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Bayesian Methods for Estimating Animal Abundance at Large Spatial Scales Using Data From Multiple Sources
Abstract
Estimating animal distributions and abundances over large regions is of primary
interest in ecology and conservation. Specifically, integrating data from reliable but
expensive surveys conducted at smaller scales with cost-effective but less reliable
data generated from surveys at wider scales, remains a central challenge in
statistical ecology. In this study, we use a Bayesian smoothing technique based on a
conditionally autoregressive (CAR) prior distribution and Bayesian regression to address
this problem. We illustrate the utility of our proposed methodology by integrating
(i) abundance estimates of tigers in wildlife reserves from intensive photographic
capture recapture methods, with (ii) estimates of tiger habitat occupancy from in-
direct sign surveys, conducted over a wider region. We also investigate whether the
random effects which represent the spatial association due to the CAR structure
have any confounding effect on the fixed effects of the regression coefficients.
... MTP scientists collaboratively developed 'abundance-occupancy models' (Royle and Nichols, 2003), to integrate camera trap-based density estimates with landscape level signencounter data to estimate overall tiger abundance. Dey et al. (2017) employed Bayesian regression and spatial smoothing under a Conditional Autoregressive (CAR) model structure, which linked rigorous tiger densities estimated from camera traps to the sign survey data from across the landscape. This study resulted in a mean abundance estimate of 391 tigers in the landscape, encompassing four distinct population clusters. ...
... An intensive study of ungulate density variations across the Nagarahole-Bandipur reserve complex, using spatial distance sampling models, also showed law enforcement intensity and regulation of human disturbances to be key determinants of ungulate densities (Kumar et al., 2021). Similar conclusions emerged from occupancy sampling studies of tigers as well as of key prey species using sign-surveys across Malenad Dey et al., 2017). A country-wide assessment ) using expert-based questionnaire surveys, highlighted the proximity of wildlife reserves as an important factor in the persistence of large mammal species at regional scales. ...
... Estimates of land and forest cover, tiger occupied habitats, densities and numbers in the population clusters of Anshi-Dandeli, Bhadra-Kudremukh, Nagarahole-Bandipur and BRT-Cauvery of Malenad landscape, India (Fig. 1) based on Karanth et al. (2011) and Dey et al. (2017). We also present our estimates of potential densities and carrying capacities under optimal management, computed as shown in the foot-note. ...
Tigers are in serious decline from anthropogenic pressures: prey depletion by human hunting, killing of tigers for conflict mitigation or trade in their body parts, and habitat loss or degradation. In spite of conservation efforts over 50 years, wild tigers now occupy <7% of their historic range. Reproducing tiger populations survive in <1% of the ~1.6 million km² potential habitat. Evaluating both successful and failed conservation efforts is critical to reversing tiger declines. Wild tiger populations are managed by governments under constraints imposed by other stake-holders. We review recoveries of tigers across a ~38,000 km² area landscape matrix in Malenad region in southwestern India, during the past five decades. We examine data and empirical observations on tiger ecology, human impacts, emergent conflicts, as well as conservation interventions made at macro-ecological scales by the non-governmental project titled Malenad Tiger Program. We estimate that between 1970 and 2015, tiger habitat occupancy remained unchanged at 14,000 km², out of ~21,000 km² of potential habitat in Malenad. However, tiger numbers rose from ~70 to ~391, only because of sporadic recoveries in a few wildlife reserves. We conclude that if tiger recovery efforts can be optimized in future, the Malenad landscape can potentially support ~1300 wild tigers. We propose pragmatic strategies that may improve success rates and cost-effectiveness of future recovery efforts directed at tigers, and, generally at other threatened large carnivores. We evaluate both challenges and opportunities that non-governmental conservation programs must address to be effective in assisting tiger recoveries in the future.
... Recent studies have reported the potential impacts of large infrastructures, particularly road development, on persistence of large mammals across the landscapes such as tigers (Carter et al., 2020) and elephants (Wadey et al., 2018). However, few studies quantified the potential impact of roads on ecological parameters such as photographic capture rates (Gubbi et al., 2012), occupancy (Chen and Koprowski, 2015), density and abundance (Fahrig and Rytwinski, 2009), and survival (Basille et al., 2013). The Government of Nepal is planning to invest US $4.7 billion in large infrastructure projects by 2030 (GoN, 2019). ...
... Yet, it is still a challenge logistically and financially to survey the entire landscape at a fine-scale to estimate leopard density. Also, to continue multi-season occupancy modeling to track distributional changes over time, we suggest that future research utilize recent advances (Dey et al., 2017;Linden et al., 2017) integrating data from multiple sources at multiple scales, such as camera trap data from core areas (Kumar et al., 2019) and landscape-wide occupancy data, as a viable way forward for estimating landscape-scale leopard density. ...
Better conservation planning requires updated information about leopard distribution to prioritize and allocate limited resources available. The long-term persistence of leopards and sympatric tigers can be compromised by linear infrastructure development such as roads that fragment habitat. We used detection and non-detection data collected along walking search paths (∼4,140 km) in 96 grid cells (each cell 15 km by 15 km) spread across potential habitat (∼13,845 km²) in the Terai Arc Landscape, Nepal. Multi-season occupancy models allowed us to make both spatial and temporal inferences between two surveys in 2009 and 2013, based on ecologically relevant covariates recorded in the field or remotely sensed. Additionally, we used 2013 data to make inferences on co-occurrence between tigers and leopards at the landscape level. We found the additive model containing deforestation and district roads negatively influenced leopard detection across the landscape. Although weak, we found anthropogenic factors such as extent of deforestation (decrease in forest cover) negatively affected leopard occupancy. Road abundance, especially for the east-west highway and district roads, also negatively (but weakly) influenced leopard occupancy. We found substantially lower occupancy in the year 2013 (0.59 (SE 0.06) than in 2009 (0.86 (SE 0.04)). Tigers and leopards co-occurred across the landscape based on the species interaction factor (SIF) estimated at 1.47 (0.13) but the amount of available habitat and the prey index mediated co-occurrence. The SIF decreased as habitat availability increased, reaching independence at large habitat patches, but leopard occupancy declined in sites with tigers, primarily in large patches. The prey index was substantially lower outside of protected areas and leopards and tigers co-occurred more strongly in small patches and at low prey indices, indicating potential attraction to the same areas when prey is scarce. Mitigation measures should focus on preventing loss of critical leopard, tiger, and prey habitat through appropriate wildlife-friendly underpasses and avoiding such habitat when building infrastructure. Leopard conservation has received lower priority than tigers, but our metrics show a large decline in leopard occupancy, thus conservation planning to reverse this decline should focus on measures to facilitate human-leopard coexistence to ensure leopard persistence across the landscape.
... Sign can also be used in an occupancy framework, where spatial replicates are used to estimate detection and has been achieved with several species of Asian bears (Das et al., 2014;Babu et al., 2015;Srivathsa et al., 2018;Bisi et al., 2019;Sharief et al., 2020). Such data may be employed in a Bayesian model that employs various measures of population status at different scales (e.g., Dey et al., 2017). Bear biologists in Asia also have relied on sign density as an indicator of differential habitat use (Ngoprasert et al., 2011), sometimes extending this to bear density (Stander, 1998). ...
Efficient and effective monitoring methods are required to assess population status and gauge efficacy of conservation actions for threatened species. Here we review the spectrum of field methods useful for monitoring distribution, occupancy, abundance, and population trend for the five species of Asian terrestrial bears. Methods reviewed include expert opinion, local knowledge, bear sign, visual observations, camera traps, DNA-based methods (hair and scat derived), and radio telemetry. We examine the application of each method in terms of realizing specific monitoring objectives, their assumptions, challenges, and advantages. Our goal is to assist researchers in matching appropriate field methods with sought-after project objectives and to highlight shortfalls and trade-offs. Methods vary greatly in terms of cost, logistics, required number and expertize of staff, and the reliability of the data they provide. Many Asian bear population assessments have relied on expert opinion, local interviews, and sign surveys to provide estimates of distribution, abundance, and trend, in part because these are inexpensive and relatively easy to employ. However, increasing use of camera traps and DNA-based methods now allow for better monitoring via occupancy or rigorous capture–recapture population estimation, with the caveat that these methods may be restricted by inadequate budgets or logistical constraints. For distribution monitoring, camera traps and DNA yield the most definitive records of presence, but in low density bear populations, sign and local knowledge may be more effective. For occupancy, camera traps and DNA are advantageous in providing definitive detections in known time periods. For abundance/density or population trend monitoring in relatively small areas (
... Conversely, boats may be able to enter areas inaccessible by plane (e.g., narrow fjords), but are unlikely to cover the same spatial extent as planes or to reach offshore areas. Bayesian hierarchical models can leverage data from both platforms to compensate for each platform's deficiencies (e.g., Dey et al. 2017, Wilson et al. 2017. ...
Wildlife managers often rely on analyses conducted prior to the widespread adoption of hierarchical models which can lead to questions about the accuracy of previous inferences. Hierarchical models allow observed data to be partitioned into factors that influenced the collection of the data such as detectability of animals (i.e., observation processes) and factors that influence the ecology of a population such as features that affect the distribution of animals (i.e., ecological processes). Population surveys for sea otters in the Aleutian Islands, Alaska, have historically been conducted by conflating the observation and ecological processes potentially leading to inaccurate population estimates. Based on boat and plane‐based sea otter survey data collected in 2017, we sought to overcome many problems of previous sea otter surveys in southwestern Alaska. We developed a spatially explicit hierarchical distance sampling model to estimate the abundance of sea otters in the Eastern Aleutians Management Unit while explicitly accounting for factors that affect the ability to detect sea otters during surveys (i.e., group size, ocean conditions). We also sought to account for the environmental factors leading to the non‐uniform distribution of sea otter groups by identifying relationships between otter group abundance and environmental attributes (i.e., ocean depth, presence of kelp, underwater substrate). Detection of sea otter groups was related to group size, and ocean conditions (e.g., ocean swell size). After accounting for detection, we estimated a mean population size of 8593 individual sea otters (95% CI: 7450–9984) which is considerably higher than previous estimates, although comparisons are difficult given divergent methodologies. Sea otter group density was negatively related to ocean depth and the presence of rock and gravel as underwater substrates. Conversely, sea otter group density was positively related to the presence of kelp and mud as an underwater substrate. Our hierarchical distance sampling model accounted for the observation process which allowed better estimates of the environmental attributes affecting otter abundance. Our research can serve as a template for other study systems requiring spatially explicit density estimates from a distance sampling framework and can help provide a new baseline for managers to gauge future population changes in sea otters.
... Geostatistical methods have been developed for modelling multiple indices of animal abundance for a single species. However, these methods were found to either use one index as a predictor for another index [49], thus violating C1 and C2, or to use multivariate kriging for all indices [50], violating C1 and C6. ...
A key requirement in studies of endemic vector-borne or zoonotic disease is an estimate of the spatial variation in vector or reservoir host abundance. For many vector species, multiple indices of abundance are available, but current approaches to choosing between or combining these indices do not fully exploit the potential inferential benefits that might accrue from modelling their joint spatial distribution. Here, we develop a class of multivariate generalized linear geostatistical models for multiple indices of abundance. We illustrate this novel methodology with a case study on Norway rats in a low-income urban Brazilian community, where rat abundance is a likely risk factor for human leptospirosis. We combine three indices of rat abundance to draw predictive inferences on a spatially continuous latent process, rattiness, that acts as a proxy for abundance. We show how to explore the association between rattiness and spatially varying environmental factors, evaluate the relative importance of each of the three contributing indices and assess the presence of residual, unexplained spatial variation, and identify rattiness hotspots. The proposed methodology is applicable more generally as a tool for understanding the role of vector or reservoir host abundance in predicting spatial variation in the risk of human disease.
... Geostatistical methods have been developed for modelling multiple indices of animal abundance for a single species. However, these methods were found to either use one index as a predictor for another index [49], thus violating C1 and C2, or to use multivariate kriging for all indices [50], violating C1 and C6. ...
A key requirement in studies of endemic vector-borne or zoonotic disease is an estimate of the spatial variation in vector or reservoir host abundance. For many vector species, multiple indices of abundance are available, but current approaches to choosing between or combining these indices do not fully exploit the potential inferential benefits that might accrue from modelling their joint spatial distribution. Here, we develop a class of multivariate generalized linear geostatistical models for multiple indices of abundance. We illustrate this novel methodology with a case study on Nor- way rats in a low-income urban Brazilian community, where rat abundance is a likely risk-factor for human leptospirosis. We combine three indices of rat abundance to draw predictive inferences on a spatially continuous latent process, rattiness, that acts as a proxy for abundance. We show how to explore the association between rattiness and spatially varying environmental factors, evaluate the relative importance of each of the three contributing indices, assess the presence of residual, unexplained spatial variation, and identify rattiness hotspots. The proposed methodology is applicable more generally as a tool for understanding the role of vector or reservoir host abundance in predicting spatial variation in the risk of human disease.
... A recent statistical study (Dey et al., 2017) combined information from occupancy surveys and camera trap surveys at multiple scales by exploiting the occupancyabundance relationship (Royle & Nichols, 2003) and accounted for spatial random effects by utilizing a Bayesian conditionally-autoregressive (CAR) prior in the analysis. This approach offers promise to estimate abundance at regional scales with reduced SOD and parameter covariances. ...
Agencies responsible for recovering populations of iconic mammals may exaggerate population trends without adequate scientific evidence. Recently, such populations were termed as “political populations” in the conservation literature. We surmise such cases are manifested when agencies are pressured to estimate population parameters at large spatial scales for elusive species. For example, India's tiger conservation agencies depend on an extrapolation method using index‐calibration models for estimating population size. A recent study demonstrated mathematically the unreliability of this approach in practical situations. However, it continues to be applied by official agencies in Asia and promoted further by global organizations working on tiger conservation. In this article, we aim to: (a) discuss the ecological oddities in the results of India's national tiger surveys, (b) contrast these survey approaches to known statistical approaches for large scale wildlife abundance estimation, (c) demystify the mathematics underlying the problems with the survey methodology, and (d) substantiate these arguments with results from India's national tiger survey of 2014. Our analyses show that the predictions of tiger abundance reported by the 2014 survey, and consequently on tiger population trends, are misleading because of the presence of high sampling‐based overdispersion and parameter covariance due to unexplained heterogeneity in detection probabilities. We plead for designing monitoring programs to answer clearly defined scientific or management questions rather than attempt to meet extraneous social or funding related expectations.
... Generally, this happens when initial study areas are centred on regions of known species occurrence and then expanded to include low density areas. Thus, despite several methodological advances, estimates of large carnivore densities at meaningfully large spatial scales are rare and can face methodological inadequacies (e.g., Dey et al., 2017). ...
Accurate assessments of the status of threatened species and their conservation planning require reliable estimation of their global populations and robust monitoring of local population trends. We assessed the adequacy and suitability of studies in reliably estimating the global snow leopard (Panthera uncia) population. We compiled a dataset of all the peer‐reviewed published literature on snow leopard population estimation. Metadata analysis showed estimates of snow leopard density to be a negative exponential function of area, suggesting that study areas have generally been too small for accurate density estimation, and sampling has often been biased towards the best habitats. Published studies are restricted to six of the 12 range countries, covering only 0.3–0.9% of the presumed global range of the species. Re‐sampling of camera trap data from a relatively large study site (c.1684 km²) showed that small‐sized study areas together with a bias towards good quality habitats in existing studies may have overestimated densities by up to five times. We conclude that current information is biased and inadequate for generating a reliable global population estimate of snow leopards. To develop a rigorous and useful baseline and to avoid pitfalls, there is an urgent need for (a) refinement of sampling and analytical protocols for population estimation of snow leopards (b) agreement and coordinated use of standardized sampling protocols amongst researchers and governments across the range, and (c) sampling larger and under‐represented areas of the snow leopard's global range.
... Results from this study yielded a composite map of the probable distribution of wolverines in northern Ontario that likely reflects the relative abundance patterns of wolverines across the study area (i.e. | 7 areas of higher abundance had a greater probability of detection) (Magoun et al., 2007;Dey et al., 2017). The HBOM framework allowed us to use survey data in a post hoc manner rather than design a broad-scale survey a priori-a reality in vast, remote areas where logistics bring many constraints and resources for inventory and monitoring are unpredictable. ...
Aim
We used data from aerial surveys of wolverine tracks collected in seven winters over a 10‐year period (2003–2012) within a 574,287 km² study area to evaluate the broad‐scale pattern of wolverine occurrence across a remote northern boreal forest region, identifying areas of high and low occupancy.
Location
Northern Ontario, Canada.
Taxon
Wolverine (Gulo gulo Linnaeus, 1758).
Methods
We collected wolverine tracks and observations in 100‐km² hexagonal survey units, making a total of 6,664 visits to 3,039 units, visiting each 1–9 times. We used hierarchical Bayesian occupancy modelling to model wolverine occurrence, and included covariates with the potential to affect detection and/or occupancy probability of wolverines.
Results
we detected wolverines on 946 visits, 14.2% of total visits. Probability of detecting a wolverine varied among years and between the two ecozones in the study area. Wolverine occupancy was negatively related to two important covariates, the geographical coordinate Easting and thawing degree‐days. A site occupancy probability map indicated that wolverine occupancy probabilities were highest, and standard error lowest, in the western and northern portions of the study area.
Main conclusions
The occupancy framework enabled us to use observation data from tracks of this elusive, wide‐ranging carnivore over a vast, remote area while explicitly considering detectability and spatial autocorrelation, yielding a map of probable wolverine distribution in northern Ontario that would not be possible using other methods of detection across a large region. With resource development pressures increasing in this globally significant region in the face of a changing climate, it is important to monitor changes in distribution of species like wolverines that have low population growth rates, large spatial requirements and sensitivity to human disturbance. This study demonstrates a relatively cost‐effective and non‐invasive alternative to monitoring based on wolverine harvest records, which have not been available since 2009 in Ontario due to changes in the provincial regulatory regime for this threatened species.
... To model the number and spatial distribution of two or more interacting species, we require a multivariate, spatial point process -specifically, a Markov point process formulated to specify pairwise interactions among individuals of each species (Cressie 1993;Högmander and Särkkä 1999;Diggle 2014). In this process the activity center of an individual is determined, at least in part, by its position relative to the activity centers of all other individuals. ...
Conservationists and managers are continually under pressure from the public, the media, and political policy makers to provide “tiger numbers,” not just for protected reserves, but also for large spatial scales, including landscapes, regions, states, nations, and even globally. Estimating the abundance of tigers within relatively small areas (e.g., protected reserves) is becoming increasingly tractable (see Chaps. 9 and 10), but doing so for larger spatial scales still presents a formidable challenge. Those who seek “tiger numbers” are often not satisfied by estimates of tiger occupancy alone, regardless of the reliability of the estimates (see Chaps. 4 and 5). As a result, wherever tiger conservation efforts are underway, either substantially or nominally, scientists and managers are frequently asked to provide putative large-scale tiger numbers based either on a total count or on an extrapolation of some sort (see Chaps. 1 and 2).
... Additionally, occupancy methods and sign surveys can be used to identify potential habitat corridors, landscape connectivity, dispersal routes, and threats faced by tigers (see Chap. 13). Furthermore, sign survey-based occupancy modeling has the potential to link occupancy data with more intensively measured abundance estimates at local scales (Royle and Nichols 2003;Conroy et al. 2008, Dey et al. 2017 to draw inference about abundance at large spatial scales. When such surveys are conducted periodically (e.g., each year) as part of a long term monitoring program, then resulting data can be used to assess the influence of landscape characteristics on metapopulation vital rates (local extinction and colonization rates) and changes in occupancy. ...
Conservation and management planning of tigers and prey species requires basic information on the spatial distribution at regional and landscape levels, at an appropriate scale (Karanth and Nichols 2000).
... Without across-model calibration, it is impossible to disentangle true changes in abundance from those that are an artifact of changes in methods. The application of more recent methods to extrapolate tiger density using occupancy data needs to be explored (Dey et al. 2017). ...
Reports claim a dramatic 22% increase in wild-tiger Panthera tigris abundance within five years (3200 to 3890 individuals; WWF 2016). Such significant population increases could potentially change the status of tigers from endangered to vulnerable on the IUCN Red List, and substantially contribute to the global target of doubling wild-tiger numbers by 2022 (GTRP 2010). While this purported increase has been attributed to improved conservation practices in India, Nepal, Bhutan and Russia, the claimed increase is questionable given unreported methodology (Russia), lack of comparable baselines (Bhutan and Russia), and failure to adjust population estimates to account for expanded survey effort and methodological changes (Nepal and India). The latter source of bias requires explanation as it accounts for a large portion of the assumed population increase. This article is protected by copyright. All rights reserved
Models for occupancy, where it is important to determine whether a particular species is present or not, have a long history. The literature is very extensive, as referred to in some reviews mentioned, and both frequentist and Bayesian methods have been developed for all kinds of population, including those of animal signs. The methods are particularly important in dealing with the incomplete detection of rare species or species with a low detection rate and with detecting the presence of an invasive species.
In addition to the basic models used in the past based on single surveys, there have been several extensions to include multiple surveys, multiple seasons, multiple species, and multistate models. The use of so-called robust models with primary and secondary surveys is discussed. The aim is to either overcome or allow for violations of traditional model assumptions such as nonclosure, non-independent replicates, heterogeneous detection, and false positives. Key assumptions are listed, and appropriate tests for them are described in two places using goodness-of-fit tests and residual plots for covariate regressions, as well as tests for occupancy differences.
As well as estimating detection probabilities, abundance estimates can be made with some models. Design questions are considered in several places, such as determining the number of visits to a site and dealing with possible dependent visits. Covariates have also been used more frequently, and these help to reduce the number of unknown parameters.
A general collection of various models is presented including an introduction to spatial models with spatial autocorrelation, time-to-detection removal models, and staggered arrival and departure times. For example, adaptive-type methods using a two-phase design or a conditional design are given for rare species. Multi-scale models are considered as species distributions depend critically on the scales at which data is used for model building.
Types of observation errors are discussed as well as combining types of detection using multiple detection devices. DNA and eDNA (environmental DNA) as genetic tags are being used increasingly. Sometimes, there is incomplete data information such as having presence-only data, but extra environmental information can often be incorporated. In addition to using specialist observers, it can be cost-effective to use “opportunistic” data such as data from volunteers, called “citizen” science.
Occupancy models are extended to allow for the possibility that individuals are being detected in one of the several states (multistate models), including disease modeling. Undetected states are considered along with the use of dynamic state models. It is shown how multiple data sources can be combined. The chapter ends with a brief discussion on fisheries and marine environments.
1. The challenges associated with monitoring low-density carnivores across large landscapes have limited the ability to implement and evaluate conservation and management strategies for such species. Noninvasive sampling techniques and advanced statistical approaches have alleviated some of these challenges and can even allow for spatially explicit estimates of density, arguably the most valuable wildlife monitoring tool. 2. For some species, individual identification comes at no cost when unique attributes (e.g., pelage patterns) can be discerned with remote cameras, while other species require viable genetic material and expensive lab processing for individual assignment. Prohibitive costs may still force monitoring efforts to use species distribution or occupancy as a surrogate for density, which may not be appropriate under many conditions. 3. Here, we used a large-scale monitoring study of fisher Pekania pennanti to evaluate the effectiveness of occupancy as an approximation to density, particularly for informing harvest management decisions. We used a combination of remote cameras and baited hair snares during 2013-2015 to sample across a 70,096 km2 region of western New York, USA. We fit occupancy and Royle-Nichols models to species detection-nondetection data collected by cameras, and spatial capture-recapture models to individual encounter data obtained by genotyped hair samples. 4. We found a close relationship between grid-cell estimates of fisher state variables from the models using detection-nondetection data and those from the SCR model, likely due to informative spatial covariates across a large landscape extent and a grid cell resolution that worked well with the movement ecology of the species. Spatially-explicit management recommendations for fisher were similar across models. We discuss design-based approaches to occupancy studies that can improve approximations to density.
The challenges associated with monitoring low‐density carnivores across large landscapes have limited the ability to implement and evaluate conservation and management strategies for such species. Non‐invasive sampling techniques and advanced statistical approaches have alleviated some of these challenges and can even allow for spatially explicit estimates of density, one of the most valuable wildlife monitoring tools.
For some species, individual identification comes at no cost when unique attributes (e.g. pelage patterns) can be discerned with remote cameras, while other species require viable genetic material and expensive laboratory processing for individual assignment. Prohibitive costs may still force monitoring efforts to use species distribution or occupancy as a surrogate for density, which may not be appropriate under many conditions.
Here, we used a large‐scale monitoring study of fisher Pekania pennanti to evaluate the effectiveness of occupancy as an approximation to density, particularly for informing harvest management decisions. We combined remote cameras with baited hair snares during 2013–2015 to sample across a 70 096‐km ² region of western New York, USA . We fit occupancy and Royle–Nichols models to species detection–non‐detection data collected by cameras, and spatial capture–recapture (SCR) models to individual encounter data obtained by genotyped hair samples. Variation in the state variables within 15‐km ² grid cells was modelled as a function of landscape attributes known to influence fisher distribution.
We found a close relationship between grid cell estimates of fisher state variables from the models using detection–non‐detection data and those from the SCR model, likely due to informative spatial covariates across a large landscape extent and a grid cell resolution that worked well with the movement ecology of the species. Fisher occupancy and density were both positively associated with the proportion of coniferous‐mixed forest and negatively associated with road density. As a result, spatially explicit management recommendations for fisher were similar across models, though relative variation was dampened for the detection–non‐detection data.
Synthesis and applications . Our work provides empirical evidence that models using detection–non‐detection data can make similar inferences regarding relative spatial variation of the focal population to models using more expensive individual encounters when the selected spatial grain approximates or is marginally smaller than home range size. When occupancy alone is chosen as a cost‐effective state variable for monitoring, simulation and sensitivity analyses should be used to understand how inferences from detection–non‐detection data will be affected by aspects of study design and species ecology.
Many data sources report related variables of interest that are also
referenced over geographic regions and time; however, there are relatively few
general statistical methods that one can readily use that incorporate these
multivariate spatio-temporal dependencies. Additionally, many multivariate
spatio-temporal areal datasets are extremely high-dimensional, which leads to
practical issues when formulating statistical models. For example, we analyze
Quarterly Workforce Indicators (QWI) published by the US Census Bureau's
Longitudinal Employer-Household Dynamics (LEHD) program. QWIs are available by
different variables, regions, and time points, resulting in millions of
tabulations. Despite their already expansive coverage, by adopting a fully
Bayesian framework, the scope of the QWIs can be extended to provide estimates
of missing values along with associated measures of uncertainty. Motivated by
the LEHD, and other applications in federal statistics, we introduce the
multivariate spatio-temporal mixed effects model (MSTM), which can be used to
efficiently model high-dimensional multivariate spatio-temporal areal datasets.
The proposed MSTM extends the notion of Moran's I basis functions to the
multivariate spatio-temporal setting. This extension leads to several
methodological contributions including extremely effective dimension reduction,
a dynamic linear model for multivariate spatio-temporal areal processes, and
the reduction of a high-dimensional parameter space using {a novel} parameter
model.
For full text please download from http://wii.gov.in/tiger_reports. The publication reports the status of tigers, copredators, and prey in India for 2014 assessment.
The advent of spatially explicit capture–recapture models is changing the way ecologists analyse capture–recapture data. However, the advantages offered by these new models are not fully exploited because they can be difficult to implement.
To address this need, we developed a user‐friendly software package, created within the R programming environment, called SPACECAP . This package implements B ayesian spatially explicit hierarchical models to analyse spatial capture–recapture data.
Given that a large number of field biologists prefer software with graphical user interfaces for analysing their data, SPACECAP is particularly useful as a tool to increase the adoption of B ayesian spatially explicit capture–recapture methods in practice.
1. Abundance is a key quantity for conservation and management strategies but remains challenging to assess in the field. Capture-recapture (CR) methods are often used to estimate abundance while correcting for imperfect detection but these methods are costly. Occupancy, sometimes considered as a surrogate for abundance, is estimated through the collection of presence/absence data and is less costly while allowing gathering of information at a large spatial scale.
2. Building on the recent pieces of work on the combination of different data sources, we showed how abundance data can be complemented by presence/absence data and can be analysed conjointly to improve abundance estimates. Our approach relies on a hierarchical model that makes explicit the link between the abundance and occupancy state variables while formally accounting for imperfect detection.
3. We used a population of Eurasian lynx in France monitored via camera-traps and a collection of presence signs as an illustration of our approach.
4. Synthesis and applications. We combined capture/recapture and occupancy data, and demonstrated that we can efficiently improve abundance estimates. Our method can be used by managers when estimates of trends in abundance lack power due to sparse data collected during an intensive survey, by simply integrating data collected during non-systematic survey. Furthermore, combining these two sampling procedures makes full use of all available data and allows the development of conservation and management strategies based on precise abundance estimates. Overall, the combination of different data sources in an integrated statistical framework has a great potential, especially for elusive species.
Although they play a critical role in shaping ecological communities, many threatened predator species are data-deficient. The Dhole Cuon alpinus is one such rare canid with a global population thought to be <2500 wild individuals. We assessed habitat occupancy patterns of dholes in the Western Ghats of Karnataka, India, to understand ecological and anthropogenic determinants of their distribution and habitat-use. We conducted spatially replicated detection/non-detection surveys of dhole signs along forest trails at two appropriate scales: the entire landscape and a single wildlife reserve. Landscape-scale habitat occupancy was assessed across 38,728 km2 surveying 206 grid cells of 188-km2 each. Finer scale habitat-use within 935 km2 Bandipur Reserve was studied surveying 92 grid cells of 13-km2 km each. We analyzed the resulting data of dhole signs using likelihood-based habitat occupancy models. The models explicitly addressed the problematic issue of imperfect detection of dhole signs during field surveys as well as potential spatial auto-correlation between sign detections made on adjacent trail segments. We show that traditional 'presence versus absence' analyses underestimated dhole habitat occupancy by 60% or 8682 km2 [naïve = 0.27; [Formula: see text](SE) = 0.68 (0.08)] in the landscape. Addressing imperfect sign detections by estimating detection probabilities [[Formula: see text](L) (SE) = 0.12 (0.11)] was critical for reliable estimation. Similar underestimation occurred while estimating habitat-use probability at reserve-scale [naïve = 0.39; [Formula: see text](SE) = 0.71 (0.06)]. At landscape scale, relative abundance of principal ungulate prey primarily influenced dhole habitat occupancy. Habitat-use within a reserve, however, was predominantly and negatively influenced by anthropogenic disturbance. Our results are the first rigorous assessment of dhole occupancy at multiple spatial scales with potential conservation value. The approach used in this study has potential utility for cost-effectively assessing spatial distribution and habitat-use in other species, landscapes and reserves.
We present a Bayesian model for area-level count data that uses Gaussian
random effects with a novel type of G-Wishart prior on the inverse
variance-covariance matrix. Specifically, we introduce a new distribution
called the negative G-Wishart distribution that has support over precision
matrices that lead to positive associations between the random effects of
neighboring regions while preserving conditional independence of
non-neighboring regions. We describe Markov chain Monte Carlo sampling
algorithms for the negative G-Wishart prior in a disease mapping context and
compare our results to Bayesian hierarchical models based on intrinsic
autoregression priors. A simulation study illustrates that using the negative
G-Wishart prior improves over the intrinsic autoregressive priors when there
are discontinuities in the disease risk surface. The new model is applied to an
analysis of cancer incidence data in Washington State
Studies of demographic processes are typically restricted to small geographic areas and short time periods due to the costs of marking and monitoring individuals. However, environmental changes are occurring at much broader spatial and temporal scales, and thus, inferences about the mechanisms governing population dynamics need to be scaled accordingly. Recently developed integrated population models (IPMs) represent an approach for doing so, by jointly analysing survey data and capture–recapture data.
Although promising, several shortcomings of conventional IPMs exist, including difficulties accounting for spatial variation in demographic, movement and detection parameters; limited ability to make spatially explicit predictions of abundance or vital rates; and a requirement that the survey data and the capture–recapture data are independent. We demonstrate how each of these limitations can be resolved by adopting a spatial population dynamics model upon which both the survey data and the capture–recapture data are conditioned.
We applied the model to 6 years of hair data collected on the threatened Louisiana black bear Ursus americanus luteolus . For years in which the hair samples were genotyped, the resulting data are information‐rich (but expensive) spatial capture–recapture (SCR) data. For the remaining years, the data are binary detection data, of the type often analysed using occupancy models. We compared estimates of demographic parameters and annual abundance using various combinations of the SCR and detection data, and found that combining the SCR data and the detection data resulted in more precise estimates of abundance relative to estimates that did not use the detection data. A simulation study provided additional evidence of increased precision, as well as evidence that the estimators of annual abundance are approximately unbiased.
The ability to combine survey data and capture–recapture data using a spatially explicit model opens many possibilities for designing cost effective studies and scaling up inferences about the demographic processes influencing spatial and temporal population dynamics.
Spatial models for point-referenced data are used for capturing spatial association and for providing spatial prediction, typically in the presence of explanatory variables. The goal of this paper is to treat the situation where there is misalignment be- tween at least one of the explanatory variables and the response variable. In this context we formalize three inference problems. One, which we call interpolation, seeks to infer about missing response at an observed explanatory location. The second, which we call prediction, seeks to infer about a response at a location with the explanatory variable unobserved. The last, which we call regression, seeks to investigate the functional rela- tionship between the response and explanatory variable through the conditional mean of the response. We treat both the case of Gaussian and binary spatial response. We adopt a Bayesian approach, providing full posterior inference for each of the above problems. We illustrate both cases using portions of a study of isopod burrows in the Negev desert in Israel.
This article presents a multivariate spatial prediction methodology in a Bayesian framework. The method is especially suited for use in environmetrics, where vector-valued responses are observed at a small set of ambient monitoring stations "(gauged sites)" at successive time points. However, the stations may have varying start-up times so that the data have a "staircase" pattern ("monotone" pattern in the terminology of Rubin and Shaffer). The lowest step corresponds to the newest station in the monitoring network. We base our approach on a hierarchical Bayes prior involving a Gaussian generalized inverted Wishart model. For given hyperparameters, we derive the predictive distribution for currently gauged sites at times before their start-up when no measurements were taken. The resulting predictive distribution is a matric t distribution with appropriate covariance parameters and degrees of freedom. We estimate the hyperparameters using the method of moments (MOM) as an easy-to-implement alternative to the more complex EM algorithm. The MOM in particular gives exact parameter estimates and involves less cumbersome calculations than the EM algorithm. Finally, we obtain the predictive distribution for unmeasured responses at "ungauged" sites. The results obtained here allow us to pool the data from different sites that measure different pollutants and also to treat cases where the observed data monitoring stations have a monotonic "staircase" structure. We demonstrate the use of this methodology by mapping PM 205 elds for Philadelphia during the period of May 1992 to September 1993. Large amounts of data missing by design make this application particularly challenging. We give empirical evidence that the method performs well.
Offers a comprehensive introduction to distance sampling, a statistical method used by many biologists and conservationists to estimate animal abundance. The text discusses point transect sampling and line transect sampling and also describes several other related techniques. There are updates on study design and field methods, laser range finders, theodolites and the GPS and advice is given on a wide range of survey methods. Analysis methods have also been generalized, through the use of various types of multiplier and exercises for students in wildlife and conservation management are included.
Nondetection of a species at a site does not imply that the species is absent unless the probability of detection is 1. We propose a model and likelihood-based method for estimating site occupancy rates when detection probabilities are 1. The model provides a flexible framework enabling covariate information to be included and allowing for missing observations. Via computer simulation, we found that the model provides good estimates of the occupancy rates, generally unbiased for moderate detection probabilities (0.3). We estimated site occupancy rates for two anuran species at 32 wetland sites in Maryland, USA, from data collected during 2000 as part of an amphibian monitoring program, Frog-watch USA. Site occupancy rates were estimated as 0.49 for American toads (Bufo amer-icanus), a 44% increase over the proportion of sites at which they were actually observed, and as 0.85 for spring peepers (Pseudacris crucifer), slightly above the observed proportion of 0.83.
Projections of future climate change caused by increasing greenhouse gases depend critically on numerical climate models coupling the ocean and atmosphere (GCMs). However, difierent models difier substantially in their projections, which raises the question of how the difierent models can best be combined into a probability distribution of future climate change. For this analysis, we have collected both current and future projected mean temperatures produced by nine climate models for 22 regions of the earth. We also have estimates of current mean temperatures from actual observations, together with standard errors, that can be used to calibrate the climate models. We propose a Bayesian analysis that allows us to combine the difierent climate models into a posterior distribution of future temperature increase, for each of the 22 regions, while allowing for the difierent climate models to have difierent variances. Two versions of the analysis are proposed, a univariate analysis in which each region is analyzed separately, and a multivariate analysis in which the 22 regions are combined into an overall statistical model. A cross-validation approach is proposed to conflrm the reasonableness of our Bayesian predictive distributions. The results of this analysis allow for a quantiflcation of the uncertainty of climate model projections as a Bayesian posterior distribution, substantially extending previous approaches to uncertainty in climate models.
This article develops methods for fitting spatial models to line transect data. These allow animal density to be related to
topographical, environmental, habitat, and other spatial variables, helping wildlife managers to identify the factors that
affect abundance. They also enable estimation of abundance for any subarea of interest within the surveyed region, and potentially
yield estimates of abundance from sightings surveys for which the survey design could not be randomized, such as surveys conducted
from platforms of opportunity. The methods are illustrated through analyses of data from a shipboard sightings survey of minke
whales in the Antarctic.
Spatial capture-recapture (SCR) methods represent a major advance over
traditional capture-capture methods because they yield explicit estimates of
animal density instead of population size within an unknown area, and they
account for heterogeneity in capture probability arising from the juxtaposition
of individuals and sample locations. However, the requirement that all
individuals can be uniquely identified excludes their use in many contexts. In
this paper, we develop models for situations in which individual recognition is
not possible, thereby allowing SCR methods to be applied in studies of unmarked
or partially-marked populations. The data required for our model are
spatially-referenced counts at a collection of closely-spaced sample units such
that individuals can be encountered at multiple locations. Our approach
utilizes the spatial correlation in counts as information about the location of
individual activity centers, which enables estimation of density and
distance-related heterogeneity in detection. A simulation study demonstrated
that while the posterior distribution of abundance or density is strongly
skewed in small samples, the posterior mode is an accurate point estimator as
long as the trap spacing is not too large relative to scale parameter of the
detection function. Marking a subset of the population can lead to substantial
reductions in posterior skew and increased posterior precision. We also fit the
model to point count data collected on the northern parula, and obtained a
density estimate of 0.38 (95% CI: 0.19, 1.64) birds/ha. Our paper challenges
sampling and analytical conventions by demonstrating that neither spatial
independence nor individual recognition is needed to estimate population
density---rather, spatial dependence induced by design can be informative about
individual distribution and density.
This article expands upon recent interest in Bayesian hierarchical models in quantitative genetics by developing spatial process models for inference on additive and dominance genetic variance within the context of large spatially referenced trial datasets of multiple traits of interest. Direct application of such multivariate models to large spatial datasets is often computationally infeasible because of cubic order matrix algorithms involved in estimation. The situation is even worse in Markov chain Monte Carlo (MCMC) contexts where such computations are performed for several thousand iterations. Here, we discuss approaches that help obviate these hurdles without sacrificing the richness in modeling. For genetic effects, we demonstrate how an initial spectral decomposition of the relationship matrices negates the expensive matrix inversions required in previously proposed MCMC methods. For spatial effects we discuss a multivariate predictive process that reduces the computational burden by projecting the original process onto a subspace generated by realizations of the original process at a specified set of locations (or knots). We illustrate the proposed methods using a synthetic dataset with multivariate additive and dominant genetic effects and anisotropic spatial residuals, and a large dataset from a scots pine (Pinus sylvestris L.) progeny study conducted in northern Sweden. Our approaches enable us to provide a comprehensive analysis of this large trial which amply demonstrates that, in addition to violating basic assumptions of the linear model, ignoring spatial effects can result in downwardly biased measures of heritability.
In studies of wild animals, one frequently encounters both count and mark-recapture-recovery data. Here, we consider an integrated Bayesian analysis of ring¿recovery and count data using a state-space model. We then impose a Leslie-matrix-based model on the true population counts describing the natural birth-death and age transition processes. We focus upon the analysis of both count and recovery data collected on British lapwings (Vanellus vanellus) combined with records of the number of frost days each winter. We demonstrate how the combined analysis of these data provides a more robust inferential framework and discuss how the Bayesian approach using MCMC allows us to remove the potentially restrictive normality assumptions commonly assumed for analyses of this sort. It is shown how WinBUGS may be used to perform the Bayesian analysis. WinBUGS code is provided and its performance is critically discussed.
Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.
The Valencia International Meetings on Bayesian Statistics, held every four years, provide the main forum for researchers in the area of Bayesian Statistics to come together to present and discuss frontier developments in the field. The resulting Proceedings provide a definitive, up-to-date overview encompassing a wide range of theoretical and applied research. This seventh Proceedings containing 23 invited articles and 31 contributed papers is no exception, and will be an indispensable reference to all statisticians.
Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence, Second Edition, provides a synthesis of model-based approaches for analyzing presence-absence data, allowing for imperfect detection. Beginning from the relatively simple case of estimating the proportion of area or sampling units occupied at the time of surveying, the authors describe a wide variety of extensions that have been developed since the early 2000s. This provides an improved insight about species and community ecology, including, detection heterogeneity; correlated detections; spatial autocorrelation; multiple states or classes of occupancy; changes in occupancy over time; species co-occurrence; community-level modeling, and more. Occupancy Estimation and Modeling: Inferring Patterns and Dynamics of Species Occurrence, Second Edition has been greatly expanded and detail is provided regarding the estimation methods and examples of their application are given. Important study design recommendations are also covered to give a well rounded view of modeling.
Monitoring abundance and distribution of organisms over large landscapes can be difficult. Because of challenges associated with logistics and data analyses uncorrected counts are often used as a proxy for abundance. We present the first statewide estimate of abundance for Florida manatees (Trichechus manatus latirostris) using an innovative approach that combines multiple sources of information. We used a combination of a double-observer protocol, repeated passes, and collection of detailed diving behavior data to account for imperfect detection of animals. Our estimate of manatee abundance was 6350 (95%CI: 5310–7390). Specifically, we estimated 2790 (95%CI: 2160–3540) manatees on the west coast (2011), and 3560 (95%CI: 2850–4410) on the east coast (2012). Unlike uncorrected counts conducted since 1991, our estimation method considered two major sources of error: spatial variation in distribution and imperfect detection. The Florida manatee is listed as endangered, but its status is currently under review; the present study may become important for the review process. Interestingly, we estimated that 70% (95%CI: 60–80%) of manatees on the east coast of Florida were aggregated in one county during our survey. Our study illustrates the value of combining information from multiple sources to monitor abundance at large scales. Integration of information can reduce cost, facilitate the use of data obtained from new technologies to increase accuracy, and contribute to encouraging coordination among survey teams from different organizations nationally or internationally. Finally, we discuss the applicability of our work to other conservation applications (e.g., risk assessment) and to other systems.
An index‐calibration experiment involves rigorous estimation of animal abundance at a small scale to calibrate a less rigorously derived index of abundance. The efficacy of such index‐calibration experiments has been a matter of much controversy. In this study, we develop theoretical models and test them with empirical data on large‐scale index‐calibration experiments on tigers Panthera tigris to advance our understanding of this controversy.
We propose two models that describe the sampling processes involved in typical index‐calibration experiments. Using analytical derivations and some simulations, we evaluate the relative roles of these sampling parameters on the (coefficient of determination) statistic – a common inferential tool used in such calibration experiments.
We make predictions about the statistic using our theoretical derivations and estimates from large‐scale occupancy surveys of tigers in India. We then compare our predictions with empirical estimates of two tiger sign index‐calibration experiments (IC‐Karanth and IC‐Jhala).
Our theoretical models show that the statistic increases when individual‐specific detection probability is high and is constant, increases when the variance‐to‐mean ratio of abundance increases, increases when precision of abundance estimates improves and declines when mean abundance increases. All predictions about the two index‐calibration experiments showed a poor performance of the statistic ( < 0·40). Inference from IC‐Karanth was extremely poor ( ) and comparable to model predictions ( P value = 0·0754). Anomalously ( P value < 0·0001), inference from IC‐Jhala was exceedingly high ( ).
Our study shows that such direct index‐calibration experiments using the statistic yield poor inferences unless all the sampling process parameters lie within a limited range. Ignoring the consequence of the effect of these parameters during survey design could result in expenditure of huge resources with little gain in ecological inference. Analysis using joint likelihood models, with appropriate survey designs, may be more fruitful than clinging on to such composite, direct ‐based models.
In spatial generalized linear mixed models (SGLMMs), covariates that are spatially smooth are often collinear with spatially smooth random effects. This phenomenon is known as spatial confounding and has been studied primarily in the case where the spatial support of the process being studied is discrete (e.g., areal spatial data). In this case, the most common approach suggested is restricted spatial regression (RSR) in which the spatial random effects are constrained to be orthogonal to the fixed effects. We consider spatial confounding and RSR in the geostatistical (continuous spatial support) setting. We show that RSR provides computational benefits relative to the confounded SGLMM, but that Bayesian credible intervals under RSR can be inappropriately narrow under model misspecification. We propose a posterior predictive approach to alleviating this potential problem and discuss the appropriateness of RSR in a variety of situations. We illustrate RSR and SGLMM approaches through simulation studies and an analysis of malaria frequencies in The Gambia, Africa. Copyright © 2015 John Wiley & Sons, Ltd.
Aerosols are tiny solid or liquid particles suspended in the atmosphere; examples of aerosols include windblown dust, sea salts, volcanic ash, smoke from wildfires, and pollution from factories. The global distribution of aerosols is a topic of great interest in climate studies since aerosols can either cool or warm the atmosphere depending on their location, type, and interaction with clouds. Aerosol concentrations are important input components of global climate models, and it is crucial to accurately estimate aerosol concentrations from remote sensing instruments so as to minimize errors “downstream” in climate models. Currently, space-based observations of aerosols are available from two remote sensing instruments on board NASA’s Terra spacecraft: the Multiangle Imaging SpectroRadiometer (MISR), and the MODerate-resolution Imaging Spectrometer (MODIS). These two instruments have complementary coverage, spatial support, and retrieval characteristics, making it advantageous to combine information from both sources to make optimal inferences about global aerosol distributions. In this article, we predict the true aerosol process from two noisy and possibly biased datasets, and we also estimate the uncertainties of these estimates. Our data-fusion methodology scales linearly and bears some resemblance to Fixed Rank Kriging (FRK), a variant of kriging that is designed for spatial interpolation of a single, massive dataset. Our spatial statistical approach does not require assumptions of stationarity or isotropy and, crucially, allows for change of spatial support. We compare our methodology to FRK and Bayesian melding, and we show that ours has superior prediction standard errors compared to FRK and much faster computational speed compared to Bayesian melding.
We investigate the 20-year-average boreal winter temperatures generated by an ensemble of six regional climate models (RCMs) in phase I of the North American Regional Climate Change Assessment Program. We use the long-run average (20-year integration) to smooth out variability and to capture the climate properties from the RCM outputs. We find that, although the RCMs capture the large-scale climate variation from coast to coast and from south to north similarly, their outputs can differ substantially in some regions. We propose a Bayesian hierarchical model to synthesize information from the ensemble of RCMs, and we construct a consensus climate signal with each RCM contributing to the consensus according to its own variability parameter. The Bayesian methodology enables us to make posterior inference on all the unknowns, including the large-scale fixed effects and the small-scale random effects in the consensus climate signal and in each RCM. The joint distributions of the consensus climate and the outputs from the RCMs are also investigated through posterior means, posterior variances and posterior spatial quantiles. We use a spatial random-effects model in the Bayesian hierarchical model and, consequently, we can deal with the large data sets of fine resolution outputs from all the RCMs. Additionally, our model allows a flexible spatial covariance structure without assuming stationary or isotropy.
Population dynamics with regard to evolution of traits has typically been studied using matrix projection models (MPMs). Recently, to work with continuous traits, integral projection models (IPMs) have been proposed. Imitating the path with MPMs, IPMs are handled first with a fitting stage, then with a projection stage. Fitting these models has so far been done only with individual-level transition data. These data are used to estimate the demographic functions (survival, growth, fecundity) that comprise the kernel of the IPM specification. Then, the estimated kernel is iterated from an initial trait distribution to project steady state population behavior under this kernel. When trait distributions are observed over time, such an approach does not align projected distributions with these observed temporal benchmarks.
The contribution here, focusing on size distributions, is to address this issue. Our concern is that the above approach introduces an inherent mismatch in scales. The redistribution kernel in the IPM proposes a mechanistic description of population level redistribution. A kernel of the same functional form, fitted to data at the individual level, would provide a mechanistic model for individual-level processes. Resulting parameter estimates and the associated estimated kernel are at the wrong scale and do not allow population-level interpretation.
Our approach views the observed size distribution at a given time as a point pattern over a bounded interval. We build a three-stage hierarchical model to infer about the dynamic intensities used to explain the observed point patterns. This model is driven by a latent deterministic IPM and we introduce uncertainty by having the operating IPM vary around this deterministic specification. Further uncertainty arises in the realization of the point pattern given the operating IPM. Fitted within a Bayesian framework, such modeling enables full inference about all features of the model. Such dynamic modeling, optimized by fitting to data observed over time, is better suited to projection.
Exact Bayesian model fitting is very computationally challenging; we offer approximate strategies to facilitate computation. We illustrate with simulated data examples as well as well as a set of annual tree growth data from Duke Forest in North Carolina. A further example shows the benefit of our approach, in terms of projection, compared with the foregoing individual level fitting.
This paper presents a theory for modeling random environmental spatial-temporal fields that allows simulated data (numerical-physical
model output) to be combined with measurements made at fixed monitoring sites. That theory involves Bayesian hierarchical
models that provide temporal forecasts and spatial predictions along with appropriate credibility intervals. A by-product
is a method for re-calibrating the simulated data to bring it into line with the measurements for certain applications. While
the approach covers a broad domain of potential applications, this paper addresses a field of particular importance, ground
level ozone concentrations over the eastern and central USA. A univariate model is developed and illustrated with hourly ozone
fields. A multivariate alternative is also provided and illustrated with daily concentration fields. The forecasts and predictions
they provide are compared with those from other approaches.
Survival models have a long history in the biomedical and biostatistical literature, and are enormously popular in the analysis of time-to-event data. Very often these data will be grouped into strata, such as clinical sites, geographic regions, and so on. Such data will often be available over multiple time periods, and for multiple diseases. In this paper, we consider hierarchical spatial process models for multivariate survival data sets which are spatio-temporally arranged. Such models must account for correlations between survival rates in neighboring spatial regions, adjacent time periods, and similar diseases (say, diierent forms of cancer). We investigate Cox semiparametric survival modeling approaches, adding spatial and temporal eeects in a hierarchical structure. Due to data limitations and computational complexity issues, we avoid geostatistical (kriging) models, and instead handle spatial correlation by placing a particular multivariate generalization of the conditionally autoregressive (CAR) distribution on the region-speciic frailties. Exempliication is provided using time-to-event data for various cancers from the National Cancer Institute's Surveillance, Epidemiology, and End Results (SEER) database.
SUMMARY Capture-recapture models are widely used in the estimation of population sizes. Based on data augmentation considerations,
we show how Gibbs sampling can be applied to calculate Bayes estimates in this setting. As a result, formulations which were
previously avoided because of analytical and numerical intractability can now be easily considered for practical application.
We illustrate this potential by using Gibbs sampling to calculate Bayes estimates for a hierarchical capture-recapture model
in a real example.
Spatial processes are important models for many environmental problems. Classical geostatistics and Fourier spectral methods are powerful tools for stuyding the spatial structure of stationary processes. However, it is widely recognized that in real applications spatial processes are rarely stationary and isotropic. Consequently, it is important to extend these spectral methods to processes that are nonstationary. In this work, we present some new spectral approaches and tools to estimate the spatial structure of a nonstationary process. More specifically, we propose an approach for the spectral analysis of nonstationary spatial processes that is based on the concept of spatial spectra, i.e., spectral functions that are space-dependent. This notion of spatial spectra generalizes the definition of spectra for stationary processes, and under certain conditions, the spatial spectrum at each Location can be estimated from a single realization of the spatial process.
The motivation for this work is the modeling and prediction of ozone concentrations over different geopolitical boundaries for assessment of compliance with ambient air quality standards.
1. Assessing spatial distributions of threatened large carnivores at landscape scales poses formidable challenges because of their rarity and elusiveness. As a consequence of logistical constraints, investigators typically rely on sign surveys. Most survey methods, however, do not explicitly address the central problem of imperfect detections of animal signs in the field, leading to underestimates of true habitat occupancy and distribution.
2. We assessed habitat occupancy for a tiger Panthera tigris metapopulation across a c. 38 000-km2 landscape in India, employing a spatially replicated survey to explicitly address imperfect detections. Ecological predictions about tiger presence were confronted with sign detection data generated from occupancy sampling of 205 sites, each of 188 km2.
3. A recent occupancy model that considers Markovian dependency among sign detections on spatial replicates performed better than the standard occupancy model (ΔAIC = 184·9). A formulation of this model that fitted the data best showed that density of ungulate prey and levels of human disturbance were key determinants of local tiger presence. Model averaging resulted in a replicate-level detection probability = 0·17 (0·17) for signs and a tiger habitat occupancy estimate of = 0·665 (0·0857) or 14 076 (1814) km2 of potential habitat of 21 167 km2. In contrast, a traditional presence-versus-absence approach underestimated occupancy by 47%. Maps of probabilities of local site occupancy clearly identified tiger source populations at higher densities and matched observed tiger density variations, suggesting their potential utility for population assessments at landscape scales.
4. Synthesis and applications. Landscape-scale sign surveys can efficiently assess large carnivore spatial distributions and elucidate the factors governing their local presence, provided ecological and observation processes are both explicitly modelled. Occupancy sampling using spatial replicates can be used to reliably and efficiently identify tiger population sources and help monitor metapopulations. Our results reinforce earlier findings that prey depletion and human disturbance are key drivers of local tiger extinctions and tigers can persist even in human-dominated landscapes through effective protection of source populations. Our approach facilitates efficient targeting of tiger conservation interventions and, more generally, provides a basis for the reliable integration of large carnivore monitoring data between local and landscape scales.
The authors develop a methodology for predicting unobserved values in a conditionally lognormal random spatial field like those commonly encountered in environmental risk analysis. These unobserved values are of two types. The first come from spatial locations where the field has never been monitored, the second, from currently monitored sites which have been only recently installed. Thus the monitoring data exhibit a monotone pattern, resembling a staircase whose highest step comes from the oldest monitoring sites. The authors propose a hierarchical Bayesian approach using the lognormal sampling distribution, in conjunction with a conjugate generalized Wishart distribution. This prior distribution allows different degrees of freedom to be fitted for individual steps, taking into account the differential amounts of information available from sites at the different steps in the staircase. The resulting hierarchical model is a predictive distribution for the unobserved values of the field. The method is demonstrated by application to the ambient ozone field for the southwestern region of British Columbia.
We describe an approach for estimating occupancy rate or the proportion of area occupied when heterogeneity in detection probability exists as a result of variation in abundance of the organism under study. The key feature of such problems, which we exploit, is that variation in abundance induces variation in detection probability. Thus, heterogeneity in abundance can be modeled as heterogeneity in detection probability. Moreover, this linkage between heterogeneity in abundance and heterogeneity in detection probability allows one to exploit a heterogeneous detection probability model to estimate the underlying distribution of abundances. Therefore, our method allows estimation of abundance from repeated observations of the presence or absence of animals without having to uniquely mark individuals in the population.
We develop and apply an approach to the spatial interpolation of a vector-valued random response field. The Bayesian approach we adopt enables uncertainty about the underlying models to be represented in expressing the accuracy of the resulting interpolants. The methodology is particularly relevant in environmetrics, where vector-valued responses are only observed at designated sites at successive time points. The theory allows space-time modelling at the second level of the hierarchical prior model so that uncertainty about the model parameters has been fully expressed at the first level. In this way, we avoid unduly optimistic estimates of inferential accuracy. Moreover, the prior model can be upgraded with any available new data, while past data can be used in a systematic way to fit model parameters. The theory is based on the multivariate normal and related joint distributions. Our hierarchical prior models lead to posterior distributions which are robust with respect to the choice of the prior (hyperparameters). We illustrate our theory with an example involving monitoring stations in southern Ontario, where monthly average levels of ozone, sulphate, and nitrate are available and between-station response triplets are interpolated. In this example we use a recently developed method for interpolating spatial correlation fields.
1. Indices of abundance offer cost effective and rapid methods for estimating abundance of endangered species across large landscapes, yet their wide usage is controversial due to their potential of being biased. Here, we assess the utility of indices for the daunting task of estimating the abundance of the endangered tiger at landscape scales.
2. We use double sampling to estimate two indices of tiger abundance (encounters of pugmarks and scats per km searched) and calibrate those indices against contemporaneous estimates of tiger densities obtained using camera-trap mark–recapture (CTMR) at 21 sites (5185 km2) in Central and North India. We use simple and multiple weighted regressions to evaluate relationships between tiger density and indices. A model for estimating tiger density from indices was validated by Jackknife analysis and precision was assessed by correlating predicted tiger density with CTMR density. We conduct power analysis to estimate the ability of CTMR and of indices to detect changes in tiger density.
3. Tiger densities ranged between 0·25 and 19 tigers 100 km−2 were estimated with an average coefficient of variation of 13·2(SE 2·5)%. Tiger pugmark encounter rates explained 84% of the observed variability in tiger densities. After removal of an outlier (Corbett), square root transformed scat encounter rates explained 82% of the variation in tiger densities.
4. A model including pugmark and scat encounters explained 95% of the variation in tiger densities with good predictive ability (PRESS R2 = 0·99). Overall, CTMR could detect tiger density changes of >12% with 80% power at α = 0·3, while the index based model had 50% to 85% power to detect >30% declines. The power of indices to detect declines increased at high tiger densities.
5. Synthesis and applications. Indices of tiger abundance obtained from across varied habitats and a range of tiger densities could reliably estimate tiger abundance. Financial and temporal costs of estimating indices were 7% and 34% respectively, of those for CTMR. The models and methods presented herein have application in evaluation of the abundance of cryptic carnivores at landscape scales and form part of the protocol used by the Indian Government for evaluating the status of tigers.
Many statisticians have had the experience of fitting a linear model with uncorrelated errors, then adding a spatially-correlated error term (random effect) and finding that the estimates of the fixed-effect coefficients have changed substantially. We show that adding a spatially-correlated error term to a linear model is equivalent to adding a saturated collection of canonical regressors, the coefficients of which are shrunk toward zero, where the spatial map determines both the canonical regressors and the relative extent of the coefficients’ shrinkage. Adding a spatially-correlated error term can also be seen as inflating the error variances associated with specific contrasts of the data, where the spatial map determines the contrasts and the extent of error-variance inflation. We show how to avoid this spatial confounding by restricting the spatial random effect to the orthogonal complement (residual space) of the fixed effects, which we call restricted spatial regression. We consider five proposed interpretations of spatial confounding and draw implications about what, if anything, one should do about it. In doing so, we debunk the common belief that adding a spatially-correlated random effect adjusts fixed-effect estimates for spatially-structured missing covariates. This article has supplementary material online.
A problem involving non-stationary, discrete-time series of counts from a Poisson process with a varying but smooth intensity function is studied. A smoothness prior for the underlying intensity process is modelled using the hierarchical Bayesian approach, which is shown to provide an AR(1) representation for the intensity process. Since conjugate priors are not assumed, analytic derivation of estimates and predictions of the Poisson series are not available. Some reasonably good approximations are given and illustrated using data on British road casualties before and after the introduction of the seatbelt law.
We develop a unified framework for jointly defining population dynamics models and measurements taken on a population. The framework is a state-space model where the population processes are modelled by the state process and measurements are modelled by the observation process. In many cases, the expected value for the state process can be represented as a generalisation of the standard population projection matrix: each sub-process within the state process may be modelled by a separate matrix and the product of these matrices is a generalised Leslie matrix. By selecting appropriate matrices and their ordering, a wide range of models may be specified. The method is fully flexible for allowing stochastic variation in the processes. Process parameters may themselves be modelled as functions of covariates. The structure accommodates effects such as density dependence, competition and predator–prey relationships, and metapopulations are readily modelled. Observations on the population enter through an observation process model, and we show how likelihood functions can be built that reflect both demographic stochasticity (which appears in the state process) and stochastic errors in the observations. Parameter estimation and estimation of state process variables can be conducted using sequential Monte Carlo procedures.
Occupancy modeling focuses on inference about the distribution of organisms over space, using temporal or spatial replication to allow inference about the detection process. Inference based on spatial replication strictly requires that replicates be selected randomly and with replacement, but the importance of these design requirements is not well understood. This paper focuses on an increasingly popular sampling design based on spatial replicates that are not selected randomly and that are expected to exhibit Markovian dependence. We develop two new occupancy models for data collected under this sort of design, one based on an underlying Markov model for spatial dependence and the other based on a trap response model with Markovian detections. We then simulated data under the model for Markovian spatial dependence and fit the data to standard occupancy models and to the two new models. Bias of occupancy estimates was substantial for the standard models, smaller for the new trap response model, and negligible for the new spatial process model. We also fit these models to data from a large-scale tiger occupancy survey recently conducted in Karnataka State, southwestern India. In addition to providing evidence of a positive relationship between tiger occupancy and habitat, model selection statistics and estimates strongly supported the use of the model with Markovian spatial dependence. This new model provides another tool for the decomposition of the detection process, which is sometimes needed for proper estimation and which may also permit interesting biological inferences. In addition to designs employing spatial replication, we note the likely existence of temporal Markovian dependence in many designs using temporal replication. The models developed here will be useful either directly, or with minor extensions, for these designs as well. We believe that these new models represent important additions to the suite of modeling tools now available for occupancy estimation in conservation monitoring. More generally, this work represents a contribution to the topic of cluster sampling for situations in which there is a need for specific modeling (e.g., reflecting dependence) for the distribution of the variable(s) of interest among subunits.
We develop a class of models for inference about abundance or density using spatial capture-recapture data from studies based on camera trapping and related methods. The model is a hierarchical model composed of two components: a point process model describing the distribution of individuals in space (or their home range centers) and a model describing the observation of individuals in traps. We suppose that trap- and individual-specific capture probabilities are a function of distance between individual home range centers and trap locations. We show that the models can be regarded as generalized linear mixed models, where the individual home range centers are random effects. We adopt a Bayesian framework for inference under these models using a formulation based on data augmentation. We apply the models to camera trapping data on tigers from the Nagarahole Reserve, India, collected over 48 nights in 2006. For this study, 120 camera locations were used, but cameras were only operational at 30 locations during any given sample occasion. Movement of traps is common in many camera-trapping studies and represents an important feature of the observation model that we address explicitly in our application.
With scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. Over the last decade, hierarchical models implemented through Markov chain Monte Carlo methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics. However, fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data are collected at frequent time points and spatiotemporal process models are used. With regard to this challenge, our contribution is to work with what we call predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower dimensional subspace, thereby reducing the computational burden. Hence, we achieve the flexibility to accommodate non-stationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large data sets. We discuss attractive theoretical properties of these predictive processes. We also provide a computational template encompassing these diverse settings. Finally, we illustrate the approach with simulated and real data sets.
Many organisms are patchily distributed, with some patches occupied at high density, others at lower densities, and others not occupied. Estimation of overall abundance can be difficult and is inefficient via intensive approaches such as capture-mark-recapture (CMR) or distance sampling. We propose a two-phase sampling scheme and model in a Bayesian framework to estimate abundance for patchily distributed populations. In the first phase, occupancy is estimated by binomial detection samples taken on all selected sites, where selection may be of all sites available, or a random sample of sites. Detection can be by visual surveys, detection of sign, physical captures, or other approach. At the second phase, if a detection threshold is achieved, CMR or other intensive sampling is conducted via standard procedures (grids or webs) to estimate abundance. Detection and CMR data are then used in a joint likelihood to model probability of detection in the occupancy sample via an abundance-detection model. CMR modeling is used to estimate abundance for the abundance-detection relationship, which in turn is used to predict abundance at the remaining sites, where only detection data are collected. We present a full Bayesian modeling treatment of this problem, in which posterior inference on abundance and other parameters (detection, capture probability) is obtained under a variety of assumptions about spatial and individual sources of heterogeneity. We apply the approach to abundance estimation for two species of voles (Microtus spp.) in Montana, USA. We also use a simulation study to evaluate the frequentist properties of our procedure given known patterns in abundance and detection among sites as well as design criteria. For most population characteristics and designs considered, bias and mean-square error (MSE) were low, and coverage of true parameter values by Bayesian credibility intervals was near nominal. Our two-phase, adaptive approach allows efficient estimation of abundance of rare and patchily distributed species and is particularly appropriate when sampling in all patches is impossible, but a global estimate of abundance is required.
In this paper a Bayesian alternative to Kriging is developed. The latter is an important tool in geostatistics. But aspects of environmetrics make it less suitable as a tool for interpolating spatial random fields which are observed successively over time. The theory presented here permits temporal (and spatial) modeling to be done in a convenient and flexible way. At the same time model misspecifications, if any, can be corrected by additional data if and when it becomes available, and past data may be used in a systematic way to fit model parameters. Finally, uncertainty about model parameters is represented in the (posterior) distributions, so unrealistically small credible regions for the interpolants are avoided. The theory is based on the multivariate normal and related distributions, but because of the hierarchical prior models adopted, the results would seem somewhat robust with respect to the choice of these distributions and associated hyperparameters.
There have been many attempts in recent years to map incidence and mortality from diseases such as cancer. Such maps usually display either relative rates in each district, as measured by a standardized mortality ratio (SMR) or some similar index, or the statistical significance level for a test of the difference between the rates in that district and elsewhere. Neither of these approaches is fully satisfactory and we propose a new approach using empirical Bayes estimation. The resulting estimators represent a weighted compromise between the SMR, the overall mean relative rate, and a local mean of the relative rate in nearby areas. The compromise solution depends on the reliability of each individual SMR and on estimates of the overall amount of dispersion of relative rates over different districts.
This paper presents a statistical approach, originally developed for mapping disease risk, to ecological regression analysis in the presence of spatial autocorrelated extra-Poisson variation. An, insight into the effect of allowing for Spatial autocorrelation on the relationship between disease rates and explanatory variables is given. Examples based on cancer frequency in Scotland and Sardinia are used to illustrate the interpretation of regression coefficient and further methodological issues.