Statistics and Probability Commons^™

Bayesian Decision Theoretic Approach To Directional Multiple Hypotheses Problems, Naveen K. Bansal, Klaus J. Miescke

Mathematics, Statistics and Computer Science Faculty Research and Publications

A multiple hypothesis problem with directional alternatives is considered in a decision theoretic framework. Skewness in the alternatives is considered, and it is shown that this skewness permits the Bayes rules to possess certain advantages when one direction of the alternatives is more important or more probable than the other direction. Bayes rules subject to constraints on certain directional false discovery rates are obtained, and their performances are compared with a traditional FDR rule through simulation. We also analyzed a gene expression data using our methodology, and compare the results to that of a FDR method.

A Bayesian Model For Cluster Detection, Jonathan Wakefield, Albert Y. Kim

Statistical and Data Sciences: Faculty Publications

The detection of areas in which the risk of a particular disease is significantly elevated, leading to an excess of cases, is an important enterprise in spatial epidemiology. Various frequentist approaches have been suggested for the detection of “clusters” within a hypothesis testing framework. Unfortunately, these suffer from a number of drawbacks including the difficulty in specifying a p-value threshold at which to call significance, the inherent multiplicity problem, and the possibility of multiple clusters. In this paper, we suggest a Bayesian approach to detecting “areas of clustering” in which the study region is partitioned into, possibly multiple, “zones” …

An Exploratory Study Of Pre-Service Middle School Teachers’ Knowledge Of Algebraic Thinking, Marta Magiera, Leigh A. Van Den Kieboom, John C. Moyer

Mathematics, Statistics and Computer Science Faculty Research and Publications

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed …

An Investigation Of Teachers' Intentions And Reflections About Using Standards-Based And Traditional Textbooks In The Classroom, Bikai Nie, Tony Freedman, Stephen Hwang, Ning Wang, John Moyer, Jinfa Cai

Mathematics, Statistics and Computer Science Faculty Research and Publications

This study analyzed teachers’ intentions for and reflections on their use of Standards-based [Connected Mathematics Program (CMP)] textbooks and traditional (non-CMP) mathematics textbooks to guide instruction. In this investigation of the interplay between textbooks and instruction, we focused on learning goals, instructional tasks, teachers’ anticipation of students’ difficulties, and their perceptions of students’ achievement of learning goals. All of these are aspects of teachers’ intentions and reflections that have proved fruitful in comparing the roles of the CMP and non-CMP mathematics textbooks in our Longitudinal Investigation of the Effect of Curriculum on Algebra Learning project. Whereas the cognitive level …

The Semigroups B2 And B0 Are Inherently Nonfinitely Based, As Restriction Semigroups, Peter R. Jones

Mathematics, Statistics and Computer Science Faculty Research and Publications

The five-element Brandt semigroup B₂ and its four-element subsemigroup B₀, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B₂ carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, ^-1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x^-1 and retaining the induced operations x ↦ x^{+ …}

Sub-Independence: An Expository Perspective, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Limit theorems as well as other well-known results in probability and statistics are often based on the distribution of the sums of independent random variables. The concept of sub-independence, which is much weaker than that of independence, is shown to be sufficient to yield the conclusions of these theorems and results. It also provides a measure of dissociation between two random variables which is much stronger than uncorrelatedness.

Characterizations Of Distributions Of Ratio Of Certain Independent Random Variables, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Various characterizations of the distributions of the ratio of two independent gamma and exponential random variables as well as that of two independent Weibull random variables are presented. These characterizations are based, on a simple relationship between two truncated moments ; on hazard function ; and on functions of order statistics.

On Generalized Gamma Convolution Distributions, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

We present here characterizations of certain families of generalized gamma convolution distributions of L. Bondesson based on a simple relationship between two truncated moments. We also present a list of well-known random variables whose distributions or the distributions of certain functions of them belong to the class of generalized gamma convolutions.

The Pitman Inequality For Exchangeable Random Vectors, J. Behboodian, Naveen K. Bansal, Gholamhossein Hamedani, Hans Volkmer

Mathematics, Statistics and Computer Science Faculty Research and Publications

In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (X₁,X₂,…,X_n) without the assumption of continuity and symmetry for each component X_i:

P(|1n∑i=1nXi|≤|∑i=1nαiXi|)≥12 ,

where all αi≥0 are special weights with ∑i=1nαi=1.

Extremal H-Colorings Of Graphs With Fixed Minimum Degree, John Engbers

Mathematics, Statistics and Computer Science Faculty Research and Publications

For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices of H that preserves adjacency. When H is composed of an edge with one looped endvertex, an H-coloring of G corresponds to an independent set in G. Galvin showed that, for sufficiently large n, the complete bipartite graph K_δ,n-δ is the n-vertex graph with minimum degree δ that has the largest number of independent sets.

In this paper, we begin the project of generalizing this …

Decoupling The Stationary Navier-Stokes-Darcy System With The Beavers-Joseph-Saffman Interface Condition, Yong Cao, Yuchuan Chu, Xiaoming He, Mingzhen Wei

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.

Toward An Mhealth Intervention For Smoking Cessation, Golam Mushih Tanimul Ahsan, Ivor D. Addo, Sheikh Iqbal Ahamed, Daniel Petereit, Shalini Kanekar, Linda Burhansstipanov, Linda U. Krebs

Mathematics, Statistics and Computer Science Faculty Research and Publications

The prevalence of tobacco dependence in the United States (US) remains alarming. Invariably, smoke-related health problems are the leading preventable causes of death in the US. Research has shown that a culturally tailored cessation counseling program can help reduce smoking and other tobacco usage. In this paper, we present a mobile health (mHealth) solution that leverages the Short Message Service (SMS) or text messaging feature of mobile devices to motivate behavior change among tobacco users. Our approach implements the Theory of Planned Behavior (TPB) and a phase-based framework. We make contributions to improving previous mHealth intervention approaches by delivering personalized …

Iterative Statistical Verification Of Probabilistic Plans, Colin M. Potts

Lawrence University Honors Projects

Artificial intelligence seeks to create intelligent agents. An agent can be anything: an autopilot, a self-driving car, a robot, a person, or even an anti-virus system. While the current state-of-the-art may not achieve intelligence (a rather dubious thing to quantify) it certainly achieves a sense of autonomy. A key aspect of an autonomous system is its ability to maintain and guarantee safety—defined as avoiding some set of undesired outcomes. The piece of software responsible for this is called a planner, which is essentially an automated problem solver. An advantage computer planners have over humans is their ability to consider and …

Time Scales Integral Inequalities For Superquadratic Functions, Josipa Barić, Rabia Bibi, Martin Bohner, Josip Pečarić

Mathematics and Statistics Faculty Research & Creative Works

In this paper, two different methods of proving Jensen's inequality on time scales for superquadratic functions are demonstrated. Some refinements of classical inequalities on time scales are obtained using properties of superquadratic functions and some known results for isotonic linear functionals. © 2013 The Korean Mathematical Society.

Analyzing And Solving Non-Linear Stochastic Dynamic Models On Non-Periodic Discrete Time Domains, Gang Cheng

Masters Theses & Specialist Projects

Stochastic dynamic programming is a recursive method for solving sequential or multistage decision problems. It helps economists and mathematicians construct and solve a huge variety of sequential decision making problems in stochastic cases. Research on stochastic dynamic programming is important and meaningful because stochastic dynamic programming reflects the behavior of the decision maker without risk aversion; i.e., decision making under uncertainty. In the solution process, it is extremely difficult to represent the existing or future state precisely since uncertainty is a state of having limited knowledge. Indeed, compared to the deterministic case, which is decision making under certainty, the stochastic …

The Torsion Angle Of Random Walks, Mu He

Masters Theses & Specialist Projects

In this thesis, we study the expected mean of the torsion angle of an n-step
equilateral random walk in 3D. We consider the random walk is generated within a confining sphere or without a confining sphere: given three consecutive vectors ^→e₁ , ^→e₂ , and ^→e₃ of the random walk then the vectors ^→e₁ and ^→e₂ define a plane and the vectors ^→e₂ and ^→e₃ define a second plane. The angle between the two planes is called the torsion angle of the three vectors. Algorithms are …

Characterizations Of Certain Recently Introduced Distributions, Gholamhossein Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Various characterizations of the Beta Weibull Geometric distribution of Bidram et al. (2011), the New Generalized Exponential distribution of Bidram et al. (2012), and the Gamma Exponentiated Weibull distribution of Pinho et al. (2012) are presented. These characterizations are based on: (i) a simple relationship between two truncated moments (ii) certain functions of the nth order statistic, and (iii) certain functions of the 1st order statistics.

Level Crossing Times In Mathematical Finance, Ofosuhene Osei

Electronic Theses and Dissertations

Level crossing times and their applications in finance are of importance, given certain threshold levels that represent the "desirable" or "sell" values of a stock. In this thesis, we make use of Wald's lemmas and various deep results from renewal theory, in the context of finance, in modelling the growth of a portfolio of stocks. Several models are employed .

Combinatorial Pooling Enables Selective Sequencing Of The Barley Gene Space, Stefano Lonardi, Denisa Duma, Matthew Alpert, Francesca Cordero, Marco Beccuti, Prasanna R. Bhat, Yonghui Wu, Gianfranco Ciardo, Burair Alsaihati, Steve Wanamaker, Josh Resnik, Serdar Bozdag, Ming-Cheng Luo, Timothy J. Close

Mathematics, Statistics and Computer Science Faculty Research and Publications

For the vast majority of species – including many economically or ecologically important organisms, progress in biological research is hampered due to the lack of a reference genome sequence. Despite recent advances in sequencing technologies, several factors still limit the availability of such a critical resource. At the same time, many research groups and international consortia have already produced BAC libraries and physical maps and now are in a position to proceed with the development of whole-genome sequences organized around a physical map anchored to a genetic map. We propose a BAC-by-BAC sequencing protocol that combines combinatorial pooling design and …

A Data Generating Review That Bops, Twists And Pulls At Misconceptions, Kimberly Gardner

Faculty Articles

Statistics is an integral part of the K-12 mathematics curriculum (age 5-18). Naturally, students construct misconceptions of what they learn. This article discusses The Bop It^© Challenge, a review activity assesses student understanding and reveals their misundertandings of statistical concepts.

Repeated Changes In Reported Sexual Orientation Identity Linked To Substance Use Behaviors In Youth, Miles Q. Ott, David Wypij, Heather L. Corliss, Margaret Rosario, Sari L. Reisner, Allegra R. Gordon, S. Bryn Austin

Statistical and Data Sciences: Faculty Publications

Purpose—Previous studies have found that sexual minority (e.g., lesbian, gay, bisexual) adolescents are at higher risk of substance use than heterosexuals, but few have examined how changes in sexual orientation over time may relate to substance use. We examined the associations between change in sexual orientation identity and marijuana use, tobacco use, and binge drinking in U.S. youth.

Methods—Prospective data from 10,515 U.S. youth ages 12-27 years in a longitudinal cohort study were analyzed using sexual orientation identity mobility measure M (frequency of change from 0 [no change] to 1 [change at every wave]) in up to five waves of …

Direct Eit Reconstructions Of Complex Admittivities On A Chest-Shaped Domain In 2-D, Sarah J. Hamilton, Jennifer L. Mueller

Mathematics, Statistics and Computer Science Faculty Research and Publications

Electrical impedance tomography (EIT) is a medical imaging technique in which current is applied on electrodes on the surface of the body, the resulting voltage is measured, and an inverse problem is solved to recover the conductivity and/or permittivity in the interior. Images are then formed from the reconstructed conductivity and permittivity distributions. In the 2-D geometry, EIT is clinically useful for chest imaging. In this work, an implementation of a D-bar method for complex admittivities on a general 2-D domain is presented. In particular, reconstructions are computed on a chest-shaped domain for several realistic phantoms including a simulated pneumothorax, …

Age-Specific Signatures Of Glioblastoma At The Genomic, Genetic, And Epigenetic Levels, Serdar Bozdag, Aiguo Li, Gregory Riddick, Yuri Kotliarov, Mehmet Baysan, Fabio M. Iwamoto, Margaret C. Cam, Svetlana Kotliarova, Howard A. Fine

Mathematics, Statistics and Computer Science Faculty Research and Publications

Age is a powerful predictor of survival in glioblastoma multiforme (GBM) yet the biological basis for the difference in clinical outcome is mostly unknown. Discovering genes and pathways that would explain age-specific survival difference could generate opportunities for novel therapeutics for GBM. Here we have integrated gene expression, exon expression, microRNA expression, copy number alteration, SNP, whole exome sequence, and DNA methylation data sets of a cohort of GBM patients in The Cancer Genome Atlas (TCGA) project to discover age-specific signatures at the transcriptional, genetic, and epigenetic levels and validated our findings on the REMBRANDT data set. We found major …

Some New Applications Of P-P Plots, Isha Dewan, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

The P-P plot is a powerful graphical tool to compare stochastically the magnitudes of two random variables. In this note, we introduce a new partial order, called P?P order based on P-P plots. For a pair of random variables (X ₁, Y₁) and (X ₂, Y ₂) one can see the relative precedence of Y ₂ over X ₂ versus that of Y ₁ over X ₁ using P-P order. We show that several seemingly very technical and difficult concepts like convex transform order and super-additive ordering can be easily explained with the …

Oscillation Of Third-Order Nonlinear Delay Differential Equations, Ravi P. Agarwal, Martin Bohner, Tongxing Li, Chenghui Zhang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we study the oscillatory behavior of a class of third-order nonlinear delay differential equations (a(t)(b(t)y′(t)) ′) ′+q(t)yγ(τ(t)) =0. Some new oscillation criteria are presented by transforming this equation to the first order delayed and advanced differential equations. Employing suitable comparison theorems, we establish new results on oscillation of the studied equation. Assumptions in our theorems are less restrictive; these criteria improve those in the recent paper [Appl. Math. Comput., 202 (2008), 102-112] and related contributions to the subject. Examples are provided to illustrate new results.

Immersed Finite Element Methods For Parabolic Equations With Moving Interface, Xiaoming He, Tao Lin, Yanping Lin, Xu Zhang

Mathematics and Statistics Faculty Research & Creative Works

This article presents three Crank-Nicolson-type immersed finite element (IFE) methods for solving parabolic equations whose diffusion coefficient is discontinuous across a time dependent interface. These methods can use a fixed mesh because IFEs can handle interface jump conditions without requiring the mesh to be aligned with the interface. These methods will be compared analytically in the sense of accuracy and computational cost. Numerical examples are provided to demonstrate features of these three IFE methods. © 2012 Wiley Periodicals, Inc.

Properties Of Higher-Order Half-Linear Functional Differential Equations With Noncanonical Operators, Chenghui Zhang, Ravi P. Agarwal, Martin Bohner, Tongxing Li

Mathematics and Statistics Faculty Research & Creative Works

Some new results are presented for the oscillatory and asymptotic behavior of higher-order half-linear differential equations with a noncanonical operator. We study the delayed and advanced equations subject to various conditions. © 2013 Zhang et al.

Mind The Gaps, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

No abstract provided.

Rattus Model Utilizing Selective Pulmonary Ischemia Induces Bronchiolitis Obliterans Organizing Pneumonia, John C. Densmore, Paul M. Jeziorczak, Anne V. Clough, Kirkwood A. Pritchard Jr, Breana Cummens, Meetha Medhora, Arjun Rao, Elizabeth R. Jacobs

Mathematics, Statistics and Computer Science Faculty Research and Publications

Bronchiolitis obliterans organizing pneumonia (BOOP), a morbid condition when associated with lung transplant and chronic lung disease, is believed to be a complication of ischemia. Our goal was to develop a simple and reliable model of lung ischemia in the Sprague-Dawley rat that would produce BOOP. Unilateral ischemia without airway occlusion was produced by an occlusive slipknot placed around the left main pulmonary artery. Studies were performed 7 days later. Relative pulmonary and systemic flow to each lung was measured by injection of technetium Tc 99m macroaggregated albumin. Histological sections were examined for structure and necrosis and scored for BOOP. …

A Graph-Theoretical Approach To The Selection Of The Minimum Tiling Path From A Physical Map, Serdar Bozdag

Mathematics, Statistics and Computer Science Faculty Research and Publications

The problem of computing the minimum tiling path (MTP) from a set of clones arranged in a physical map is a cornerstone of hierarchical (clone-by-clone) genome sequencing projects. We formulate this problem in a graph theoretical framework, and then solve by a combination of minimum hitting set and minimum spanning tree algorithms. The tool implementing this strategy, called FMTP, shows improved performance compared to the widely used software FPC. When we execute FMTP and FPC on the same physical map, the MTP produced by FMTP covers a higher portion of the genome, and uses a smaller number of clones. For …