The study of cardiac action potentials has many medical applications. Dr. Dennis Noble first used mathematical models to study cardiac action potentials in the 1960s. We begin our study of cardiac action potentials with one form of the Fitzhugh-Nagumo partial differential equation. We use the non-classical method to produce a closed form solution for the decoupled Fitzhugh Nagumo equation. Using voltage recording data of action potentials in a cardiac myocyte and in purkinje fibers, we estimate parameter values for the closed form solution with standard linear and non-linear regression methods. Results are limited, thus leading us to perturb the solution to obtain a better fit. We turn to singular perturbation theory to justify our pole-based approach. Finally, we test our model on independent action potential data sets to evaluate our model and to draw conclusions on how our model can be applied.

In this paper, we study a new family of random variables, that arise as the distribution of extrema of a random number N of independent and identically distributed random variables X1,X2, ..., XN, where each Xi has a common continuous distribution with support on [0,1]. The general scheme is first outlined, and SUG and CSUG models are introduced in detail where Xi is distributed as U[0,1]. Some features of the proposed distributions can be studied via its mean, variance, moments and moment-generating function. Moreover, we make some other choices for the continuous random variables such as Arcsine, Topp-Leone, and N is chosen to be Geometric or Zipf. Wherever appropriate, we estimate of the parameter in the one-parameter family in question and test the hypotheses about the parameter. In the last section, two permutation distributions are introduced and studied.

Traditional approaches to predicting financial market dynamics tend to be linear and stationary, whereas financial time series data is increasingly nonlinear and non-stationary. Lately, advances in dynamical systems theory have enabled the extraction of complex dynamics from time series data. These developments include theory of time delay embedding and phase space reconstruction of dynamical systems from a scalar time series. In this thesis, a time delay embedding approach for predicting intraday stock or stock index movement is developed. The approach combines methods of nonlinear time series analysis with those of causality testing, theory of dynamical systems and machine learning (artificial neural networks). The approach is then applied to the Standard and Poors Index, and the results from our method are compared to traditional methods applied to the same data set.

Other research reported that genetic mechanism plays a major role in the development process of biological shapes. The primary goal of this dissertation is to develop novel statistical models to investigate the quantitative relationships between biological shapes and genetic variants. However, these problems can be extremely challenging to traditional statistical models for a number of reasons: 1) the biological phenotypes cannot be effectively represented by single-valued traits, while traditional regression only handles one dependent variable; 2) in real-life genetic data, the number of candidate genes to be investigated is extremely large, and the signal-to-noise ratio of candidate genes is expected to be very high. In order to address these challenges, we propose three statistical models to handle multivariate, functional, and multilevel functional phenotypes, with applications to biological shape data using different shape descriptors. To the best of our knowledge, there is no statistical model developed for multilevel functional phenotypes. Even though multivariate regressions have been well-explored and these approaches can be applied to genetic studies, we show that the model proposed in this dissertation can outperform other alternatives regarding variable selection and prediction through simulation examples and real data examples. Although motivated ultimately by genetic research, the proposed models can be used as general-purpose machine learning algorithms with far-reaching applications.

The continued study of asymptomatic Ebolavirus infection is necessary to develop a more complete understanding of Ebola transmission dynamics. This paper conducts a meta-analysis of eight studies that measure seroprevalence (the number of subjects that test positive for anti-Ebolavirus antibodies in their blood) in subjects with household exposure or known case-contact with Ebola, but that have shown no symptoms. In our two random effects Bayesian hierarchical models, we find estimated seroprevalences of 8.76% and 9.72%, significantly higher than the 3.3% found by a previous meta-analysis of these eight studies. We also produce a variation of this meta-analysis where we exclude two of the eight studies. In this model, we find an estimated seroprevalence of 4.4%, much lower than our first two Bayesian hierarchical models. We believe a random effects model more accurately reflects the heterogeneity between studies and thus asymptomatic Ebola is more seroprevalent than previously believed among subjects with household exposure or known case-contact. However, a strong conclusion cannot be reached on the seriousness of asymptomatic Ebola without an international testing standard and more data collection using this adopted standard.

It is estimated that Periodontal Diseases effects up to 90% of the adult population. Given the complexity of the host environment, many factors contribute to expression of the disease. Age, Gender, Socioeconomic Status, Smoking Status, and Race/Ethnicity are all known risk factors, as well as a handful of known comorbidities. Certain vitamins and minerals have been shown to be protective for the disease, while some toxins and chemicals have been associated with an increased prevalence. The role of toxins, chemicals, vitamins, and minerals in relation to disease is believed to be complex and potentially modified by known risk factors. A large comprehensive dataset from 1999-2003 from the National Health and Nutrition Examination Survey (NHANES) contains full and partial mouth examinations on subjects for measurement of periodontal diseases as well as patient demographic information and approximately 150 environmental variables. In this dissertation, a Feasible Solution Algorithm (FSA) will be used to investigate statistical interactions of these various chemical and environmental variables related to periodontal disease. This sequential algorithm can be used on traditional statistical modeling methods to explore two and three way interactions related to the outcome of interest. FSA can also be used to identify unique subgroups of patients where periodontitis is most (or least) prevalent. In this dissertation, FSA is used to explore the NHANES data and suggest interesting relationships between the toxins, chemicals, vitamins, minerals and known risk factors that have not been previously identified.

I consider statistical modelling of data gathered by photographic identification in mark-recapture studies and propose a new method that incorporates the inherent uncertainty of photographic identification in the estimation of abundance, survival and recruitment. A hierarchical model is proposed which accepts scores assigned to pairs of photographs by pattern recognition algorithms as data and allows for uncertainty in matching photographs based on these scores. The new models incorporate latent capture histories that are treated as unknown random variables informed by the data, contrasting past models having the capture histories being fixed. The methods properly account for uncertainty in the matching process and avoid the need for researchers to confirm matches visually, which may be a time consuming and error prone process. Through simulation and application to data obtained from a photographic identification study of whale sharks I show that the proposed method produces estimates that are similar to when the true matching nature of the photographic pairs is known. I then extend the method to incorporate auxiliary information to predetermine matches and non-matches between pairs of photographs in order to reduce computation time when fitting the model. Additionally, methods previously applied to record linkage problems in survey statistics are borrowed to predetermine matches and non-matches based on scores that are deemed extreme. I fit the new models in the Bayesian paradigm via Markov Chain Monte Carlo and custom code that is available by request.

Methods of estimating parametric and nonparametric components, as well as properties of the corresponding estimators, have been examined in partially linear models by Wahba [1987], Green et al. [1985], Engle et al. [1986], Speckman [1988], Hu et al. [2004], Charnigo et al. [2015] among others. These models are appealing due to their flexibility and wide range of practical applications including the electricity usage study by Engle et al. [1986], gum disease study by Speckman [1988], etc., wherea parametric component explains linear trends and a nonparametric part captures nonlinear relationships. The compound estimator (Charnigo et al. [2015]) has been used to estimate the nonparametric component of such a model with multiple covariates, in conjunction with linear mixed modeling for the parametric component. These authors showed, under a strict orthogonality condition, that parametric and nonparametric component estimators could achieve what appear to be (nearly) optimal rates, even in the presence of subject-specific random effects. We continue with research on partially linear models with subject-specific random intercepts. Inspired by Speckman [1988], we propose estimators of both parametric and nonparametric components of a partially linear model, where consistency is achievable under an orthogonality condition. We also examine a scenario without orthogonality to find that bias could still exist asymptotically. The random intercepts accommodate analysis of individuals on whom repeated measures are taken. We illustrate our estimators in a biomedical case study and assess their finite-sample performance in simulation studies. Jump points have often been found within the domain of nonparametric models (Muller [1992], Loader [1996] and Gijbels et al. [1999]), which may lead to a poor fit when falsely assuming the underlying mean response is continuous. We study a specific type of change-point where the underlying mean response is continuous on both left and right sides of the change-point. We identify the convergence rate of the estimator proposed in Liu [2017] and illustrate the result in simulation studies.

Finite Mixture model has been studied for a long time, however, traditional methods assume that the variables are measured without error. Mixtures-of-regression model with measurement error imposes challenges to the statisticians, since both the mixture structure and the existence of measurement error can lead to inconsistent estimate for the regression coefficients. In order to solve the inconsistency, We propose series of methods to estimate the mixture likelihood of the mixtures-of-regressions model when there is measurement error, both in the responses and predictors. Different estimators of the parameters are derived and compared with respect to their relative efficiencies. The simulation results show that the proposed estimation methods work well and improve the estimating process.

A practical problem often encountered with observed count data is the presence of excess zeros. Zero-inflation in count data can easily be handled by zero-inflated models, which is a two-component mixture of a point mass at zero and a discrete distribution for the count data. In the presence of predictors, zero-inflated Poisson (ZIP) regression models are, perhaps, the most commonly used. However, the fully parametric ZIP regression model could sometimes be restrictive, especially with respect to the mixing proportions. Taking inspiration from some of the recent literature on semiparametric mixtures of regressions models for flexible mixture modeling, we propose a semiparametric ZIP regression model. We present an "EM-like" algorithm for estimation and a summary of asymptotic properties of the estimators. The proposed semiparametric models are then applied to a data set involving clandestine methamphetamine laboratories and Alzheimer's disease.