Mathematics ThesesCopyright (c) 2014 Georgia State University All rights reserved.
http://scholarworks.gsu.edu/math_theses
Recent documents in Mathematics Thesesen-usSun, 07 Dec 2014 01:35:05 PST3600Discrepancy 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Wenyan ZhouClassification For 2011-2012 Bangladesh Integrated Household Survey By Iterative Clustering Technique
http://scholarworks.gsu.edu/math_theses/130
http://scholarworks.gsu.edu/math_theses/130Wed, 24 Jul 2013 08:16:53 PDT
In this project, the raw data from a survey, 2011-2012 Bangladesh Integrated Household Survey, is cleaned. Based on the research purpose of the collaborator, important variables are extracted and principal component analysis is used to form a new data set. The iterative clustering technique is applied to the new data set to classify the households involved in the survey into different categories. The categories are interpreted as reflecting the different economic activities in Bangladesh.
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Wen ZhouJackknife Empirical Likelihood for the Variance in the Linear Regression Model
http://scholarworks.gsu.edu/math_theses/129
http://scholarworks.gsu.edu/math_theses/129Wed, 24 Jul 2013 08:16:52 PDT
The variance is the measure of spread from the center. Therefore, how to accurately estimate variance has always been an important topic in recent years. In this paper, we consider a linear regression model which is the most popular model in practice. We use jackknife empirical likelihood method to obtain the interval estimate of variance in the regression model. The proposed jackknife empirical likelihood ratio converges to the standard chi-squared distribution. The simulation study is carried out to compare the jackknife empirical likelihood method and standard method in terms of coverage probability and interval length for the confidence interval of variance from linear regression models. The proposed jackknife empirical likelihood method has better performance. We also illustrate the proposed methods using two real data sets.
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Hui-Ling LinAntiretroviral Regimens in HIV-Infected Adults Receiving Medical Care in the United States: Medical Monitoring Project, 2009
http://scholarworks.gsu.edu/math_theses/128
http://scholarworks.gsu.edu/math_theses/128Fri, 26 Apr 2013 12:12:09 PDT
Effective antiretroviral therapy (ART) is essential for viral suppression (VS) in HIV-infected patients. However, there is a lack of nationally representative data on types of ART regimens used and their impact on VS. This thesis used self-reported interview and abstracted medical record from 2009 Medical Monitoring Project (MMP) to study ART regimen type and related health outcomes. Results showed that 88.6% of HIV-infected adults in care was prescribed ART, and about half took regimens designated as ‘preferred’ according to U.S ART guidelines. Among MMP participants prescribed ART, 62.7% achieved durable VS, 77.8% achieved recent VS, 83.5% were 100% dose-adherent, and 17.1% reported side effects. Multivariate regression analyses revealed that although ART was critical for VS, there were minor differences in health outcomes among the major ART classes in the U.S. ART guidelines or six most-commonly used regimens. This study could be potentially useful for future strategic planning of HIV care.
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Yunfeng TieSmall Improvement to the Kolmogorov-Smirnov Test
http://scholarworks.gsu.edu/math_theses/127
http://scholarworks.gsu.edu/math_theses/127Fri, 26 Apr 2013 12:07:53 PDT
The Kolmogorov-Smirnov (K-S) test is widely used as a goodness-of-fit test. This thesis consists of two parts to describe ways to improve the classical K-S test in both 1-dimensional and 2-dimensional data. The first part is about how to improve the accuracy of the classical K-S goodness-of-fit test in 1-dimensional data. We replace the p-values estimated by the asymptotic distribution with near-exact p-values. In the second part, we propose two new methods to increase power of the widely used 2-dimensional two-sample Fasano and Franceschini test. Simulation studies show the new methods are significantly more powerful than the Fasano and Franceschini’s test.
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Xing DongIdentification of Differential Gene Pathways with Sparse Principal Component Analysis
http://scholarworks.gsu.edu/math_theses/126
http://scholarworks.gsu.edu/math_theses/126Fri, 26 Apr 2013 12:07:51 PDT
The development of the technology makes it possible to measure large amount of genes expressions simultaneously. Since biological functions are mostly coordinated by multiple genes, called “gene pathway”, it is interesting to identify differential gene pathways which are associated with clinical phenotype. Principal component analysis has been proposed to identify differential gene pathways in several literatures, while sparse principal component analysis (SPCA) has not drawn any attention. We proposed to use SPCA to identify differential gene pathways. The results show that, comparing to PCA, SPCA could identify more differential expressed gene pathways, especially when the higher-order interactions among genes are considered.
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Yichao YinRacial Differences in the Prevalence of Depressive Disorders Among U.S. Adult Population
http://scholarworks.gsu.edu/math_theses/125
http://scholarworks.gsu.edu/math_theses/125Fri, 26 Apr 2013 12:07:49 PDT
Depression is a common but serious illness, which may have significant impact on a person’s daily life. This study examined racial/ethnic disparities on the prevalence of lifetime diagnosis of depression and current depression according to 2010 Behavioral Risk Factor Surveillance System (BRFSS) data. Logistic regression models were constructed to calculate odds ratios for comparing prevalence rates of depression across racial/ethnic groups. Significant racial/ethnic differences were found in both the prevalence of lifetime diagnosis of depression and current depression. Regional disparities were also found to be significant in the prevalence of depressive disorders across racial/ethnic groups.
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Fang-Di YangConfidence Interval Estimation for Coefficient of Variation
http://scholarworks.gsu.edu/math_theses/124
http://scholarworks.gsu.edu/math_theses/124Fri, 26 Apr 2013 12:07:46 PDT
The coefficient of variation (CV) is a helpful quantity to describe the variation in evaluating results from different populations. There are many papers discussing methods of constructing confidence intervals for a single CV, such as exact method and approximation methods for CV when the underlying distribution is a normal distribution. However, the exact method is computationally cumbersome, and approximation methods can't be applied when the underlying distribution is unknown. In this thesis, we propose the generalized confidence interval for CV when the underlying distribution is normal and three empirical likelihood-based non-parametric intervals for CV when the underlying distribution is unknown. Simulation studies are conducted to compare the relative performances of these intervals based on the coverage probability and average interval length. Finally, the application of the proposed methods is demonstrated by using some real examples.
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Shuang Liu