with lower therapeutic bound as a predictor. The response variable was INR. Machines were compared using posterior predictive distributions of the absolute distance outside a patient's therapeutic range. For the beta-binomial models, the machine-identical model had the lower DIC, meaning that POC device was not a strong predictor of success in keeping a patient in-range. The machine-specific lognormal model with a term for lower therapeutic bound had the lowest DIC of the three lognormal models, implying that the additional information of distance out of range revealed differences among the POC devices. Three of the machines had more favorable out-of-range distributions than the other two. Both Bayesian analyses provided useful information for medical practice in managing INR.

University students often complete evaluations of their courses and instructors. The evaluation tool typically contains questions about the course and the instructor on an ordinal Likert scale. We assess instructor effectiveness while adjusting for known confounders. We present a probit regression model with a latent variable to measure the instructor effectiveness accounting for student specific covariates, such as student grade in the course, high school and university GPA, and ACT score.

Knowing the class to which a life distribution belongs gives us an idea about the aging of the device or system the life distribution represents, and enables us to compare the aging properties of different systems. This research intends to establish several new nonparametric classes of life distributions defined by the concept of inactivity time of a unit with a guaranteed minimum life length. These classes play an important role in the study of reliability theory, survival analysis, maintenance policies, economics, actuarial sciences and many other applied areas.