Mathematics ThesesCopyright (c) 2015 Georgia State University All rights reserved.
http://scholarworks.gsu.edu/math_theses
Recent documents in Mathematics Thesesen-usThu, 10 Dec 2015 01:45:42 PST3600Numerical Solutions to Two-Dimensional Integration Problems
http://scholarworks.gsu.edu/math_theses/151
http://scholarworks.gsu.edu/math_theses/151Tue, 08 Dec 2015 08:37:13 PST
This paper presents numerical solutions to integration problems with bivariate integrands. Using equally spaced nodes in Adaptive Simpson's Rule as a base case, two ways of sampling the domain over which the integration will take place are examined. Drawing from Ouellette and Fiume, Voronoi sampling is used along both axes of integration and the corresponding points are used as nodes in an unequally spaced degree two Newton-Cotes method. Then the domain of integration is triangulated and used in the Triangular Prism Rules discussed by Limaye. Finally, both of these techniques are tested by running simulations over heavily oscillatory and monomial (up to degree five) functions over polygonal regions.
]]>
Alexander CarstairsAccurate Approximation Series for Optimal Targeting Regions in a Neural Growth Model with a Low –branching Probability
http://scholarworks.gsu.edu/math_theses/150
http://scholarworks.gsu.edu/math_theses/150Thu, 03 Dec 2015 08:22:14 PST
Understanding the complex growth process of dendritic arbors is essential for the medical field and disciplines like Biology and Neurosciences. The establishment of the dendritic patterns has received increasing attention from experimental researchers that seek to determine the cellular mechanisms that play a role in the growth of neural trees. Our goal in this thesis was to prove the recurrence formula for the probability distribution of all possible neural trees, as well as the formulas of the expected number of active branches and their variances. We also derived formulas for the spatial locations of the optimal targeting region for a tree with branching probability. These formulas were necessary for the simplified stochastic computational model that Osan et al have developed in order to examine how changes in branching probability influence the success of targeting neurons located at different distances away from a starting point.
]]>
Bernardo NietoTrends in Herpes Zoster Incidence from 1940 to 2008 Using a Cross-sectional Survey
http://scholarworks.gsu.edu/math_theses/149
http://scholarworks.gsu.edu/math_theses/149Mon, 30 Nov 2015 08:32:07 PST
Previous healthcare-based studies have reported increasing herpes zoster (HZ) incidence over time; however, this could be an artifact of increased healthcare utilization. This study is a cross-sectional analysis of 15,103 respondents in the 2008 wave of the Health and Retirement Study (HRS) to evaluate changes in HZ incidence from 1940 to 2008. Negative binomial regression is used to model the effect of calendar year, age of onset of HZ, gender and race/ethnicity on HZ incidence. A nonparametric method based on B-spline basis expansion is used to model the effect of calendar year to avoid imposing a predetermined functional form and produce flexible and accurate estimates. This study demonstrates increasing HZ incidence from 1940 to 2008 using self-reported HZ. Although the reason for this increase remains unknown, this study supports the assertion that this trend is real and not an artifact of increasing healthcare utilization for HZ over time.
]]>
Craig HalesDietary Sodium Intake and Mortality among US Older Adults: The Third National Health and Nutrition Examination Survey
http://scholarworks.gsu.edu/math_theses/148
http://scholarworks.gsu.edu/math_theses/148Tue, 25 Aug 2015 09:16:54 PDT
Strong evidence has linked dietary sodium intake to blood pressure, but the effects of sodium intake on cardiovascular diseases (CVD) outcomes remain elusive, especially for older population. We examined the association between estimated usual sodium intake and CVD and all-cause mortality in a nationally representative sample of 4068 US adults aged 51 and older surveyed in 1988-1994. After a mean follow-up of 12.9 years from 1988 to 2006, 1680 participants died: 734 from CVD; 392 from ischemic heart disease (IHD); and 144 from stroke. In the Cox proportional models adjusted for sociodemographic variables and CVD risk factors, sodium intake was not significantly associated with all-cause, CVD, IHD and stroke mortality. No significant interactions were observed between sodium intake and sex, race/ethnicity, hypertension status, body mass index or physical activity for any of the outcomes studied. However, among Mexican-Americans sodium intake was significantly and linearly associated with CVD mortality.
]]>
Lixia ZhaoMultinomial Logistic Regression Analysis Of Varicella Vaccination - 2011 National Immunization Survey (NIS) – Teen Survey Data
http://scholarworks.gsu.edu/math_theses/147
http://scholarworks.gsu.edu/math_theses/147Tue, 28 Apr 2015 09:37:13 PDT
The varicella-zoster virus (VZV) causes chickenpox or varicella, a disease primarily in children, and Herpes Zoster (HZ) or zoster or shingles, a disease that affects adults. A 2-dose Varicella vaccination is recommended in the United States, the first dose at age 15-18 months and the second dose at 4 to 6 years.In this study, we used multinomial logistic regression to analysis data from the 2011 National Immunization Survey-Teen (NIS-Teen) to identify factors that have a significant impact on the number of doses (0-dose, 1-dose, or 2-dose) a teen will have. We evaluate Varicella vaccination coverage stratified by Census region and assessed factors independently associated with varicella vaccination.
]]>
Benjamin FreduaFrobenius-Like Permutations and Their Cycle Structure
http://scholarworks.gsu.edu/math_theses/146
http://scholarworks.gsu.edu/math_theses/146Fri, 24 Apr 2015 11:57:04 PDT
Polynomial functions over finite fields are a major tool in computer science and electrical engineering and have a long history. Some of its aspects, like interpolation and permutation polynomials are described in this thesis. A complete characterization of subfield compatible polynomials (f in E[x] such that f(K) is a subset of L, where K,L are subfields of E) was recently given by J. Hull. In his work, he introduced the Frobenius permutation which played an important role. In this thesis, we fully describe the cycle structure of the Frobenius permutation. We generalize it to a permutation called a monomial permutation and describe its cycle factorization. We also derive some important congruences from number theory as corollaries to our work.
]]>
Adil B. ViraniEmpirical Likelihood Confidence Intervals for the Population Mean Based on Incomplete Data
http://scholarworks.gsu.edu/math_theses/145
http://scholarworks.gsu.edu/math_theses/145Fri, 24 Apr 2015 07:17:18 PDT
The use of doubly robust estimators is a key for estimating the population mean response in the presence of incomplete data. Cao et al. (2009) proposed an alternative doubly robust estimator which exhibits strong performance compared to existing estimation methods. In this thesis, we apply the jackknife empirical likelihood, the jackknife empirical likelihood with nuisance parameters, the profile empirical likelihood, and an empirical likelihood method based on the influence function to make an inference for the population mean. We use these methods to construct confidence intervals for the population mean, and compare the coverage probabilities and interval lengths using both the ``usual'' doubly robust estimator and the alternative estimator proposed by Cao et al. (2009). An extensive simulation study is carried out to compare the different methods. Finally, the proposed methods are applied to two real data sets.
]]>
Jose Manuel Valdovinos AlvarezDiscrepancy Principle and Stable Parameter Estimation in Avian Influenza
http://scholarworks.gsu.edu/math_theses/144
http://scholarworks.gsu.edu/math_theses/144Fri, 05 Dec 2014 10:36:50 PST
In the case of a linear ill-posed problem with noisy data, a version of an a posteriori parameter selection discrepancy principle (DP) is justified for an arbitrary regularization strategy under very general assumptions on the operator and the stabilizer. Its efficiency is demonstrated for a practically important inverse problem in avian influenza. We refer to our result as an abstract discrepancy principle (ADP), which shows that applicability of the DP largely depends on the level of noise in the data rather than the method used for the construction of a specific regularization procedure.
]]>
Linda DeCampData Mining Analysis of the Parkinson's Disease
http://scholarworks.gsu.edu/math_theses/143
http://scholarworks.gsu.edu/math_theses/143Fri, 05 Dec 2014 07:56:46 PST
Biological research is becoming increasingly database driven and statistical learning can be used to discover patterns in the biological data. In the thesis, the supervised learning approaches are utilized to analyze the Oxford Parkinson’s disease detection data and build models for prediction or classification. We construct predictive models based on training set, evaluate their performance by applying these models to an independent test set, and find the best methods for predicting whether people have Parkinson’s disease. The proposed artificial neural network procedure outperforms with the best and highest prediction accuracy, while the logistic and probit regressions are preferred statistical models which can offer better interpretation with the higher prediction accuracy compared to other proposed data mining approaches.
]]>
Xiaoyuan WangA Mathematical Model For Population Dynamics of Antibiotic Treatment
http://scholarworks.gsu.edu/math_theses/142
http://scholarworks.gsu.edu/math_theses/142Thu, 04 Dec 2014 12:17:56 PST
The objective of the thesis is to model the behavior of the reaction between two species of bacteria and antibiotics by building an ordinary differential equation (ODE) system under a list of assumptions. With the ODE, we analyze equilibrium points and the stability of these equilibrium points to forecast the trend of each species of bacteria and antibiotics. We test the validity of the model assumptions. Based on these outcomes, we show that: 1. Both equilibrium points and eigenvalues differ in orders of magnitude. 2. Some figures which were generated using different initial values do not make any sense. 3. There were abnormal values of the variables sensitivity.
]]>
siyu tianJackknife Empirical Likelihood Inference For The Pietra Ratio
http://scholarworks.gsu.edu/math_theses/140
http://scholarworks.gsu.edu/math_theses/140Wed, 03 Dec 2014 08:22:00 PST
Pietra ratio (Pietra index), also known as Robin Hood index, Schutz coefficient (Ricci-Schutz index) or half the relative mean deviation, is a good measure of statistical heterogeneity in the context of positive-valued data sets. In this thesis, two novel methods namely "adjusted jackknife empirical likelihood" and "extended jackknife empirical likelihood" are developed from the jackknife empirical likelihood method to obtain interval estimation of the Pietra ratio of a population. The performance of the two novel methods are compared with the jackknife empirical likelihood method, the normal approximation method and two bootstrap methods (the percentile bootstrap method and the bias corrected and accelerated bootstrap method). Simulation results indicate that under both symmetric and skewed distributions, especially when the sample is small, the extended jackknife empirical likelihood method gives the best performance among the six methods in terms of the coverage probabilities and interval lengths of the confidence interval of Pietra ratio; when the sample size is over 20, the adjusted jackknife empirical likelihood method performs better than the other methods, except the extended jackknife empirical likelihood method. Furthermore, several real data sets are used to illustrate the proposed methods.
]]>
Yueju SuA Comparison of Two Modeling Techniques in Customer Targeting For Bank Telemarketing
http://scholarworks.gsu.edu/math_theses/139
http://scholarworks.gsu.edu/math_theses/139Tue, 02 Dec 2014 13:37:18 PST
Customer targeting is the key to the success of bank telemarketing. To compare the flexible discriminant analysis and the logistic regression in customer targeting, a survey dataset from a Portuguese bank was used. For the flexible discriminant analysis model, the backward elimination of explanatory variables was used with several rounds of manual re-defining of dummy variables. For the logistic regression model, the automatic stepwise selection was performed to decide which explanatory variables should be left in the final model. Ten-fold stratified cross validation was performed to estimate the model parameters and accuracies. Although employing different sets of explanatory variables, the flexible discriminant analysis model and the logistic regression model show equally satisfactory performances in customer classification based on the areas under the receiver operating characteristic curves. Focusing on the predicted “right” customers, the logistic regression model shows slightly better classification and higher overall correct prediction rate.
]]>
Hong TangClassification of Genotype and Age by Spatial Aspects of RPE Cell Morphology
http://scholarworks.gsu.edu/math_theses/138
http://scholarworks.gsu.edu/math_theses/138Mon, 21 Jul 2014 06:37:28 PDT
Age related macular degeneration (AMD) is a public health concern in an aging society. The retinal pigment epithelium (RPE) layer of the eye is a principal site of pathogenesis for AMD. Morphological characteristics of the cells in the RPE layer can be used to discriminate age and disease status of individuals. In this thesis three genotypes of mice of various ages are used to study the predictive abilities of these characteristics. The disease state is represented by two mutant genotypes and the healthy state by the wild-type. Classification analysis is applied to the RPE morphology from the different spatial regions of the RPE layer. Variable reduction is accomplished by principal component analysis (PCA) and classification analysis by the k-nearest neighbor (k-NN) algorithm. In this way the differential ability of the spatial regions to predict age and disease status by cellular variables is explored.
]]>
Michael BoringMRI Signal Intensity Analysis of Novel Protein-based MRI Contrast Agents
http://scholarworks.gsu.edu/math_theses/137
http://scholarworks.gsu.edu/math_theses/137Mon, 21 Jul 2014 06:12:31 PDT
Contrast agents are of great importance in clinical applications of Magnetic Resonance Imaging (MRI) to improve the contrast of internal body structures and to obtain tissue-specific image. However, current approved contrast agents still have limitations including low relaxivity, low specificity and uncontrolled blood circulation time, which motivated researchers to develop novel contrast agents with higher relaxivity, improved targeting abilities and optimal retention time. This thesis uses animal experimental data from Dr. Jenny J. Yang’s lab at the Department of Chemistry in Georgia State University to study effects of a class of newly designed protein-based MRI contrast agents (ProCAs). Models for the longitudinal data on MRI intensity are constructed to evaluate the efficiency of different MRI contrast agents. Statistically significant results suggest that ProCA1B14 has the great potential to be a tumor specific contrast agent and ProCA32 could be a promising MRI contrast agent for the liver imaging in clinical applications.
]]>
Yan QianInfluence Function-based Empirical Likelihood Inferences for Lorenz Curve
http://scholarworks.gsu.edu/math_theses/136
http://scholarworks.gsu.edu/math_theses/136Fri, 25 Apr 2014 11:02:31 PDT
In this thesis, an empirical likelihood method based on influence function is developed and used to construct confidence intervals for the Lorenz ordinates. This method is defined under the simple random sampling and the limiting distribution of the proposed empirical likelihood ratio statistic is a standard Chi-square distribution. Extensive simulation studies are conducted to evaluate the proposed empirical likelihood-based confidence intervals for the Lorenz ordinates. Finally, this method is used on a real income data as an application.
]]>
Bing LiuJackknife Empirical Likelihood Inferences for the Skewness and Kurtosis
http://scholarworks.gsu.edu/math_theses/135
http://scholarworks.gsu.edu/math_theses/135Fri, 25 Apr 2014 10:32:26 PDT
Skewness and kurtosis are measures used to describe shape characteristics of distributions. In this thesis, we examine the interval estimates about the skewness and kurtosis by using jackknife empirical likelihood (JEL), adjusted JEL, extended JEL, traditional bootstrap, percentile bootstrap, and BCa bootstrap methods. The limiting distribution of the JEL ratio is the standard chi-squared distribution. The simulation study of this thesis makes a comparison of different methods in terms of the coverage probabilities and interval lengths under the standard normal distribution and exponential distribution. The proposed adjusted JEL and extended JEL perform better than the other methods. Finally we illustrate the proposed JEL methods and different bootstrap methods with three real data sets.
]]>
Yan ZhangJackknife Empirical Likelihood-Based Confidence Intervals for Low Income Proportions with Missing Data
http://scholarworks.gsu.edu/math_theses/134
http://scholarworks.gsu.edu/math_theses/134Tue, 10 Dec 2013 06:27:16 PST
The estimation of low income proportions plays an important role in comparisons of poverty in different countries. In most countries, the stability of the society and the development of economics depend on the estimation of low income proportions. An accurate estimation of a low income proportion has a crucial role for the development of the natural economy and the improvement of people's living standards. In this thesis, the Jackknife empirical likelihood method is employed to construct confidence intervals for a low income proportion when the observed data had missing values. Comprehensive simulation studies are conducted to compare the relative performances of two Jackknife empirical likelihood based confidence intervals for low income proportions in terms of coverage probability. A real data example is used to illustrate the application of the proposed methods.
]]>
YANAN YINGeneralized Confidence Intervals for Partial Youden Index and its Corresponding Optimal Cut-Off Point
http://scholarworks.gsu.edu/math_theses/133
http://scholarworks.gsu.edu/math_theses/133Tue, 10 Dec 2013 06:27:14 PST
In the field of diagnostic test studies, the accuracy of a diagnostic test is essential in evaluating the performance of the test. The receiver operating characteristic (ROC) curve and the area under the curve (AUC) are widely used in such evaluation procedures. Meanwhile, the Youden index is also introduced into practice to measure the accuracy of the diagnostic test from another aspect. The Youden index maximizes the sum of sensitivity and specificity, assuring decent true positive and negative rates. It draws one's attention due to its merit of finding the optimal cut-off points of biomarkers. Similar to Partial ROC, a new index, called "Partial Youden index" can be defined as an extension of Youden's Index. It is more meaningful than regular Youden index since the regular one is just a special case of the Partial Youden Index. In this thesis, we focus on construction of generalized confidence intervals for the Partial Youden Index and its corresponding optimal cut-off points. Extensive simulation studies are conducted to evaluate the finite sample performances of the new intervals.
]]>
Chenxue LiJackknife Empirical Likelihood Inference for the Absolute Mean Deviation
http://scholarworks.gsu.edu/math_theses/132
http://scholarworks.gsu.edu/math_theses/132Wed, 24 Jul 2013 08:16:55 PDT
In statistics it is of interest to find a better interval estimator of the absolute mean deviation. In this thesis, we focus on using the jackknife, the adjusted and the extended jackknife empirical likelihood methods to construct confidence intervals for the mean absolute deviation of a random variable. The empirical log-likelihood ratio statistics is derived whose asymptotic distribution is a standard chi-square distribution. The results of simulation study show the comparison of the average length and coverage probability by using jackknife empirical likelihood methods and normal approximation method. The proposed adjusted and extended jackknife empirical likelihood methods perform better than other methods for symmetric and skewed distributions. We use real data sets to illustrate the proposed jackknife empirical likelihood methods.
]]>
xueping mengThe Minimum Rank of Sign Pattern Matrices with a 1-Separation
http://scholarworks.gsu.edu/math_theses/131
http://scholarworks.gsu.edu/math_theses/131Wed, 24 Jul 2013 08:16:54 PDT
Given a sign pattern matrix M composed of two sub-patterns A and B connected by a 1-separation, we provide a formula that relates the minimum rank of M to the minimum rank of some small variations of A and B.
]]>
Wenyan Zhou