Physical Sciences and Mathematics Commons^™

A Brief On Optimal Transport, Austin G. Vandegriffe

Graduate Student Research & Creative Works

Optimal transport is an interesting and exciting application of measure theory to optimization and analysis. In the following, I will bring you through a detailed treatment of random variable couplings, transport plans, basic properties of transport plans, and finishing with the Wasserstein distance on spaces of probability measures with compact support. No detail is left out in this presentation, but some results have further generality and more intricate consequences when tools like measure disintegration are used. But this is left for future work.

A Brief On Characteristic Functions, Austin G. Vandegriffe

Graduate Student Research & Creative Works

Characteristic functions (CFs) are often used in problems involving convergence in distribution, independence of random variables, infinitely divisible distributions, and stochastics. The most famous use of characteristic functions is in the proof of the Central Limit Theorem, also known as the Fundamental Theorem of Statistics. Though less frequent, CFs have also been used in problems of nonparametric time series analysis and in machine learning. Moreover, CFs uniquely determine their distribution, much like the moment generating functions (MGFs), but the major difference is that CFs always exists, whereas MGFs can fail, e.g. the Cauchy distribution. This makes CFs more robust in …

New Proper Orthogonal Decomposition Approximation Theory For Pde Solution Data, Sarah Locke, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 852- 876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for nonorthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD …

A Natural Frenet Frame For Null Curves On The Lightlike Cone In Minkowski Space ℝ⁴₂, Nemat Abazari, Martin Bohner, Ilgin Sağer, Alireza Sedaghatdoost, Yusuf Yayli

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we investigate the representation of curves on the lightlike cone ℚ³₂ in Minkowski space ℝ⁴₂ by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone ℚ³₂ in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function κ₂ = 0 , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.

An Explainable And Statistically Validated Ensemble Clustering Model Applied To The Identification Of Traumatic Brain Injury Subgroups, Dacosta Yeboah, Louis Steinmeister, Daniel B. Hier, Bassam Hadi, Donald C. Wunsch, Gayla R. Olbricht, Tayo Obafemi-Ajayi

Electrical and Computer Engineering Faculty Research & Creative Works

We present a framework for an explainable and statistically validated ensemble clustering model applied to Traumatic Brain Injury (TBI). The objective of our analysis is to identify patient injury severity subgroups and key phenotypes that delineate these subgroups using varied clinical and computed tomography data. Explainable and statistically-validated models are essential because a data-driven identification of subgroups is an inherently multidisciplinary undertaking. In our case, this procedure yielded six distinct patient subgroups with respect to mechanism of injury, severity of presentation, anatomy, psychometric, and functional outcome. This framework for ensemble cluster analysis fully integrates statistical methods at several stages of …

Student Preclass Preparation By Both Reading The Textbook And Watching Videos Online Improves Exam Performance In A Partially Flipped Course, Kaleb Bassett, Gayla R. Olbricht, Katie Shannon

Mathematics and Statistics Faculty Research & Creative Works

The flipped classroom has the potential to improve student performance. Because flipping involves both preclass preparation and problem solving in the classroom, the means by which increased learning occurs and whether the method of delivering content matters is of interest. In a partially flipped cell biology course, students were assigned online videos before the flipped class and textbook reading before lectures. Low-stakes assessments were used to incentivize both types of preclass preparation. We hypothesized that more students would watch the videos than read the textbook and that both types of preparation would positively affect exam performance. A multiple linear regression …

Evaluation Of Standard And Semantically-Augmented Distance Metrics For Neurology Patients, Daniel B. Hier, Jonathan Kopel, Steven U. Brint, Donald C. Wunsch, Gayla R. Olbricht, Sima Azizi, Blaine Allen

Electrical and Computer Engineering Faculty Research & Creative Works

Background: Patient distances can be calculated based on signs and symptoms derived from an ontological hierarchy. There is controversy as to whether patient distance metrics that consider the semantic similarity between concepts can outperform standard patient distance metrics that are agnostic to concept similarity. The choice of distance metric can dominate the performance of classification or clustering algorithms. Our objective was to determine if semantically augmented distance metrics would outperform standard metrics on machine learning tasks.

Methods: We converted the neurological findings from 382 published neurology cases into sets of concepts with corresponding machine-readable codes. We calculated patient distances by …

Energy Stable Numerical Schemes For Ternary Cahn-Hilliard System, Wenbin Chen, Cheng Wang, Shufen Wang, Xiaoming Wang, Steven M. Wise

Mathematics and Statistics Faculty Research & Creative Works

We present and analyze a uniquely solvable and unconditionally energy stable numerical scheme for the ternary Cahn-Hilliard system, with a polynomial pattern nonlinear free energy expansion. One key difficulty is associated with presence of the three mass components, though a total mass constraint reduces this to two components. Another numerical challenge is to ensure the energy stability for the nonlinear energy functional in the mixed product form, which turns out to be non-convex, non-concave in the three-phase space. to overcome this subtle difficulty, we add a few auxiliary terms to make the combined energy functional convex in the three-phase space, …

On The Noisy Gradient Descent That Generalizes As Sgd, Jingfeng Wu, Wenqing Hu, Haoyi Xiong, Jun Huan, Vladimir Braverman, Zhanxing Zhu

Mathematics and Statistics Faculty Research & Creative Works

The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the gradient matrix and a …

Vanishing Porosity Limit Of The Coupled Stokes-Brinkman System, Mingwen Fei, Dongjuan Niu, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We investigate the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. in particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9) with the constant of proportionality independent of the curvature of the interface.

Handling Missing Data For Unsupervised Learning With An Application On A Fitbir Traumatic Brain Injury (Tbi) Dataset, Louis Steinmeister, Dacosta Yeboah, Gayla Olbricht, Tayo Obafemi-Ajayi, Bassam Hadi, Daniel Hier, Donald C. Wunsch

Mathematics and Statistics Faculty Research & Creative Works

"The problem of missing data and imputation have been widely discussed amongst specialists. However, many data scientists and applied statisticians fail to appropriately consider this issue. Often, it seems intuitive to discard observations containing missing data or simply to substitute means. This can lead to disastrous consequences, particularly in an era of exponentially increasing data volumes. In the following, we show how inappropriate handling of missing data and an insufficient analysis of the censoring mechanism can lead to a bias, overconfidence in the estimation of parameters, could challenge the reproducibility of obtained results, and may distort the structure of the …

Quantifying Effects Of Sleep Deprivation On Cognitive Performance, Quang Nghia Le

Masters Theses

“The most commonly used metric for evaluating alertness and vigilance is the Psychomotor Vigilance Test (PVT), previous studies have indicated that alertness and vigilance can be affected by the lack of sleep as a function of sleep loss. This study explores methods to predict median psychomotor vigilance reaction times. The data used in this study comes from a series of tests and surveys conducted on volunteer students. The data set contains many potential predictors of PVT and one aspect of the study was to identify variables that are useful in prediction. The performances of various prediction methods that allow for …

Fuzzy Logistic Regression For Detecting Differential Dna Methylation Regions, Tarek M. Bubaker Bennaser

Doctoral Dissertations

“Epigenetics is the study of changes in gene activity or function that are not related to a change in the DNA sequence. DNA methylation is one of the main types of epigenetic modifications, that occur when a methyl chemical group attaches to a cytosine on the DNA sequence. Although the sequence does not change, the addition of a methyl group can change the way genes are expressed and produce different phenotypes. DNA methylation is involved in many biological processes and has important implications in the fields of biomedicine and agriculture.

Statistical methods have been developed to compare DNA methylation at …

The Application Of Machine Learning Models In The Concussion Diagnosis Process, Sujit Subhash

Masters Theses

“Concussions represent a growing health concern and are challenging to diagnose and manage. Roughly four million concussions are diagnosed every year in the United States. Although research into the application of advanced metrics such as neuroimages and blood biomarkers has shown promise, they are yet to be implemented at a clinical level due to cost and reliability concerns. Therefore, concussion diagnosis is still reliant on clinical evaluations of symptoms, balance, and neurocognitive status and function. The lack of a universal threshold on these assessments makes the diagnosis process entirely reliant on a physician’s interpretation of these assessment scores. This study …