Mathematical Sciences
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Mathematical research at MSU is focused primarily on related topics in pure and applied mathematics. Research programs complement each other and are often applied to problems in science and engineering. Research in statistics encompasses a broad range of theoretical and applied topics. Because the statisticians are actively engaged in interdisciplinary work, much of the statistical research is directed toward practical problems. Mathematics education faculty are active in both qualitative and quantitative experimental research areas. These include teacher preparation, coaching and mentoring for in-service teachers, online learning and curriculum development.
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Item 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, KristinaWhite-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.Item Statistical assessment on determining local presence of rare bat species(Wiley, 2022-06) Irvine, Kathryn M.; Banner, Katharine M.; Stratton, Christian; Ford, William M.; Reichert, Brian E.Surveying cryptic, sparsely distributed taxa using autonomous recording units,although cost-effective, provides imperfect knowledge about species presence.Summertime bat acoustic surveys in North America exemplify the challenges with characterizing sources of uncertainty: observation error, inability to census populations, and natural stochastic variation. Statistical uncertainty, if not considered thoroughly, hampers determining rare species presence accurately and/or estimating range wide status and trends with suitable precision. Bat acoustic data are processed using an automated workflow in which proprietary or open-source algorithms assign a species label to each recorded high-frequency echolocation sequence. A false-negative occurs, if a species is actually present but not recorded and/or all recordings from the species are of such poor quality that a correct species identity cannot be assigned to any observation. False positives for a focal species are a direct result of the presence and incorrect identification of a recording from another species. We compare four analytical approaches in terms of parameter estimation and their resulting(in)correct decisions regarding species presence or absence using realistic data-generating scenarios for bat acoustic data within a simulation study. The cur-rent standard for deciding species presence or absence uses a multinomial likelihood-ratio test p value (maximum likelihood estimate [MLE]-metric) that accounts for known species misidentifications, but not imperfect detection and only returns a binary outcome (evidence of presence or not). We found that the MLE-metric had estimated median correct decisions less than 60% for presence and greater than 85% for absence. Alternatively, a multispecies count detection model was equivalent to or better than the MLE-metric for correct claims of rare species presence or absence using the posterior probability a species was present at a site and, importantly, provided unbiased estimates of relative activity and probability of occurrence, creating opportunities for reducing posterior uncertainty through the inclusion of meaningful covariates. Single-species occupancy models with and without false-positive detections removed were insufficient for determining local presence because of substantially biased occurrence and detection probabilities. We propose solutions to potential barriers for integrating local, short-term and range wide, long-term acoustic surveys within a cohesive statistical framework that facilitates determining local species presence with uncertainty concurrent with estimating species–environment relationships.Item Coupling validation effort with in situ bioacoustic data improves estimating relative activity and occupancy for multiple species with cross‐species misclassifications(Wiley, 2022-03) Stratton, Christian; Irvine, Kathryn M.; Banner, Katharine M.; Wright, Wilson J.; Lausen, Cori; Rae, JasonThe increasing complexity and pace of ecological change requires natural resource managers to consider entire species assemblages. Acoustic recording units (ARUs) require minimal cost and effort to deploy and inform relative activity, or encounter rates, for multiple species simultaneously. ARU-based surveys require post-processing of the recordings via software algorithms that assign a species label to each recording. The automated classification process can result in cross-species misidentifications that should be accounted for when employing statistical modelling for conservation decision-making. Using simulation and ARU-based detection counts from 17 bat species in British Columbia, Canada, we investigate three strategies for adjusting statistical inference for species misclassification: (a) ‘coupling’ ambiguous and unambiguous detections by validating a subset of survey events post-hoc, (b) using a calibration dataset on the software algorithm's (in)accuracy for species identification or (c) specifying informative Bayesian priors on classification probabilities. We explore the impact of different Bayesian prior specifications for the classification probabilities on posterior estimation. We then consider how the quantity of data validated post-hoc impacts model convergence and resulting inferences for bat species relative activity as related to nightly conditions and yearly site occupancy after accounting for site-level environmental variables. Coupled methods resulted in less bias and uncertainty when estimating relative activity and species classification probabilities relative to calibration approaches. We found that species that were difficult-to-detect and those that were often inaccurately identified by the software required more validation effort than more easily detected and/or identified species. Our results suggest that, when possible, acoustic surveys should rely on coupled validated detection information to account for false-positive detections, rather than uncoupled calibration datasets. However, if the assemblage of interest contains a large number of rarely detected or less prevalent species, an intractable amount of effort may be required, suggesting there are benefits to curating a calibration dataset that is representative of the observation process. Our findings provide insights into the practical challenges associated with statistical analyses of ARU data and possible analytical solutions to support reliable and cost-effective decision-making for wildlife conservation/management in the face of known sources of observation errors.Item Resilience to fire and resistance to annual grass invasion in sagebrush ecosystems of US National Parks(Elsevier BV, 2021-08) Rodhouse, Thomas J.; Lonneker, Jeffrey; Bowersock, Lisa; Popp, Diana; Thompson, Jamela C.; Dicus, Gordon H.; Irvine, Kathryn M.Western North American sagebrush shrublands and steppe face accelerating risks from fire-driven feedback loops that transition these ecosystems into self-reinforcing states dominated by invasive annual grasses. In response, sagebrush conservation decision-making is increasingly done through the lens of resilience to fire and annual grass invasion resistance. Operationalizing resilience and resistance concepts requires place-based understanding of resilience and resistance variation among landscapes over time. Place-based insights allow for landscape prioritization in targeted areas of significance such as protected-area sagebrush ecosystems that exhibit inherently low resilience and are therefore at high risk of loss. We used a multi-scale approach to evaluate sagebrush resiliency and strategic planning across 1) the US National Park system, 2) a regional suite of five parks, and 3) for two specific park case studies. First, we summarized broad patterns of relative resilience to fire and resistance to annual grass invasion across all parks with sagebrush ecosystems. We found that national parks represented ~11% of US protected-area sagebrush ecosystems and reflected a similar low-resilience bias that occurs across the biome, broadly. Climate change is likely to shift both low- and high-resilience park sagebrush ecosystems towards moderate resiliency, creating new opportunities and constraints for park conservation. Approximately seventy park units include at least some sagebrush shrublands or steppe, but we identified 40 parks with substantial amounts (>20% of park area) that can be included in an agency-wide conservation strategy. Second, we examined detailed patterns of resilience and resistance, fire history and fire risk, cheatgrass (Bromus tectorum) invasion, and sagebrush shrub (Artemisia spp.) persistence in five national park units in Columbia Basin and Snake River Plain sagebrush steppe, contextualized by the broader summary. In these five parks, fire frequency and size increased in recent decades. Cheatgrass invasion and sagebrush persistence correlated strongly with resilience, burn frequency (0–3 fires since ~1940), and burn probability, but with important variation, in part mediated by local-scale topography. Third, we used these insights to assemble strategic sagebrush ecosystem fire protection mapping scenarios in two additional parks – Lava Beds National Monument and Great Basin National Park. Readily available and periodically updated geospatial data including soil surveys, fire histories, vegetation inventories, and long-term monitoring support resiliency-based adaptive management through tactical planning of pre-fire protection, post-fire restoration, and triage. Our assessment establishes the precarious importance of the US national park system to sagebrush ecosystem conservation and an operational strategy for place-based and science-supported conservation.Item Post-Fire Vegetation Response in a Repeatedly Burned Low-Elevation Sagebrush Steppe Protected Area Provides Insights About Resilience and Invasion Resistance(Frontiers Media SA, 2020-11) Rodhouse, Thomas J.; Irvine, Kathryn M.; Bowersock, LisaSagebrush steppe ecosystems are threatened by human land-use legacies, biological invasions, and altered fire and climate dynamics. Steppe protected areas are therefore of heightened conservation importance but are few and vulnerable to the same impacts broadly affecting sagebrush steppe. To address this problem, sagebrush steppe conservation science is increasingly emphasizing a focus on resilience to fire and resistance to non-native annual grass invasion as a decision framework. It is well-established that the positive feedback loop between fire and annual grass invasion is the driving process of most contemporary steppe degradation. We use a newly developed ordinal zero-augmented beta regression model fit to large-sample vegetation monitoring data from John Day Fossil Beds National Monument, USA, spanning 7 years to evaluate fire responses of two native perennial foundation bunchgrasses and two non-native invasive annual grasses in a repeatedly burned, historically grazed, and inherently low-resilient protected area. We structured our model hierarchically to support inferences about variation among ecological site types and over time after also accounting for growing-season water deficit, fine-scale topographic variation, and burn severity. We use a state-and-transition conceptual diagram and abundances of plants listed in ecological site reference conditions to formalize our hypothesis of fire-accelerated transition to ecologically novel annual grassland. Notably, big sagebrush (Artemisia tridentata) and other woody species were entirely removed by fire. The two perennial grasses, bluebunch wheatgrass (Pseudoroegneria spicata) and Thurber's needlegrass (Achnatherum thurberianum) exhibited fire resiliency, with no apparent trend after fire. The two annual grasses, cheatgrass (Bromus tectorum) and medusahead (Taeniatherum caput-medusae), increased in response to burn severity, most notably medusahead. Surprisingly, we found no variation in grass cover among ecological sites, suggesting fire-driven homogenization as shrubs were removed and annual grasses became dominant. We found contrasting responses among all four grass species along gradients of topography and water deficit, informative to protected-area conservation strategies. The fine-grained influence of topography was particularly important to variation in cover among species and provides a foothold for conservation in low-resilient, aridic steppe. Broadly, our study demonstrates how to operationalize resilience and resistance concepts for protected areas by integrating empirical data with conceptual and statistical models.Item Identifying occupancy model inadequacies: can residuals separately assess detection and presence?(2019-04) Wright, Wilson J.; Irvine, Kathryn M.; Higgs, Megan D.Occupancy models are widely applied to estimate species distributions, but few methods exist for model checking. Thorough model assessments can uncover inadequacies and allow for deeper ecological insight by exploring structure in the observed data not accounted for by a model. We introduce occupancy model residual definitions that utilize the posterior distribution of the partially latent occupancy states. Residual-based assessments are valuable because they can target specific assumptions and identify ways to improve a model, such as adding spatial correlation or meaningful covariates. Our approach defines separate residuals for occupancy and detection, and we use simulation to examine whether missing structure for modeling detection probabilities can be distinguished from that for occupancy probabilities. In many scenarios, our residual diagnostics were able to separate inadequacies at the different model levels successfully, but we describe other situations when this may not be the case. Applying Moran\'s I residual diagnostics to assess models for silver-haired (Lasionycteris noctivagans) and little brown (Myotis lucifugus) bats only provided evidence of residual spatial correlation among detections. Targeting specific model assumptions using carefully chosen residual diagnostics is valuable for any analysis, and we remove previous barriers for occupancy analyses-lack of examples and practical advice.