Other Applied Mathematics Commons^™

Optimal First Order Methods For Reducing Gradient Norm In Unconstrained Convex Smooth Optimization, Yunheng Jiang

All Theses

In this thesis, we focus on convergence performance of first-order methods to compute an $\epsilon$-approximate solution of minimizing convex smooth function $f$ at the $N$-th iteration.

In our introduction of the above research question, we first introduce the gradient descent method with constant step size $h=1/L$. The gradient descent method has a $\mathcal{O}(L^2\|x_0-x^*\|^2/\epsilon)$ convergence with respect to $\|\nabla f(x_N)\|^2$. Next we introduce Nesterov’s accelerated gradient method, which has an $\mathcal{O}(L\|x_0-x^*\|\sqrt{1/\epsilon})$ complexity in terms of $\|\nabla f(x_N)\|^2$. The convergence performance of Nesterov’s accelerated gradient method is much better than that of the gradient descent method but still not optimal. We also …

Computational Models To Detect Radiation In Urban Environments: An Application Of Signal Processing Techniques And Neural Networks To Radiation Data Analysis, Jose Nicolas Gachancipa

Beyond: Undergraduate Research Journal

Radioactive sources, such as uranium-235, are nuclides that emit ionizing radiation, and which can be used to build nuclear weapons. In public areas, the presence of a radioactive nuclide can present a risk to the population, and therefore, it is imperative that threats are identified by radiological search and response teams in a timely and effective manner. In urban environments, such as densely populated cities, radioactive sources may be more difficult to detect, since background radiation produced by surrounding objects and structures (e.g., buildings, cars) can hinder the effective detection of unnatural radioactive material. This article presents a computational model …

Fine-Tuning A 𝑘-Nearest Neighbors Machine Learning Model For The Detection Of Insurance Fraud, Alliyah Stout

Honors Theses

Billions of dollars are lost within insurance companies due to fraud. Large money losses force insurance companies to increase premium costs and/or restrict policies. This negatively affects a company’s loyal customers. Although this is a prevalent problem, companies are not urgently working toward bettering their machine learning algorithms. Underskilled workers paired with inefficient computer algorithms make it difficult to accurately and reliably detect fraud.

The goal of this study is to understand the idea of -Nearest Neighbors ( -NN) and to use this classification technique to accurately detect fraudulent auto insurance claims. Using -NN requires choosing a value and a …

A Network Analysis Of Covid-19 In The United States, Joseph C. Mcguire

Master's Theses

Through methods in network theory and time-series analysis, we will analyze the spread of COVID-19 in the United States by determining trends in state-by-state daily cases through a network construction. Previous researchers have found frameworks for approximating the spread of the COVID-19 pandemic and identifying potential rises in cases by a network construction based on correlation of cases between regions [1]. Applying this network construction we determine how this network and its structure act as a predictor for overall COVID-19 cases in the United States by preforming a trend analysis on a variety of network statistics and US COVID-19 cases.

Unique Signed Minimal Wiring Diagrams And The Stanley-Reisner Correspondence, Vanessa Newsome-Slade

Master's Theses

Biological systems are commonly represented using networks consisting of interactions between various elements in the system. Reverse engineering, a method of mathematical modeling, is used to recover how the elements in the biological network are connected. These connections are encoded using wiring diagrams, which are directed graphs that describe how elements in a network affect one another. A signed wiring diagram provides additional information about the interactions between elements relating to activation and inhibition. Due to cost concerns, it is optimal to gain insight into biological networks with as few experiments and data as possible. Minimal wiring diagrams identify the …

A Novel Correction For The Adjusted Box-Pierce Test, Sidy Danioko, Jianwei Zheng, Kyle Anderson, Alexander Barrett, Cyril S. Rakovski

Mathematics, Physics, and Computer Science Faculty Articles and Research

The classical Box-Pierce and Ljung-Box tests for auto-correlation of residuals possess severe deviations from nominal type I error rates. Previous studies have attempted to address this issue by either revising existing tests or designing new techniques. The Adjusted Box-Pierce achieves the best results with respect to attaining type I error rates closer to nominal values. This research paper proposes a further correction to the adjusted Box-Pierce test that possesses near perfect type I error rates. The approach is based on an inflation of the rejection region for all sample sizes and lags calculated via a linear model applied to simulated …

Cooccurrr - Applying Glcm Analysis Techniques To Non-Image Objects Such As Fitness Landscapes, Steph J. Owen

Biology and Medicine Through Mathematics Conference

No abstract provided.

Empirical And Computational Approaches To Explore The Underlying Neuromodulatory Mechanisms Of Social Status Regulation On Zebrafish Motor Circuits, Sungwoo Ahn, Choongseok Park, Fadi Issa

Biology and Medicine Through Mathematics Conference

No abstract provided.

Age-Dependent Ventilator-Induced Lung Injury, Quintessa Hay, Christopher Grubb, Rebecca L. Heise, Sarah Minucci, Michael S. Valentine, Jennifer Van Mullekom, Angela M. Reynolds

Biology and Medicine Through Mathematics Conference

No abstract provided.

Generating A Dataset For Comparing Linear Vs. Non-Linear Prediction Methods In Education Research, Jack Mauro, Elena Martinez, Anna Bargagliotti

Honors Thesis

Machine learning is often used to build predictive models by extracting patterns from large data sets. Such techniques are increasingly being utilized to predict outcomes in the social sciences. One such application is predicting student success. Machine learning can be applied to predicting student acceptance and success in academia. Using these tools for education-related data analysis, may enable the evaluation of programs, resources and curriculum. Currently, research is needed to examine application, admissions, and retention data in order to address equity in college computer science programs. However, most student-level data sets contain sensitive data that cannot be made public. To …

Data And Algorithmic Modeling Approaches To Count Data, Andraya Hack

Honors College Theses

Various techniques are used to create predictions based on count data. This type of data takes the form of a non-negative integers such as the number of claims an insurance policy holder may make. These predictions can allow people to prepare for likely outcomes. Thus, it is important to know how accurate the predictions are. Traditional statistical approaches for predicting count data include Poisson regression as well as negative binomial regression. Both methods also have a zero-inflated version that can be used when the data has an overabundance of zeros. Another procedure is to use computer algorithms, also known as …

Advancements In Gaussian Process Learning For Uncertainty Quantification, John C. Nicholson

All Dissertations

Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. The research presented provides advancements in uncertainty quantification using Gaussian processes from two distinct perspectives. The first provides a more fundamental means of constructing Gaussian processes which take on arbitrary linear operator constraints in much more general framework than its predecessors, and the other from the perspective of calibration of state-aware parameters in computer models. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to …

Mathematical Analysis Of An Sir Disease Model With Non-Constant Transmission Rate, Emma Bollinger, Tayler Valdez, Swarup Ghosh, Sunil Giri

Student Research

• Epidemiology: A branch of medicine that studies causes, transmission, and control methods of diseases at the population level.
• Mathematical epidemiology deals with creating a model for a disease through the study of incidence and distribution of the disease throughout a population.
• Here, we have examined the behavior of a measles-like disease[2] that is characterized by a non-constant transmission rate.

Vertical Take-Off And Landing Control Via Dual-Quaternions And Sliding Mode, Joshua Sonderegger

Doctoral Dissertations and Master's Theses

The landing and reusability of space vehicles is one of the driving forces into renewed interest in space utilization. For missions to planetary surfaces, this soft landing has been most commonly accomplished with parachutes. However, in spite of their simplicity, they are susceptible to parachute drift. This parachute drift makes it very difficult to predict where the vehicle will land, especially in a dense and windy atmosphere such as Earth. Instead, recent focus has been put into developing a powered landing through gimbaled thrust. This gimbaled thrust output is dependent on robust path planning and controls algorithms. Being able to …

Quadratic Neural Network Architecture As Evaluated Relative To Conventional Neural Network Architecture, Reid Taylor

Senior Theses

Current work in the field of deep learning and neural networks revolves around several variations of the same mathematical model for associative learning. These variations, while significant and exceptionally applicable in the real world, fail to push the limits of modern computational prowess. This research does just that: by leveraging high order tensors in place of 2nd order tensors, quadratic neural networks can be developed and can allow for substantially more complex machine learning models which allow for self-interactions of collected and analyzed data. This research shows the theorization and development of mathematical model necessary for such an idea to …

On Efficacy And Effectiveness Of Vaccines: A Mathematical Approach Based On Conditional Probability With Applications To The Covid-19 Context, Flavius Guias

Spora: A Journal of Biomathematics

This paper presents a mathematically formalized approach which points out the relation between efficacy and effectiveness of vaccines. The first term denotes the relative degree of protection in clinical trials or under ideal conditions, while the latter is based on observed real-life data. We define the efficacy by a similar formula to the effectiveness, but the probabilities involved in the relative risk are conditional with respect to the exposure to the virus. If exposure and vaccination status are independent, the two quantities are equal. Otherwise, the observed value of the effectiveness is a biased one, as it could be seen …

Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper

Honors Program: Senior Projects (Public)

Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …

An Axiomatic And Contextual Review Of The Armitage And Doll Model Of Carcinogenesis, W. Zane Billings, Justin Clifton, Josh Hiller, Tommy Meek, Andrew Penland, Wesley Rogers, Gabriella Smokovich, Andrew Velasquez-Berroteran, Eleni Zamagias

Spora: A Journal of Biomathematics

In 1954, Armitage and Doll published one of the most influential papers in the history of mathematical epidemiology. However, when one examines the literature one finds that there are in fact at least three distinct mathematical models attributed to the 1954 paper. In this study, we examine this important paper and the mathematical derivation of their model. We find, very surprisingly, that no stochastic process can account for all the assumptions of the model and that many of the models in the literature use a consistent subset of the assumptions used in Armitage and Doll's paper.

Strengthening A Linear Reformulation Of The 0-1 Cubic Knapsack Problem Via Variable Reordering, Richard Forrester, Lucas Waddell

Faculty Journal Articles

The 0-1 cubic knapsack problem (CKP), a generalization of the classical 0-1 quadratic knapsack problem, is an extremely challenging NP-hard combinatorial optimization problem. An effective exact solution strategy for the CKP is to reformulate the nonlinear problem into an equivalent linear form that can then be solved using a standard mixed-integer programming solver. We consider a classical linearization method and propose a variant of a more recent technique for linearizing 0-1 cubic programs applied to the CKP. Using a variable reordering strategy, we show how to improve the strength of the linear programming relaxation of our proposed reformulation, which ultimately …

An Exploration In Health Analytics: Pediatric Burns, Care Policy Assessment And Interrupted Time Series, Chao Wang

2022

Healthcare systems globally face multiple challenges in the face of population growth and changes in disease pathology. With regard to the rising demand of the healthcare and the global threats of the pandemic, the medical datasets can be trained further to develop preventive methods. Meanwhile, policy reforms of health systems could be a critical aspect to deal with the public crisis and concerns. However, two basic problems must be addressed first: identification of key factors on a priority basis and evaluation of changes.

Thus, the paper presents a series of trials on the application of data analytics to health-related problems, …