A general science-based framework for dynamical spatio-temporal models

Bayesian hierarchical modeling and inference for mechanistic systems in industrial hygiene

A series of experiments in stationary and moving passenger rail cars were conducted to measure removal rates of particles in the size ranges of SARS-CoV-2 viral aerosols and the air changes per hour provided by existing and modified air handling systems. Such methods for exposure assessments are customarily based on mechanistic models derived from physical laws of particle movement that are deterministic and do not account for measurement errors inherent in data collection. The resulting analysis compromises on reliably learning about mechanistic factors such as ventilation rates, aerosol generation rates, and filtration efficiencies from field measurements. This manuscript develops a Bayesian state-space modeling framework that synthesizes information from the mechanistic system as well as the field data. We derive a stochastic model from finite difference approximations of differential equations explaining particle concentrations. Our inferential framework trains the mechanistic system using the field measurements from the chamber experiments and delivers reliable estimates of the underlying physical process with fully model-based uncertainty quantification. Our application falls within the realm of the Bayesian "melding" of mechanistic and statistical models and is of significant relevance to environmental hygienists and public health researchers working on assessing the performance of aerosol removal rates for rail car fleets.

Revealing the extent of sea otter impacts on bivalve prey through multi-trophic monitoring and mechanistic models

Sea otters are apex predators that can exert considerable influence over the nearshore communities they occupy. Since facing near extinction in the early 1900s, sea otters are making a remarkable recovery in Southeast Alaska, particularly in Glacier Bay, the largest protected tidewater glacier fjord in the world. The expansion of sea otters across Glacier Bay offers both a challenge to monitoring and stewardship and an unprecedented opportunity to study the top-down effect of a novel apex predator across a diverse and productive ecosystem. Our goal was to integrate monitoring data across trophic levels, space, and time to quantify and map the predator-prey interaction between sea otters and butter clams Saxidomus gigantea, one of the dominant large bivalves in Glacier Bay and a favoured prey of sea otters. We developed a spatially-referenced mechanistic differential equation model of butter clam dynamics that combined both environmental drivers of local population growth and estimates of otter abundance from aerial survey data. We embedded this model in a Bayesian statistical framework and fit it to clam survey data from 43 intertidal and subtidal sites across Glacier Bay. Prior to substantial sea otter expansion, we found that butter clam density was structured by an environmental gradient driven by distance from glacier (represented by latitude) and a quadratic effect of current speed. Estimates of sea otter attack rate revealed spatial heterogeneity in sea otter impacts and a negative relationship with local shoreline complexity. Sea otter exploitation of productive butter clam habitat substantially reduced the abundance and altered the distribution of butter clams across Glacier Bay, with potential cascading consequences for nearshore community structure and function. Spatial variation in estimated sea otter predation processes further suggests that community context and local environmental conditions mediate the top-down influence of sea otters on a given prey. Overall, our framework provides high-resolution insights about the interaction among components of this food web and could be applied to a variety of other systems involving invasive species, epidemiology or migration.

Artificial Intelligence to Aid Glaucoma Diagnosis and Monitoring: State of the Art and New Directions

Recent developments in the use of artificial intelligence in the diagnosis and monitoring of glaucoma are discussed. To set the context and fix terminology, a brief historic overview of artificial intelligence is provided, along with some fundamentals of statistical modeling. Next, recent applications of artificial intelligence techniques in glaucoma diagnosis and the monitoring of glaucoma progression are reviewed, including the classification of visual field images and the detection of glaucomatous change in retinal nerve fiber layer thickness. Current challenges in the direct application of artificial intelligence to further our understating of this disease are also outlined. The article also discusses how the combined use of mathematical modeling and artificial intelligence may help to address these challenges, along with stronger communication between data scientists and clinicians.

A Mechanistic Model of Annual Sulfate Concentrations in the United States

Understanding how individual pollution sources contribute to ambient sulfate pollution is critical for assessing past and future air quality regulations. Since attribution to specific sources is typically not encoded in spatial air pollution data, we develop a mechanistic model which we use to estimate, with uncertainty, the contribution of ambient sulfate concentrations attributable specifically to sulfur dioxide (SO₂) emissions from individual coal-fired power plants in the central United States. We propose a multivariate Ornstein-Uhlenbeck (OU) process approximation to the dynamics of the underlying space-time chemical transport process, and its distributional properties are leveraged to specify novel probability models for spatial data that are viewed as either a snapshot or time-averaged observation of the OU process. Using US EPA SO₂ emissions data from 193 power plants and state-of-the-art estimates of ground-level annual mean sulfate concentrations, we estimate that in 2011 - a time of active power plant regulatory action - existing flue-gas desulfurization (FGD) technologies at 66 power plants reduced population-weighted exposure to ambient sulfate by 1.97 μg/m³ (95% CI: 1.80 - 2.15). Furthermore, we anticipate future regulatory benefits by estimating that installing FGD technologies at the five largest SO₂-emitting facilities would reduce human exposure to ambient sulfate by an additional 0.45 μg/m³ (95% CI: 0.33 - 0.54).

Smoothing spatio-temporal data with complex missing data patterns

We consider spatio-temporal data and functional data with spatial dependence, characterized by complicated missing data patterns. We propose a new method capable to efficiently handle these data structures, including the case where data are missing over large portions of the spatio-temporal domain. The method is based on regression with partial differential equation regularization. The proposed model can accurately deal with data scattered over domains with irregular shapes and can accurately estimate fields exhibiting complicated local features. We demonstrate the consistency and asymptotic normality of the estimators. Moreover, we illustrate the good performances of the method in simulations studies, considering different missing data scenarios, from sparse data to more challenging scenarios where the data are missing over large portions of the spatial and temporal domains and the missing data are clustered in space and/or in time. The proposed method is compared to competing techniques, considering predictive accuracy and uncertainty quantification measures. Finally, we show an application to the analysis of lake surface water temperature data, that further illustrates the ability of the method to handle data featuring complicated patterns of missingness and highlights its potentiality for environmental studies.

Bridging implementation gaps to connect large ecological datasets and complex models

Merging robust statistical methods with complex simulation models is a frontier for improving ecological inference and forecasting. However, bringing these tools together is not always straightforward. Matching data with model output, determining starting conditions, and addressing high dimensionality are some of the complexities that arise when attempting to incorporate ecological field data with mechanistic models directly using sophisticated statistical methods. To illustrate these complexities and pragmatic paths forward, we present an analysis using tree-ring basal area reconstructions in Denali National Park (DNPP) to constrain successional trajectories of two spruce species (Picea mariana and Picea glauca) simulated by a forest gap model, University of Virginia Forest Model Enhanced-UVAFME. Through this process, we provide preliminary ecological inference about the long-term competitive dynamics between slow-growing P. mariana and relatively faster-growing P. glauca. Incorporating tree-ring data into UVAFME allowed us to estimate a bias correction for stand age with improved parameter estimates. We found that higher parameter values for P. mariana minimum growth under stress and P. glauca maximum growth rate were key to improving simulations of coexistence, agreeing with recent research that faster-growing P. glauca may outcompete P. mariana under climate change scenarios. The implementation challenges we highlight are a crucial part of the conversation for how to bring models together with data to improve ecological inference and forecasting.

Recursive Bayesian computation facilitates adaptive optimal design in ecological studies

Optimal design procedures provide a framework to leverage the learning generated by ecological models to flexibly and efficiently deploy future monitoring efforts. At the same time, Bayesian hierarchical models have become widespread in ecology and offer a rich set of tools for ecological learning and inference. However, coupling these methods with an optimal design framework can become computationally intractable. Recursive Bayesian computation offers a way to substantially reduce this computational burden, making optimal design accessible for modern Bayesian ecological models. We demonstrate the application of so-called prior-proposal recursive Bayes to optimal design using a simulated data binary regression and the real-world example of monitoring and modeling sea otters in Glacier Bay, Alaska. These examples highlight the computational gains offered by recursive Bayesian methods and the tighter fusion of monitoring and science that those computational gains enable.

Diffusion modeling reveals effects of multiple release sites and human activity on a recolonizing apex predator

BACKGROUND

Reintroducing predators is a promising conservation tool to help remedy human-caused ecosystem changes. However, the growth and spread of a reintroduced population is a spatiotemporal process that is driven by a suite of factors, such as habitat change, human activity, and prey availability. Sea otters (Enhydra lutris) are apex predators of nearshore marine ecosystems that had declined nearly to extinction across much of their range by the early 20th century. In Southeast Alaska, which is comprised of a diverse matrix of nearshore habitat and managed areas, reintroduction of 413 individuals in the late 1960s initiated the growth and spread of a population that now exceeds 25,000.

METHODS

Periodic aerial surveys in the region provide a time series of spatially-explicit data to investigate factors influencing this successful and ongoing recovery. We integrated an ecological diffusion model that accounted for spatially-variable motility and density-dependent population growth, as well as multiple population epicenters, into a Bayesian hierarchical framework to help understand the factors influencing the success of this recovery.

RESULTS

Our results indicated that sea otters exhibited higher residence time as well as greater equilibrium abundance in Glacier Bay, a protected area, and in areas where there is limited or no commercial fishing. Asymptotic spread rates suggested sea otters colonized Southeast Alaska at rates of 1-8 km/yr with lower rates occurring in areas correlated with higher residence time, which primarily included areas near shore and closed to commercial fishing. Further, we found that the intrinsic growth rate of sea otters may be higher than previous estimates suggested.

CONCLUSIONS

This study shows how predator recolonization can occur from multiple population epicenters. Additionally, our results suggest spatial heterogeneity in the physical environment as well as human activity and management can influence recolonization processes, both in terms of movement (or motility) and density dependence.

Linking mosquito surveillance to dengue fever through Bayesian mechanistic modeling

Our ability to effectively prevent the transmission of the dengue virus through targeted control of its vector, Aedes aegypti, depends critically on our understanding of the link between mosquito abundance and human disease risk. Mosquito and clinical surveillance data are widely collected, but linking them requires a modeling framework that accounts for the complex non-linear mechanisms involved in transmission. Most critical are the bottleneck in transmission imposed by mosquito lifespan relative to the virus' extrinsic incubation period, and the dynamics of human immunity. We developed a differential equation model of dengue transmission and embedded it in a Bayesian hierarchical framework that allowed us to estimate latent time series of mosquito demographic rates from mosquito trap counts and dengue case reports from the city of Vitória, Brazil. We used the fitted model to explore how the timing of a pulse of adult mosquito control influences its effect on the human disease burden in the following year. We found that control was generally more effective when implemented in periods of relatively low mosquito mortality (when mosquito abundance was also generally low). In particular, control implemented in early September (week 34 of the year) produced the largest reduction in predicted human case reports over the following year. This highlights the potential long-term utility of broad, off-peak-season mosquito control in addition to existing, locally targeted within-season efforts. Further, uncertainty in the effectiveness of control interventions was driven largely by posterior variation in the average mosquito mortality rate (closely tied to total mosquito abundance) with lower mosquito mortality generating systems more vulnerable to control. Broadly, these correlations suggest that mosquito control is most effective in situations in which transmission is already limited by mosquito abundance.

Fundamental Flaws with the Fundamental Niche

For more than 70 years, Hutchinson's concept of the fundamental niche has guided ecological research. Hutchinson envisioned the niche as a multidimensional hypervolume relating the fitness of an organism to relevant environmental factors. Here, we challenge the utility of the concept to modern ecologists, based on its inability to account for environmental variation and phenotypic plasticity. We have ample evidence that the frequency, duration, and sequence of abiotic stress influence the survivorship and performance of organisms. Recent work shows that organisms also respond to the spatial configuration of abiotic conditions. Spatiotemporal variation of the environment interacts with the genotype to generate a unique phenotype at each life stage. These dynamics cannot be captured adequately by a multidimensional hypervolume. Therefore, we recommend that ecologists abandon the niche as a tool for predicting the persistence of species and embrace mechanistic models of population growth that incorporate spatiotemporal dynamics.

Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data

Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications.

Quantifying spatio-temporal variation of invasion spread

- The spread of invasive species can have far-reaching environmental and ecological consequences. Understanding invasion spread patterns and the underlying process driving invasions are key to predicting and managing invasions. - We combine a set of statistical methods in a novel way to characterize local spread properties and demonstrate their application using simulated and historical data on invasive insects. Our method uses a Gaussian process fit to the surface of waiting times to invasion in order to characterize the vector field of spread. - Using this method, we estimate with statistical uncertainties the speed and direction of spread at each location. Simulations from a stratified diffusion model verify the accuracy of our method. - We show how we may link local rates of spread to environmental covariates for two case studies: the spread of the gypsy moth ( Lymantria dispar), and hemlock woolly adelgid ( Adelges tsugae) in North America. We provide an R-package that automates the calculations for any spatially referenced waiting time data.

Approximate Bayes learning of stochastic differential equations

We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions, and estimates are calculated directly from dense data sets using Gaussian process regression. We develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation.

Machine Learning Using Hyperspectral Data Inaccurately Predicts Plant Traits Under Spatial Dependency

Spectral, temporal and spatial dimensions are difficult to model together when predicting in situ plant traits from remote sensing data. Therefore, machine learning algorithms solely based on spectral dimensions are often used as predictors, even when there is a strong effect of spatial or temporal autocorrelation in the data. A significant reduction in prediction accuracy is expected when algorithms are trained using a sequence in space or time that is unlikely to be observed again. The ensuing inability to generalise creates a necessity for ground-truth data for every new area or period, provoking the propagation of “single-use” models. This study assesses the impact of spatial autocorrelation on the generalisation of plant trait models predicted with hyperspectral data. Leaf Area Index (LAI) data generated at increasing levels of spatial dependency are used to simulate hyperspectral data using Radiative Transfer Models. Machine learning regressions to predict LAI at different levels of spatial dependency are then tuned (determining the optimum model complexity) using cross-validation as well as the NOIS method. The results show that cross-validated prediction accuracy tends to be overestimated when spatial structures present in the training data are fitted (or learned) by the model.

Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

Monitoring dynamic spatio-temporal ecological processes optimally

Population dynamics vary in space and time. Survey designs that ignore these dynamics may be inefficient and fail to capture essential spatio-temporal variability of a process. Alternatively, dynamic survey designs explicitly incorporate knowledge of ecological processes, the associated uncertainty in those processes, and can be optimized with respect to monitoring objectives. We describe a cohesive framework for monitoring a spreading population that explicitly links animal movement models with survey design and monitoring objectives. We apply the framework to develop an optimal survey design for sea otters in Glacier Bay. Sea otters were first detected in Glacier Bay in 1988 and have since increased in both abundance and distribution; abundance estimates increased from 5 otters to >5,000 otters, and they have spread faster than 2.7 km/yr. By explicitly linking animal movement models and survey design, we are able to reduce uncertainty associated with forecasting occupancy, abundance, and distribution compared to other potential random designs. The framework we describe is general, and we outline steps to applying it to novel systems and taxa.

Evaluation of spatio-temporal Bayesian models for the spread of infectious diseases in oil palm

In the field of epidemiology, studies are often focused on mapping diseases in relation to time and space. Hierarchical modeling is a common flexible and effective tool for modeling problems related to disease spread. In the context of oil palm plantations infected by the fungal pathogen Ganoderma boninense, we propose and compare two spatio-temporal hierarchical Bayesian models addressing the lack of information on propagation modes and transmission vectors. We investigate two alternative process models to study the unobserved mechanism driving the infection process. The models help gain insight into the spatio-temporal dynamic of the infection by identifying a genetic component in the disease spread and by highlighting a spatial component acting at the end of the experiment. In this challenging context, we propose models that provide assumptions on the unobserved mechanism driving the infection process while making short-term predictions using ready-to-use software.

When mechanism matters: Bayesian forecasting using models of ecological diffusion

Ecological diffusion is a theory that can be used to understand and forecast spatio-temporal processes such as dispersal, invasion, and the spread of disease. Hierarchical Bayesian modelling provides a framework to make statistical inference and probabilistic forecasts, using mechanistic ecological models. To illustrate, we show how hierarchical Bayesian models of ecological diffusion can be implemented for large data sets that are distributed densely across space and time. The hierarchical Bayesian approach is used to understand and forecast the growth and geographic spread in the prevalence of chronic wasting disease in white-tailed deer (Odocoileus virginianus). We compare statistical inference and forecasts from our hierarchical Bayesian model to phenomenological regression-based methods that are commonly used to analyse spatial occurrence data. The mechanistic statistical model based on ecological diffusion led to important ecological insights, obviated a commonly ignored type of collinearity, and was the most accurate method for forecasting.

The basis function approach for modeling autocorrelation in ecological data

Analyzing ecological data often requires modeling the autocorrelation created by spatial and temporal processes. Many seemingly disparate statistical methods used to account for autocorrelation can be expressed as regression models that include basis functions. Basis functions also enable ecologists to modify a wide range of existing ecological models in order to account for autocorrelation, which can improve inference and predictive accuracy. Furthermore, understanding the properties of basis functions is essential for evaluating the fit of spatial or time-series models, detecting a hidden form of collinearity, and analyzing large data sets. We present important concepts and properties related to basis functions and illustrate several tools and techniques ecologists can use when modeling autocorrelation in ecological data.

An integrated data model to estimate spatiotemporal occupancy, abundance, and colonization dynamics

Ecological invasions and colonizations occur dynamically through space and time. Estimating the distribution and abundance of colonizing species is critical for efficient management or conservation. We describe a statistical framework for simultaneously estimating spatiotemporal occupancy and abundance dynamics of a colonizing species. Our method accounts for several issues that are common when modeling spatiotemporal ecological data including multiple levels of detection probability, multiple data sources, and computational limitations that occur when making fine-scale inference over a large spatiotemporal domain. We apply the model to estimate the colonization dynamics of sea otters (Enhydra lutris) in Glacier Bay, in southeastern Alaska.

A Four Dimensional Spatio-Temporal Analysis of an Agricultural Dataset

While a variety of statistical models now exist for the spatio-temporal analysis of two-dimensional (surface) data collected over time, there are few published examples of analogous models for the spatial analysis of data taken over four dimensions: latitude, longitude, height or depth, and time. When taking account of the autocorrelation of data within and between dimensions, the notion of closeness often differs for each of the dimensions. Here, we consider a number of approaches to the analysis of such a dataset, which arises from an agricultural experiment exploring the impact of different cropping systems on soil moisture. The proposed models vary in their representation of the spatial correlation in the data, the assumed temporal pattern and choice of conditional autoregressive (CAR) and other priors. In terms of the substantive question, we find that response cropping is generally more effective than long fallow cropping in reducing soil moisture at the depths considered (100 cm to 220 cm). Thus, if we wish to reduce the possibility of deep drainage and increased groundwater salinity, the recommended cropping system is response cropping.