Analyzing animal movements using Brownian bridges

By studying animal movements, researchers can gain insight into many of the ecological characteristics and processes important for understanding population-level dynamics. We developed a Brownian bridge movement model (BBMM) for estimating the expected movement path of an animal, using discrete location data obtained at relatively short time intervals. The BBMM is based on the properties of a conditional random walk between successive pairs of locations, dependent on the time between locations, the distance between locations, and the Brownian motion variance that is related to the animal's mobility. We describe two critical developments that enable widespread use of the BBMM, including a derivation of the model when location data are measured with error and a maximum likelihood approach for estimating the Brownian motion variance. After the BBMM is fitted to location data, an estimate of the animal's probability of occurrence can be generated for an area during the time of observation. To illustrate potential applications, we provide three examples: estimating animal home ranges, estimating animal migration routes, and evaluating the influence of fine-scale resource selection on animal movement patterns.     

Estimating behavioral parameters in animal movement models using a state-augmented particle filter

Data on fine-scale animal movement are being collected worldwide, with the number of species being tagged and the resolution of data rapidly increasing. In this study, a general methodology is proposed to understand the patterns in these high-resolution movement time series that relate to marine animal behavior. The approach is illustrated with dive data from a northern fur seal (Callorhinus ursinus) tagged on the Pribilof Islands, Alaska, USA. We apply a state-space model composed of a movement model and corresponding high-resolution vertical movement data. The central goal is to estimate parameters of this movement model, particularly their variation on appropriate time scales, thereby providing a direct link to behavior. A particle filter with state augmentation is used to jointly estimate the movement parameters and the state. A multiple iterated filter using overlapping data segments is implemented to match the parameter time scale with the behavioral inference. The time variation in the auto-covariance function facilitates identification of a movement model, allows separation of observation and process noise, and provides for validation of results. The analysis yields fitted parameters that show distinct time-evolving changes in fur seal behavior over time, matching well what is observed in the original data set.     

A mechanistic model of GPS collar location data: Implications for analysis and bias mitigation

Global positioning system (GPS) collars have revolutionized the collection of animal location data; however, it is well-recognized that considerable bias can be present in these data due to habitat or behavior-induced obstruction of satellite signals resulting in inaccurate or missing locations. To date, no explicit theoretical framework of GPS fix acquisition specific to animal telemetry has been presented, and studies make differing assumptions regarding factors influencing GPS fix acquisition and how these data should be analyzed. Inappropriate statistical models have been used, interaction effects have been misunderstood, and the implementation of bias mitigation techniques has been problematic. Herein we outline current conceptual and analytical problems in the GPS animal telemetry literature, and subsequently present a theoretical model-based framework for GPS fix acquisition that clarifies the single and interactive effects of habitat and behavioral obstruction, fix interval, and collar model on GPS collar performance. By recognizing that GPS fix acquisition is a Bernoulli process, it becomes apparent that all forms of obstruction inherently interact with each other, making generalizations across study areas, study species, and collar models problematic. Stationary collar tests to determine the probability of fix acquisition (PFA), location accuracy, and the response to sources of obstruction are thus of limited applicability to animal-deployed collars. Bias mitigation techniques that extrapolate PFA models across samples, especially those using stationary collar tests to correct animal-deployed collars, are theoretically unsound. It is also demonstrated that nonlinearities in the relationships between sources of obstruction and PFA complicate PFA modeling with limited data and that even slight model misspecification can lead to considerable errors in correction factors, especially when using inverse weighting to mitigate bias. By emphasizing the importance of GPS collar sensitivity and ephemeris retention, the theoretical framework predicts that newer, more sensitive GPS collars will be less severely biased by sources of obstruction than reported for the older, less sensitive collars that have been used in the majority of GPS performance studies to date and we expect this trend to continue. This heuristic modeling exercise should be of value to researchers planning and analyzing studies using GPS collars and it also establishes a starting point for future theoretical investigations into GPS collar performance and bias mitigation.     

Measurement error causes scale-dependent threshold erosion of biological signals in animal movement data.

Recent advances in telemetry technology have created a wealth of tracking data available for many animal species moving over spatial scales from tens of meters to tens of thousands of kilometers. Increasingly, such data sets are being used for quantitative movement analyses aimed at extracting fundamental biological signals such as optimal searching behavior and scale-dependent foraging decisions. We show here that the location error inherent in various tracking technologies reduces the ability to detect patterns of behavior within movements. Our analyses endeavored to set out a series of initial ground rules for ecologists to help ensure that sampling noise is not misinterpreted as a real biological signal. We simulated animal movement tracks using specialized random walks known as Lévy flights at three spatial scales of investigation: 100-km, 10-km, and 1-km maximum daily step lengths. The locations generated in the simulations were then blurred using known error distributions associated with commonly applied tracking methods: the Global Positioning System (GPS), Argos polar-orbiting satellites, and light-level geolocation. Deviations from the idealized Lévy flight pattern were assessed for each track after incrementing levels of location error were applied at each spatial scale, with additional assessments of the effect of error on scale-dependent movement patterns measured using fractal mean dimension and first-passage time (FPT) analyses. The accuracy of parameter estimation (Lévy mu, fractal mean D, and variance in FPT) declined precipitously at threshold errors relative to each spatial scale. At 100-km maximum daily step lengths, error standard deviations of > or = 10 km seriously eroded the biological patterns evident in the simulated tracks, with analogous thresholds at the 10-km and 1-km scales (error SD > or = 1.3 km and 0.07 km, respectively). Temporal subsampling of the simulated tracks maintained some elements of the biological signals depending on error level and spatial scale. Failure to account for large errors relative to the scale of movement can produce substantial biases in the interpretation of movement patterns. This study provides researchers with a framework for understanding the limitations of their data and identifies how temporal subsampling can help to reduce the influence of spatial error on their conclusions.     

An evaluation of three statistical methods used to model resource selection

The performance of statistical methods for modeling resource selection by animals is difficult to evaluate with field data because true selection patterns are unknown. Simulated data based on a known probability distribution, though, can be used to evaluate statistical methods. Models should estimate true selection patterns if they are to be useful in analyzing and interpreting field data. We used simulation techniques to evaluate the effectiveness of three statistical methods used in modeling resource selection. We generated 25 use locations per animal and included 10, 20, 40, or 80 animals in samples of use locations. To simulate species of different mobility, we generated use locations at four levels according to a known probability distribution across DeSoto National Wildlife Refuge (DNWR) in eastern Nebraska and western Iowa, USA. We either generated 5 random locations per use location or 10,000 random locations (total) within 4 predetermined areas around use locations to determine how the definition of availability and the number of random locations affected results. We analyzed simulated data using discrete choice, logistic-regression, and a maximum entropy method (Maxent). We used a simple linear regression of estimated and known probability distributions and area under receiver operating characteristic curves (AUC) to evaluate the performance of each method. Each statistical method was affected differently by number of animals and random locations used in analyses, level at which selection of resources occurred, and area considered available. Discrete-choice modeling resulted in precise and accurate estimates of the true probability distribution when the area in which use locations were generated was ≥ the area defined to be available. Logistic-regression models were unbiased and precise when the area in which use locations were generated and the area defined to be available were the same size; the fit of these models improved with increased numbers of random locations. Maxent resulted in unbiased and precise estimates of the known probability distribution when the area in which use locations were generated was small (home-range level) and the area defined to be available was large (study area). Based on AUC analyses, all models estimated the selection distribution better than random chance. Results from AUC analyses, however, often contradicted results of the linear regression method used to evaluate model performance. Discrete-choice modeling was best able to estimate the known selection distribution in our study area regardless of sample size or number of random locations used in the analyses, but we recommend further studies using simulated data over different landscapes and different resource metrics to confirm our results. Our study offers an approach and guidance for others interested in assessing the utility of techniques for modeling resource selection in their study area.     

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R². We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).     

Estimating age-specific hazards from wildlife telemetry data

We illustrate 2 techniques for estimating age-specific hazards with wildlife telemetry data: Siler’s (Ecology 60:750–757, 1979) competing risk model fit using maximum likelihood and a penalized likelihood estimate that only assumes the hazard varies smoothly with age. In most telemetry studies, animals enter at different points in time (and at different ages), leading to data that are left-truncated. In addition, death times may only be known to occur within an interval of time (interval-censoring). Observations may also be right-censored (e.g., due to the end of the study, radio-collar failure, or emigration from the study area). It is important to consider the observation process, since the contribution of each individual’s data to the likelihood will depend on whether data are left-truncated or censored. We estimate age-specific hazards using telemetry data collected in two Phases during a 13-year study of white-tailed deer (Odocoileus virginianus) in northern Minnesota. The hazards estimated from the two methods were similar for the full data set that included 302 adults and 76 neonates (followed since or shortly after birth). However, estimated hazards for early-aged individuals differed considerably for subsets of the data that did not include neonates. We discuss the advantages and disadvantages of these two modeling approaches and also compare the estimators using a short simulation study.     

Habitat selection models to account for seasonal persistence in radio telemetry data

Models for the analysis of habitat selection data incorporate covariates in an independent multinomial selections model (McCracken et al. 1998) Ramsey and Usner 2003 and an extension of that model to include a persistence parameter (2003). In both cases, all parameters are assumed to be fixed through time. Radio telemetry data collected for habitat selection studies typically consist of animal relocations through time, suggesting the need for an extension to these models. We use a Bayesian approach that allows for the habitat selection probabilities, persistence parameter, or both, to change with season. These extensions are particularly important when movement patterns are expected to differ seasonally and/or when availabilities of habitats change throughout the study period due to weather or migration. We implement and compare the models using radio telemetry data for westslope cutthroat trout in two streams in eastern Oregon.     

Spatial memory, habitat auto-facilitation and the emergence of fractal home range patterns

Animals interact with their habitat in a manner which involves both negative and positive feedback mechanisms. We apply a specific modeling approach, “multi-scaled random walk”, for the scenario where a spatially explicit positive feedback process emerges from a combination of a spatial memory-dependent tendency to return to familiar patches and a consequently objective or subjective improvement of the quality of these patches (habitat auto-facilitation). In addition to the potential for local resource improvement from physically altering a patch, primarily known from the ecology of grazing ungulates, auto-facilitation from site fidelity may also embed more subtle subjective, individual-specific advantages from patch familiarity. Under the condition of resource superabundance, fitness gain from intra-home range patch fidelity creates a self-reinforcing use of the preferred patches on expense of a broader foraging in a priori equally favorable patches. Through this process, our simulations show that a spatially fractal dispersion of accumulated locations of the individual will emerge under the given model assumptions. Based on a conjecture that intra-home range patch fidelity depends on spatial memory we apply the multi-scaled random walk model to construct a spatially explicit habitat suitability parameter H_ij, which quantifies the dispersion of the generally most constraining resource from the individual's perspective. An intra-home range set of observed H-scores, H_obs, can then be estimated from a simple 2-scale calculation that is derived from the local dispersion of fixes. We show how the spatially explicit habitat utilization index H_obs not necessarily correlates positively with the local density fluctuations of fixes. The H-index solves some well-known problems from using the pattern of local densities of telemetry fixes - the classic utilization distribution - as a proxy variable for relative intra-home range habitat quality and resource selection. A pilot study on a set of telemetry fixes collected from a herd of free-ranging domestic sheep with overlapping summer home ranges illustrates how the H-index may be estimated and interpreted as a first-level approach towards a more extensive analysis of intra-home range habitat resource availability and patch preferences. Spatial memory in combination with site fidelity requires a modeling framework that explicitly describes the property of positive feedback mechanism under auto-facilitation in a spatio-temporally explicit manner.     

An animal movement model incorporating home range and habitat selection

Wildlife biologists are often interested in how an animal uses space and the habitat resources within that space. We propose a single model that estimates an animal’s home range and habitat selection parameters within that range while accounting for the inherent autocorrelation in frequently sampled telemetry data. The model is applied to brown bear telemetry data in southeast Alaska. This article is based on a portion of this author’s Ph.D. dissertation completed in 2003 at the University of Iowa.     

Modeling resource selection using polytomous logistic regression and kernel density estimates

Wildlife resource selection studies typically compare used to available resources; selection or avoidance occurs when use is disproportionately greater or less than availability. Comparing used to available resources is problematic because results are often greatly influenced by what is considered available to the animal. Moreover, placing relocation points within resource units is often difficult due to radiotelemetry and mapping errors. Given these problems, we suggest that an animal’s resource use be summarized at the scale of the home range (i.e., the spatial distribution of all point locations of an animal) rather than by individual points that are considered used or available. To account for differences in use-intensity throughout an animal’s home range, we model resource selection using kernel density estimates and polytomous logistic regression. We present a case study of elk (Cervus elaphus) resource selection in South Dakota to illustrate the procedure. There are several advantages of our proposed approach. First, resource availability goes undefined by the investigator, which is a difficult and often arbitrary decision. Instead, the technique compares the intensity of animal use throughout the home range. This technique also avoids problems with classifying locations rigidly as used or unused. Second, location coordinates do not need to be placed within mapped resource units, which is problematic given mapping and telemetry error. Finally, resource use is considered at an appropriate scale for management because most wildlife resource decisions are made at the level of the patch. Despite the advantages of this use-intensity procedure, future research should address spatial autocorrelation and develop spatial models for ordered categorical variables.     

Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

Mapping risk for nest predation on a barrier island

Barrier islands and coastal beach systems provide nesting habitat for marine and estuarine turtles. Densely settled coastal areas may subsidize nest predators. Our purpose was to inform conservation by providing a greater understanding of habitat-based risk factors for nest predation, for an estuarine turtle. We expected that habitat conditions at predated nests would differ from random locations at two spatial extents. We developed and validated an island-wide model for the distribution of predated Diamondback terrapin nests using locations of 198 predated nests collected during exhaustive searches at Fisherman Island National Wildlife Refuge, USA. We used aerial photographs to identify all areas of possible nesting habitat and searched each and surrounding environments for nests, collecting location and random-point microhabitat data. We built models for the probability of finding a predated nest using an equal number of random points and validated them with a reserve set (N?=?67). Five variables in 9 a priori models were used and the best selected model (AIC weight 0.98) reflected positive associations with sand patches near marshes and roadways. Model validation had an average capture rate of predated nests of 84.14 % (26.17–97.38 %, Q1 77.53 %, median 88.07 %, Q3 95.08 %). Microhabitat selection results suggest that nests placed at the edges of sand patches adjacent to upland shrub/forest and marsh systems are vulnerable to predation. Forests and marshes provide cover and alternative resources for predators and roadways provide access; a suggestion is to focus nest protection efforts on the edges of dunes, near dense vegetation and roads.     

Predicting the variation in detection function in line transect sampling through random parameter model

A method for calibrating (localizing) detection function models in line transect sampling is proposed. The method is based on a random parameter model which supplies localized predictions of detection function parameters utilizing a few sample data points from the concerned location(s). The method has the clear advantage of being able to provide density estimates based on very few observations from a location which would be impossible through traditional methods. The method is successfully illustrated using census data on sambar (Cervus unicolor) from a set of wildlife sanctuaries in Kerala, India. The need for further research in this direction is indicated.     

Kernel-based home range method for data with irregular sampling intervals

Studies of habitat selection and movements often use radio-tracking data for defining animal home ranges. Home ranges (HR) can be approximated by a utilization density distribution (UD) that instead of assuming uniform use of areas within HR boundary provides a probabilistic measure of animal space use. In reality, radio-tracking data contain periods of frequent autocorrelated observations interspersed with temporally more independent observations. Using such temporally irregular data directly may result in biased UD estimates, because areas that have been sampled intensively receive too much weight. The problem of autocorrelation has been tackled by resampling data with an appropriate time interval. However, resampling may cause a large reduction in the data set size along with a loss of information. Evidently, biased UD estimates or reduction in data may prejudice the results on animal habitat selection and movement. We introduce a new method for estimating UDs with temporally irregular data. The proposed method, called the time kernel, accounts for temporal aggregation of observations and gives less weight to temporally autocorrelated observations. A further extension of the method accounts also for spatially aggregated observations with relatively low weights given to observations that are both temporally and spatially aggregated. We test the behaviour of the time kernel method and its spatiotemporal version using simulated data. In addition, the method is applied to a data set of brown bear locations.     

Using hidden Markov chains and empirical Bayes change-point estimation for transect data

Consider a lattice of locations in one dimension at which data are observed. We model the data as a random hierarchical process. The hidden process is assumed to have a (prior) distribution that is derived from a two-state Markov chain. The states correspond to the mean values (high and low) of the observed data. Conditional on the states, the observations are modelled, for example, as independent Gaussian random variables with identical variances. In this model, there are four free parameters: the Gaussian variance, the high and low mean values, and the transition probability in the Markov chain. A parametric empirical Bayes approach requires estimation of these four parameters from the marginal (unconditional) distribution of the data and we use the EM-algorithm to do this. From the posterior of the hidden process, we use simulated annealing to find the maximum a posteriori (MAP) estimate. Using a Gibbs sampler, we also obtain the maximum marginal posterior probability (MMPP) estimate of the hidden process. We use these methods to determine where change-points occur in spatial transects through grassland vegetation, a problem of considerable interest to plant ecologists.     

Integration of distance, direction and habitat into a predictive migratory movement model for blue-winged teal (Anas discors)

Historically, the migration of birds has been poorly understood in comparison to other life stages during the annual cycle. The goal of our research is to present a novel approach to predict the migratory movement of birds. Using a blue-winged teal case study, our process incorporates not only constraints on habitat (temperature, precipitation, elevation, and depth to water table), but also approximates the likely bearing and distance traveled from a starting location. The method allows for movement predictions to be made from unsampled areas across large spatial scales. We used USGS’ Bird Banding Laboratory database as the source of banding and recovery locations. We used recovery locations from banding sites with multiple within-30-day recoveries were used to build core maximum entropy models. Because the core models encompass information regarding likely habitat, distance, and bearing, we used core models to project (or forecast) probability of movement from starting locations that lacked sufficient data for independent predictions. The final model for an unsampled area was based on an inverse-distance weighted averaged prediction from the three nearest core models. To illustrate this approach, three unsampled locations were selected to probabilistically predict where migratory blue-wing teals would stopover. These locations, despite having little or none data, are assumed to have populations. For the blue-winged teal case study, 104 suitable locations were identified to generate core models. These locations ranged from 20 to 228 within-30-day recoveries, and all core models had AUC scores greater than 0.80. We can infer based on model performance assessment, that our novel approach to predicting migratory movement is well-grounded and provides a reasonable approximation of migratory movement.