2025

March

March 6: Jun Li, Ph.D. (University of Dayton): The Shape of Proof: Manifolds, Mappings and Machine Provers

Abstract: In this talk, we aim to bridge seemingly remote fields — geometry and topology — with formalized mathematics, tracing back to Leibniz’s dream of a “machine that can calculate ideas.” We begin with a survey of manifolds and their invariants, accessible to a broad audience. Then, we explore surface mapping theory following Smale, Earle-Eells and Thurston-Penner, along with my joint works extending these ideas to four-dimensional symplectic manifolds. Finally, we discuss ongoing efforts to formalize parts of this theory, involving students in research. This journey will highlight not only the challenges but also the potential of integrating machine-assisted proof systems into core mathematics courses to attract people from general STEM fields.

February

Feb. 13: Michael Goldberg (University of Cincinnati): Differentiability of Fourier Restrictions

Abstract: We want to explore the connection between two well-known properties of the Fourier transform: 1. The Fourier transform of an integrable function may be continuous, but it typically is not differentiable at all. 2. If you look at the values of the Fourier transform along a surface, better things happen if the surface is curved instead of flat. The end result is that we're able to extract some semblance of a derivative of the Fourier transform on curved surfaces, even when it's not differentiable in the classical sense. In one special case we can show that the classical partial derivative actually exists at most points. The talk will introduce some old and new results, and try to describe why they were all a bit surprising upon first discovery.

January

Jan. 30: Jon Brown (University of Dayton): Differences in Reconstruction Theorems between group C*-algebras and group rings

Abstract: Given a group G one can construct the group ring (over the complex numbers) and the group C*-algebra. Given a ring (or a C^*-algebra) the set of invertible elements forms a group. This talk explores when the set of invertible elements of a group ring or the group C*-algebra recovers the group you started with. 

2024

December

Dec. 2: Abdullah Alhelali (University of Dayton graduate student): Analysis of Functional and Neutral Differential Equations via Lyapunov Functionals

Abstract: We employ Lyapunov types functions and functionals and obtain sufficient conditions that guarantee the boundedness and the exponential decay of solutions, stability and exponential stability of the zero solution in nonlinear delay and neutral differential systems.

November

Nov. 14: Reza Bidar, Ph.D. (University of Dayton): When is the exponential map Quasi-Isometry?

Abstract: Let G be a real connected Lie group with a left invariant metric d, g its Lie algebra, exp : g → G be the Lie exponential map. When the exponential map is diffeomorphism, an interesting question would be asking if there are conditions under which the exponential map is quasi-isometry. A quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. This is obviously true if G is isomorphic to Rn. In this presentation, Dr. Bidar uses the adjoint formula for |d expx(y)|, x, y ∈ g to find out when the exponential map is a quasi-isometry. It is proved that the exponential map is quasi-isometry only when G is isomorphic to Rn.

Nov. 7: Benjamin Akers (Air Force Institute of Technology): On Ripples: Bifurcations of Bimodal Traveling Waves

Abstract: In the early 1900s, several authors presented expansions to approximate ripples with harmonic resonance at Bond numbers where Stokes' expansion is singular. These waves are now called Wilton ripples. Their procedure is effectively a Lyapunov-Schmidt reduction, and has been employed numerous times to asymptotically approximate, numerically compute and prove existence of these waves. Historically solvability conditions in the expansion are satisfied using corrections to the wave speed and the ratio of the two modes. In the classic expansion, the ratio of the two modes can take only special values. Recently another expansion has been used to study the bifurcation of small amplitude bimodal ripples, which shows that waves bifurcate with almost all ratios of the two modes provided the problem is expanded in Bond number. In this talk, the history of both expansions will be presented and the connection between the two will be explained.

October

Oct. 31: Robert Viator, Ph.D. (Denison University): Steklov eigenvalues in nearly-(hyper)spherical domains

Abstract: The Steklov eigenvalue problem is a spectral boundary value problem for the Laplacian where the spectral parameter appears as part of a boundary condition, rather than in the interior of the domain. Applications for the Steklov problem range from SONAR to medical imaging, and more besides. We will explore the problem of shape optimization of Steklov eigenvalues under various geometric constraints (fixed volume, fixed perimeter/surface area). We will begin with a review of well-known, relatively classical results of R. Weinstock and F. Brock for the first non-zero Steklov eigenvalue. We will then explore further pursuits by myself, coauthors, and other colleagues toward (local) shape optimizers for higher Steklov eigenvalues in two and higher dimensions using spectral perturbation theory and hyperspherical harmonics.

Oct. 10: Dr. Thilini Jayasinghe (UD): Convolutional Neural Network for Fake Review Detection on Amazon

Abstract: This study focuses on Amazon, a leading e-commerce platform in the United States, where fake reviews have become a significant concern. Given the limited availability of authentic datasets for analysis, we propose a novel methodology to differentiate between genuine and fraudulent reviews across verified and non-verified purchases. We present a comprehensive framework integrating Convolutional Neural Networks with word embedding and emotion-mining techniques through Natural Language Processing. Our method demonstrates exceptional performance, achieving an accuracy rate of over 96% in distinguishing fake reviews from user reviews. This study contributes to the growing research on online review authenticity and offers practical implications for e-commerce platforms, regulatory bodies and consumers.

Oct. 3: Dr. Jonathan Brown (UD): Intermediate Subalgebras of Quasi-Cartan Inclusions

Abstract: Given a group action (an a ring) one can construct an algebra. This talk explores when subalgebras of this algebra can be described using the sub actions of the given group action. 

August

Aug. 22: Dr. Paul Eloe (UD Professor Emeritus): Nonlinear Interpolation and Boundary Value Problems

Abstract: In the tradition of uniqueness of solutions implies existence of solutions, it is assumed that solutions of two point boundary value problems for nonlinear ordinary differential equations are unique if they exist. It is then shown that solutions do exist. Rather than apply the Pontryagin maximum principle to obtain best interval lengths for the uniqueness of solutions, a monotonicity condition is imposed on the nonlinear term to obtain uniqueness of solutions. Thus, the existence results obtained here are global.

2024

July

July 2: Joseph Tiller (UD Applied Mathematics graduate student): Semiclassical Resolvent Estimates for Repulsive Potentials.

Abstract: We study a class of semi-classical Schrödinger operators with repulsive potentials. We obtain optimal limiting absorption resolvent estimates in dimensions five and higher using energy methods. For potentials which are also radially symmetric and short range, we use separation of variables to obtain optimal resolvent estimates in dimensions three and four. The focus of the talk will be on the development of the energy techniques, which are widely applicable. In the radial case, we also explain how to use ODE tools after separating variables.

April

April 23: Dr. Marina Mancuso (UD alum), Doctor of Philosophy, Applied Mathematics, Arizona State University: Mathematical Modeling of Infectious Diseases

Abstract: This talk will discuss two applications of mechanistic mathematical modeling on infectious disease dynamics. The first part of the talk introduces a mathematical model for COVID-19, specifically designed to assess the impact of vaccine-induced, cross-protective efficacy of COVID-19 transmission in the United States. We present conditions for achieving vaccine-derived herd immunity and results from global sensitivity analysis under different transmissibility and cross-protection scenarios. The second part of the talk relates to modeling West Nile Virus (WNV), which is the most common vector-borne disease in the continental U.S. The model includes time and temperature dependence on demographic and epidemiological processes, and considers time-since-infection structure for vector and host populations. We describe how we connect experimental infection and epidemiological data to infection-age dependent processes, and parameterize the model to human case data. We further show a range of scenario projections under climate change to quantify the increased risk and variability for future WNV prevalence. This talk will be accessible to graduate students and undergraduate students with familiarity of differential equations.

April 16: Jungmi McBride, UD Applied Mathematics graduate student: Difference Equation SIR model for Spread of Disease

Abstract: In this project, we will investigate linearizing non-linear system and add perturbation to study the equilibrium solution. We will begin this by observing the prey and predator model, SI model and SIR model.

April 2: Dr. Adam Waterbury, Denison University (Host: Matt Wascher): Large Deviations for Empirical Measures of Self-Interacting Markov Chains

Abstract: Self-interacting Markov chains arise in a range of models and applications. For example, they can be used to approximate the quasi-stationary distributions of irreducible Markov chains and to model random walks with edge or vertex reinforcement. The term self-interacting Markov chain is something of a misnomer, as such processes interact with their full path history at each time instant, and therefore are non-Markovian. Under conditions on the self-interaction mechanism, we establish a large deviation principle for the empirical measure of self-interacting chains on finite spaces. In this setting, the rate function takes a strikingly different form than the classical Donsker-Varadhan rate function associated with the empirical measure of a Markov chain; the rate function for self-interacting chains is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function. This is based on joint work with Amarjit Budhiraja and Pavlos Zoubouloglou.

March

March 14: Jeff Neugebauer: P-periodic solutions of a q-integral equation with finite delay

Abstract: A Volterra type integral equation with a finite delay is considered on a discrete nonadditive time scale domain q^{N_0}. The existence of periodic solutions of this equation, which we call a q-integral equation, are shown employing the contraction mapping principle and a fixed point theorem due to Krasnosel’skii.

February

Feb. 27: Dr. Alexander Sistko (Algebra faculty candidate): An Introduction to Representation Theory over the Field with One Element

Abstract: To any finite quiver Q, we may associate its category of finite-dimensional representations over the so-called "field with one element" F1. This category acts like a combinatorial degeneration of the more familiar versions over fields, and in particular, it admits a well-defined Hall algebra. In this talk, we discuss recent progress towards understanding this category and its Hall algebra. In particular, we outline a recent result which stratifies finite connected quivers by the asymptotic growth of their nilpotent indecomposable F1-representations. Time permitting, we will also discuss open problems, future directions of research, and other instances of representation theory over F1.

Feb. 20: Dr. Thanh Thai Nguyen; Symbolic Powers: Degree Bounds, Containment Problem  and Beyond

Abstract: One of the fundamental questions in polynomial interpolation is the following: What is the smallest degree of a homogeneous polynomial that vanishes to order at least on a given set of points? This question is wide open and answer to it has many applications and implications in many other fields. Even lower bounds for such degree play a crucial role in many contexts, such as Nagata’s counterexamples to Hilbert’s fourteenth problem, or the Schwarz exponent in complex analysis to name a few. A classical result by Zariski and Nagata tells us the set of such polynomials is precisely an algebraic object called the math symbolic power of the defining ideal of the set of points. Symbolic power is one of the central objects which has a long history in commutative algebra. To study the above interpolation question, one can study containment between symbolic powers and ordinary powers of I. The talk will be an introduction to this subject. Some recent progresses on studying the lower bounds on the degree and containment problem based on our joint projects with Sankhaneel Bisui, Eloísa Grifo and Huy Tài Hà will be presented.

Feb. 15: Dr. Alessandra Costantini; Rees algebras: an algebraic tool to study singularities

Abstract: How many tangent lines does a plane curve have at a given point? How can we find the implicit equations of a parametric curve? These seemingly unrelated problems from algebraic geometry can be solved using tools from abstract algebra, through the notion of Rees algebras. In this talk, I will discuss the fundamental role of Rees algebras in the study of commutative rings and of the singularities of algebraic varieties. I will then give an overview of how one can use methods from algebraic combinatorics to understand the algebraic properties of Rees algebras.

Feb. 13: Reza Bidar, University of Dayton; Estimates for the Norm of the Derivative of Lie Exponential Map for Connected Lie Groups (new results)

2023

November

Nov. 16: Reza Bidar, University of Dayton; Estimates for the Norm of the Derivative of Lie Exponential Map for Connected Lie Groups

May

May 10: Mohammadreza Bidar, Michigan State University; Blocking problems and the derivative of the exponential maps in connected Lie groups

Abstract: In the first part of the research presentation, I talk about the connection blocking property which originates from a billiard orbit problem. I present a brief preliminary and a brief summary of my Ph.D. dissertation work. Then I proceed to my more recent work on Connection blocking in a particular Lie group called Sol and the differential of the exponential map on Lie groups.

May 8: Tavish Dunn, Oxford College of Emory University; Properties of Generalized Inverse Limits

Abstract: Inverse limits are an interesting object that allows us to glue compact metric spaces together via continuous functions to form more exotic spaces. Properties like compactness and connectedness of the resulting space, as well as the structure of closed sets, can be determined by the component spaces and the individual bonding functions. When a single bonding function is used, the dynamics of the function determines qualities of the resulting inverse limit and vice-versa. More recently, set-valued functions have been employed to construct a wider variety of inverse limits. However, many basic results, such as the inverse limit of continua being itself a continua, do not generalize to this new setting. We discuss conditions on set-valued bonding functions that are sufficient for the inverse limit to be connected and have the full-projection property. Next, we show that the existence of points of period not a power of 2 implies the existence of an indecomposable subcontinuum of the inverse limit. We show the conditions are sharp by way of constructing a function that does not meet all the criteria yet does have points of all periods and whose inverse limit is hereditarily decomposable.

April

April 28: Mohammed Almalki, University of Dayton; Exponential Stability and Instability Via Lyapunov Functionals

Abstract: In this project we display a Lyapunov functional to obtain exponential  stability of the zero solution of an integro-diffetential equation with multiple delays. We extend our method to show instability of the zero solution under slightly different conditions.

April 28: Sharmina Yasmin, University of Dayton; Forecasting Categorical Time Series Using a Combination of Logistic Regression and ARIMA Models

Abstract: In this research, we explore a categorical time series data that changes with time and other input variables using a combination of Logistic Regression, and ARIMA model. We use an Electroencephalogram (EEG) dataset with two states of the response variable (closed or open state of the eye). Using EEG sensor values as input, we use Logistic Regression model to obtain the predictive probability to classify the eye state. Due to the autocorrelation among the residuals and to time dependence, the Logistic Regression model can be improved using the ARIMA model to produce better results. This will help making the residuals a white noise. This work is developed further using a Transfer Function model that produces an even better result. 

April 27: Nur Saglam, School of Mathematics, Georgia Tech; Geography and Botany Problem of Symplectic 4-Manifolds

Abstract: The world of 4-manifolds is very interesting in many senses. For example, a closed topological n-manifold X^n has exactly one smooth structure if n≤3 and has at most finitely many smooth structures if n>4. However, there are many simply-connected closed 4-manifolds which admit infinitely many smooth structures. Also, classification problems for smooth, simply connected 4-manifolds have still not fully been understood. For example, the generalized Poincaré conjecture is still open for dimension 4. In this talk, we will talk about the construction of symplectic 4-manifolds homeomorphic but not diffeomorphic to some well-known manifolds with given topological invariants. We will mostly focus on the simply-connected symplectic 4-manifolds, describe the topological invariants, and give examples.

April 25: Challita Jabbour, University of Dayton; Using neural networks for European option pricing in Heston model and for implied volatility

Abstract: This project aims to investigate option pricing in the Black-Scholes-Merton (BSM) model and the Heston model using neural networks. These two models are well-known for their significance in modern finance theory, particularly when it comes to option pricing. Analytical formulas were examined, and neural networks were tweaked to efficiently produce accurate results. Apple stock was used for the BSM model. The Heston model takes stochastic volatility into consideration, and a combining technique between the two models was utilized to generate implied volatility (both smile and surface). MatLab was used  for the implementation.

April 25: Ibrahim Guediri, University of Dayton; A Numerical Solution of Coupled Drinfeld-Sokolov-Wilson System Using Meshless Method of Lines

Abstract: For this project, we reproduce the meshless method of lines (MOL) numerical solution of coupled Drinfeld-Sokolov-Wilson system. This method uses radial basis functions (RBFs) for spatial collocation. Time integration of the resulting system of ODEs will be solved using fourth order of Runge-Kutta method. Accuracy (L2 and L∞) will be compared with the results from other methods available in the literature.

April 25: Richard Buckalew, Wilmington College; From the cell cycle to the redistricting cycle, feedback mechanisms in complex systems lead to emergent dynamics (5 p.m.)

Abstract: A common theme of my research is studying feedback mechanisms in complex dynamical systems: small-scale interactions between individuals that lead to interesting, and sometimes surprising, behavior at the population scale. A Dynamical Systems approach provides a common framework for understanding many different phenomena, including the metabolic cell cycle in yeast, the early development of the fruit fly embryo, potassium homeostasis in the brain, the movement of actors during a warm-up exercise, and the self-sorting of voters.

April 20: Heshan Aravinda Pathirannehelage, University of Florida; Discrete Log-Concave Distributions, Properties, and Applications

Abstract: Log-concavity appears naturally in combinatorics, algebra, analysis, geometry, computer science, probability and statistics. In the context of probability, log-concave assumption provides a broad and flexible, yet natural, convolution-stable class of distributions on integers. Examples include Bernoulli, independent Bernoulli sums, geometric, binomial, negative binomial and Poisson. While log-concave measures, their geometry, and properties are well understood in the classical setting, consideration of discrete log-concavity in the probabilistic setting is very limited. In this talk, I will talk about several results concerning these distributions. First, we study a structured class within discrete log-concave distributions, namely ultra-log concave, and prove that all ultra log-concave sequences exhibit Poisson-type concentration. As an application, we derive concentration bounds for so-called “intrinsic volumes” of a convex body, which generalizes and improves a result of Lotz-McCoy-Nourdin-Peccati-Tropp (2019). Next, we show that a strengthened version of a conjecture of Feige holds for the class of log-concave distributions. Finally, we explore information-theoretic properties. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance, improving the work of Bobkov-Marsiglietti-Melbourne (2022). Our approach is based on a localization-type machinery, a technique that reduces these problems to some extremal cases, which we manage to identify. This talk is based on joint works with Arnaud Marsiglietti, James Melbourne, and Abdulmajeed Alqasem. 

April 18: Charbel Al Bacha and Soulayma Saba, University of Dayton; Neural Networks for European Option Pricing Using Jump Diffusion Models

Abstract: This project aims to examine Neural Networks (NN) for pricing European options and compare to other numerical methods. We consider European options written on one and two stocks where the stock prices follow jump-diffusion models. The work intends to investigate the efficiency and time needed to calculate the option price. The analytical formula of the jump diffusion model for one stock and the Monte-Carlo simulation for two stocks are used respectively to generate a large data set which is used to train the Neural Network. Neural Networks efficiency is directly related to the size of the data set, the number of nodes and layers, the optimizer model, and the number of inputs. Thus, the same data sets have been analyzed in different ways to compare the results with the analytical formula and the Monte-Carlo simulation. Our experiments show that the well trained NN can reduce the computational time significantly.

April 11: Charles Destefani, University of Dayton; Using Neural Networks for Option Pricing in Regime-Switching Models

Abstract: We studied the application of neural networks (NNs) to European-style regime-switching option pricing models. We discuss this model's construction and operation in detail (including a comparison to the Black-Scholes model) and summarize the analytical and semi-Monte Carlo simulation approaches used for pricing. Further, we produced Matlab codes that implement these methods efficiently and generally. We conclude that NNs are appropriate for this pricing model and begin to expand our consideration to American-style options.

April 4: Serge Alhalbi, University of Dayton; Multivariate Time Series Transfer Function Models

Abstract: A time series is a sequence of data points that occur over time. Time series forecasting is the process of using a model to predict future values based on past observations. While regression analysis is commonly used to examine the relationship between time series, it does not analyze the relationship between different time points within a single series. Time series examples include weather data, temperature readings, heart rate monitoring, and stock prices. The number of inputs studied determines whether the time series is univariate or multivariate. Each case may require different models and theories. Multivariate time series problems can be addressed using transfer function models, which explain the output variable in terms of the input variables using cross-correlation. In this work, a multivariate transfer function model will be constructed and applied to forecast the Vanguard Real Estate Index Fund.

March

March 28: Jun Li, University of Dayton; Isotopy of the Base Class of a Ruled 4-Manifold

Abstract: In this talk, we will begin by reviewing the geometry and topology of closed 2-dimensional surfaces. We will then move on to discuss the importance of studying 4-dimensional manifolds, which are much more complex and difficult to understand than their lower-dimensional counterparts. Specifically, we will focus on ruled 4-manifolds, which can be constructed by fibering a 2-sphere over a base, which is a 2-dimensional surface. We will also introduce a symplectic form on that and explain why they are of particular interest to mathematicians. Our main result in this talk is the symplectic isotopy of the base class of a ruled 4-manifold, which describes a continuous transformation of the base class that preserves certain important properties.

March 21: Kyle Helfrich, University of Dayton; Applications of DNNs

Abstract: Deep Neural Networks (DNNs) have grown in popularity over the last decade. Despite the popularity of DNNs, there are still misconceptions on what a DNN is and how they work. The purpose of this talk is to provide a high-level overview of DNNs and is designed to be accessible for those unfamiliar with DNNs. I will also be focusing on several applications of DNNs including Natural Language Processing (NLP) in the context of ChatGPT, solving PDEs, and image infill.

March 7: Ying-Ju (Tessa) Chen, Jun Li, George Todd, Matthew Wascher (Facilitator, Aparna Higgins); Exploring ChatGPT

Abstract: According to ChatGPT itself, “ChatGPT is an AI language model developed by OpenAI, which is capable of generating human-like text based on the input it is given.” ChatGPT can create some impressive paragraphs of writing, but it can also give completely incorrect answers to some factual questions. Especially since anyone can explore ChatGPT freely at this time, it will surely influence our teaching – including what questions we ask and whether we can give unsupervised assignments with access to the internet. However, ChatGPT may also provide us with opportunities in many of our teaching and research activities. Four of our departmental colleagues will share with us some of their explorations on ChatGPT, giving us an idea of what they tried to do, and how satisfied they were with the results. Dr. Ying-Ju (Tessa) Chen will tell us about using ChatGPT to brainstorm ideas for the Data Analytics Major and how ChatGPT can be (mis)used in SPC (statistical process control) Practice, Education, and Research; Dr. Jun Li will show us how he may use some of ChatGPT’s responses in his MTH 342, Set Theory, course; Dr. George Todd will tell us about trying to get ChatGPT to write an algorithm and how that went; and Dr. Mathew Wascher will tell us about trying to get ChatGPT to answer test questions from his courses. We hope to have some time at the end for questions and comments from the audience.

February

Feb. 28: David Sivakoff, Ohio State University; Recurrent epidemics on networks

Abstract: The contact process is a stochastic dynamical system modeling a recurring disease in a population represented by a network or a spatial grid. I will introduce the model and highlight some of its history and recent results. As time allows, I will also discuss a variant of the model wherein healthy individuals attempt to avoid those who are infected. I intend for this talk to be accessible to students, and will give a brief advertisement for the Statistics Ph.D. program at Ohio State. Based on joint work with Shirshendu Chatterjee and Matthew Wascher.

Feb. 21: Paul Eloe, University of Dayton (emeritus); Maximum and anti-maximum principles in neighborhoods of simple eigenvalues

Abstract: Under suitable hypotheses, for boundary value problems of the form Ly+ay=f, BCy=0, where L is a linear ordinary or partial differential operator, a is real and BC denotes a linear boundary operator, it can be shown that there exists A > 0 such that if 0< |a| < A, then Ly +ay =f, BCy = 0 has a unique solution for each f in an appropriate space. Moreover, f > 0 implies ay > 0. In particular, for -A < a < 0, y < 0, and for 0 < a < A, y > 0. We shall provide suitable hypotheses so that this behavior is valid for a linear fractional differential operator L of Riemann Liouville type. Two examples will be presented. If time permits, an application of this behavior to a boundary value problem for a nonlinear fractional differential equation will be given.

January

Jan. 31: Yanxi Li, University of Kentucky; Some Modeling Considerations Involving the Exponentially-Modified Gaussian (EMG) Distribution

Abstract: Fitts' law is often employed as a predictive model for human movement, especially in the field of humancomputer interaction. Models with an assumed Gaussian error structure are usually adequate when applied to data collected from controlled studies. However, observational data (often referred to as data gathered "in the wild") typically display noticeable positive skewness relative to a mean trend as users do not routinely try to minimize their task completion time. As such, the exponentially-modified Gaussian (EMG) regression model has been applied to aimed movements data. However, it is also of interest to reasonably characterize those regions where a user likely was not trying to minimize their task completion time. In this paper, we propose a novel model with a two-component mixture structure -- one Gaussian and one exponential -- on the errors to identify such a region. An expectation-conditional-maximization (ECM) algorithm is developed for estimation of such a model and some properties of the algorithm are established. The efficacy of the proposed model, as well as its ability to inform model-based clustering, are addressed in this work through extensive simulations and an insightful analysis of a human aiming performance study.

Jan. 24: Dr. Thilini Jayasinghe, Wittenberg University; Regression models using the LINEX loss to predict lower bounds for the number of points for approximating planar contour shapes and LINEX loss to fit SIR model

Abstract: Researchers in statistical shape analysis often analyze outlines of objects in two dimensions, which can be modeled as planar contours. However, even though these objects are infinite-dimensional in theory, they must be discretized in practice. When discretizing, it is important to reduce the number of sampling points considerably to reduce computational costs but not to use too few sampling points to result in too much approximation error. Unfortunately, determining the minimum number of sampling points needed to achieve sufficiently low approximation error is computationally expensive. As such, we fit linear regression models to predict these lower bounds using characteristics of the contours that are easy to calculate as predictor variables. However, least squares regression is inadequate for this task because it treats overestimation and underestimation equally, but underestimation is far more serious since the response variable is a lower bound. Thus, the LINEX loss function was used to fit the regression models, which allows penalizing underestimation at an exponential rate while penalizing overestimation only linearly. A novel, data-driven approach to select the shape parameter of the loss function and tools for analyzing how well the model fits the data that are analogous to least squares regression. Through validation methods, we show that the LINEX regression models work well for limiting the amount of underestimation for the lower bounds for the number of sampling points. The usage of the LINEX loss function is not only limited to regression analysis. We further used the LINEX loss function in dynamical data fitting in different applications rather than the Least squares method to reduce the underestimation. A basic SIR model was used to expand the studies on the usage of the LINEX loss function for infectious disease data to reduce the underestimation of the model parameters based on the data.

Jan. 19: Yafan Guo, University of Kentucky; Approximate Tolerance Intervals for Semiparametric Regression Models

Abstract: Tolerance intervals in regression allow the user to quantify, with a specified degree of confidence, bounds for a specified proportion of the sampled population when conditioned on a set of covariate values. While methods are available for tolerance intervals in fully-parametric regression settings, the construction of tolerance intervals for semiparametric regression models has been treated in a limited capacity. This paper fills this gap and develops likelihood-based approaches for the construction of pointwise one-sided and two-sided tolerance intervals for semiparametric regression models. A numerical approach is also presented for constructing simultaneous tolerance intervals. An appealing facet of this work is that the resulting methodology is consistent with what is done for fully-parametric regression tolerance intervals. Extensive coverage studies are presented, which demonstrate very good performance of the proposed methods. The proposed tolerance intervals are calculated and interpreted for analyses involving a fertility dataset and a triceps measurement dataset.