Scholarly Work - Mathematical Sciences

Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/8719

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Accelerated Gibbs sampling of normal distributions using matrix splittings and polynomials
(2017-11) Fox, Colin; Parker, Albert E.
Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geometric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibbs sampling, REGS sampling, and generalizations. Hence, efficient deterministic stationary relaxation schemes lead to efficient generalizations of Gibbs sampling. The technique of polynomial acceleration that significantly improves the convergence rate of an iterative solver derived from a symmetric matrix splitting may be applied to accelerate the equivalent generalized Gibbs sampler. Identicality of error polynomials guarantees convergence of the inhomogeneous Markov chain, while equality of convergence factors ensures that the optimal solver leads to the optimal sampler. Numerical examples are presented, including a Chebyshev accelerated SSOR Gibbs sampler applied to a stylized demonstration of low-level Bayesian image reconstruction in a large 3-dimensional linear inverse problem.
Advanced college-level students' categorization and use of mathematical definitions
(2012) Dickerson, David S.; Pitman, Damien J.
This qualitative study of five undergraduate mathematics majors found that some students, (even students at an advanced level of undergraduate mathematical study) have a mathematician’s perspective neither on the concept of mathematical definition nor on the structure of mathematics as a whole. Participants in this study were likely to reason from incomplete concept images rather than from concept definitions and were likely to perceive that definitions (like theorems) need to be verified. The results of this study have implications for college-level mathematics instruction.
Adversary decision-making using Markov models
(SPIE, 2023-06) Andreas, Elizabeth; Dorismond, Jessica; Gamarra, Marco
This study conducts three experiments on adversary decision-making modeled as a graph. Each experiment has the overall goal to understand how to exploit an adversary’s decision-making in order to obtain desired outcomes, as well as specific goals unique to each experiment. The first experiment models adversary decision-making using an Absorbing Markov chain (AMC). A sensitivity analysis of states (nodes in the graph) and actions (edges in the graph) is conducted which informs how downstream adversary decisions could be manipulated. The next experiment uses a Markov decision process (MDP). Assuming the adversary is initially blind to the rewards they will receive when they take an action, a Q´learning algorithm is used to determine the sequence of actions that maximizes the adversary rewards (called an optimum policy). This experiment gives insight in the possible decision-making of an adversary. Lastly, in the third experiment a two-player Markov game is developed, played by an agent (friend) and the adversary (foe). The agents goal is to decrease the overall rewards the adversary receives when it follows optimum policy. All experiments are demonstrated using specific examples.
Advising caution in studying seasonal oscillations in crime rates
(2017-09) Dong, Kun; Yunbai, Cao; Wilber, Matthew; McCalla, Scott G.
Most types of crime are known to exhibit seasonal oscillations, yet the annual variations in the amplitude of this seasonality and their causes are still uncertain. Using a large collection of data from the Houston and Los Angeles Metropolitan areas, we extract and study the seasonal variations in aggravated assault, break in and theft from vehicles, burglary, grand theft auto, rape, robbery, theft, and vandalism for many years from the raw daily data. Our approach allows us to see various long term and seasonal trends and aberrations in crime rates that have not been reported before. We then apply an ecologically motivated stochastic differential equation to reproduce the data. Our model relies only on social interaction terms, and not on any exigent factors, to reproduce both the seasonality, and the seasonal aberrations observed in our data set. Furthermore, the stochasticity in the system is sufficient to reproduce the variations seen in the seasonal oscillations from year to year. Researchers should be very careful about trying to correlate these oscillations with external factors.
Agent-Based Models for Collective Animal Movement: Proximity-Induced State Switching
(2021-08) Hoegh, Andrew; van Manen, Frank T.; Haroldson, Mark
Animal movement is a complex phenomenon where individual movement patterns can be influenced by a variety of factors including the animal’s current activity, available terrain and habitat, and locations of other animals. Motivated by modeling grizzly bear movement in the Greater Yellowstone Ecosystem, this article presents an agent-based model represented in a state-space framework for collective animal movement. The novel contribution of this work is a collective animal movement model that captures interactions between animals that can trigger changes in movement patterns, such as when a dominant grizzly bear may cause another subordinate bear to temporarily leave an area. The modeling framework enables learning different movement patterns through a state-space representation with particle-MCMC methods for fully Bayesian model fitting and the prediction of future animal movement behaviors.Supplementary materials accompanying this paper appear online.
An Alternative to the Carnegie Classifications: Identifying Similar Institutions with Structural Equation Models and Clustering
(2019-01) Harmon, Paul; McKnight, Sarah; Hildreth, Laura; Godwin, Ian; Greenwood, Mark C.
The Carnegie Classification of Institutions of Higher Education is a commonly used framework for institutional classification that classifies doctoral-granting schools into three groups based on research productivity. Despite its wide use, the Carnegie methodology involves several shortcomings, including a lack of thorough documentation, subjectively placed thresholds between institutions, and a methodology that is not completely reproducible. We describe the methodology of the 2015 and 2018 updates to the classification and propose an alternative method of classification using the same data that relies on structural equation modeling (SEM) of latent factors rather than principal component-based indices of productivity. In contrast to the Carnegie methodology, we use SEM to obtain a single factor score for each school based on latent metrics of research productivity. Classifications are then made using a univariate model-based clustering algorithm as opposed to subjective thresholding, as is done in the Carnegie methodology. Finally, we present a Shiny web application that demonstrates sensitivity of both the Carnegie Classification and SEM-based classification of a selected university and generates a table of peer institutions in line with the stated goals of the Carnegie Classification.
Authors and editors assort on gender and geography in high-rank ecological publications
(2018-02) Manlove, Kezia R.; Belou, Rebecca M.
Peer-reviewed publication volume and caliber are widely-recognized proxies for academic merit, and a strong publication record is essential for academic success and advancement. However, recent work suggests that publication productivity for particular author groups may also be determined in part by implicit biases lurking in the publication pipeline. Here, we explore patterns of gender, geography, and institutional rank among authors, editorial board members, and handling editors in high-impact ecological publications during 2015 and 2016. A higher proportion of lead authors had female first names (33.9%) than editorial board members (28.9%), and the proportion of female first names among handling editors was even lower (21.1%). Female editors disproportionately edited publications with female lead authors (40.3% of publications with female lead authors were handled by female editors, though female editors handled only 34.4% of all studied publications). Additionally, ecological authors and editors were overwhelmingly from countries in the G8, and high-ranking academic institutions accounted for a large portion of both the published work, and its editorship. Editors and lead authors with female names were typically affiliated with higher-ranking institutions than their male peers. This description of author and editor features provides a baseline for benchmarking future trends in the ecological publishing culture.
Batalin-Vilkovisky quantization and the algebraic index
(2017-09) Grady, Ryan E.; Li, Qin; Li, Si
Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the Batalin–Vilkovisky (BV) quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.
Beginning High School Teachers’ Organization of Students for Learning and Methods for Teaching Mathematics
(Editorial de la Universidad de Granada, 2020-11) Williams, Derek; Cudd, Michele; Hollebrands, Karen; Lee, Hollylynne
We observed eight beginning secondary mathematics teachers’ classrooms to investigate which they organized students for learning, uses of instructional methods, and how these may differ based on the level of course being taught. We found that beginning teachers frequently organize their students to learn collaboratively – either in small groups or as a whole class – coupled with an abundance of teacher directed instruction. Differences in organizations, teaching methods, and associated learning opportunities between course levels also exist. Implications for supporting practicing teachers and preparing prospective teachers to establish collaborative learning environments and utilize student centered teaching methods are discussed.
Bioenergy and Breast Cancer: A report on tumor growth and metastasis
(2016) Running, Alice; Greenwood, Mark C.; Hildreth, Laura; Schmidt, Jade
As many as 80% of the 296,000 women and 2,240 men diagnosed with breast cancer in the United States will seek out complementary and alternative medicine (CAM) treatments. One such therapy is Healing Touch (HT), recognized by the National Center for Complementary and Integrative Health (NCCIH) as a treatment modality. Using a multiple experimental groups design, fifty-six six- to eight-week-old Balb/c mice were injected with 4T1 breast cancer tumor cells and randomly divided into intervention and positive control groups. Five days after tumor cell injection, mice in the intervention groups received HT either daily or every other day for 10 minutes by one HT practitioner. At 15 days after tumor cell injection, tumor size was measured, and metastasis was evaluated by a medical pathologist after necropsy. Tumor size did not differ significantly among the groups (F(3,52) = 0.75, p value = 0.53). The presence of metastasis did not differ across groups (chi-square(3) = 3.902, p = 0.272) or when compared within an organ (liver: chi-square(3) = 2.507, p = 0.474; lungs: chi-square(3) = 3.804, ; spleen: chi-square(3) = 0.595, p = 0.898). However, these results did indicate a moderate, though insignificant, positive impact of HT and highlight the need for continued research into dose, length of treatment, and measurable outcomes (tumor size, metastasis) to provide evidence to suggest application for nursing care.
Calibration of an Adaptive Genetic Algorithm for Modeling Opinion Diffusion
(MDPI AG, 2022-01) Johnson, Kara Layne; Carnegie, Nicole Bohme
Genetic algorithms mimic the process of natural selection in order to solve optimization problems with minimal assumptions and perform well when the objective function has local optima on the search space. These algorithms treat potential solutions to the optimization problem as chromosomes, consisting of genes which undergo biologically-inspired operators to identify a better solution. Hyperparameters or control parameters determine the way these operators are implemented. We created a genetic algorithm in order to fit a DeGroot opinion diffusion model using limited data, making use of selection, blending, crossover, mutation, and survival operators. We adapted the algorithm from a genetic algorithm for design of mixture experiments, but the new algorithm required substantial changes due to model assumptions and the large parameter space relative to the design space. In addition to introducing new hyperparameters, these changes mean the hyperparameter values suggested for the original algorithm cannot be expected to result in optimal performance. To make the algorithm for modeling opinion diffusion more accessible to researchers, we conduct a simulation study investigating hyperparameter values. We find the algorithm is robust to the values selected for most hyperparameters and provide suggestions for initial, if not default, values and recommendations for adjustments based on algorithm output.
Characterization of synovial fluid metabolomic phenotypes of cartilage morphological changes associated with osteoarthritis
(2019-08) Carlson, Alyssa K.; Rawle, Rachel A.; Wallace, Cameron W.; Brooks, Ellen G.; Adams, Erik; Greenwood, Mark C.; Olmer, Merissa; Lotz, Martin K.; Bothner, Brian; June, Ronald K.
"Objective Osteoarthritis (OA) is a multifactorial disease with etiological heterogeneity. The objective of this study was to classify OA subgroups by generating metabolomic phenotypes from human synovial fluid. Design: Post mortem synovial fluids (n = 75) were analyzed by high performance-liquid chromatography mass spectrometry (LC-MS) to measure changes in the global metabolome. Comparisons of healthy (grade 0), early OA (grades I-II), and late OA (grades III-IV) donor populations were considered to reveal phenotypes throughout disease progression. Results: Global metabolomic profiles in synovial fluid were distinct between healthy, early OA, and late OA donors. Pathways differentially activated among these groups included structural deterioration, glycerophospholipid metabolism, inflammation, central energy metabolism, oxidative stress, and vitamin metabolism. Within disease states (early and late OA), subgroups of donors revealed distinct phenotypes. Synovial fluid metabolomic phenotypes exhibited increased inflammation (early and late OA), oxidative stress (late OA), or structural deterioration (early and late OA) in the synovial fluid. Conclusion: These results revealed distinct metabolic phenotypes in human synovial fluid, provide insight into pathogenesis, represent novel biomarkers, and can move toward developing personalized interventions for subgroups of OA patients.
The chemostat with lateral gene transfer
(2010-11) De Leenheer, Patrick; Dockery, Jack D.; Gedeon, Tomas; Young, T.
We investigate the standard chemostat model when lateral gene transfer is taken into account. We will show that when the different genotypes have growth rate functions that are sufficiently close to a common growth rate function, and when the yields of the genotypes are sufficiently close to a common value, then the population evolves to a globally stable steady state, at which all genotypes coexist. These results can explain why the antibiotic-resistant strains persist in the pathogen population.
Coding Code: Qualitative Methods for Investigating Data Science Skills
(Informa UK Limited, 2023-11) Theobold, Allison S.; Wickstrom, Megan H.; Hancock, Stacey A.
Despite the elevated importance of Data Science in Statistics, there exists limited research investigating how students learn the computing concepts and skills necessary for carrying out data science tasks. Computer Science educators have investigated how students debug their own code and how students reason through foreign code. While these studies illuminate different aspects of students’ programming behavior or conceptual understanding, a method has yet to be employed that can shed light on students’ learning processes. This type of inquiry necessitates qualitative methods, which allow for a holistic description of the skills a student uses throughout the computing code they produce, the organization of these descriptions into themes, and a comparison of the emergent themes across students or across time. In this article we share how to conceptualize and carry out the qualitative coding process with students’ computing code. Drawing on the Block Model to frame our analysis, we explore two types of research questions which could be posed about students’ learning. Supplementary materials for this article are available online.
Coliform Contamination in private well water on the Crow Reservation
(2018-10) Powell, Michaela
Emery Three Irons is a master’s level graduate student in the Department of Land Resources and Environmental Sciences (LRES) at Montana State University (MSU). Advised by Scott Powell, Ph.D., who is an Associate Professor in the Department of LRES at MSU, Emery is working on his master’s thesis which explores coliform contamination in private well water on the Crow Reservation. He has requested the assistance of the Statistical Consulting and Research Services in deciding on an appropriate analysis for the data he collected.
Combinatorial Representation of Parameter Space for Switching Networks
(2016-11) Cummins, Bree; Harker, Shaun; Mischaikow, Konstantin; Mok, Kafung; Gedeon, Tomas
We describe the theoretical and computational framework for the Dynamic Signatures Generated by Regulatory Networks (DSGRN) database. The motivation stems from an urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and have decades of parameters. The input to the database computations is a regulatory network, i.e., a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate) and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piecewise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type of information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5 nodes and a 29-dimensional parameter space.
Combining Dynamic Bayesian Networks and Continuous Time Bayesian Networks for Diagnostic and Prognostic Modeling
(IEEE, 2022-08) Schupbach, Jordan; Pryor, Elliott; Webster, Kyle; Sheppard, John
The problem of performing general prognostics and health management, especially in electronic systems, continues to present significant challenges. The low availability of failure data, makes learning generalized models difficult, and constructing generalized models during the design phase often requires a level of understanding of the failure mechanism that elude the designers. In this paper, we present a new, generalized approach to PHM based on two commonly available probabilistic models, Bayesian Networks and Continuous-Time Bayesian Networks, and pose the PHM problem from the perspective of risk mit-igation rather than failure prediction. We describe the tools and process for employing these tools in the hopes of motivating new ideas for investigating how best to advance PHM in the aerospace industry.
Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models
(2019-04) Crawford-Kahrl, Peter; Cummins, Bree; Gedeon, Tomas
Modeling the dynamics of biological networks introduces many challenges, among them the lack of first principle models, the size of the networks, and difficulties with parameterization. Discrete time Boolean networks and related continuous time switching systems provide a computationally accessible way to translate the structure of the network to predictions about the dynamics. Recent work has shown that the parameterized dynamics of switching systems can be captured by a combinatorial object, called a Dynamic Signatures Generated by Regulatory Networks (DSGRN) database, that consists of a parameter graph characterizing a finite parameter space decomposition, whose nodes are assigned a Morse graph that captures global dynamics for all corresponding parameters. We show that for a given network there is a way to associate the same type of object by considering a continuous time ODE system with a continuous right-hand side, which we call an L-system. The main goal of this paper is to compare the two DSGRN databases for the same network. Since the L-systems can be thought of as perturbations (not necessarily small) of the switching systems, our results address the correspondence between global parameterized dynamics of switching systems and their perturbations. We show that, at corresponding parameters, there is an order preserving map from the Morse graph of the switching system to that of the L-system that is surjective on the set of attractors and bijective on the set of fixed-point attractors. We provide important examples showing why this correspondence cannot be strengthened.
Competitive resource allocation to metabolic pathways contributes to overflow metabolisms and emergent properties in cross-feeding microbial consortia
(2018-04) Carlson, Ross P.; Beck, Ashley E.; Phalak, Poonam; Fields, Matthew W.; Gedeon, Tomas; Hanley, Luke; Harcombe, W. R.; Henson, Michael A.; Heys, Jeffrey J.
Resource scarcity is a common stress in nature and has a major impact on microbial physiology. This review highlights microbial acclimations to resource scarcity, focusing on resource investment strategies for chemoheterotrophs from the molecular level to the pathway level. Competitive resource allocation strategies often lead to a phenotype known as overflow metabolism; the resulting overflow byproducts can stabilize cooperative interactions in microbial communities and can lead to cross-feeding consortia. These consortia can exhibit emergent properties such as enhanced resource usage and biomass productivity. The literature distilled here draws parallels between in silico and laboratory studies and ties them together with ecological theories to better understand microbial stress responses and mutualistic consortia functioning.
Considerations for assessing model averaging of regression coefficients
(2016-08) Banner, Katharine M.; Higgs, Megan D.
"Model choice is usually an inevitable source of uncertainty in model-based statistical analyses. While the focus of model choice was traditionally on methods for choosing a single model, methods to formally account for multiple models within a single analysis are now accessible to many researchers. The specific technique of model averaging was developed to improve predictive ability by combining predictions from a set of models. However, it is now often used to average regression coefficients across multiple models with the ultimate goal of capturing a variable\'s overall effect. This use of model averaging implicitly assumes the same parameter exists across models so that averaging is sensible. While this assumption may initially seem tenable, regression coefficients associated with particular explanatory variables may not hold equivalent interpretations across all of the models in which they appear, making explanatory inference about covariates challenging. Accessibility to easily implementable software, concerns about being criticized for ignoring model uncertainty, and the chance to avoid having to justify choice of a final model have all led to the increasing popularity of model averaging in practice. We see a gap between the theoretical development of model averaging and its current use in practice, potentially leaving well-intentioned researchers with unclear inferences or difficulties justifying reasons for using (or not using) model averaging. We attempt to narrow this gap by revisiting some relevant foundations of regression modeling, suggesting more explicit notation and graphical tools, and discussing how individual model results arecombined to obtain a model averaged result. Our goal is to help researchers make informed decisions about model averaging and to encourage question-focused modeling over method-focused modeling. "