### Introduction

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

Abundance is a parameter of central interest to animal ecologists because it affects virtually all other aspects of biology or conservation of the species of interest. Consequently, estimating abundance is a fundamental goal of many animal sampling problems, and it forms the basis of a vast body of literature on statistical methods in animal ecology as exemplified in the synthetic treatments by Seber (1982), Borchers, Buckland & Zucchini (2002), and Williams, Nichols & Conroy (2002). The main consideration in estimating abundance of most animal populations is that individuals cannot be observed or detected perfectly. That is, the probability of encountering or detecting an animal is less than 1. A myriad of techniques have been devised for dealing with imperfect detection so that estimates of *N* may be obtained from observations of individuals (Williams *et al*. 2002).

Capture–recapture methods in particular are frequently applied to estimate abundance using encounter history data from arrays of traps or passive detection devices. These include classical animal trapping grids, general photography (Goswami, Madhusudan & Karanth 2007), camera traps and DNA collected from dung or ‘hair snares’ (Waits 2004). It is natural to view the resulting capture–recapture data as being generated by a closed population provided the sample occasions are close enough together in time to preclude mortality, recruitment and movement. However, lacking a physical barrier around the sampled area, there is almost always some movement of individuals onto and off the area during the survey – referred to as an absence of geographic closure (e.g. White *et al*. 1982). This phenomenon in the simplest case (‘random temporary emigration’, Kendall, Nichols & Hines 1997) biases *p̂* low, hence *N̂* high, with respect to the nominal area over which traps are physically situated. While we may be able to obtain an estimate of *N* that is relevant to a real population of individuals, we do not know the area from which animals were sampled. In addition to movement of individuals complicating the interpretation of abundance estimates, another problem that arises in capture–recapture studies based on spatially organized trap arrays is that the probabilistic models of the capture process (e.g. Otis *et al*. 1978) used to estimate *N* are no longer strictly appropriate. Specifically, heterogeneity in capture probability can arise as a result of individual locations relative to the array location. That is, some individuals experience more exposure to trapping than others. These problems have received considerable attention in the literature (Dice 1938, 1941; Stickel 1954; Smith *et al*. 1971; Otis *et al*. 1978; Wilson & Anderson 1985) and a large number of *ad hoc* inference procedures for spatial capture–recapture data have been proposed. Conversely, formal model-based inference procedures have only recently been developed (Efford 2004; Borchers & Efford 2008; Royle & Young 2008).

We propose a spatial-explicit capture–recapture model for estimating density from trapping arrays that yield encounter histories of individuals in addition to auxiliary spatial information – the traps in which each individual is captured. Motivated by a similar problem (Royle & Young 2008), we formulate a hierarchical model for the observations (the locations of trapping) that is specified conditional on a set of latent variables (random effects). These latent variables are the hypothetical home range or activity centres of every individual in the population. The objective is to estimate the absolute density of home range centres in the region containing the trap array. Classical inference under such models is achieved by removing the latent variables from the conditional likelihood by integration (see Borchers & Efford 2008). Here, we adopt an approach based on Bayesian analysis of the hierarchical model using Markov chain Monte Carlo (MCMC). In particular, we make use of the method of data augmentation (Royle, Dorazio & Link 2007), which yields a simple treatment of this problem as a ‘missing data’ problem, where the home range centres are the missing data (Royle & Young 2008). Formally, data augmentation yields a parameterization of the model that can be implemented in the freely available software WinBUGS (Gilks, Thomas & Spiegelhalter 1994) with little more than a few lines of ‘pseudo-code’ to describe the model.

### Model development

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

Suppose that an array of *R* traps is established, having locations (**x**_{j}; *j* = 1, 2, ... , *R*). We denote the set of trap locations as *χ*. Suppose that a population consists of *N* individuals potentially exposed to sampling, and that each individual in the population has a centre of activity, which we denote by *s*_{i} for individuals *i* = 1, 2, ... , *N*. The centre of activity is the two-dimensional coordinate **s**_{i} = (*s*_{1i}, *s*_{2i}). This construction has been previously adopted in similar contexts (Efford 2004; Royle & Young 2008). We assume that the activity centres are fixed for the period over which the population is sampled. Thus, individuals may move around their activity centre, but their activity centre does not move.

The model is developed conditional on the individual activity centres *s*_{i} which are unknown. Following Efford (2004), Borchers & Efford (2008) and Royle & Young (2008), we regard these as random effects, and assume that they are uniformly distributed over some region *S* that contains the traps. We express this assumption by:

Thus, the probability mass function for *s*_{i} is constant on *S*. The inference objective is estimation of the density of activity centres located within *S*. Alternatively, we may be interested in estimating population size or density over some polygon or formal subset of *S*. For example, a polygon might describe the boundaries of a National Park or a Reserve. Obviously, estimating the density with *S* or an arbitrary subset is equivalent (by a linear transformation) to estimating the *number* of activity centres within that polygon. In other words, if *P* is any polygon having area *A*(*P*), and *N*(*P*) is the number of activity centres in *P*, then the density is *D*(*P*) =*N*(*P*)/*A*(*P*).

The observations are denoted by *h*_{it}, which are integers *h*_{it} ∈ (0, 1, 2, ... , *R*), where *h*_{it} = 0 indicates no capture of individual *i* at sampling occasion *t*, and nonzero values indicate the trap location at which the animal is caught at occasion *t*. For example, if *T* = 5 and *R* = 10 traps, then the encounter history **h**_{i} = (0, 8, 9, 0, 8) indicates an individual not captured in occasions 1 and 4, and captured in trap number 8 (occasions 2 and 5) and number 9 (occasion 3). We assume that *h*_{it};* t* = 1, 2, ... , *T* are independent observations of a categorical random variable with probabilities **π**_{i} which depend on the latent variable *s*_{i} and **x**_{j}. Note that is **π**_{i} is a vector of length *R* + 1, with one of the cells corresponding to the event ‘not captured.’ The model can be thought of as describing the result of rolling an *R* + 1-sided die. We denote the observation model assumption by:

*h*_{it}|*s*_{i}, θ ~ Categorical(**π**_{i}(*s*_{i}, θ))(eqn 1)

As indicated here, the cell probabilities are assumed to depend on some parameter(s) θ, in a manner described subsequently. There is an equivalence between a categorical random variable and a multinomial trial. If we create a vector of 0s, of length *R* + 1, and insert a single 1 into that vector in position *h*_{it} + 1, then the resulting vector is a multinomial trial.

That we assume *h*_{it} to be categorical random variables that are independent among survey occasions for each individual *and* among individuals has several implications. In particular, it implies that multiple individuals may be captured in the same trap during any occasion. For camera traps this provides an accurate characterization of the method since cameras operate continuously (under normal circumstances). Also, the model implies that capture in traps for each individual are mutually exclusive events. This means: (i) individuals cannot be captured more than one time in the same trap for each period; and (ii) individuals cannot be captured in more than one trap. While the first is technically incorrect, since cameras (usually) operate continuously between checks, as an operational matter when the sampling interval is short (e.g. daily), data are processed into single trap-specific encounters of individuals for each sampling occasion (i.e., in Karanth & Nichols (1998, 2000, 2002)). Thus, the first implication is effectively satisfied under the current manner in which the data are processed. With regard to the second implication, in the data we analyse below, there are instances of individuals captured in multiple traps during the same daily sampling interval. Historically, (Karanth & Nichols 1998, 2000, 2002) multiple captures have been used to estimate the distance moved for calculation of the buffer around the trap array, but not in the definition of the encounter histories. In our analysis, we regard the multiple captures as *iid* multinomial trials with the belief that, for low capture rates, this should be reasonably efficient as the multiple captures are providing information on the movement process (see below). Moreover, given the scarcity of the data (low number of captures of each individual), this is preferred to discarding data.

### Modelling exposure to trapping

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

Our development of the specific form of the cell probabilities **π** is based on thinking about exposure to trapping that results from animal movements about their activity centre **s**. To motivate this ‘movement/exposure process’ in the context of trapping arrays, we draw on a related model described by Royle & Young (2008; henceforth ‘RY’). RY described a survey of lizards based on repeated searches of quadrats using a team of observers. They described the encounter process explicitly in terms of individual movements. In fact, in camera- trapping problems, the linkage between encounter and movement/exposure is imprecise because exposure to trapping and encounter does not occur independently. Technical ‘exposure’ to a camera trap necessarily yields a detection (unless the camera malfunctions, which is not admitted explicitly in the encounter history data). Nevertheless, the underlying movement of individuals is fundamental to the encounter process, and thus, we motivate construction of the **π**_{i} probabilities in equation 1 using that model.

RY assumes that individuals move around their activity centre *s*_{i} according to a bivariate normal distribution. That is, locations on consecutive days (**x**_{it}) are independent draws from a normal distribution having mean *s*_{i}and standard deviation σ:

**x**_{it} ~ Normal(**s**_{i}, σ^{2})

If we conceptualize a ‘trap’ as having a physical area over which individuals are exposed, say *χ*, then the probability that individual *i* is exposed to capture during a survey is

- (eqn 2)

where *k*() is the bivariate normal pdf.

This movement model provides a concise description of exposure to capture but does not provide a characterization of whether an individual is captured or not. Individual probability of capture *p*_{i} is naturally expressed conditional on exposure:

*p*_{i} = Pr(capture individual *i*) = Pr(capture|exposed) Pr(exposed).

This provides a partitioning of the capture process into information about detectability, Pr(capture|exposed), and spatial information, Pr(exposed), resulting from the juxtaposition of individuals with the trap array. RY assumed uniform capture intensity, so that

Here there is individual heterogeneity in the probability of exposure due to the location of individuals relative to the trap array. RY provided a Bayesian analysis of this model by MCMC, and thus, the integral equation 1 is not computed explicitly. Rather, it is evaluated by Monte Carlo integration within the general framework for inference provided by a fully Bayesian analysis (described below).

The conceptual adoption of the movement/exposure model to trapping arrays leads to a specific formulation of the cell probabilities in equation 1. Let *χ*_{j} denote the region around trap *j* within which an animal may be exposed. The probability that individual *i* is exposed to trap *j* is the integral:

To a reasonable approximation for small *χ*_{j},

- φ
_{ij} = α × *k*(**x**_{j}; **s**_{i}, σ),

where α = area(*χ*_{j}). If exposure to traps is independent and mutually exclusive, then the total exposure is:

- (eqn 3)

where *k*_{ij} = *k*(**x**_{j}; **s**_{i}, σ). We refer to as the total exposure of individual *i* to the trap array, since it relates directly to φ_{i}. As such, we see that the total exposure of an individual to trapping, resulting from its movements, can be described as an additive function of trap-specific contributions involving the movement kernel *k*().

The probability of capture for individual *i* is therefore

This form suggests that the two constants *r* and α are not uniquely identifiable. However, we can uniquely estimate their product. Note that while 0 ≤ *r* ≤ 1, the product may be greater than 1. Thus, to retain the interpretation of this parameter as a probability, we scale equation 4 appropriately. In particular, since φ_{i} ≤ 1

Define . Therefore, multiply and divide equation 4 by *E*_{max}, and define

- (eqn 5)

Then, *p*_{0} ∈ [0, 1], and *p*_{0} has the interpretation as the probability of capture for the hypothetical ‘most exposed’ individual.

- (eqn 6)

The movement-induced exposure to traps that motivated equation 6 describes the probability of capture in traps as a decreasing function of the distance between activity centre *s* and the trap locations. This basic phenomenon (detection decreasing with distance) is shared with the formulation of the model by Efford (2004). The main distinction is conceptual – here we have motivated the form as arising under explicit biological processes (movement-based exposure to sampling), whereas the construction by Efford (2004) was more phenomenological in origin. As a practical matter, this distinction may not be very important.

### Effective sample area

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

An estimand that is a by-product of the model specification (Royle & Young 2008) is the effective sample area, say *A*_{e}. An intuitive way to describe effective sample area is to think of a discrete version of *S*, consisting of *G* pixels *s*_{1}, ... , *s*_{G}. Imagine that we were to sample *S* by flipping a coin for every pixel and then sampling that pixel if the coin comes up ‘heads.’ In that case, we will survey that pixel, thereby exposing individuals on that pixel to sampling. If the coin has probability π_{i} of coming up heads for pixel *i* then we would expect to sample an area of size

In the context of our spatial capture–recapture model, individuals are flipping the coins, with probabilities

- π
_{i} = 1 − (1 −φ_{i})^{T},

which we can only evaluate under certain conditions (since we have absorbed the constant α into *p*_{0} in equation 4). It might be reasonable to assert that the most exposed individual has φ_{i} = 1 (i.e., it is always susceptible to capture). This is setting (from equation 3) α = 1/*E*_{max}. Whether or not this is reasonable depends on the typical movements of individuals relative to the extent of the trap array. The assumption that φ = 1 when *s* coincides with a trap location seems reasonable in some cases.

### Example

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

A simulated realization under the model is shown in Fig. 1 for *N* = 120 individuals, subjected to capture *T* = 6 times by a 10 × 10 array of traps having unit spacing, using the normal exposure kernel with σ^{2} = 1 and with *p*_{0} = 0·50. The individual activity centres were uniformly distributed over a larger square of 17 × 17 units. Individuals are connected to traps where they were captured by black lines. The probability of exposure, φ, of an individual as a function of **s** is shown in Fig. 2. We see that individuals within the array have very high φ, which decreases rapidly as distance from the outer band of traps increases.

The summary statistics for *p*_{i} for the 120 individuals were (0·003, 0·063, 0·181, 0·3981) for first quartile, median, mean, and third quartile, respectively. Evidently, there is extreme heterogeneity in detection probabilities under this model, with one-half of the population having *p*_{i} < 0·063 while the mean is 0·181. For the simulated *T* = 6 study, 55 unique individuals were captured a total of 118 times, the detection frequencies being 16, 21, 12, 6 individuals captured 1, 2, 3, 4 times, respectively (none captured 5 or 6 times). An analysis of this simulated data set can be found in Supporting Material Appendix S1.

### Conditional formulation

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

The model is a spatial version of the standard null model referred to as Model *M*_{0}. A reformulation of the model is convenient for extending the model to, for example, allow for modelling covariates that influence detection probability, *p*_{0}. In particular, we express the observation model as the product of two components: (i) the probability that an individual is captured (‘at all’), and (ii) the probability of capture in each trap conditional on capture. That is, given that an individual is captured, what are the trap-specific probabilities of capture? Let *y*_{it} be a binary indicator of capture for individual *i*. Thus, (*y*_{i}_{1}, *y*_{i}_{2}, ... , *y*_{iT}) is a traditional encounter history. Then,

where *p*_{i} is the individual probability of capture, equation 4. Then, let γ_{ij} be the probability of capture in trap *j* given that *y*_{it} = 1. From equation 6, we obtain

where *k*_{ij} is the bivariate normal kernel.

Thus, conditional on capture, the trap-of-capture is a categorical random variable having probabilities γ_{ij} for *j* = 1, 2, ... , *R*. This is convenient for modelling detection covariates because it separates (in the model) the parameter *p*_{0} from the spatial information contained in the trap identities. Thus, the model for the traditional encounter histories is:

It is possible to model temporal effects in *p*_{0t} such as time effects, behavioural effects, etc.

### Bayesian analysis by data augmentation

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

We note that this model is conceptually similar to a broad class of capture–recapture models known as individual covariate models (Williams *et al*. 2002; p. 301). In the present situation, the activity centres *s*_{i} represent an individual covariate that is observed with error. Indeed, the model is most structurally similar to distance sampling with distance measurement error (Royle & Dorazio 2008, Ch. 7). Bayesian analysis of classical individual covariate models that motivates the following analysis can be found in Royle (2008). We adopt that basic strategy here.

To conduct a Bayesian analysis of the model requires that we describe prior distributions for the parameters *N*, *p*_{0}, and σ. A natural choice of priors that reflect the absence of information about these parameters is to assume a discrete uniform prior for *N* on the integers 0, ... ,* M* for some large value of *M *[i.e., *N *∼ Du(0, *M*)]. A customary vague prior for *p*_{0}, which, recall has support on [0, 1], is the uniform (0, 1) prior. Choice of *M* is not a critical consideration except that it must be chosen large enough so as to not truncate the posterior distribution of *N* which can be checked after a trial analysis. For σ, we use a proper uniform prior distribution with a large upper-bound.

In principle, the model could be analysed under this prior specification by conventional MCMC methods for sampling from the posterior distribution (Link *et al*. 2002). However, we note that, because *N* is unknown, the dimension of the parameter space (i.e., the number of random effects *s*_{i}) is also unknown. As such, each time that a new draw of *N* is made from the posterior distribution, the number of random effects (activity centres) changes. Properly updating parameters in this setting has proved to be a challenging problem of some technical complexity. This technical problem motivated an approach to the analysis of such models using the method of data augmentation in Royle *et al*. (2007). We provide a brief description of data augmentation here. See Royle *et al*. (2007) and Royle & Dorazio (2008) for technical details and applications.

A heuristic description of data augmentation is that it arises by simply adding excess ‘all zero’ encounter histories to the data set. That is, for *M* sufficiently large, then we can augment the data set with *M* − *n* all-zero encounter histories. We then recognize that the resulting model for the augmented data is a zero-inflated version of the model for the complete data set (i.e. as if *N* were known). In models with individual effects, data augmentation is a convenient framework because it allows us to retain a maximal set of random effects in the (augmented) data set, and their values are updated at each iteration of the MCMC algorithm.

Formally, data augmentation is justified as a re-parameterization of the model that arises under the discrete uniform prior for *N*. In particular, note that the discrete uniform prior for *N* can be constructed by specifying a binomial prior for *N*: *N *∼ Bin(*M*, ψ), and then placing a uniform(0, 1) prior on ψ. When ψ is removed from the binomial component by integration, the result is *N *∼*Du*(0, *M*). While this may seem only a mathematical curiosity, it suggests a convenient implementation of Bayesian analysis for such models. Namely, we can think of the discrete uniform prior as suggesting a super-population consisting of *M* individuals, where *M* − *n* of them have corresponding ‘all-zero’ encounter histories (because they were not captured). We recognize that some of the *M* individuals are fixed zeros, whereas some of them are sampling zeros – they correspond to individuals in the population that were not captured. This can be formalized by the introduction of a set of latent indicator variables *z*_{1}, *z*_{2}, ... , *z*_{M} such that *z*_{i} = 1 if individual *i* is a member of the population and *z*_{i} = 0 if individual *i* is a fixed zero. We assume *z*_{i} ~ Bernoulli(ψ). To implement data augmentation, we augment the *n* observed encounter histories with *M* − *n*‘all-zero’ histories, and then specify the model for the augmented data set in terms of the zero-inflated version of the ‘known-*N*,’ model. For most capture–recapture models (including the present), the model for the total number of detections, *y*_{i} (see Conditional formulation), is

*y*_{i} ~ Bin(*p*_{i}) if *z*_{i} = 1,

Under data augmentation, the parameter ψ formally replaces the parameter *N*, the two being related by the prior specification *N *∼ Bin(*M*, ψ).

Because the dimension of the parameter space is fixed, this facilitates a formal analysis by standard methods of MCMC. While developing the MCMC algorithm for analysis of the augmented data is straightforward under this model, we avoid those technical details because the model can also be implemented in the freely available software package WinBUGS (Gilks *et al*. 1994) as demonstrated in Supporting Material Appendix S1 (see also Gardner, Royle & Wegan 2008).

### Analysis of the tiger camera-trapping data

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

The tiger population of Nagarahole reserve in the state of Karnataka, southwestern India, has been studied via camera-trap methods from 1991 until the present (e.g. Karanth 1995; Karanth & Nichols 1998; Karanth *et al*. 2006). Here we present analyses of data collected during 24 January–16 March 2006 from 120 trap stations (Fig. 3). Two camera traps (unambiguous identification requires photographs of both flanks) were placed at each station. Not all 120 trap stations were operated simultaneously. Instead, the reserve was subdivided into four blocks of approximately 30 trapping stations each, and each block was run for 12 consecutive days. Then, cameras were moved to the next block for another 12 days and the process repeated until all four blocks were sampled. This design follows sample design 4 of Karanth and Nichols (2002, p. 133).

This study resulted in encounter history data on 45 individuals. Each encounter history was a vector of length *T* = 48, with a 0 recorded for each occasion where the animal was not detected and the location (station number) recorded for each capture. For instances where an individual was captured > 1 time in the same survey period, the additional captures were regarded as distinct recaptures. While our motivation was purely practical for doing this, we believe it is statistically efficient when the capture rate is low.

We refer to the 120 trap locations across the four blocks as ‘traps’, recognizing that the physical cameras were absent from most of the locations at any particular time. As a result of this trap movement, individuals could not be detected at approximately three-quarters of the traps during any particular interval. This is simple to handle in the model by defining a binary indicator of trap operation during each day, say *o*_{tj} for survey *t* and trap *j*, and then redefining exposure accordingly. For example, equation 4 becomes

Simply put, this accumulates exposure to trapping based only on traps that are in operation at time *t*.

For the analysis of these data, we excluded areas that were judged to be non-habitat within a 15-km buffer area containing the trap array (Fig. 3). This renders the definition of a uniform prior for the activity centres difficult. As such, we described this region of suitable habitat by a fine grid of 9961 equally spaced points, each representing approximately 0·336 km^{2} over the buffered region. Of these, 4898 (1645·7 km^{2}) were judged to represent suitable habitat. The activity centres *s*_{i} were assumed to be uniformly distributed over these 4898 points. We developed an implementation of the model for this discrete-*S* situation in the r software (Ihaka & Gentleman 1996). Bayesian analysis of the model was conducted using data augmentation, augmenting the data set with 800 ‘all zero’ encounter histories. The MCMC algorithm was run for 52 000 iterations, the first 2000 were discarded, and posterior summaries were computed from the remaining 50 000 iterations. The coordinate system was scaled so that a standard unit was 5 km, and thus also are the units of σ. Posterior summaries are provided in Table 1.

Table 1. Posterior summaries of model parameters for the tiger camera-trapping data based on *n* = 45 observed individuals. Here, *A*_{e} is the area exposed to trapping or the effective sample area of the 120-trap array. It is a derived parameter under the model. *N* is the number of tiger activity centres in the population exposed to sampling and *D* is the density per 100 km^{2}, ψ is the data augmentation parameters, *p*_{0} is detection probability, and σ is the parameter in the bivariate normal pdf Parameter | Mean | SD | 2·5% | Median | 97·5% |
---|

ψ | 0·278 | 0·056 | 0·182 | 0·272 | 0·404 |

σ | 0·407 | 0·050 | 0·325 | 0·402 | 0·517 |

*p*_{0} | 0·083 | 0·018 | 0·052 | 0·081 | 0·123 |

*N* | 233·93 | 45·20 | 158·00 | 229·00 | 337·00 |

*D* | 14·300 | 2·800 | 9·600 | 14·000 | 20·500 |

*A*_{e} | 725·62 | 34·53 | 666·88 | 722·91 | 800·36 |

The posterior median of *N*, the population size for the 1645·7 km^{2} area, is 229. This is the number of individuals existing within the region over which **s** was assumed uniformly distributed (i.e., suitable habitat). We see that the posterior mean of ψ (the data augmentation parameter) is approximately 0·28 and the posterior mass is concentrated well away from the boundary ψ = 1. This indicates that the posterior of *N* was not truncated as a result of too few all-zero encounter histories. In Table 1, *D* is the density per 100 km^{2} of suitable habitat, and the posterior mean is 14·3 with a 95% posterior interval of (9·6, 20·5). Based on earlier work on tiger ecology and knowledge of the relationship of tiger density to prey abundance (Karanth 1995; Karanth *et al*. 2004b), as well as from estimates of tiger densities at this site over multiple years (Karanth *et al*. 2006), we believe the density estimate generated in this study is reasonable. Finally, we note that the estimate *p*_{0} = 0·083 (posterior mean) corresponds to the detectability of some hypothetical individual having maximal exposure to the trap array.

Putting the abundance estimate in context, fitting a classical closed-population model with heterogeneous detection probabilities (so-called ‘model *M*_{h}’), we obtain *N̂* = 111.7. It is impossible to compare this directly with the posterior mean of *N* produced under our model because the latter corresponds to a precise area over which the activity centres **s** were assumed to be uniformly distributed. Conversely, the estimate of *N* obtained under *M*_{h} is, at best, an estimate of the number of individuals exposed to sampling, for which we do not know the precise area. However, our model produces an estimate of the effective sample area (the posterior mean is *c*. 725 km^{2}, see Table 1). The probability of exposure as a function of location is shown in Fig. 4. We might hope that the estimate of *N* under model *M*_{h} is relevant to this area. If so, the comparable estimate of density would be approximately 15·4 individuals per 100 km^{2}.

### Discussion

- Top of page
- Summary
- Introduction
- Model development
- Modelling exposure to trapping
- Effective sample area
- Example
- Conditional formulation
- Bayesian analysis by data augmentation
- Analysis of the tiger camera-trapping data
- Discussion
- Acknowledgments
- References
- Supporting Information

Capture–recapture studies of many species, based on camera traps, DNA sampling, and traditional traps that physically capture individuals, produce individual encounter history data with auxiliary spatial information related to location of capture. It has long been recognized that this information could be used to produce unbiased estimates of density when the population is not geographically closed due to movement of individuals (Dice 1938; Smith *et al*. 1971; Wilson & Anderson 1985; Efford 2004).

For camera-trap studies, the standard method for density estimation, as exemplified by Karanth & Nichols (1998, 2000, 2002), Trolle & Kéry (2003), and other recent studies, has been based on the use of conventional closed-population estimators with a buffer strip estimated from observed movements of individuals. In this study, we developed a hierarchical model for spatial capture–recapture data which integrates the detection information provided by the encounter history data as well as the auxiliary spatial information contained in the trap locations of capture. The model describes both the distribution of individuals in space (i.e. their activity centres), which are latent parameters regarded as random effects in the model, as well as the encounters of individuals that result from their movement-induced exposure to trapping (Royle & Young 2008). We adopt a Bayesian analysis of the hierarchical model, facilitated by the use of data augmentation (Royle *et al*. 2007).

Observations from camera-trap studies have a number of distinctive characteristics compared to conventional ‘traps’ that physically capture or kill individuals. Importantly, camera traps can detect multiple individuals during a single survey occasion (a night in our example). Our model accommodates this by the assumption that capture of individuals are independent multinomial trials (equation 1). Secondly, a camera trap can, conceivably, detect the same individual multiple times during a sample occasion. However, current data processing protocols (in our study) render the data into unique binary detection events. This is probably desirable when the sampling interval is short (daily) since within-day movements are likely to be highly correlated. Finally, an individual can be encountered in multiple traps during a sampling occasion. Historically, multiple captures have been used to estimate the distance moved for calculation of the buffer around the trap array, but not in the definition of the encounter histories. Our model does not properly admit such multiple captures. However, in our analysis of the tiger data, we regarded such multiple captures as ordinary recaptures (*iid* multinomial trials). This is more efficient than discarding such data and, we believe, at low capture rates, there is little loss of efficiency treating the data in this manner.

Despite this shortcoming, our formulation offers several practical advantages to studies that involve non-invasive animal ‘captures’ from photographs, DNA or other types of individual identification. First, our model explicitly includes a biological explanation for detection heterogeneity in terms of proximity of individuals to the trap array. Thus, it possibly addresses some concerns (Link 2003) regarding identifiability of abundance using standard models. More importantly, because the point process is parameterized explicitly in the hierarchical model, we believe that this component of the model can, in the future, be generalized considerably and in substantive, meaningful directions. For example, in carnivore populations, it is likely that the ‘activity centres’ are not independent and uniformly distributed over *S*. Individuals probably interact with neighbours and their density varies in response to habitat conditions. Both of these features can be integrated into the framework which we have described. In particular, in the discrete-*S* formulation of the model used in the analysis of the tiger data, one may associate covariates with each pixel, and model density as a function of those covariates. To model interactions among individuals, it seems natural to formulate the point process in terms of Markovian dependence, wherein successive points are conditioned on the nearness and even ‘type’ (male, female; social status; age) of neighbours. Finally, the possibility exists of formulating point process models that admit explicit spatio-temporal structure, representing the survival and recruitment of individuals. As such, we can make efficient use of available data to potentially yield great improvements in the precision of density estimates. We are pursuing a number of extensions of the model in the context of camera-trapping studies.

We noted that our model is structurally similar to the model described by Efford (2004) for which he devised a simulation-based method of fitting based on inverse prediction. We regard our development here as a formalization of the inference framework, using a Bayesian hierarchical model. We note that formalization of the inference framework for such models was also recently addressed by Borchers & Efford (2008), who developed an integrated likelihood framework for inference. Our formulation is distinct with regard to formulation of the model, the manner in which inference is carried out, and implementation of the model. With regard to formulation, our development was motivated by exposure of individuals induced by movement about their activity centres. While it is more heuristic in this case (in contrast to Royle & Young 2008), since the processes are not separately identifiable, the explicit movement model provides a conceptual linkage between detection probability and ‘temporary emigration’ (Kendall *et al*. 1997) which we find appealing. One technical distinction with the Borchers & Efford (2008) formulation is that we condition on the population size *N*, and then analyse what is often referred to (Borchers *et al*. 2002) as the ‘joint likelihood’, using data augmentation. Conversely, Borchers & Efford (2008) adopt the conditional likelihood approach under which the likelihood is formulated conditional on *n* (Sanathanan 1972). We believe that there are important advantages of the explicit conditioning on *N*, and the hierarchical formulation of the model that this allows. In particular, as we demonstrated (Supporting Material Appendix S1), analysis of the Bayesian hierarchical formulation based on data augmentation can be achieved in the flexible and freely available software WinBUGS requiring little or no specialized knowledge. As such, implementation and extension are widely accessible to practitioners. More importantly, as analysis of the model is conditional on the individual point locations, this component of the model can be the focus of additional model structure describing the distribution of point locations, or relationships among point locations (e.g. as described in the previous paragraph). We believe that it would be generally difficult to carry out the integration necessary to estimate parameters for more complex point-process models based on integrated likelihood as developed by Borchers & Efford (2008). Indeed, parametric inference under such models is largely simulation-based at the present time for this reason (e.g. see Møller & Waagepetersen 2004). We expect such extensions to be the focus of considerable research in the future, especially as passive methods of DNA sampling and technological advances render spatial capture–recapture data easier and cheaper to obtain.