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High-dimensional variable selection has emerged as one of the prevailing statistical challenges in the big data revolution. Many variable selection methods have been adapted for identifying single nucleotide polymorphisms (SNPs) linked to phenotypic variation in genome-wide association studies. We develop a Bayesian variable selection regression model for identifying SNPs linked to phenotypic variation. We modify our Bayesian variable selection regression models to control the false discovery rate of SNPs using a knockoff variable approach. We reduce spurious associations by regressing the phenotype of interest against a set of basis functions that account for the relatedness of individuals. Using a restricted regression approach, we simultaneously estimate the SNP-level effects while removing variation in the phenotype that can be explained by population structure. We also accommodate the spatial structure among causal SNPs by modeling their inclusion probabilities jointly with a reduced rank Gaussian process. In a simulation study, we demonstrate that our spatial Bayesian variable selection regression model controls the false discovery rate and increases power when the relevant SNPs are clustered. We conclude with an analysis of Arabidopsis thaliana flowering time, a polygenic trait that is confounded with population structure, and find the discoveries of our method cluster near described flowering time genes., Comment: 32 pages and 6 figures

Mechanistic statistical models are commonly used to study the flow of biological processes. For example, in landscape genetics, the aim is to infer spatial mechanisms that govern gene flow in populations. Existing statistical approaches in landscape genetics do not account for temporal dependence in the data and may be computationally prohibitive. We infer mechanisms with a Bayesian hierarchical dyadic model that scales well with large data sets and that accounts for spatial and temporal dependence. We construct a fully-connected network comprising spatio-temporal data for the dyadic model and use normalized composite likelihoods to account for the dependence structure in space and time. We develop a dyadic model to account for physical mechanisms commonly found in physical-statistical models and apply our methods to ancient human DNA data to infer the mechanisms that affected human movement in Bronze Age Europe., Comment: 46 pages, 7 figures, 3 appendices, code on GitHub

Measurement error in multinomial data is a well-known and well-studied inferential problem that is encountered in many fields, including engineering, biomedical and omics research, ecology, finance, and social sciences. Surprisingly, methods developed to accommodate measurement error in multinomial data are typically equipped to handle false negatives or false positives, but not both. We provide a unified framework for accommodating both forms of measurement error using a Bayesian hierarchical approach. We demonstrate the proposed method's performance on simulated data and apply it to acoustic bat monitoring data., Comment: 22 pages, 4 figures

Methods for population estimation and inference have evolved over the past decade to allow for the incorporation of spatial information when using capture-recapture study designs. Traditional approaches to specifying spatial capture-recapture (SCR) models often rely on an individual-based detection function that decays as a detection location is farther from an individual's activity center. Traditional SCR models are intuitive because they incorporate mechanisms of animal space use based on their assumptions about activity centers. We modify the SCR model to accommodate a wide range of space use patterns, including for those individuals that may exhibit traditional elliptical utilization distributions. Our approach uses underlying Gaussian processes to characterize the space use of individuals. This allows us to account for multimodal and other complex space use patterns that may arise due to movement. We refer to this class of models as geostatistical capture-recapture (GCR) models. We adapt a recursive computing strategy to fit GCR models to data in stages, some of which can be parallelized. This technique facilitates implementation and leverages modern multicore and distributed computing environments. We demonstrate the application of GCR models by analyzing both simulated data and a data set involving capture histories of snowshoe hares in central Colorado, USA.

To study population dynamics, ecologists and wildlife biologists use relative abundance data, which are often subject to temporal preferential sampling. Temporal preferential sampling occurs when sampling effort varies across time. To account for preferential sampling, we specify a Bayesian hierarchical abundance model that considers the dependence between observation times and the ecological process of interest. The proposed model improves abundance estimates during periods of infrequent observation and accounts for temporal preferential sampling in discrete time. Additionally, our model facilitates posterior inference for population growth rates and mechanistic phenometrics. We apply our model to analyze both simulated data and mosquito count data collected by the National Ecological Observatory Network. In the second case study, we characterize the population growth rate and abundance of several mosquito species in the Aedes genus., Comment: 29 pages, 5 figures, 1 table

The Alaskan landscape has undergone substantial changes in recent decades, most notably the expansion of shrubs and trees across the Arctic. We developed a dynamic statistical model to quantify the impact of climate change on the structural transformation of ecosystems using remotely sensed imagery. We used latent trajectory processes in a hierarchical framework to model dynamic state probabilities that evolve annually, from which we derived transition probabilities between ecotypes. Our latent trajectory model accommodates temporal irregularity in survey intervals and uses spatio-temporally heterogeneous climate drivers to infer rates of land cover transitions. We characterized multi-scale spatial correlation induced by plot and subplot arrangement in our study system. We also developed a Polya-Gamma sampling strategy to improve computation. Our model facilitates inference on the response of ecosystems to shifts in the climate and can be used to predict future land cover transitions under various climate scenarios.

Ecologists increasingly rely on Bayesian methods to fit capture-recapture models. Capture-recapture models are used to estimate abundance while accounting for imperfect detectability in individual-level data. A variety of implementations exist for such models, including integrated likelihood, parameter-expanded data augmentation, and combinations of those. Capture-recapture models with latent random effects can be computationally intensive to fit using conventional Bayesian algorithms. We identify alternative specifications of capture-recapture models by considering a conditional representation of the model structure. The resulting alternative model can be specified in a way that leads to more stable computation and allows us to fit the desired model in stages while leveraging parallel computing resources. Our model specification includes a component for the capture history of detected individuals and another component for the sample size which is random before observed. We demonstrate this approach using three examples including simulation and two data sets resulting from capture-recapture studies of different species.

In many applications, a signal is deformed by well-understood dynamics before it can be measured. For example, when a pollutant enters a river, it immediately begins dispersing, flowing, settling, and reacting. If the pollutant enters at a single point, its concentration can be measured before it enters the complex dynamics of the river system. However, in the case of a non-point source pollutant, it is not clear how to efficiently measure its source. One possibility is to record concentration measurements in the river, but this signal is masked by the fluid dynamics of the river. Specifically, concentration is governed by the advection-diffusion-reaction PDE, with an unknown source term. We propose a method to statistically reconstruct a source term from these PDE-deformed measurements. Our method is general and applies to any linear PDE. This method has important applications in the study of environmental DNA and non-point source pollution., Comment: 28 pages, 8 figures, 2 tables

We propose a multistage method for making inference at all levels of a Bayesian hierarchical model (BHM) using natural data partitions to increase efficiency by allowing computations to take place in parallel form using software that is most appropriate for each data partition. The full hierarchical model is then approximated by the product of independent normal distributions for the data component of the model. In the second stage, the Bayesian maximum {\it a posteriori} (MAP) estimator is found by maximizing the approximated posterior density with respect to the parameters. If the parameters of the model can be represented as normally distributed random effects then the second stage optimization is equivalent to fitting a multivariate normal linear mixed model. This method can be extended to account for common fixed parameters shared between data partitions, as well as parameters that are distinct between partitions. In the case of distinct parameter estimation, we consider a third stage that re-estimates the distinct parameters for each data partition based on the results of the second stage. This allows more information from the entire data set to properly inform the posterior distributions of the distinct parameters. The method is demonstrated with two ecological data sets and models, a random effects GLM and an Integrated Population Model (IPM). The multistage results were compared to estimates from models fit in single stages to the entire data set. Both examples demonstrate that multistage point and posterior standard deviation estimates closely approximate those obtained from fitting the models with all data simultaneously and can therefore be considered for fitting hierarchical Bayesian models when it is computationally prohibitive to do so in one step., Comment: 32 pages, 4 figures, 1 table

Agent-based methods allow for defining simple rules that generate complex group behaviors. The governing rules of such models are typically set a priori and parameters are tuned from observed behavior trajectories. Instead of making simplifying assumptions across all anticipated scenarios, inverse reinforcement learning provides inference on the short-term (local) rules governing long term behavior policies by using properties of a Markov decision process. We use the computationally efficient linearly-solvable Markov decision process to learn the local rules governing collective movement for a simulation of the self propelled-particle (SPP) model and a data application for a captive guppy population. The estimation of the behavioral decision costs is done in a Bayesian framework with basis function smoothing. We recover the true costs in the SPP simulation and find the guppies value collective movement more than targeted movement toward shelter.

A suite of statistical methods are used to study animal movement. Most of these methods treat animal telemetry data in one of three ways: as discrete processes, as continuous processes, or as point processes. We briefly review each of these approaches and then focus in on the latter. In the context of point processes, so-called resource selection analyses are among the most common way to statistically treat animal telemetry data. However, most resource selection analyses provide inference based on approximations of point process models. The forms of these models have been limited to a few types of specifications that provide inference about relative resource use and, less commonly, probability of use. For more general spatio-temporal point process models, the most common type of analysis often proceeds with a data augmentation approach that is used to create a binary data set that can be analyzed with conditional logistic regression. We show that the conditional logistic regression likelihood can be generalized to accommodate a variety of alternative specifications related to resource selection. We then provide an example of a case where a spatio-temporal point process model coincides with that implied by a mechanistic model for movement expressed as a partial differential equation derived from first principles of movement. We demonstrate that inference from this form of point process model is intuitive (and could be useful for management and conservation) by analyzing a set of telemetry data from a mountain lion in Colorado, USA, to understand the effects of spatially explicit environmental conditions on movement behavior of this species.

Binary regression models are ubiquitous in virtually every scientific field. Frequently, traditional generalized linear models fail to capture the variability in the probability surface that gives rise to the binary observations and novel methodology is required. This has generated a substantial literature comprised of binary regression models motivated by various applications. We describe a novel organization of generalizations to traditional binary regression methods based on the familiar three-part structure of generalized linear models (random component, systematic component, link function). This new perspective facilitates both the comparison of existing approaches, and the development of novel, flexible models with interpretable parameters that capture application-specific data generating mechanisms. We use our proposed organizational structure to discuss some concerns with certain existing models for binary data based on quantile regression. We then use the framework to develop several new binary regression models tailored to occupancy data for European red squirrels (Sciurus vulgaris).

Multivariate compositional count data arise in many applications including ecology, microbiology, genetics, and paleoclimate. A frequent question in the analysis of multivariate compositional count data is what values of a covariate(s) give rise to the observed composition. Learning the relationship between covariates and the compositional count allows for inverse prediction of unobserved covariates given compositional count observations. Gaussian processes provide a flexible framework for modeling functional responses with respect to a covariate without assuming a functional form. Many scientific disciplines use Gaussian process approximations to improve prediction and make inference on latent processes and parameters. When prediction is desired on unobserved covariates given realizations of the response variable, this is called inverse prediction. Because inverse prediction is mathematically and computationally challenging, predicting unobserved covariates often requires fitting models that are different from the hypothesized generative model. We present a novel computational framework that allows for efficient inverse prediction using a Gaussian process approximation to generative models. Our framework enables scientific learning about how the latent processes co-vary with respect to covariates while simultaneously providing predictions of missing covariates. The proposed framework is capable of efficiently exploring the high dimensional, multi-modal latent spaces that arise in the inverse problem. To demonstrate flexibility, we apply our method in a generalized linear model framework to predict latent climate states given multivariate count data. Based on cross-validation, our model has predictive skill competitive with current methods while simultaneously providing formal, statistical inference on the underlying community dynamics of the biological system previously not available., Comment: 20 pages, 5 figures, 2 tables

Bayesian models provide recursive inference naturally because they can formally reconcile new data and existing scientific information. However, popular use of Bayesian methods often avoids priors that are based on exact posterior distributions resulting from former studies. Two existing Recursive Bayesian methods are: Prior- and Proposal-Recursive Bayes. Prior-Recursive Bayes uses Bayesian updating, fitting models to partitions of data sequentially, and provides a way to accommodate new data as they become available using the posterior from the previous stage as the prior in the new stage based on the latest data. Proposal-Recursive Bayes is intended for use with hierarchical Bayesian models and uses a set of transient priors in first stage independent analyses of the data partitions. The second stage of Proposal-Recursive Bayes uses the posteriors from the first stage as proposals in an MCMC algorithm to fit the full model. We combine Prior- and Proposal-Recursive concepts to fit any Bayesian model, and often with computational improvements. We demonstrate our method with two case studies. Our approach has implications for big data, streaming data, and optimal adaptive design situations.

Vital rates such as survival and recruitment have always been important in the study of population and community ecology. At the individual level, physiological processes such as energetics are critical in understanding biomechanics and movement ecology and also scale up to influence food webs and trophic cascades. Although vital rates and population-level characteristics are tied with individual-level animal movement, most statistical models for telemetry data are not equipped to provide inference about these relationships because they lack the explicit, mechanistic connection to physiological dynamics. We present a framework for modeling telemetry data that explicitly includes an aggregated physiological process associated with decision making and movement in heterogeneous environments. Our framework accommodates a wide range of movement and physiological process specifications. We illustrate a specific model formulation in continuous-time to provide direct inference about gains and losses associated with physiological processes based on movement. Our approach can also be extended to accommodate auxiliary data when available. We demonstrate our model to infer mountain lion (in Colorado, USA) and African buffalo (in Kruger National Park, South Africa) recharge dynamics.

The analysis of animal tracking data provides an important source of scientific understanding and discovery in ecology. Observations of animal trajectories using telemetry devices provide researchers with information about the way animals interact with their environment and each other. For many species, specific geographical features in the landscape can have a strong effect on behavior. Such features may correspond to a single point (e.g., dens or kill sites), or to higher-dimensional subspaces (e.g., rivers or lakes). Features may be relatively static in time (e.g., coastlines or home-range centers), or may be dynamic (e.g., sea ice extent or areas of high-quality forage for herbivores). We introduce a novel model for animal movement that incorporates active selection for dynamic features in a landscape. Our approach is motivated by the study of polar bear (Ursus maritimus) movement. During the sea ice melt season, polar bears spend much of their time on sea ice above shallow, biologically productive water where they hunt seals. The changing distribution and characteristics of sea ice throughout the late spring through early fall means that the location of valuable habitat is constantly shifting. We develop a model for the movement of polar bears that accounts for the effect of this important landscape feature. We introduce a two-stage procedure for approximate Bayesian inference that allows us to analyze over 300,000 observed locations of 186 polar bears from 2012--2016. We use our proposed model to answer a particular question posed by wildlife managers who seek to cluster polar bears from the Beaufort and Chukchi seas into sub-populations., Comment: Correction to caption of Figure 4

We clarify relationships between conditional (CAR) and simultaneous (SAR) autoregressive models. We review the literature on this topic and find that it is mostly incomplete. Our main result is that a SAR model can be written as a unique CAR model, and while a CAR model can be written as a SAR model, it is not unique. In fact, we show how any multivariate Gaussian distribution on a finite set of points with a positive-definite covariance matrix can be written as either a CAR or a SAR model. We illustrate how to obtain any number of SAR covariance matrices from a single CAR covariance matrix by using Givens rotation matrices on a simulated example. We also discuss sparseness in the original CAR construction, and for the resulting SAR weights matrix. For a real example, we use crime data in 49 neighborhoods from Columbus, Ohio, and show that a geostatistical model optimizes the likelihood much better than typical first-order CAR models. We then use the implied weights from the geostatistical model to estimate CAR model parameters that provides the best overall optimization., Comment: 18 pages, 4 figures

Animals often exhibit changes in their behavior during migration. Telemetry data provide a way to observe geographic position of animals over time, but not necessarily changes in the dynamics of the movement process. Continuous-time models allow for statistical predictions of the trajectory in the presence of measurement error and during periods when the telemetry device did not record the animal's position. However, continuous-time models capable of mimicking realistic trajectories with sufficient detail are computationally challenging to fit to large data sets and basic models lack realism in their ability to capture nonstationary dynamics. We present a unified class of animal movement models that are computationally efficient and provide a suite of approaches for accommodating nonstationarity in continuous trajectories due to migration and interactions among individuals. We show how to nest convolution models to incorporate interactions among migrating individuals to account for nonstationarity and provide inference about dynamic migratory networks. We demonstrate these approaches in two case studies involving migratory birds. Specifically, we used process convolution models with temporal deformation to account for heterogeneity in individual greater white-fronted goose migrations in Europe and Iceland and we used nested process convolutions to model dynamic migratory networks in sandhill cranes in North America. The approach we present accounts for various forms of temporal heterogeneity in animal movement and is not limited to migratory applications. Furthermore, our models rely on well-established principles for modeling dependent data and leverage modern approaches for modeling dynamic networks to help explain animal movement and social interaction.

Population dynamics varies 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 per year. By explicitly linking animal movement models and survey design, we were able to reduce uncertainty associated with predicted occupancy, abundance, and distribution. The framework we describe is general, and we outline steps to applying it to novel systems and taxa.

The analysis of telemetry data is common in animal ecological studies. While the collection of telemetry data for individual animals has improved dramatically, the methods to properly account for inherent uncertainties (e.g., measurement error, dependence, barriers to movement) have lagged behind. Still, many new statistical approaches have been developed to infer unknown quantities affecting animal movement or predict movement based on telemetry data. Hierarchical statistical models are useful to account for some of the aforementioned uncertainties, as well as provide population-level inference, but they often come with an increased computational burden. For certain types of statistical models, it is straightforward to provide inference if the latent true animal trajectory is known, but challenging otherwise. In these cases, approaches related to multiple imputation have been employed to account for the uncertainty associated with our knowledge of the latent trajectory. Despite the increasing use of imputation approaches for modeling animal movement, the general sensitivity and accuracy of these methods have not been explored in detail. We provide an introduction to animal movement modeling and describe how imputation approaches may be helpful for certain types of models. We also assess the performance of imputation approaches in a simulation study. Our simulation study suggests that inference for model parameters directly related to the location of an individual may be more accurate than inference for parameters associated with higher-order processes such as velocity or acceleration. Finally, we apply these methods to analyze a telemetry data set involving northern fur seals (Callorhinus ursinus) in the Bering Sea.

Gaussian processes are a fundamental statistical tool used in a wide range of applications. In the spatio-temporal setting, several families of covariance functions exist to accommodate a wide variety of dependence structures arising in different applications. These parametric families can be restrictive and are insufficient in some situations. In contrast, process convolutions represent a flexible, interpretable approach to defining the covariance of a Gaussian process and have modest requirements to ensure validity. We introduce a generalization of the process convolution approach that employs multiple convolutions sequentially to form a "process convolution chain." In our proposed multi-stage framework, complex dependencies that arise from a combination of different interacting mechanisms are decomposed into a series of interpretable kernel smoothers. We demonstrate an application of process convolution chains to model killer whale movement, in which the paths taken by multiple individuals are not independent, but reflect dynamic social interactions within the population. Our proposed model for dependent movement provides inference for the latent dynamic social structure in the study population. Additionally, by leveraging the positive dependence among individual paths, we achieve a reduction in uncertainty for the estimated locations of the whales, compared to a model that treats paths as independent.

Lake sediment charcoal records are used in paleoecological analyses to reconstruct fire history including the identification of past wildland fires. One challenge of applying sediment charcoal records to infer fire history is the separation of charcoal associated with local fire occurrence and charcoal originating from regional fire activity. Despite a variety of methods to identify local fires from sediment charcoal records, an integrated statistical framework for fire reconstruction is lacking. We develop a Bayesian point process model to estimate probability of fire associated with charcoal counts from individual-lake sediments and estimate mean fire return intervals. A multivariate extension of the model combines records from multiple lakes to reduce uncertainty in local fire identification and estimate a regional mean fire return interval. The univariate and multivariate models are applied to 13 lakes in the Yukon Flats region of Alaska. Both models resulted in similar mean fire return intervals (100-350 years) with reduced uncertainty under the multivariate model due to improved estimation of regional charcoal deposition. The point process model offers an integrated statistical framework for paleo-fire reconstruction and extends existing methods to infer regional fire history from multiple lake records with uncertainty following directly from posterior distributions., Comment: 26 pages, 6 figures, 1 table

New methods for modeling animal movement based on telemetry data are developed regularly. With advances in telemetry capabilities, animal movement models are becoming increasingly sophisticated. Despite a need for population-level inference, animal movement models are still predominantly developed for individual-level inference. Most efforts to upscale the inference to the population-level are either post hoc or complicated enough that only the developer can implement the model. Hierarchical Bayesian models provide an ideal platform for the development of population-level animal movement models but can be challenging to fit due to computational limitations or extensive tuning required. We propose a two-stage procedure for fitting hierarchical animal movement models to telemetry data. The two-stage approach is statistically rigorous and allows one to fit individual-level movement models separately, then resample them using a secondary MCMC algorithm. The primary advantages of the two-stage approach are that the first stage is easily parallelizable and the second stage is completely unsupervised, allowing for a completely automated fitting procedure in many cases. We demonstrate the two-stage procedure with two applications of animal movement models. The first application involves a spatial point process approach to modeling telemetry data and the second involves a more complicated continuous-time discrete-space animal movement model. We fit these models to simulated data and real telemetry data arising from a population of monitored Canada lynx in Colorado, USA.

Analyzing ecological data often requires modeling the autocorrelation created by spatial and temporal processes. Many of the statistical methods used to account for autocorrelation can be viewed as regression models that include basis functions. Understanding the concept of basis functions enables ecologists to modify commonly used ecological models to account for autocorrelation, which can improve inference and predictive accuracy. Understanding the properties of basis functions is essential for evaluating the fit of spatial or time-series models, detecting a hidden form of multicollinearity, 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.

Advances in satellite-based data collection techniques have served as a catalyst for new statistical methodology to analyze these data. In wildlife ecological studies, satellite-based data and methodology have provided a wealth of information about animal space use and the investigation of individual-based animal-environment relationships. With the technology for data collection improving dramatically over time, we are left with massive archives of historical animal telemetry data of varying quality. While many contemporary statistical approaches for inferring movement behavior are specified in discrete time, we develop a flexible continuous-time stochastic integral equation framework that is amenable to reduced-rank second-order covariance parameterizations. We demonstrate how the associated first-order basis functions can be constructed to mimic behavioral characteristics in realistic trajectory processes using telemetry data from mule deer and mountain lion individuals in western North America. Our approach is parallelizable and provides inference for heterogeneous trajectories using nonstationary spatial modeling techniques that are feasible for large telemetry data sets.

Network modeling techniques provide a means for quantifying social structure in populations of individuals. Data used to define social connectivity are often expensive to collect and based on case-specific, ad hoc criteria. Moreover, in applications involving animal social networks, collection of these data is often opportunistic and can be invasive. Frequently, the social network of interest for a given population is closely related to the way individuals move. Thus telemetry data, which are minimally-invasive and relatively inexpensive to collect, present an alternative source of information. We develop a framework for using telemetry data to infer social relationships among animals. To achieve this, we propose a Bayesian hierarchical model with an underlying dynamic social network controlling movement of individuals via two mechanisms: an attractive effect, and an aligning effect. We demonstrate the model and its ability to accurately identify complex social behavior in simulation, and apply our model to telemetry data arising from killer whales. Using auxiliary information about the study population, we investigate model validity and find the inferred dynamic social network is consistent with killer whale ecology and expert knowledge.