Scholarly Work - Mathematical Sciences

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    Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory
    (Springer Science and Business Media LLC, 2024-01) Davey, Blair; Vega Garcia, Mariana Smit
    This paper continues the study initiated in Davey (Arch Ration Mech Anal 228:159–196, 2018), where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. In this article, we extend these ideas to the variable-coefficient setting. This generalized technique is demonstrated through new proofs of three important theorems for variable-coefficient heat operators, one of which establishes a result that is, to the best of our knowledge, also new. Specifically, we give new proofs of L2 → L2 Carleman estimates and the monotonicity of Almgren-type frequency functions, and we prove a new monotonicity of Alt–Caffarelli–Friedman-type functions. The proofs in this article rely only on their related elliptic theorems and a limiting argument. That is, each parabolic theorem is proved by taking a high-dimensional limit of a related elliptic result.
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    Joint Spatial Modeling Bridges the Gap Between Disparate Disease Surveillance and Population Monitoring Efforts Informing Conservation of At-risk Bat Species
    (Springer Science and Business Media LLC, 2024-02) Stratton, Christian; Irvine, Kathryn M.; Banner, Katharine M.; Almberg, Emily S.; Bachen, Dan; Smucker, Kristina
    White-Nose Syndrome (WNS) is a wildlife disease that has decimated hibernating bats since its introduction in North America in 2006. As the disease spreads westward, assessing the potentially differential impact of the disease on western bat species is an urgent conservation need. The statistical challenge is that the disease surveillance and species response monitoring data are not co-located, available at different spatial resolutions, non-Gaussian, and subject to observation error requiring a novel extension to spatially misaligned regression models for analysis. Previous work motivated by epidemiology applications has proposed two-step approaches that overcome the spatial misalignment while intentionally preventing the human health outcome from informing estimation of exposure. In our application, the impacted animals contribute to spreading the fungus that causes WNS, motivating development of a joint framework that exploits the known biological relationship. We introduce a Bayesian, joint spatial modeling framework that provides inferences about the impact of WNS on measures of relative bat activity and accounts for the uncertainty in estimation of WNS presence at non-surveyed locations. Our simulations demonstrate that the joint model produced more precise estimates of disease occurrence and unbiased estimates of the association between disease presence and the count response relative to competing two-step approaches. Our statistical framework provides a solution that leverages disparate monitoring activities and informs species conservation across large landscapes. Stan code and documentation are provided to facilitate access and adaptation for other wildlife disease applications.
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    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.
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    Ribosome Abundance Control in Prokaryotes
    (Springer Science and Business Media LLC, 2023-10) Shea, Jacob; Davis, Lisa; Quaye, Bright; Gedeon, Tomas
    Cell growth is an essential phenotype of any unicellular organism and it crucially depends on precise control of protein synthesis. We construct a model of the feedback mechanisms that regulate abundance of ribosomes in E. coli, a prototypical prokaryotic organism. Since ribosomes are needed to produce more ribosomes, the model includes a positive feedback loop central to the control of cell growth. Our analysis of the model shows that there can be only two coexisting equilibrium states across all 23 parameters. This precludes the existence of hysteresis, suggesting that the ribosome abundance changes continuously with parameters. These states are related by a transcritical bifurcation, and we provide an analytic formula for parameters that admit either state.
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    Leveraging social networks for identification of people living with HIV who are virally unsuppressed
    (Wolters Kluwer Health, Inc., 2023-10) Cummins, Breschine; Johnson, Kara; Schneider, John A.; Del Vicchio, Natasha; Moshiri, Niema; Wertheim, Joel O.; Goyal, Ravi; Skaathun, Britt
    Objectives: This study investigates primary peer-referral engagement (PRE) strategies to assess which strategy results in engaging higher numbers of people living with HIV (PLWH) who are virally unsuppressed. Design: We develop a modeling study that simulates an HIV epidemic (transmission, disease progression, and viral evolution) over 6 years using an agent-based model followed by simulating PRE strategies. We investigate two PRE strategies where referrals are based on social network strategies (SNS) or sexual partner contact tracing (SPCT). Methods: We parameterize, calibrate, and validate our study using data from Chicago on Black sexual minority men to assess these strategies for a population with high incidence and prevalence of HIV. For each strategy we calculate the number of PLWH recruited who are undiagnosed or out-of-care and the number of direct or indirect transmissions. Results: SNS and SPCT identified 256.5 (95% C.I.: [234,279]) and 15 (95% C.I.: [7,27]) PLWH, respectively. Of these, SNS identified 159 (95% C.I.: [142,177]) PLWH out-of-care and 32 (95% C.I.: [21, 43]]) PLWH undiagnosed compared to 9 (95% C.I.: [3,18]) and 2 (95% C.I.: [0,5]) for SPCT. SNS identified 15.5 (95% C.I.: [6,25]) and 7.5 (95% C.I.: [2, 11]]) indirect and direct transmission pairs, while SPCT identified 6 (95% C.I.: [0,8]) and 5 (95% C.I.: [0,8]), respectively. Conclusions: With no testing constraints, SNS is the more effective strategy to identify undiagnosed and out-of-care PLWH. Neither strategy is successful at identifying sufficient indirect or direct transmission pairs to investigate transmission networks.
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    Detecting punctuated evolution in SARS-CoV-2 over the first year of the pandemic
    (Frontiers Media SA, 2023-02) Surya, Kevin; Gardner, Jacob D.; Organ, Chris L.
    The Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) evolved slowly over the first year of the Coronavirus Disease 19 (COVID-19) pandemic with differential mutation rates across lineages. Here, we explore how this variation arose. Whether evolutionary change accumulated gradually within lineages or during viral lineage branching is unclear. Using phylogenetic regression models, we show that ~13% of SARS-CoV-2 genomic divergence up to May 2020 is attributable to lineage branching events (punctuated evolution). The net number of branching events along lineages predicts ~5% of the deviation from the strict molecular clock. We did not detect punctuated evolution in SARS-CoV-1, possibly due to the small sample size, and in sarbecovirus broadly, likely due to a different evolutionary process altogether. Punctuation in SARS-CoV-2 is probably neutral because most mutations were not positively selected and because the strength of the punctuational effect remained constant over time, at least until May 2020, and across continents. However, the small punctuational contribution to SARS-CoV-2 diversity is consistent with the founder effect arising from narrow transmission bottlenecks. Therefore, punctuation in SARS-CoV-2 may represent the macroevolutionary consequence (rate variation) of a microevolutionary process (transmission bottleneck).
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    Lifetime alcohol consumption patterns and young-onset breast cancer by subtype among Non-Hispanic Black and White women in the Young Women’s Health History Study
    (Springer Nature, 2023-10) Hirko, Kelly A.; Lucas, Darek R.; Pathak, Dorothy R.; Hamilton, Ann S.; Post, Lydia M.; Ihenacho, Ugonna; Carnegie, Nicole Bohme; Houang, Richard T.; Schwartz, Kendra; Velie, Ellen M.
    Purpose. The role of alcohol in young-onset breast cancer (YOBC) is unclear. We examined associations between lifetime alcohol consumption and YOBC in the Young Women’s Health History Study, a population-based case–control study of breast cancer among Non-Hispanic Black and White women < 50 years of age. Methods. Breast cancer cases (n = 1,812) were diagnosed in the Metropolitan Detroit and Los Angeles County SEER registry areas, 2010–2015. Controls (n = 1,381) were identified through area-based sampling and were frequency-matched to cases by age, site, and race. Alcohol consumption and covariates were collected from in-person interviews. Weighted multivariable logistic regression was conducted to calculate adjusted odds ratios (aOR) and 95% confidence intervals (CI) for associations between alcohol consumption and YOBC overall and by subtype (Luminal A, Luminal B, HER2, or triple negative). Results. Lifetime alcohol consumption was not associated with YOBC overall or with subtypes (all ptrend ≥ 0.13). Similarly, alcohol consumption in adolescence, young and middle adulthood was not associated with YOBC (all ptrend ≥ 0.09). An inverse association with triple-negative YOBC, however, was observed for younger age at alcohol use initiation (< 18 years vs. no consumption), aOR (95% CI) = 0.62 (0.42, 0.93). No evidence of statistical interaction by race or household poverty was observed. Conclusions. Our findings suggest alcohol consumption has a different association with YOBC than postmenopausal breast cancer—lifetime consumption was not linked to increased risk and younger age at alcohol use initiation was associated with a decreased risk of triple-negative YOBC. Future studies on alcohol consumption in YOBC subtypes are warranted.
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    Using physical simulations to motivate the use of differential equations in models of disease spread
    (Informa UK Limited, 2023-09) Arnold, Elizabeth G.; Burroughs, Elizabeth A.; Burroughs, Owen; Carlson, Mary Alice
    The SIR model is a differential equations based model of the spread of an infectious disease that compartmentalises individuals in a population into one of three states: those who are susceptible to a disease (S), those who are infected and can transmit the disease to others (I), and those who have recovered from the disease and are now immune (R). This Classroom Note describes how to initiate teaching the SIR model with two concrete physical simulations to provide students with first-hand experience with some of the nuanced behaviour of how an infectious disease spreads through a closed population. One simulation physically models disease spread by the exchange of fluids, using pH to simulate infection. A second simulation incorporates randomness through the use of a probability game to keep track of the state of each individual at each time step. Both simulations invite students to ask questions about what factors influence disease spread. The concrete experience from the physical simulations enables students to make connections to the abstract mathematical representation of the SIR model and discuss the sources of stochasticity present in the spread of an infectious disease.
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    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
    (The Royal Society, 2023-07) Liu, Yuxuan; McCalla, Scott G.; Schaeffer, Hayden
    Particle dynamics and multi-agent systems provide accurate dynamical models for studying and forecasting the behaviour of complex interacting systems. They often take the form of a high-dimensional system of differential equations parameterized by an interaction kernel that models the underlying attractive or repulsive forces between agents. We consider the problem of constructing a data-based approximation of the interacting forces directly from noisy observations of the paths of the agents in time. The learned interaction kernels are then used to predict the agents’ behaviour over a longer time interval. The approximation developed in this work uses a randomized feature algorithm and a sparse randomized feature approach. Sparsity-promoting regression provides a mechanism for pruning the randomly generated features which was observed to be beneficial when one has limited data, in particular, leading to less overfitting than other approaches. In addition, imposing sparsity reduces the kernel evaluation cost which significantly lowers the simulation cost for forecasting the multi-agent systems. Our method is applied to various examples, including first-order systems with homogeneous and heterogeneous interactions, second-order homogeneous systems, and a new sheep swarming system.
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    Estimating contact network properties by integrating multiple data sources associated with infectious diseases
    (Wiley, 2023-07) Goyal, Ravi; Carnegie, Nicole; Slipher, Sally; Turk, Philip; Little, Susan J.; De Gruttola, Victor
    To effectively mitigate the spread of communicable diseases, it is necessary to understand the interactions that enable disease transmission among individuals in a population; we refer to the set of these interactions as a contact network. The structure of the contact network can have profound effects on both the spread of infectious diseases and the effectiveness of control programs. Therefore, understanding the contact network permits more efficient use of resources. Measuring the structure of the network, however, is a challenging problem. We present a Bayesian approach to integrate multiple data sources associated with the transmission of infectious diseases to more precisely and accurately estimate important properties of the contact network. An important aspect of the approach is the use of the congruence class models for networks. We conduct simulation studies modeling pathogens resembling SARS-CoV-2 and HIV to assess the method; subsequently, we apply our approach to HIV data from the University of California San Diego Primary Infection Resource Consortium. Based on simulation studies, we demonstrate that the integration of epidemiological and viral genetic data with risk behavior survey data can lead to large decreases in mean squared error (MSE) in contact network estimates compared to estimates based strictly on risk behavior information. This decrease in MSE is present even in settings where the risk behavior surveys contain measurement error. Through these simulations, we also highlight certain settings where the approach does not improve MSE.
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    Resource allocation accounts for the large variability of rate-yield phenotypes across bacterial strains
    (eLife Sciences Publications, Ltd, 2023-05) Baldazzi, Valentina; Ropers, Delphine; Gouzé, Jean-Luc; Gedeon, Tomas; de Jong, Hidde
    Different strains of a microorganism growing in the same environment display a wide variety of growth rates and growth yields. We developed a coarse-grained model to test the hypothesis that different resource allocation strategies, corresponding to different compositions of the proteome, can account for the observed rate-yield variability. The model predictions were verified by means of a database of hundreds of published rate-yield and uptake-secretion phenotypes of Escherichia coli strains grown in standard laboratory conditions. We found a very good quantitative agreement between the range of predicted and observed growth rates, growth yields, and glucose uptake and acetate secretion rates. These results support the hypothesis that resource allocation is a major explanatory factor of the observed variability of growth rates and growth yields across different bacterial strains. An interesting prediction of our model, supported by the experimental data, is that high growth rates are not necessarily accompanied by low growth yields. The resource allocation strategies enabling high-rate, high-yield growth of E. coli lead to a higher saturation of enzymes and ribosomes, and thus to a more efficient utilization of proteomic resources. Our model thus contributes to a fundamental understanding of the quantitative relationship between rate and yield in E. coli and other microorganisms. It may also be useful for the rapid screening of strains in metabolic engineering and synthetic biology.
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    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.
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    Numerical analysis of a time filtered scheme for a linear hyperbolic equation inspired by DNA transcription modeling
    (Elsevier BV, 2023-09) Boatman, K.; Davis, L.; Pahlevani, F.; Rajan, T. Susai
    The focus of this paper is the development and analysis of a time filtering process for a linear hyperbolic equation motivated by the modeling of the transcription of ribosomal RNA in bacteria Davis et al. (2021). We demonstrate that a time filter technique can be combined with the classical upwind to produce a new explicit scheme with virtually no dissipation introduced by the method, and the filter can be implemented with minimal computational cost. The analysis shows that the filtered scheme gives the practitioner the ability to adjust the filtering so the dissipation can be made arbitrarily small over a range of time step choices. The analysis also indicates that the filtered scheme has a smaller local truncation error when compared to that of the original upwind method. A CFL condition for the new algorithm is derived, and it is shown to depend explicitly on the filter parameter. Numerical computations illustrate stability and convergence as well as dissipation and dispersion assessments of the filtered upwind scheme.
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    The integrated nested Laplace approximation applied to spatial log-Gaussian Cox process models
    (Informa UK Limited, 2023-04) Flagg, Kenneth; Hoegh, Andrew
    Spatial point process models are theoretically useful for mapping discrete events, such as plant or animal presence, across space; however, the computational complexity of fitting these models is often a barrier to their practical use. The log-Gaussian Cox process (LGCP) is a point process driven by a latent Gaussian field, and recent advances have made it possible to fit Bayesian LGCP models using approximate methods that facilitate rapid computation. These advances include the integrated nested Laplace approximation (INLA) with a stochastic partial differential equations (SPDE) approach to sparsely approximate the Gaussian field and an extension using pseudodata with a Poisson response. To help link the theoretical results to statistical practice, we provide an overview of INLA for point process data and then illustrate their implementation using freely available data. The analyzed datasets include both a completely observed spatial field and an incomplete data situation. Our well-commented R code is shared in the online supplement. Our intent is to make these methods accessible to the practitioner of spatial statistics without requiring deep knowledge of point process theory.
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    The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves
    (American Mathematical Society, 2022-12) Frăţilă, Dragoş; Gunningham, Sam; Li, Penghui
    We study the moduli stack of degree $0$ semistable $G$-bundles on an irreducible curve $E$ of arithmetic genus $1$, where $G$ is a connected reductive group in arbitrary characteristic. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups $H$ of $G$ (the $E$-pseudo-Levi subgroups), where each stratum is computed in terms of $H$-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where $E$ has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra $\mathfrak {g}$ (respectively, algebraic group $G$). We also provide a Tannakian description of these moduli stacks and use it to show that if $E$ is not a supersingular elliptic curve, the moduli of framed unipotent bundles on $E$ are equivariantly isomorphic to the unipotent cone in $G$. Finally, we classify the $E$-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.
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    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.
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    Extending Combinatorial Regulatory Network Modeling to Include Activity Control and Decay Modulation
    (Society for Industrial & Applied Mathematics, 2022-09) Cummins, Bree; Gameiro, Marcio; Gedeon, Tomas; Kepley, Shane; Mischaikow, Konstantin; Zhang, Lun
    Understanding how the structure of within-system interactions affects the dynamics of the system is important in many areas of science. We extend a network dynamics modeling platform DSGRN, which combinatorializes both dynamics and parameter space to construct finite but accurate summaries of network dynamics, to new types of interactions. While the standard DSGRN assumes that each network edge controls the rate of abundance of the target node, the new edges may control either activity level or a decay rate of its target. While motivated by processes of post-transcriptional modification and ubiquitination in systems biology, our extension is applicable to the dynamics of any signed directed network.
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    Quantifying the role of airborne transmission in the spread of COVID-19
    (American Institute of Mathematical Sciences, 2022-01) Hayden, Matthew; Morrow, Bryce; Yang, Wesley; Wang, Jin
    There is an ongoing debate on the different transmission modes of SARS-CoV-2 and their relative contributions to the pandemic. In this paper, we employ a simple mathematical model, which incorporates both the human-to-human and environment-to-human transmission routes, to study the transmission dynamics of COVID-19. We focus our attention on the role of airborne transmission in the spread of the disease in a university campus setting. We conduct both mathematical analysis and numerical simulation, and incorporate published experimental data for the viral concentration in the air to fit model parameters. Meanwhile, we compare the outcome to that of the standard SIR model, utilizing a perturbation analysis in the presence of multiple time scales. Our data fitting and numerical simulation results show that the risk of airborne transmission for SARS-CoV-2 strongly depends on how long the virus can remain viable in the air. If the time for this viability is short, the airborne transmission route would be inconsequential in shaping the overall transmission risk and the total infection size. On the other hand, if the infectious virus can persist in aerosols beyond a few hours, then airborne transmission could play a much more significant role in the spread of COVID-19.
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    Encountering ideas about teaching and learning mathematics in undergraduate mathematics courses
    (Springer Science and Business Media LLC, 2023-01) Burroughs, Elizabeth A.; Arnold, Elizabeth G.; Álvarez, James A. M.; Kercher, Andrew; Tremaine, Rachel; Fulton, Elizabeth; Turner, Kyle
    We study the ideas about teaching and learning mathematics that undergraduate students generate when they encounter tasks designed to embed approximations of teaching practice in mathematics courses taken by a general population of students. These tasks attend to the dual goals of developing an understanding of mathematics content and an understanding of how teachers provide classroom experiences that foster mathematics learning. The study employs a qualitative, multiple-case study methodology, with four cases bounded by the content areas of abstract algebra, single variable calculus, discrete mathematics, and introductory statistics. The data for the study come from undergraduate students’ written work on mathematical tasks, interviews with a subset of students from each course, and interviews with each instructor throughout the term during which they implemented the tasks. Our findings indicate that students identified the broad applicability of teaching skills (discussed by 32 of the 61 interviewed students), recognized the value of examining hypothetical learners’ mathematical work (discussed by 59 of the 61 interviewed students), and reported empathy for hypothetical learners (discussed by 38 of the 61 interviewed students). These findings persisted across the course content and course levels we studied, leading us to conclude that our findings can transfer to additional mathematics courses in secondary mathematics teacher preparation.
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    Extremal event graphs: A (stable) tool for analyzing noisy time series data
    (American Institute of Mathematical Sciences, 2023-01) Belton, Robin; Cummins, Bree; Gedeon, Tomáš; Fasy, Brittany Terese
    Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.
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