Regional Mathematics and Statistics Conference

Accepted Abstracts

Name(s): Kingsley Timilehin AKINFE
Affiliation(s): University of North Carolina at Charlotte (UNC Charlotte)
Title: A reliable analytic technique for the modified prototypical Kelvin–Voigt viscoelastic fluid model by means of the hyperbolic tangent function
Abstract:
Inspired by the extensive applicability of solitons in nonlinear optics, advanced telecommunication industry, trans-continental, and trans-oceanic systems, coupled with the copious functionality of viscoelastic models in highway engineering pavement theory, civil engineering, and solid mechanics, unique closed-form solutions have been obtained for a highly nonlinear model, which describes the dynamics of the incompressible viscoelastic Kelvin–Voigt fluid viz: the modified (1+1) dimension Oskolkov equation. Abundant solitary and traveling wave solutions to the prototypical Kelvin– Voigt incompressible model have been elucidated and put forward by means of the hyperbolic tangent method using the tangent function, thus, depicted breather, cusp, rouge, compacton, and kink waveforms via graphical illustrations for arbitrary intervals.

This present research yields new traveling wave profiles and structures with holomorphic terms, precise with minimized computational work, capturing all solitonic features of the Kelvin–Voigt model, contrary to the existing results with cumbersome and longwinded solution profiles, thus, the present research novelty established. The dynamical analyses investigated, and the solution framework presented here is unprecedented and will be of substantial benefit to the geosciences, ocean, plasma, and, material sciences, civil engineering, nanotechnology, signaling, and signal processing to effectively tackle and convey theoretical explorations on cogent equations arising in these afore-mentioned fields.

Name(s): Matt Brunet
Affiliation(s): Winthrop University
Title: Magic and Antimagic Vertex Labelings of Water Wiggler Graphs
Abstract:
A graph with $m$ edges is said to have a magic labeling if its edges can be labeled with distinct numbers from 1 to $m$ so that the sum of the edge labels at each vertex is the same for all vertices. This notion is a generalization of the standard “magic square”. Alternatively, a graph with $m$ edges has an antimagic labeling if its edges can be labeled with distinct numbers from 1 to $m$ so that the sums of the edge labels at each vertex are distinct. Antimagic and magic labelings of graphs (and their many generalizations) have been extensively studied.

This talk will focus on a family of graphs we call “water wiggler graphs”. A water wiggler graph can be obtained by starting with a cycle on $r$ vertices and then duplicating and subdividing edges as much as you like (with every duplicate edge subdivided at least once). We will discuss some results on whether or not graphs in this family have antimagic/magic labelings. If time permits, variations of antimagic/magic-type labelings will be discussed for this family of graphs.

Name(s): Alexander Bryan
Affiliation(s): University of North Carolina at Wilmington
Title: Variant of the Grundy Domination Number
Abstract:
A legal dominating sequence $S$ of a graph $G$ is a sequence of elements from $V(G)$ such that for each $v_i$ in $S$, there exists some vertex $v’\in N[v_i]$ with $v’$ not previously dominated by $S$. The Grundy domination number $\gamma_{gr}(G)$ is the length of the longest legal dominating sequence.

A long-standing conjecture in Grundy domination is the Grundy domination number of the strong product of two graphs, $H$ and $G$, which proposes
\[
\gamma_{gr}(H\boxtimes G) = \gamma_{gr}(H)\gamma_{gr}(G).
\]
In hopes of making progress on this conjecture, we propose and investigate a variant of the Grundy domination number, the $k$-Grundy domination number of a graph $G$, $\gamma_{kgr}(G)$. In $k$-Grundy domination, we allow vertices to be dominated up to $k$ times. We find the $k$-Grundy domination number exactly based on the normal Grundy domination number, and will discuss loosely how we hope this will give us insight into the strong product conjecture.

Name(s): Nishant Chinnasami
Affiliation(s): University of South Carolina Salkehatchie
Title: A Study of Tricolor Towers of Hanoi
Abstract:
In 2015, Chaugule studied the bicolor Towers of Hanoi problems. That inspired us to study the tricolor Towers of Hanoi. There might be several ways to play with this new version, but we focus on: how to move three towers with different colors to its adjacent peg, following the same rule of a regular Tower of Hanoi problem. This is an on-going project. In this talk, we will introduce some patterns we observed.

Name(s): Noah Couch
Affiliation(s): University of North Carolina at Greensboro
Title: Evaluating the EM algorithm in recombination model for constructing phylogenetic trees
Abstract:
The study of phylogenetic trees is integral to the field of biology, especially in its application to the study of evolution and research into genetic markers for diseases. To this end, statistical methods have been applied to DNA sequences to assist in construction of such trees. Past research has studied the effect of genetic mutations over generations, creating an ancestral mixture model to estimate ancestral DNA proportions. More recent research has taken genetic recombination into account, proposing a hierarchical estimator based on Markov Chain Composite Likelihood. However, there has not yet been discussion on the accuracy of this estimator. The purpose of this paper is to compare the hierarchical estimator to the EM algorithm in terms of accuracy and computation time, for small lengths L of DNA sequences. We find that, in general, the EM algorithm is more accurate while the hierarchical estimator is significantly more efficient. In cases where the distribution can be represented by a Markov Chain, the hierarchical estimator performs about as well as the EM algorithm.

Name(s): Owen Deen
Affiliation(s): University of North Carolina Wilmington
Title: Iterative Algorithms for Matrix Separation
Abstract:
Matrix separation is a fundamental problem in various fields, specifically for image analysis, signal processing, and computer vision. The goal is to separate a matrix into a low rank and sparse component to de-noise the given data. We will present an approach for solving the matrix separation problem under specific constraints. Two possible techniques we will study are the singular value decomposition and the alternating direction method of multipliers. We provide a Python based implementation of these two techniques, with emphasis on efficiency and scalability, as well a preliminary investigation of the mathematical framework. We will then compare and test this performance on simulated data and real-world applications such as facial recognition and video surveillance.

Name(s): Jesse Dimino
Affiliation(s): CUNY Graduate Center
Title: Topological Classification of Firn Data
Abstract:
Glaciers are formed through continual snowfall. As more snow falls, the deeper layers are put under more pressure until it solidifies into ice. The upper layers of granular snow that haven’t been compressed into ice are referred to as firn. This process imposes distinct structure on firn as a function of its depth. This naturally lends itself to analyzing how the topology of firn varies across depth, and we propose a classifier to determine depth based on topological structure through the use of techniques from topological data analysis. Results are compared to traditional image classification techniques such as convolutional neural networks. This topological approach has advantages in terms of generalizability, invariance against rotation, and interpretability while providing comparable classification accuracy to other methods for our test set.

Name(s): Chidinma Ezugwu
Affiliation(s): University of North Carolina Greensboro
Title: Reducing the Number of Node Expansions by DIBBS
Abstract:
This paper discovers and proves various interesting and useful properties of a new Bidirectional Heuristic Search (BHS) algorithm that dynamically improves the bounds during its execution. The algorithm, Dynamically Improved Bounds Bidirectional Search (DIBBS), has the property that it always terminates on or before the forward search meets the backward search. It also has the ability to solve certain problems while expanding fewer nodes than prominent BHS algorithms like GBFHS and MMe. Ultimately, we present a theorem that can be used in reducing the number of nodes expanded by DIBBS in solving a shortest path problem.

Name(s): C. Matthew Farmer
Affiliation(s): UNC Greensboro
Title: Problems regarding the Randić Index of a Graph
Abstract:
A topological index is a function from the set of all finite graphs to the field of complex number which is invariant under isomorphism. A classic example is the largest eigenvalue of the adjacency matrix of a graph, since isomorphic graphs have similar matrices. The Randić ( pronounced “rand-eech” ) index of a graph a sum of weight imposed on the edges of a graph given by the degrees of the incident vertices which originated from the field of organic chemistry. Mathematicians have studied this and similar indices for their combinatorial properties and have posed several significant results regarding the extremal problems that these indices present. We will present some of our findings on extremal problems with the Randić index and spectral radii of two matrices one can obtain from graphs.

Name(s): Liam Gilmore-Greiss and Gleb Gribovskii
Affiliation(s): University of North Carolina at Greensboro
Title: A game-theoretic analysis of COVID-19 dynamics with self-isolation and vaccination behavior
Abstract:
We construct a mathematical model of COVID-19 transmission with quarantine and hospitalization coupled with a dynamic game model of adaptive human behavior. We adopt the epidemiological model and dynamic game model of self-isolation behavior from Agusto et al. (2022) and construct a new model of dynamic vaccination behavior. Symptomatically infected individuals choose to either self-isolate or not-self isolate based on their sensitivity to social isolation and potential burden to the society when infecting other individuals. Susceptible individuals choose to either vaccinate or not vaccinate based on the morbidity risk of the vaccine and the morbidity risk of the disease coupled with the probability of getting infected. We analyze the complex feedback loop between the epidemiological model, which determines the optimal behavioral strategies, and the behavioral models, which affect the disease transmission. We also examine the impact of vaccine scare and vaccine confidence on the model outcomes.

Name(s): Ayaan Hafer
Affiliation(s): UNC Charlotte
Title: Solutions in Overfitting of Generative Models
Abstract:
Image generative AI models are a type of model that generates new images that are created from a large dataset of images. Two current systems of collecting training data for these models are either via a file containing images with associated captions that describe the image, or an algorithm that scrapes the internet, using existing captions or associated words in which the image is found. For example, one who googles “paris” will have several thousand pictures that are already tagged with at least one word to describe them, that being the search term itself. However, such models are still in their infancy, and can generate copies of an image within the dataset, particularly if that image becomes too influential in generation. My research proposes using a process called pixel cluster analysis to help reduce this kind of overfitting within generative models. Pixel cluster analysis, in essence, creates a matrix of values dependent on pixel values that is unique to the individual image. Depending on time and ability I may also propose using a similar method that may be useful for text generation as well. Limitations of application, as well as solutions to such limitations will also be discussed.

Name(s): Melissa Hall and Bailey Meche
Affiliation(s): UNCG and University of Louisiana at Lafayette
Title: Effect of Unequal Variance on Statistical Tests for Mixed Paired and Two-sample Designs
Abstract:
Choosing the right statistical test is critical for designing a study, especially when equal variance cannot be assumed. The standard statistics for mixed paired and two-sample designs include the paired and independent t-tests – both of which do not incorporate all observations. Several improved statistics have been proposed to utilize all available data. However, most of these improved approaches have not been tested under the effect of unequal variance. Several tests were compared under this condition. To address the limitation of the assumption of equal variance, this work develops a test for the effect of unequal variance and compares its performance with existing tests, contributing to a more complete understanding of statistical methods in the presence of unequal variance. Two tests are recommended under certain conditions after the performance is observed.

Name(s): Shelby Horth
Affiliation(s): Wake Forest University
Title: Modeling Multiple Capillary Layers in the Human Retina
Abstract:
Impaired vascular factors have been associated with many ocular diseases, including glaucoma. For example, decreased capillary density has been observed in both early and advanced stages of the disease. Mathematical modeling can be used to assess the impact of vascular factors on retinal blood flow and oxygenation. The retina has a complex three-layer capillary network; our study extends a previous model to incorporate these three layers in five possible orientations. This model includes flow regulation and oxygen transport mechanisms and is used to predict how venous oxygen saturation and retinal blood flow are affected by changes in capillary density. Simulating a 10\% capillary density reduction (as observed clinically), our model predicted a 5.6\% decrease in venous saturation. Combining this simulation with elevated intraocular pressure led to a 9.8\% decrease in venous saturation. Ultimately, our model demonstrates the importance of considering distinct capillary layers for realistic retinal oxygenation predictions. These modeled capillary layers will be integrated into an existing model accounting for arteriolar vascular network heterogeneity to predict tissue regions at high risk of the low oxygenation characteristic of glaucoma.

Name(s): Lucas Alland, Yerim Kone, Amy Liu
Affiliation(s): Swarthmore College
Title: Steklov Eigenvalues of Nearly Circular Domains
Abstract:
Our research this summer was in the domain of spectral geometry and partial differential equations. Specifically, we analyze a particular boundary-value problem, known as the Steklov Eigenvalue problem, on the unit disk in 2D. Some properties are known: the eigenvalues are nonnegative and form a sequence which increases to infinity. We ask the following questions: Given a differentiable perturbation of the boundary of the disk, what can be said about the change in the eigenvalues associated with the PDE? Which eigenvalues are maximized, if only locally, by the disk? We answer these questions by parameterizing the perturbation, and then obtaining a Taylor Series representation of the eigenvalues and eigenfunctions in the given parameter. We obtain explicit formulas for the first order terms in the expansion, which we use to show that the disk is a stationary point for an infinite subset of Steklov eigenvalues. Time permitting, we will explore preliminary results in the second order behavior as well.

Name(s): Mingyan Li
Affiliation(s): University of North Carolina at Greensboro
Title: An improved residual based random forest for robust prediction
Abstract:
Data contamination is common in real life data, and it is not surprising to have around 10% contaminated observations in a data set, argued by Hample et al. Random forest (RF) model, introduced by Breiman, is a robust method such that it is very unlikely to overfit data with low signal-to-noise ratio, although it could be true that RF deteriorates if mean-shift contamination presents in the training set. We introduce a residual based solution, the penalized weighted random forest (PWRF) method, which modifies random forest (RF) model to improve robustness over systematic or trend contamination. This method dictates the impact from contamination in the training set based on the squared residual of each training observation, which provides the flexibility to deal with different types of data. The numerical experiment and the real data analysis with stock return data suggest that PWRF provides competent if not better results than two robust RF methods and the original RF method.

Name(s): Jamie Lin and Francisco Carrillo-Villagrana
Affiliation(s): UNC Wilmington and UNC Asheville
Title: AN OPTIONAL QUANTITATIVE MIXTURE RRT MODEL THAT ACCOUNTS FOR LACK OF TRUST
Abstract:
Randomized Response Technique (RRT) allows respondents to provide scrambled response when the survey question is sensitive. RRT has be used to combat Social Desirability Bias (SDB) which can lead to a reduction in untruthful responding and a promotion in response rate. Even so, if respondents find the traditional RRT model untrustworthy, untruthful responds will increase leading to significant bias within the estimation. Previous quantitative RRT model, Gupta, et al. (2022), has used Warner’s additive and Diana and Perry (2011) to win the trust of those respondents who do not trust the basic Warner additive model. We want to see if the trust level will improve by adding more scrambling options. We propose an Optional Quantitative Mixture RRT Model that Accounts for Lack of Trust by including Greenberg, et al. (1971) model for those that do not trust Diana and Perry (2011) model.

Name(s): Jiachen Liu
Affiliation(s): Wake Forest University
Title: A study on Fractional Order Mandelbrot Set
Abstract:
The Mandelbrot set is a well-known mathematical fractal with a number of interesting properties. In recent years, researchers have considered a modification of this set, called the Fractional Order Mandelbrot set, or FOM. In this poster, we discuss some properties of the FOM. We show that the map which generates the FOM does not have fixed points. We also show that as q goes to 0, the FOM will approach the ordinary Mandelbrot map. Finally, we approximate the stability region of the FOM map and make some conjectures based on our numerical results.

Name(s): Changzhi Ma
Affiliation(s): The University of North Carolina at Greensboro
Title: Order Estimation on Long-memory Processes
Abstract:
Long memory, also known as Long-range dependence (LRD), is a fundamental phenomenon encountered in the analysis of spatial and time series data. It signifies a correlation structure where the dependence diminishes at a slower rate than an exponential decay. Long memory has found applications in diverse fields, including econometrics, hydrology, and earth sciences.

The Hurst parameter (H) is a measure that quantifies the extent of long memory. This parameter assumes values within the range of 0 to 1, with 0.5 denoting the absence of long memory. Some researchers employ the parameter ‘d’ (d = H – 0.5) to describe the order of long memory, where 0 < d < 0.5. Within the domain of long-memory research, estimating order 'd' represents a topic to study.

This presentation commences with an in-depth exploration of the fundamental attributes of long memory processes. It proceeds to review the various methodologies employed for estimating the parameter of order 'd'. Subsequently, we address the estimating 'd' through regression techniques within two distinct long memory processes. Furthermore, this presentation extends its examination to the estimation of 'd' within non-stationary processes. The research is ongoing.

Name(s): Bailey Meche
Affiliation(s): University of Louisiana at Lafayette
Title: The Effect of Measurement Error on Binary RRT Models
Abstract:
This study introduces the effect of measurement error on binary Randomized Response Technique (RRT) models. We discuss a method for estimating and accounting for measurement error in two basic models and one comprehensive model. Both theoretical and empirical results show that not accounting for measurement error leads to inaccurate estimates. We introduce estimators that account for the effect of measurement error. Furthermore, we introduce a new measure of model privacy using an odds ratio statistic which offers better interpretability than traditional methods.

Name(s): Abigail Mervine
Affiliation(s): Winthrop University
Title: Time Series Forecasting Models for Local Light Pollution
Abstract:
This study builds a forecasting model to describe local trends in light pollution: the brightening of the night sky as a result of anthropogenic, artificial light sources. Satellite-based radiance measurements can be used as a proxy for light pollution levels. Specifically, a sample from a dataset representing radiance (nW/(cm2 sr)) values from 2012-2022, collected by the VIIRS-DNB sensors on the SNPP satellite, along with time series forecasting, was used to predict future radiance values for Rock Hill, SC. Autocorrelation plots and the augmented Dickey-Fuller test were utilized to select parameters for an ARIMA forecasting model. The accuracy of this model, quantified by an AIC value, was compared to that of models built by Python’s auto_arima package. The model with the lowest AIC was chosen. A test-train split was then performed on the dataset to cross-validate the chosen model. After cross-validation, the chosen model was used to generate an 8-year forecast with 95% confidence intervals, which appears to show a decrease in radiance values. We conclude with a discussion on the degree to which these radiance values could correlate with changes in light pollution, including the impact of adopting LED technology over the period of time used in the training data.

Name(s): Eric Ng’eno
Affiliation(s): Biodiversity Institute, University of Kansas
Title: Relationship between patterns of antimicrobial exposure risk conditions and carriage of a low prevalence ceftazidime-resistant Escherichia coli in a human settlement.
Abstract:
Knowledge of antimicrobial exposure-resistant bacteria carriage patterns relationship can inform prediction of resistance occurrence risk distribution in community settings for targeted interventions. We contrasted environmental conditions where individuals carrying Escherichia coli resistant to ampicillin, trimethoprim-sulfamethoxazole, and ceftazidime lived and baseline conditions derived from random sampling of the areas surveyed in a densely populated human settlement. We used five-year data on antimicrobial prescriptions from a population-based infectious diseases surveillance clinic to members of georeferenced households and constructed raster layers representing exposure risk conditions in defined daily doses (DDD) per 1000 inhabitants-days-of-observation (IDO). We generated layers summarizing household distances to water, sanitation and other amenities and used resistance data obtained from stool cultures of asymptomatic individuals from randomly selected households. Relationships were assessed using PERMANOVA and univariate analyses approaches. Prevalence of trimethoprim and sulfamethoxazole, ampicillin and ceftazidime resistant E. coli carriage (n =1341 individuals) was 90.0%, 85.2% and 4.3% respectively. Overall antimicrobial exposure risks were highest for trimethoprim-sulfamethoxazole (1.81 DDD per 1000 IDO) followed by amoxicillin (0.74 DDD per 1000 IDO). Exposure conditions for twenty-two assessed antimicrobials varied across locations surveyed. Ceftazidime-resistant E. coli carriage occurred commonly in areas with increased exposure risk of trimethoprim-sulfamethoxazole, cloxacillin, and erythromycin. We did not observe relationship between carriage patterns of ampicillin or trimethoprim-sulfamethoxazole-resistant E. coli and assessed environmental conditions. We observed specific antimicrobial exposure-resistance carriage patterns relationship for an E. coli strain with relatively low carriage prevalence, suggesting resistant E. coli carriage risk is nonrandom and distribution can be predicted before high levels of carriage.

Name(s): Camilla De Oliveira Fonseca and Maria Pasaylo
Affiliation(s): University of São Paulo and University of Florida
Title: Applying Computer Vision for Out-of-Stock Detection in Retail Stores
Abstract:
Retail stores today have difficulty keeping track of the product inventory on shelves due to the current inventory management system that involves manual scanning and counting of products which is time-consuming and error prone. Computer vision enables us to automate inventory management by using cameras to capture real-time data on product availability. However, most of the current work done in this field focus on detecting and extracting information from in-stock products. In this paper, we focus on developing a novel approach to analyze out-of-stock (OOS) items. Specifically, we make three contributions. First, we provided new annotated datasets of images of retail store shelves. Second, we use deep learning models to detect OOS items and price tags on the shelves. Lastly, we implement an algorithm to match the OOS item to the appropriate price tag and extract its barcode and textual information to successfully identify the OOS items.

Name(s): Omar Rodriguez garcia
Affiliation(s): University of Texas at Austin
Title: Determination of Firn Image Depth Through Curvature Features
Abstract:
Ice cores are vertical samples of glaciers and ice sheets used in geology to gather information about Earth from geologic time scale eras. Ice cores consist of sections of snow, granular ice, firn, and glacial ice. Firn refers to the layer of compacted and recrystallized ice that accumulates in polar regions. The distribution and shape of air and ice in firn layers carries important information about historical atmospheric conditions. Our dataset consists of micro ct scans of firn at various depths, which clearly display a change in the shape of the pore space as depth increases. Our goal is to quantify this change using geometric methods and demonstrate how it can be used to predict the depth of individual scans. To accomplish this, we binarize the images and measure the curvature at the boundary between ice and air. Our main finding is a statistically significant linear relationship between depth and mean curvature. We also design a depth classifier using the full range of observed curvature values, which can be applied to other ice cores in the future.

Name(s): Wenhao Shou
Affiliation(s): UNCG
Title: Kernel Density Estimation of a Sensitive Variable in the Presence of Auxiliary Information
Abstract:
Auxiliary information is widely used in various studies to improve the precision of estimation. In this article, we extend the application of auxiliary information within the context of randomized response techniques (RRT), building upon the prior research on kernel density estimation (KDE) under additive RRT models. Inspired by Mostafa and Ahmad (2019), we proposed a kernel density estimator that incorporates an auxiliary variable to enhance the accuracy of estimating the distribution of a sensitive variable. Extensive simulations are conducted to evaluate the performance of this proposed methodology, highlighting the advantages of utilizing auxiliary information and the impact of factors such as noise levels, sample size, and the correlation between the auxiliary variable and the sensitive variable.

Name(s): Dasia Singleton
Affiliation(s): Elizabeth City State University
Title: Eigenface Image Processing Using Linear Algebra
Abstract:
Image processing is a powerful technique used to enhance images and extract useful information from them, with applications ranging from medical imaging to facial recognition. One popular approach to facial recognition is Eigenfaces, which uses appearance-based methods to capture the unique differences between a group of face images. In this presentation, I will provide an overview of image processing, including its history and various applications. I then explain the relationship between eigenvectors and eigenvalues, and demonstrate how they are used in the Eigenface method. Finally, I present the results using MATLAB to show the application of Eigenface through matrices, eigenvectors, and eigenvalues. This analysis illustrates the effectiveness of the Eigenface method in facial recognition and highlights the importance of linear algebra in this process. Overall, this presentation offers valuable insights into the use of image processing methods for facial recognition and presents a useful guide for researchers and practitioners in this field.

Name(s): Sunia Tanweer
Affiliation(s): Michigan State University
Title: Establishing a Topology-Driven Framework for Phenomenological Bifurcations in Stochastic Systems
Abstract:
Modifications in the parameters of dynamical systems can shift the system’s state, causing transitions between distinct qualitative regimes. These shifts, termed bifurcations, have paramount importance since they indicate possible adverse alterations in the system’s behaviour. In stochastic systems, emphasis is placed on Phenomenological (P) bifurcations, which encompass transitions from mono-stable states to multi-stable states, emergence of stochastic limit cycles, and other complex changes in the probability density function (PDF) of the system. Existing methodologies to detect such changes have their limitations, and often fall short in detecting the full spectrum of potential PDF variations, while requiring human intervention for visual discernment of changes in the PDF. In contrast, we introduce a novel approach based in Topological Data Analysis (TDA), which leverages superlevel persistence to quantitatively detect P-bifurcations in by means of a “homological bifurcation plot” that graphically depicts the changing ranks of the 0th and 1st homology groups. Through the use of TDA and these plots, we demonstrate the effective detection of P-bifurcations in stochastic oscillators when given their analytical PDFs. Furthermore, we elaborate the process for the same, when given a kernel density estimate (KDE).

Name(s): Xiaohuan Xue
Affiliation(s): UNC Greensboro
Title: Constructing Covariance Functions for Axially Symmetric Processes on the Sphere
Abstract:
Covariance functions are used to characterize dependency in spatial statistics and the construction of covariance functions is critical when performing prediction or ”kriging”. In this talk, we will discuss the construction of covariance models for axially symmetric processes on the sphere. We will first review some of the recent development in this area, and we propose our preliminary results, then conclude the presentation with a discussion of future work.