A Spatially Explicit Capture–Recapture Model for Partially Identified Individuals When Trap Detection Rate Is Less than One

Abstract

Spatially explicit capture–recapture (SECR) models have gained enormous popularity to solve abundance estimation problems in ecology. In this study, we develop a novel Bayesian SECR model that disentangles two processes: one is the process of animal arrival within a detection region, and the other is the process of recording this arrival by a given set of detectors. We integrate this complexity into an advanced version of a recent SECR model involving partially identified individuals (Royle JA. Spatial capture-recapture with partial identity. arXiv preprint arXiv:1503.06873, 2015). We assess the performance of our model over a range of realistic simulation scenarios and demonstrate that estimates of population size N improve when we utilize the proposed model relative to the model that does not explicitly estimate trap detection probability (Royle JA. Spatial capture-recapture with partial identity. arXiv preprint arXiv:1503.06873, 2015). We confront and investigate the proposed model with a spatial capture–recapture dataset from a camera trapping survey of tigers (Panthera tigris) in Nagarahole study area of southern India. Detection probability is estimated at 0.489 (with 95% credible interval (CI) [0.430, 0.543]) which implies that the camera traps are performing imperfectly and thus justifying the use of our model in real world applications. We discuss possible extensions, future work and relevance of our model to other statistical applications beyond ecology.

AMS classification codes: 62F15, 92D40

Keywords

Capture-recapture survey detection probability hierarchical bayes SECR model

1 Introduction

Understanding the dynamics of wildlife populations is central to answering ecological questions and forms the basis for conservation. However, owing to sampling problems (primarily imperfect detection and spatial sampling)^{1, 2} it is a major challenge to accurately characterize wildlife populations from field data. The challenge is greater when the species is cryptic, occurs at low density and often elusive, as with large carnivores^{3, 4} and rare ungulates.⁵ This problem has motivated the development of several tailor-made statistical estimators over the years.^{1, 6, 7, 8}

More recently, such classes of ecological problems have been addressed elegantly using hierarchical models, where a distinction between the “state process” (the true state of the ecological system that is of main interest) and the “observation process” (the way in which observations occur during sampling) is explicitly defined in the modelling.^{9, 10} Based on this philosophy, the development of spatially explicit capture–recapture models (hereafter SECR models)^11–13 for estimating animal abundance has witnessed explosive growth⁸. Under this approach, observation data about individuals are recorded by spatial array of detectors (such as camera traps, hair snares and fixed traps) within an area of interest over a fixed time period. SECR models utilize the spatial locations of animal detections to explicitly enable inference about the spatial distribution of animals in addition to estimating animal abundance and has seen wide application for globally threatened species.^{8, 14}

However, all these inferences are drawn from data emanating when animals pass through a spatial array of detectors. Currently, SECR models do not disentangle the process of animal arrival within a detection region from the process of recording this arrival by a given set of detectors. Furthermore, local detector-level effects may explain whether an animal will pass through the detection region or not. For example, workers often use baits to attract animals to trap stations when animals are in the vicinity and investigators are often interested to understand animal response to such detectors. However, an animal passing through a detection region will not necessarily mean that the detectors will record this event perfectly. The variables affecting detection and animal arrival are generally quite different. Different types of detectors (e.g., hair snare versus camera) perform differently under similar conditions. Further, temperature levels at the detection region can also have substantial effect on detector performance (e.g., on passive infrared cameras). When detection rates in recorded samples are low due to failure or malfunction of the detectors, these can result in data with uncertain identities or “partially identified individuals”.¹⁵ Recently, Augustine et al.,¹⁶ more specifically describing paired camera trap surveys, utilizes separate parameters to model the relative encounter frequencies when both cameras function versus when only one of the two cameras functions. As an improvement, in this article, we provide a probabilistic approach that describes the exact mechanism by which we obtain these different events. If we assume that over a fixed number of detection attempts at a location we will detect an animal with certainty, then a newer development¹⁷ can be used to address such problem. However, this is a restrictive assumption to meet in the real world.

Such surveys also result in partially identified individuals that require reconciliation in the modelling process. This area has received attention in the recent past, for example, McClintock et al.¹⁸ has developed a model for photographic and genetic capture–recapture survey, Wimmer et al.¹⁹ has developed a model that deals with partially identified individuals in the context of live captures. More recently, Royle JA¹⁵ and Augustine et al.¹⁶ have developed a model for partially identified sample where they demonstrate that spatial locations of captures assist in improved reconciliation of partially identified individuals.

Study objectives. In this study: (a) we develop an SECR estimator that disentangles the process of animal arrival within a detection region from the process of device performance by utilizing information from recorded captures on multiple devices at sampled locations. Further, we integrate this complexity into an advanced version of the SECR model involving partial identifications of individuals.^{15, 16} (b) We assess the performance of our model over a range of realistic simulation scenarios typically faced in field ecological studies of large, charismatic, wildlife species. (c) We confront this model with a spatial capture–recapture dataset from a long-term camera trapping survey on tigers.^{20, 21} (d) We discuss possible extensions, future work and relevance of our model to other statistical applications beyond ecology.

2 Methods

In typical photographic capture–recapture surveys^{13, 22} an array consisting of camera trap stations is placed to sample a species of interest. Each station comprises of two cameras (detectors) facing each other and they are meant to independently capture both flank images of animals. If the species is naturally marked, individuals can be identified by their unique markings. While this example motivated our specific model development, we can also envision many scenarios where more than one detector and different detection types may be used to extract features of individual identity of animals.¹⁴

2.1 Modelling Approach

We utilize the hierarchical modelling philosophy⁹ to formulate a model to address the problem of imperfect detection of detectors in spatial capture–recapture models. A list of notations used in this article is provided in Table 1 (and another table of notations is given in Appendix Table 11).

2.1.1 State Process

Consider a population of individuals of certain species that reside within a bounded geographic region $V (\subset ℝ^{2})$ that has scientific or operational relevance. Each individual is assumed to be located following a point process¹² by having an activity centre located at $s (\in V$ ). Let S denote an array of latent variables defining the locations of the N (unknown) animals in the study. For the ease of computation and other technical advantages (described later), we define $N \sim binomial (M, ψ)$ , where M represents the maximum possible number of individuals present within $V$ and $ψ$ is a thinning parameter to indicate the proportion of M that represent the real population. It should be noted here, that the N animals located at S are assumed to move around S according to some prescribed density kernel during the period of sampling. However, previous SECR models consider this movement inherently as part of the observation process.^{12, 13, 15}

2.1.2 Observation Process

We suppose that a spatial array of J trap stations are placed in the state space $V$ . We consider the situation where two detectors are deployed at each of these trap stations and are kept active for K sampling occasions. Often nightly time intervals of fixed length (e.g., 6 pm to 6 am) are taken as sampling occasions with sufficient time gap in between two such successive time intervals. This time gap is kept to meet the independence assumption of the capture observations made in any two successive sampling occasions. We assume that each detector captures some mutually exclusive attribute of an individual.

Let $y_{ijk}^{(1)}$ and $y_{ijk}^{(2)}$ represent the binary capture outcomes for an individual i at trap station xj on sampling occasion k for detectors 1 and 2, respectively. Here the indices 1 and 2 correspond to two distinct features of an individual's identity. For example, in a camera trap survey, detector 1 can correspond to left flank image and detector 2 to right flank image. We note here that both the flank patterns of a tiger are distinct and a set of single flank detections (e.g., only left flank detections) from a tiger can itself constitute a detection history. It is possible to ascertain the identity of an individual without a doubt only when both the detectors record the individual simultaneously at the same trap station on at least one occasion during the survey and then the individual can be called as “fully identified”. This means, a tiger is fully identified when both of its flanks are photographed simultaneously at the same trap station on at least one occasion during the survey. Camera trap devices also record the date and time of the photographs as metadata. This information enables the investigators to tell whether the flanks captured within a single occasion belong to the same individual or not. We will suppose that at the end of this survey n individuals are captured and fully identified. Thus, the recorded observations obtained by detectors 1 and 2 are individual-specific detection histories, $Y_{obs}^{(1)} = ((y_{ijk}^{(1)}))$ , $Y_{obs}^{(2)} = ((y_{ijk}^{(2)}))$ , respectively. This implies that i indices of $Y_{obs}^{(1)}$ and $Y_{obs}^{(2)}$ are in the same order and each of the two arrays are of dimension n × J × K. Further, the paired binary outcomes $y_{ijk} = (y_{ijk}^{(1)}, y_{ijk}^{(2)})$ give rise to bilateral capture–recapture data for each individual i at location xj on occasion k. So, for an individual i, we denote the bilateral capture history by $Y_{i, obs} = (Y_{i, obs}^{(1)}, Y_{i, obs}^{(2)}) = ((y_{ijk}^{(1)}, y_{ijk}^{(2)}))_{1 \leq j \leq J, 1 \leq k \leq K}$ , which is of dimension 2 × J × K.

Example 2.1. In a survey, consider paired detectors (1 and 2), deployed at each of 3 (= J) trap stations and active for 4 (= K) sampling occasions. From this survey, we suppose that 2 (= n) distinct individuals were fully identified since we obtained at least one simultaneous capture (caught at the same time in both detectors) during the survey. Detection histories are thus presented in Table 2. For each of the fully identified individuals, the dimension of the detection history data set is 2 × 3 × 4.

The observation process described above with the example entails two problems that need to be addressed simultaneously: (a) determining whether an animal passes through the detection region (in trap station) in the face of imperfect detection of detectors and (b) reconciling partially-identified individuals. While the second problem has been recently addressed by Royle JA¹⁵ and Augustine et al.¹⁶, our emphasis in this article is to address the first and integrate it into the solution of the second.

Table 1.

Notations of data which are used in this article

Data	Definition
$x_{j} = (x_{j 1}, x_{j 2})^{'}$	$j^{th}$ trap station for detectors.
$y_{ijk}^{(1)}$	$y_{ijk}^{(1)} = 1$ if individual i is detected in detector 1 at trap station
	xj on occasion k, $y_{ijk}^{(1)} = 0$ if not detected in detector 1.
$y_{i \cdot \cdot}^{(1)} = \sum_{j = 1}^{J} \sum_{k = 1}^{K} y_{i j k}^{(1)}$	Number of times individual i got detected
	in detector 1 over J trap stations and K occasions.
$y_{ijk}^{(2)}$	$y_{ijk}^{(2)} = 1$ if individual i is detected in detector 2 at trap station
	xj on occasion k, $y_{ijk}^{(2)} = 0$ if not detected in detector 2.
$y_{i \cdot \cdot}^{(2)} = \sum_{j = 1}^{J} \sum_{k = 1}^{K} y_{ijk}^{(2)}$	Number of times individual i got detected
	in detector 2 over J trap stations and K occasions.
n	Number of fully identified individuals, each of them is captured
	by both the detectors on at least one occasion.
$Y_{obs}^{(1)} = ((y_{ijk}^{(1)}))$	Array of individual specific capture histories obtained by
	detector 1 (dimension n × J × K).
$Y_{obs}^{(2)} = ((y_{ijk}^{(2)}))$	Array of individual specific capture histories obtained by
	detector 2 (dimension n × J × K).
$Y^{(1)}$	Zero augmented array of individual specific capture histories
	corresponding to detector 1 (dimension M × J × K).
$Y^{(2)}$	Zero augmented array of individual specific capture histories
	corresponding to detector 2 (dimension M × J × K).
$u_{obs} (\subset u)$	Vector of “recorded” binary observations on sexes of the
	captured individuals.
$Y^{(2 *)}$	Reordered $Y^{(2)}$ according to L (dimension M × J × K).
$n_{ij} = \sum_{k = 1}^{K} I (y_{ijk}^{(1)} + y_{ijk}^{(2)} > 0)$	Number of times individual i got detected at trap j
	on at least one of its sides over K occasions.
$n_{i \cdot} = \sum_{j = 1}^{J} nij$	Number of times individual i got detected on at least
	one of its sides over J traps and K occasions.

Source: The authors.

Note: Bold symbols represent collections (vectors) of parameters.

Table 2.

An Example of Detection Histories Generated from Equation (2.1)

	Occasion Trap	Detector 1				Occasion Trap	Detector 2
	Occasion Trap	1	2	3	4	Occasion Trap	1	2	3	4
Fully-identified individual 1	1	0	①	0	0	1	1	①	0	0
	2	0	0	0	0	2	0	1	1	0
	3	1	0	0	0	3	0	1	0	0
Fully-identified individual 2	1	0	0	0	①	1	1	1	0	①
	2	1	0	0	1	2	0	0	0	0
	3	0	1	0	0	3	1	0	1	0
Partially-identified individual	1	0	1	0	1	1	—	—	—	—
	2	1	0	1	0	2	—	—	—	—
	3	0	0	0	1	3	—	—	—	—
Partially-identified individual	1	—	—	—	—	1	1	0	1	0
	2	—	—	—	—	2	0	1	0	0
	3	—	—	—	—	3	1	0	1	0

Source: The authors.

Note: The survey yields 2 fully-identified individuals and detection histories of partially-identified individuals. The circled outcome corresponds to the detection event that assists in the reconciliation of an individual identity. We note that it is necessary for the experimental design to ensure that there are other sources of metadata available (e.g., exact time of detection) to ensure such a reconciliation. For example, individual 1 was fully-identified owing to the simultaneous capture event at trap 1 on occasion 2 by both the detectors. Due to mutual exclusivity of capture events in the detection histories of the partially-identified individuals, we are uncertain about whether these histories correspond to two different individuals or to the same individual.

2.2 Model Development

2.2.1 Disentangling Animal Entry to Trap Station and Detection in Spatially Explicit Capture–Recapture Models

We note that for an animal to be observed by a detector at a given location and occasion, the animal (a) has to pass through the detection region and (b) has to be captured by the detector(s). We aim to disentangle these two processes by utilizing a hierarchical model. From Example 2.1, there are four types of detection histories observable at a given trap station on a given sampling occasion: “11” (observed by both detectors), “10” (observed by detector 1 but not by detector 2), “01” (not observed by detector 1 but observed by detector 2) and “00” (not observed by either detector). The first three histories (“11”, “10” and “01”) conclusively state that the animal passed through detection region in the trap station since we have one observation. But in the fourth case (“00”), we are presented with two possibilities: (a) the animal passed through the detection region and both detectors failed to record this event or (b) the animal did not pass through the detection region.

Defining the state process of animal entry to trap station. Let E_ijk be a latent variable that indicates whether individual i has entered a trap station xj on occasion k $(E_{ijk} = 1)$ or not $(E_{ijk} = 0)$ . Further, let $π_{ij} = P (E_{ijk} = 1)$ be the probability of the corresponding event of trap entrance. We model the probability that an individual i passes through a trap station xj as a decreasing function of distance between its activity centre si and trap station xj. A typical model to describe “trap entry probability” $π_{ij}$ is the Gaussian form of the type $π_{ij} = p_{0} exp (- d_{ij}^{2} / (2 σ^{2}))$ , where $d_{ij} = d (s_{i}, x_{j}) = ‖ s_{i} - x_{j} ‖$ is the Euclidean distance between points si and xj, p₀ is called “baseline trap entry probability” and σ quantifies the rate of decline in trap entry probability as the distance between individual activity centre si and trap station xj increases. We note with interest, that previous SECR models regard this modelling structure as part of the observation process such that p₀ is instead regarded as the “baseline encounter probability” and σ is instead regarded as the rate of decline in detection probability as the distance between individual activity centre si and trap station xj increases.^{12, 13} As in most SECR models, it is assumed that trap entry at location j on occasion k does not depend on presence or absence anywhere on any previous occasions.

We proceed with the Gaussian form in our development, while recognizing that there can be many other options to define the rate of decline in animal trap entry probability to represent other realities. Further, it is often the case that sex acts as an important covariate to define the extent of animal movement.^{14, 23, 24} For example, often males and females have different extents of spatial movement, defined by the parameter σ in our development. We then define σ as the following: $σ (u_{i}) = σ_{m}$ , if $u_{i} = 1$ , that is, individual i is a male; $σ (u_{i}) = σ_{f}$ , if $u_{i} = 0$ , that is, individual i is a female. Here each u_i is assumed to follow the Bernoulli distribution with parameter θ, θ being the probability that an arbitrary individual in the population is male. Additionally, the explicit recognition of these sex effects will, later, be very helpful in synchronizing the partially identified individuals as seen in Example 2.1 because we can utilize the fact that sex is ascertained for each individual i and we constrain the sychronization to probabilistically linking partially identified individuals of only the same sex.

Defining the observation process at trap stations. Here, we introduce the detection probabilities for our observation model conditional on the entry at a trap station. Let $p_{j}^{(12)}$ be the probability of detection by both detectors simultaneously at xj on a sampling occasion, $p_{j}^{(1 \bar{2})}$ be the probability of detection only by detector 1 at xj on a sampling occasion and $p_{j}^{(\bar{1} 2)}$ be the probability of detection only by detector 2. Collection of these different probabilities is denoted by $p^{(12)} = (p_{1}^{(12)}, p_{2}^{(12)}, \dots, p_{J}^{(12)})$ , $p^{(1 \bar{2})} = (p_{1}^{(1 \bar{2})}, p_{2}^{(1 \bar{2})}, \dots, p_{J}^{(1 \bar{2})})$ , $p^{(\bar{1} 2)} = (p_{1}^{(\bar{1} 2)}, p_{2}^{(\bar{1} 2)}, \dots, p_{J}^{(\bar{1} 2)})$ and $p = (p^{(12)}, p^{(1 \bar{2})}, p^{(\bar{1} 2)})$ . As described in Section 2.1, $y_{ijk}^{(1)}$ , $y_{ijk}^{(2)}$ are binary responses corresponding to detections on detectors 1 and 2, respectively, defined only when $E_{ijk} = 1$ . However, when $E_{ijk} = 0$ , both $y_{ijk}^{(1)}$ , $y_{ijk}^{(2)}$ have degenerate distributions at 0.

2.2.2 Development of the Joint Posterior Density of All Parameters

Joint posterior density to disentangle animal trap entry to trap stations and trap detections. When $E_{ijk} = 1$ , the conditional probability of detection in detector 1 is $p_{j}^{(1)} = p_{j}^{(12)} + p_{j}^{(1 \bar{2})}$ and the conditional probability of detection in detector 2 is $p_{j}^{(2)} = p_{j}^{(12)} + p_{j}^{(\bar{1} 2)}$ . If we assume both the detectors are of the same quality, $p_{j}^{(1)} = p_{j}^{(2)} = ϕ_{j}$ for each j. Consequently, we have $p_{j}^{(1 \bar{2})} = p_{j}^{(\bar{1} 2)} = ϕ_{j} - p_{j}^{(12)}$ . We note here that, $E_{ijk} = 1$ if $y_{ijk}^{(1)} + y_{ijk}^{(2)} > 0$ . In contrast, E_ijk is unobserved if $(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 0)$ . We can then construct probability arguments to compute probabilities of various data outcomes for an individual i at trap station xj on sampling occasion k:

$\begin{matrix} P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (1, 0)] = P [E_{ijk} = 1] P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (1, 0) | E_{ijk} = 1] \\ = π_{ij} (ϕ_{j} - p_{j}^{(12)}) = P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 1)], \end{matrix}$

$\begin{matrix} P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (1, 1)] = P [E_{ijk} = 1] P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (1, 1) | E_{ijk} = 1] = π_{ij} p_{j}^{(12)}, \end{matrix}$

$\begin{matrix} P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 0)] = P [E_{ijk} = 0] P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 0) | E_{ijk} = 0] + \\ P [E_{ijk} = 1] P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 0) | E_{ijk} = 1] \\ = (1 - π_{ij}) + π_{ij} (1 - (2 ϕ_{j} - p_{j}^{(12)})) = 1 - π_{ij} (2 ϕ_{j} - p_{j}^{(12)}) . \end{matrix}$

Note that, $E_{ijk} = 1$ in the first two cases (i) and (ii), as $y_{ijk}^{(1)} + y_{ijk}^{(2)} > 0$ in both the cases. But in the third case (iii), we are unsure of E_ijk because it is unobserved as the individual i has not been observed by either detector 1 or 2. Recall that, the detection by one detector is independent of detection by the other detector when an animal enters the common detection region. Therefore $p_{j}^{(12)} = p_{j}^{(1)} p_{j}^{(2)} = ϕ_{j}^{2}$ and $p_{j}^{(1 \bar{2})} = p_{j}^{(\bar{1} 2)} = ϕ_{j} (1 - ϕ_{j})$ , for each j (also see Section 1). The data outcomes above then become:

$\begin{matrix} P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (1, 0)] = ϕ_{j} (1 - ϕ_{j}) π_{ij} = P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 1)], \end{matrix}$

$\begin{matrix} P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (1, 1)] = ϕ_{j}^{2} π_{ij}, \end{matrix}$

$\begin{matrix} P [(y_{ijk}^{(1)}, y_{ijk}^{(2)}) = (0, 0)] = (1 - π_{ij}) + (1 - ϕ_{j})^{2} π_{ij} = 1 - ϕ_{j} (2 - ϕ_{j}) π_{ij} . \end{matrix}$

For each observed individual i, the detection history $Y_{i, obs} = (Y_{i, obs}^{(1)}, Y_{i, obs}^{(2)}) = ((y_{ijk}^{(1)}, y_{ijk}^{(2)}))_{1 \leq j \leq J, 1 \leq k \leq K}$ occurs with probability

\begin{matrix} f & (Y_{i, obs}^{(1)}, Y_{i, obs}^{(2)} | ϕ, p_{0}, σ_{m}, σ_{f}, s_{i}) = \prod_{j = 1}^{J} \prod_{k = 1}^{K} f (y_{ijk}^{(1)}, y_{ijk}^{(2)} | ϕ_{j}, p_{0}, σ_{m}, σ_{f}, s_{i}) \\ = \prod_{j = 1}^{J} \prod_{k = 1}^{K} {P (E_{ijk} = 0) f (y_{ijk}^{(1)}, y_{ijk}^{(2)} | E_{ijk} = 0, ϕ_{j}, p_{0}, σ_{m}, σ_{f}, s_{i}) + \\ P (E_{ijk} = 1) f (y_{ijk}^{(1)}, y_{ijk}^{(2)} | E_{ijk} = 1, ϕ_{j}, p_{0}, σ_{m}, σ_{f}, s_{i})} \\ = \prod_{j = 1}^{J} \prod_{k = 1}^{K} {(1 - π_{ij}) I (y_{ijk}^{(1)} + y_{ijk}^{(2)} = 0) + π_{ij} ϕ_{j}^{y_{ijk}^{(1)} + y_{ijk}^{(2)}} (1 - ϕ_{j})^{2 - (y_{ijk}^{(1)} + y_{ijk}^{(2)})}} . \end{matrix}

Note that the population size N, which is a parameter of major interest, is an unknown quantity. Due to this, the number of some other variables including some latent variables is unknown and therefore the dimension of the parameter space is also unknown. This is one of the main difficulties in analysing the proposed SECR model. We consider the method of data augmentation¹³ for analysing the proposed SECR model to handle this difficulty. This is implemented by choosing a large integer M to bound N and augmenting the two observed data sets with a large number of “all-zero” encounter histories. We denote the zero-augmented data sets by $Y^{(1)}$ and $Y^{(2)}$ corresponding to detectors 1 and 2, respectively, each of these is now of dimension M × J × K:

(Y^{(1)}, Y^{(2)}) = [(\begin{matrix} Y_{obs}^{(1)} \\ Y_{rest}^{(1)} \end{matrix}) (\begin{matrix} Y_{obs}^{(2)} \\ Y_{rest}^{(2)} \end{matrix})] .

Here $Y_{rest}^{(1)}$ denotes the rest of the $Y^{(1)}$ components. The dimensions of $Y_{obs}^{(1)}$ and $Y_{rest}^{(1)}$ are n × J × K and $(M - n) \times J \times K$ , respectively. Similarly for $Y^{(2)}$ . A vector of M latent binary variables $z = (z_{1}, \dots, z_{M})^{'}$ is introduced. When $z_{i} = 1$ , it implies that individual i is a member of the population. We assume that each z_i is Bernoulli with parameter $ψ$ . Thus, the true population size N follows the binomial distribution with parameters M and $ψ$ (see Section 2.1.1). The augmented latent vector on sex category is denoted by u. Let $u_{obs}$ be a vector of binary observation (length $n \times 1$ ) on sex category of the captured individuals: $u_{i} (\in u_{obs})$ takes the value 1 when individual i is a male, takes the value 0 if its female. The vector of latent missing observations in u is denoted by $u_{0}$ (length $(M - n) \times 1$ ). Here we assume that the sex category is known for all the n fully identified individuals. However, as we show in the data analysis later in Section 3.2 it is straightforward to permit for the situation where sex category for some of the fully identified individuals is unknown (length of $u_{obs} < n$ ). Assuming that the i indices of the detection histories coming from detector 1 and 2 (viz., $Y^{(1)}, Y^{(2)}$ ) are in the same order and covariate information (partially observed) on individual sex category is available for each real individual (with $z_{i} = 1$ ), the joint density of $(Y^{(1)}, Y^{(2)}) = ((y_{ijk}^{(1)}, y_{ijk}^{(2)}))$ and u is the following:

\begin{matrix} f (Y^{(1)}, Y^{(2)}, u_{obs} | z, u_{0}, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S) = \prod_{i = 1}^{M} [f (Y_{i, obs}^{(1)}, Y_{i, obs}^{(2)} | z_{i}, ϕ, p_{0}, σ_{m}, σ_{f}, s_{i}) g (u_{i} | z_{i}, θ)] \\ = \prod_{i = 1}^{M} [\prod_{j = 1}^{J} \prod_{k = 1}^{K} {(1 - π_{ij}) I (y_{ijk}^{(1)} + y_{ijk}^{(2)} = 0) + π_{ij} ϕ_{j}^{(y_{ijk}^{(1)} + y_{ijk}^{(2)})} (1 - ϕ_{j})^{2 - (y_{ijk}^{(1)} + y_{ijk}^{(2)})}} θ^{u_{i}} (1 - θ)^{1 - u_{i}}]^{z_{i}} . \end{matrix}

(2.1)

It is straightforward to handle the latent missing observations in u, denoted by $u_{0}$ , using a Bayesian MCMC analysis.⁹ For simplicity, we can assume $ϕ_{j} = ϕ$ , for each j. The posterior density of parameters ${z$ , $u_{0}$ , $ψ$ , θ, ϕ, p₀, $σ_{m}$ , $σ_{f}$ , $S}$ can be obtained as follows:

\begin{matrix} g (z, u_{0}, ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S ∣ Y^{(1)}, Y^{(2)}, u_{obs}) \\ \propto f (Y^{(1)}, Y^{(2)}, u_{obs} | z, ϕ, p_{0}, σ_{m}, σ_{f}, S) g (u_{0} | z, θ) g (z | ψ) g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S) \\ = \prod_{i = 1}^{M} [\prod_{j = 1}^{J} \prod_{k = 1}^{K} {(1 - π_{ij}) I (y_{ijk}^{(1)} + y_{ijk}^{(2)} = 0) + π_{ij} ϕ^{(y_{ijk}^{(1)} + y_{ijk}^{(2)})} (1 - ϕ)^{2 - (y_{ijk}^{(1)} + y_{ijk}^{(2)})}}^{z_{i}} \\ \times θ^{z_{i} u_{i}} (1 - θ)^{z_{i} (1 - u_{i})} ψ^{z_{i}} (1 - ψ)^{1 - z_{i}}] \times g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S), \end{matrix}

(2.2)

where $g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S)$ is the prior density for the parameters $ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S$ . Here $g (u_{0} | z, θ) = \prod_{i : u_{i} \in u_{0}} g (u_{i} | z_{i}, θ)$ , $g (u_{i} | z_{i}, θ)$ is the conditional prior density of u_i, which is density of the Bernoulli distribution with parameter θ when z_i takes the value 1. The conditional prior density of z_i is denoted by $g (z_{i} | ψ)$ , which is density of the Bernoulli distribution with parameter $ψ$ .

Joint posterior density to include bilateral synchronization complexity. Equation (2.2), however, does not deal with the problem of synchronizing data from the partially identified individuals as described in Table 2. Ignoring this problem may result in overestimation of abundance, underestimation of standard errors and poor coverage for credible interval (CI) estimates. We integrate the solution used by Royle JA¹⁵ into our problem formulation (2.2).

Accordingly, the two lists of capture histories generated as in Table 2 essentially come from the same population and therefore there must be a unique association between the two lists. As noted earlier, we are particularly interested to form the associations for the “partially identified” individuals. Accordingly, we treat the true identity of a partially identified individual as a latent variable. We then probabilistically link individuals from the two lists obtained from detector 1 and detector 2, respectively, by introducing a latent identity variable $L = (L_{1}, L_{2}, \dots, L_{M})^{'}$ . L is a permutation of ${1, 2, \dots, M}$ which re-orders the set of individuals from detector 2 to correspond with the set of individuals from detector 1.

More details on the synchronization procedure can be found in Royle JA¹⁵. Without loss of generality, we define the true identity of each individual in the population to be in the row-order of capture histories of detector 1. Then we reorder the rows of detector 2 dataset $Y^{(2)}$ as indicated by L to synchronize with the individuals of the detector 1 dataset $Y^{(1)}$ . We denote this newly ordered detector 2 dataset as $Y^{(2 *)}$ . Now these two synchronized datasets can be used in the SECR model (2.2).

An individual i will be called “detected” if there exists a non-zero observation $y_{ijk}^{(1)}$ or $y_{ijk}^{(2 *)}$ for some j and k; that is, if $y_{i \cdot \cdot}^{(1)} + y_{i \cdot \cdot}^{(2 *)} > 0$ . Thus, if we obtain detection history observations $Y_{obs}^{(1)}$ and $Y_{obs}^{(2)}$ from two detectors during a spatial capture–recapture survey, they may not be synchronized (see Example 2.1). Our aim will be to use the latent vector L to synchronize $Y_{obs}^{(1)}$ and $Y_{obs}^{(2)}$ . Accordingly, by integrating this synchronization complexity into the joint posterior density (2.2), we obtain the new combined posterior of parameters ${z, u_{0}, ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S, L}$ as follows:

\begin{matrix} g (z, u_{0}, ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S, L | Y^{(1)}, Y^{(2 *)}, u_{obs}) \\ \propto f (Y^{(1)}, Y^{(2 *)}, u_{obs} | z, u_{0}, ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S, L) g (z | ψ) g (u_{0} | θ) g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S, L) \end{matrix}

\begin{matrix} = \prod_{i = 1}^{M} [\prod_{j = 1}^{J} \prod_{k = 1}^{K} {(1 - π_{ij}) I (y_{ijk}^{(1)} + y_{ijk}^{(2 *)} = 0) + π_{ij} ϕ^{(y_{ijk}^{(1)} + y_{ijk}^{(2 *)})} (1 - ϕ)^{2 - (y_{ijk}^{(1)} + y_{ijk}^{(2 *)})}}^{z_{i}} \\ \times θ^{z_{i} u_{i}} (1 - θ)^{z_{i} (1 - u_{i})} ψ^{z_{i}} (1 - ψ)^{1 - z_{i}}] \times g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S) \\ = \prod_{i = 1}^{M} [{ψ θ^{u_{i}} (1 - θ)^{1 - u_{i}} ϕ^{(y_{i \cdot \cdot}^{(1)} + y_{i \cdot \cdot}^{(2 *)})} (1 - ϕ)^{2 n_{i \cdot} - (y_{i \cdot \cdot}^{(1)} + y_{i \cdot \cdot}^{(2 *)})} \prod_{j = 1}^{J} π_{ij}^{n_{ij}} {(1 - π_{ij}) + π_{ij} (1 - ϕ)^{2}}^{K - n_{ij}}}^{z_{i}} \\ \times (1 - ψ)^{1 - z_{i}}] \times g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S, L), \end{matrix}

(2.3)

where $n_{ij} = \sum_{k = 1}^{K} I (y_{ijk}^{(1)} + y_{ijk}^{(2 *)} > 0)$ and $g (ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, s, L)$ is the joint prior density for the parameters $ψ, θ, ϕ, p_{0}, σ_{m}, σ_{f}, S, L$ . The MCMC algorithm used to sample from this posterior density (2.3) is detailed in Appendix B.

2.2.3 Identifiability of Model Parameters

It is necessary to check for issues of identifiability when new models and estimators such as ours are proposed. Inherent identifiability issues in the model give rise to problems of variance inflation, estimation biases and also false specification of the number of true parameters in penalized methods of model selection.²⁵ We evaluate the identifiability concerns of two important pairs of parameters in our SECR model, $(ϕ, π)$ and $(p_{0}, σ)$ .

Identifiability between $ϕ$ and $π$ . The relevant probability statements describing the probability of the data conditional on the parameters for detection probability ϕ of a detector and the trap entry probability $π$ , is given by

\begin{matrix} f & (y^{(1)}, y^{(2)} | ϕ, π) = (1 - π) I (y^{(1)} + y^{(2)} = 0) + π ϕ^{(y^{(1)} + y^{(2)})} (1 - ϕ)^{2 - (y^{(1)} + y^{(2)})} . \end{matrix}

(2.4)

Note that Equation (2.4) is a four cell multinomial model, where the cells are “00”, “01”, “10” and “11”. This model is identifiable, provided both ϕ and $π$ lie strictly between 0 and 1. The formal proof is derived in Appendix A.1. However, even with this condition, it is always possible that the given data (mostly due to inadequate sample size) may appear to only arrive in the form of “11” and “00” pairs. In such a case as well, we will have issues of non-identifiability.

Identifiability between $p_{0}$ and $σ$ . The trap entry probability $π$ is modelled as a decreasing function of distance between location of activity centre of an individual and a trap station. The two parameters in the model for trap entry probability $π$ are: (a) the baseline trap entry probability parameter p₀ and (b) the scale parameter σ. This pair of parameters $(p_{0}, σ)$ is identifiable under the condition that there exist two observation indices $(i_{1}, j_{1})$ and $(i_{2}, j_{2})$ such that $z_{i_{1}} > 0$ , $z_{i_{2}} > 0$ and $d (s_{i_{1}}, x_{j_{1}}) \neq d (s_{i_{2}}, x_{j_{2}})$ . Here, $(i, j)$ represents the indices of the pair individual i, trap station j. It is sufficient if the index of $(s_{i_{1}}, x_{j_{1}})$ is different from the index of $(s_{i_{2}}, x_{j_{2}})$ , implying that we achieve identifiability if $s_{i_{1}} \neq s_{i_{2}}$ or $x_{j_{1}} \neq x_{j_{2}}$ or both, $s_{i_{1}} \neq s_{i_{2}}$ and $x_{j_{1}} \neq x_{j_{2}}$ as long as $d (s_{i_{1}}, x_{j_{1}}) \neq d (s_{i_{2}}, x_{j_{2}})$ . This condition is proved in Appendix A.2.

2.2.4 Posterior Propriety

Link²⁶ discussed an important and often overlooked aspect of posterior impropriety during Bayesian analysis of estimation problems in ecology and stresses the need for practitioners to ensure that posteriors are proper. More recently, Gopalaswamy and Delampady²⁷ indirectly suggest the use of defensibly informed or bounded priors to ensure posterior propriety in such problems and indicate the close association between posterior impropriety and identifiability. Accordingly, in this article, we implement bounded priors based on ecologically justifiable upper limits for all the parameters used in our model. The assumed proper prior distributions for these parameters along with other model parameters and latent variables are as follows: a uniform distribution over the interval $(0, 1)$ for the probability parameters ϕ, p₀, $ψ$ and θ; a uniform distribution over the interval $(0, R)$ for parameters $σ_{m}$ and $σ_{f}$ where R is high enough to expect that it would be impossible for animals to exhibit movement as widely as this scale during sampling. In our study, we have taken the value of R to be 10 km (see Section 2.4.1 for details). L has a Uniform distribution over the permutation space of ${1, 2, \dots, M}$ . Each z_i follows a Bernoulli $(ψ)$ distribution and each u_i follows a Bernoulli $(θ)$ distribution. Each $s_{i} = (s_{i 1}, s_{i 2})^{'}$ follows uniform distribution over the state space $(V)$ . All the parameters are distributed independently of each other.

Prior robustness is an important issue in Bayesian analysis, and therefore a prior sensitivity analysis is necessary when a fully specified subjective prior is not used. We have indeed performed our computations with the beta prior. Even though there is some information available on the (prior) mean of some of the parameters, the variability is uncertain. Having seen no real differences in the estimates based on a range of beta priors, we have shown only results for the uniform prior, assuming that it is best to report inferences based on a flat prior which is expected to be a robust choice.²⁸

2.2.5 Use of Covariates

The advantage of the estimator we have developed in this study will only be realized effectively if trap-specific covariates are provided as explanatory variables for the ecological process parameter, p₀, as well as observation process parameter, ϕ. In practical wildlife surveys using camera traps,²² investigators may be interested to assess the movement ecology of animals and assess what factors drive animals to visit particular trap stations or not. For example, investigators might be interested to test the effectiveness of various lures/baits at trap stations or identify local site characteristics that attract or repel animals. These explanatory variables may suitably describe the variation in trap entry probability, p₀. However, such covariates are likely to have little influence on whether the cameras installed at trap stations work effectively or not. Instead some other covariates may better describe factors influencing how well the cameras fire and capture records of animals passing by. In real landscapes, heterogeneity in habitat type can induce spatial variation in detection efficiency. For example, detectors placed in regions with high canopy cover may perform more efficiently than detectors placed in open habitats in tropical forests because cooler temperatures under canopy cover enable detection of warm blooded animals better. Similarly, in search-encounter surveys,^{29, 30} individuals may get differentially detected, spatially, based on the amount of shrub cover. Hence, such covariates can adequately describe the detection probability of the detector, ϕ. It is common practice in ecology, to permit for such covariates in the model using a logit-link for p₀ and/or ϕ.

2.3 Assessment of Model Performance

2.3.1 Simulation Design

For a high dimensional problem such as this, it would be infeasible to assess model performance for an exhaustive range of parameters simply owing to the number of combinations and computation time. We conducted simulations for 70 scenarios (provided in Appendix Table 1) grouped into 2 equal sized sets, to assess the performance of the model proposed here. We set $σ_{m}$ = 0.3 and $σ_{f}$ = 0.15 for the first set of 35 scenarios, $σ_{m}$ = 0.4 and $σ_{f}$ = 0.2 for the second set of 35 scenarios. The simulation design was aimed to highlight the importance of identifying the pair of parameters (p₀ and ϕ) and its effect on the robustness of estimates of other parameters (especially N). We set p₀ = 0.005, 0.01, 0.03, 0.05, 0.07; ϕ = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, which gives us 35 different scenarios for each of the two sets corresponding to the values taken by p₀ and ϕ. We assumed that a total of 100 individuals are residing inside the state space of which 40 are male. Each of the simulation experiments is conducted within a rectangular state space of dimension 5 unit $\times$ 7 unit (Appendix Figure 1), after setting a buffer of 1 unit in both horizontal and vertical directions, a $10 \times 16$ trapping array of total $J = 160$ trap stations has been set (trap spacing is 0.3 unit on X axis and 0.3125 unit on Y axis). Each of the traps remains active for $K = 50$ sampling occasions simultaneously. For parameter estimation, we set the maximum possible number of individuals present in the population (M) at 400 for all the scenarios. The MCMC chains for each of the parameters are obtained (each of length 30,000) and the estimates were computed using those chains with a burn-in of 10,000.

2.3.2 Comparison with “Unidentified” Model

Often practitioners are interested to know about robustness of estimates of particular parameters of interest under violations of model assumptions. For example, ecologists are very interested in N and will often base the choice of their models based on robustness of estimates of N in the face of model violations. Motivated by this concern we also performed a parallel simulation study of the partial identification model proposed by Royle JA¹⁵. The mechanisms of extracting information from the recorded data sets are different in our proposed model and Royle JA¹⁵. We view our approach as a natural extension of Royle JA¹ that disentangles the state and the observation processes at the detection region. And is thus not an approach evolved from Augustine et al.¹⁶ Hence, we assess the performance of our model only relative to the more restrictive model.¹⁵ Specifically, we modelled an extra source of variation for the recorded (0,0) events, viz., uncaptured given animal entry in a trap station and uncaptured due to absence of animal in trap station.

Table 3 provides an illustrative example to demonstrate the need for practitioners to use the model we have proposed in this article by indicating the biases in estimates of N and other parameters relative to the reduced, unidentified, model. For ease of comparison we preserve, as before, $N = 100$ and $N_{Male} = 40$ .

2.4 Application to Tiger Camera Trapping Data from Nagarahole

2.4.1 Sampling Design

We have considered a specific application of modelling the bilateral capture–recapture data from a single season camera trapping study on tigers in Nagarahole study area of southern India (area = 1,134 km²). The study area extends from 596,626.7 m to 641,533.9 m longitudinally and 1,301,307.5 m to 1,371,205.7 m latitudinally. The coordinates are in Universal Transverse Mercator (UTM) unit system. The trapping array (Appendix Figure 12) consisted of 162 dual camera stations (where two opposite cameras are installed facing each other in each trap station) with a mean spacing of 1.5 km and the survey lasted 50 days (26 November, 2014 to 13 January, 2015), resulting in 7,364 trap nights of effort. We used Panthera branded passive motion sensor cameras (Model: V4, Manufacturer: Panthera) at each trap stations in our study. Usually, the cameras are placed 3–4 m away from the centre of the trail. This setup ensures that each camera can clearly obtain independent flank images.¹⁴

Our use of the Gaussian function implies that the buffer around the trapping array should, theoretically, be set at infinity. However, for practical reasons, this is usually set large enough so that individuals have a near zero probability of being exposed to the trapping array beyond such a buffer.¹³ Accordingly, we set a buffer of 10 km (aiming for a width $> 3 \hat{σ}$ , where $\hat{σ}$ is a reliable estimate of σ obtained from past estimates from the same study area^{13, 21}) around the trapping array for analysing the tiger data. For new study systems, when a reliable estimate of σ is generally unknown, we recommend R to be $10 \tilde{σ}$ , where $\tilde{σ}$ is a reasonable assumption on σ based on the prior experience of the investigators. Surprisingly, practitioners often misunderstand the reason for deciding on a buffer width. For example, in a recent camera trap SECR study of tigers³¹ in the same landscape, but at a different site, the authors set an arbitrarily small buffer (buffer width $< 0.75 \hat{σ}$ ), which is both statistically and ecologically indefensible.

Tigers can be individually identified by matching the unique patterns of flanks on both left and right sides. Researchers use software³² to assist in matching flank patterns from photographs and consequently obtain individual specific detection histories in standard spatial capture–recapture format.¹³ However, since flank patterns are not identical on both sides of a tiger, at least one simultaneous detection of both side flanks over the course of camera trapping survey is needed to identify a tiger. A “simultaneous detection” is defined for an individual when the event time recorded by passive motion sensor cameras matches exactly (to the minute) for either flanks of an individual. Data were arranged in the format described by the sampling structure defined in Table 2.

2.4.2 Data Summary

In our field experiment, we could identify 65 tigers (22 male, 33 female, 10 of unknown sex). This meant that we recorded at least one simultaneous capture of both flanks for each of the above set of tigers. In addition, we obtained 14 partially identified left flank only detection histories (6 male, 5 female, 3 of unknown sex) and 17 partially identified right flank detection histories (7 male, 4 female, 6 of unknown sex). Overall, we obtained 123 simultaneous detections, 126 left flank only detections and 137 right flank only detections.

2.4.3 Analysis

We used the covariate information on sexes for the detected individuals. As male and female tigers do not share the same σ, that is, do not have the same home-range size, we modelled σ as a function of this covariate. We fitted the model described in Section 2.2.1 and augmented the detection histories by all-zero detections to make them of the same dimension. We ran one chain of 50,000 iterations and discarded first 25,000 as burn-in. Further, we assessed the quality of the parameter estimates by computing the coverage probabilities. Here, we fixed the parameters at the values estimated in the data analysis and simulated 100 data sets under the same conditions (i.e., state space, trap deployment) as in the case of the Nagarahole study. Coverage probabilities are computed as the proportion of times when the estimated 95 per cent CIs contain the true value of the parameter. In these simulations, the true values are defined as the posterior mean estimates from the results of the field experiment of the following parameters: N, $ψ$ , N_Male, θ, ϕ, p₀, $σ_{m}$ , $σ_{f}$ .

3 Results and Conclusions

3.1 Assessment of Model Performance

3.1.1 Simulation Results

Here we summarize the main findings of the simulation study over different simulation scenarios as mentioned in Section 2.3.1. The detailed discussion of the study is provided in Appendix C and the simulation results are presented in Appendix Tables 3–10. We observe that, the quality of the estimates of different parameters substantially improves when trap entry probability p₀ increases. The scenarios in which p₀ is set to values greater than 0.03 had performed reasonably well. This is noted by the manner in which root mean square error (RMSE) values shrink substantially as p₀ increases. Whereas when the trap entry probability p₀ is set at low values (below 0.01), in most of those scenarios the posterior estimates of parameters are inaccurate with wide 95 per cent CIs. This outcome may be explained by the poor information content emerging when individuals rarely enter trap stations. The boxplots (Appendix Figure 2) of N, obtained by using the MCMC samples, show signs of positive skewness in most of the scenarios. Also, the bias and posterior standard deviation (SD) of N are influenced by the conditional detection probability ϕ (indicating detector performance) in a similar manner to how p₀ influences model performance. That is, both bias and posterior SD decrease as the value of ϕ increases.

The scenarios with $σ_{m}$ = 0.4 and $σ_{f}$ = 0.2 performed better than scenarios with $σ_{m}$ = 0.3 and $σ_{f}$ = 0.15 while estimating N, in the sense of having lesser RMSE estimate ( $\approx 51.46$ for the former setting as compared to $\approx 24.86$ for the latter setting). This is perhaps associated with the fact that with less movement, both the number of detections and the number of distance classes recorded in data decrease. We would envisage the trap station layout also plays an important role in this assessment.³³ The estimated posterior correlations between ϕ and p₀ lie between $- 0.3$ and 0 for scenarios where both p₀ and ϕ take high values, that is, $p_{0} \in {0.05, 0.07}$ and $ϕ \in {0.7, 0.8, 0.9}$ . In comparison, for scenarios where ϕ takes small values, the posterior correlation estimates are between $- 0.7$ and $- 0.5$ , whereas the RMSE estimates of N are $\approx 11.18$ . Here posterior mean estimates of N show signs of robustness, even though a moderate amount of covariation is present between MCMC samples of ϕ and p₀.

The posterior mean estimates of p₀ have a decreasing trend on bias as ϕ increases. In a similar manner, posterior mean estimates of ϕ also have a decreasing trend on bias as p₀ increases. This simulation outcome is indicative of poor information content in the data and consequently reflects on the identifiability of the parameter estimates. We surmise that these correlations will play an important role during model selection and inference.

3.1.2 Comparison with “Unidentified” Model

In both the scenarios, ${ϕ = 0.4, p_{0} = 0.05, σ_{m} = 0.3, σ_{f} = 0.15}$ and ${ϕ = 0.3, p_{0} = 0.05, σ_{m} = 0.4, σ_{f} = 0.2}$ , we see a substantial bias in the estimates of N (see Table 3) corresponding to the model which does not disentangle the parameters p₀ and ϕ.¹⁵ In these two scenarios the estimated posterior correlation between ϕ and p₀ are $- 0.518$ and $- 0.689$ , respectively. Furthermore, we observe that the estimates of ϕ and p₀ are not unbiased, but the estimate of N stays robust. The estimates of $λ_{0}$ in the unidentified model are 0.023 and 0.016 corresponding to the two scenarios which are close to the product of the true values of ϕ and p₀; also, the estimates of N have larger RMSEs. These indicate that this model involves an over-simplification of the true model assumptions.

Table 3.

Posterior estimates for two different models are compared: (i) Unidentified model where detection probability parameter is defined as $p_{ij} = λ_{0} exp (- \frac{d (s_{i}, x_{j})^{2}}{2 σ^{2}})$ , (ii) Identified model where trap entrance probability parameter is defined as $π_{ij} = p_{0} exp (- \frac{d (s_{i}, x_{j})^{2}}{2 σ^{2}})$ , detection probability conditional on trap entrance is defined as ϕ. The movement parameter is $σ_{m}$ or $σ_{f}$ , depending on whether individual is male or female, respectively

	(i) Unidentified Model		(ii) Identified Model
	(p₀ and ϕ not identified)		(p₀ and ϕ identified)
Parameters	Mean	RMSE	Mean	RMSE
Scenario: $ϕ = 0.4$ , $p_{0} = 0.05$ , $σ_{m} = 0.3$ , $σ_{f} = 0.15$
N	89	12.743	99	10.060
N_Male	36	5.726	40	4.109
ϕ	—	—	0.352	0.059
p ₀	—	—	0.060	0.013
$λ_{0}$	0.023	0.027	—	—
$σ_{m}$	0.287	0.017	0.307	0.016
$σ_{f}$	0.157	0.010	0.144	0.011
Scenario: $ϕ = 0.3$ , $p_{0} = 0.05$ , $σ_{m} = 0.4$ , $σ_{f} = 0.2$
N	93	9.615	99	7.632
N_Male	43	4.405	40	3.464
ϕ	—	—	0.283	0.035
p ₀	—	—	0.047	0.007
$λ_{0}$	0.016	0.033	—	—
$σ_{m}$	0.364	0.039	0.407	0.018
$σ_{f}$	0.199	0.010	0.216	0.021

Source: The authors.

3.2 Application to Tiger Camera Trapping Data from Nagarahole

3.2.1 Data Analysis

The posterior estimates of parameters are provided in Table 4. The posterior mean estimate of population size (over the state space) is 133 with a 95 per cent CI of (117, 152). The density of tiger is estimated at 11.73 tigers per 100 km² in our study area. The posterior mean of $σ_{m}$ (1.970) is estimated to be higher than that of $σ_{f}$ (1.209). The estimates of $σ_{m}$ and $σ_{f}$ also confirm that the buffer we had set (10 km) was sufficiently large enough. The number of male tigers in the population is estimated at 41 with a 95 per cent CI (33, 50), and hence the number of female tigers is estimated at 92. The sex ratio was estimated to be 2.24 females to 1 male.

The scatter plot provided in Appendix Figure 15 shows that there is moderate amount of correlation between ϕ and p₀ ( $\approx - 0.41$ ) present in the MCMC samples, which also matches the simulation results for relatively smaller values of ϕ. Sample correlation between the pairs (p₀, $σ_{m}$ ) ( $\approx - 0.44$ ) and (p₀, $σ_{f}$ ) ( $\approx - 0.48$ ) indicate identifiability issues between those parameters, but is not expected to effect the estimate of the other parameters of interest, viz., N, N_Male. As we discussed in Section 2.2.3, the accuracy and precision of p₀ and σs depend on dispersion of distances between individuals’ activity centres and trap locations. Higher dispersion in these distances is likely to make the estimates of p₀, $σ_{m}$ and $σ_{f}$ more accurate and precise.

3.2.2 Inference

The detection probability ϕ in the analysis of Nagarahole capture–recapture dataset on tigers is estimated at 0.489 (see Table 4). This implies that each camera records a clear flank image in a little less than 50 per cent of the cases. This is not surprising to us as a clear “valid sample” depends on many other factors, such as quality of the traps, camera malfunctions, ambient temperature etc. in typical field conditions.

The simulation study was designed to reflect a typical field study, so that performance of the model can be evaluated based on different values taken by the model parameters in a practical setup. Accordingly, p₀ is the most dominant parameter which influences the performance of the model while obtaining posterior summaries of the other parameters. Furthermore, the estimates corresponding to the scenarios where p₀ is set to 0.05 or 0.07 perform fairly well as compared to the scenarios where p₀ is set to smaller values, viz., 0.005, 0.01. In the field study p₀ is estimated at 0.041 with a 95 per cent CI (10.32, 13.41) (see Table 4).

We estimated the tiger density to be 11.73 tigers per 100 km² with 95 per cent CI (10.32, 13.41) in our study area. This is comparable to estimates of tiger density from a similar study²¹ in this area using a different version of SECR models. The 95 per cent CI width of 3.09 from our study is only 26.3 per cent of its corresponding density estimate. In comparison²¹, had 95 per cent CI widths as 42.5 per cent of their density estimate (11.3 tigers per 100 km² with 95% CI [9.1, 13.9]). Clearly our model provides more precise estimates for tiger density. The estimated posterior density map of tigers (per sq. km.) over the study area is provided in Figure 1.

We found that coverage probabilities of all the continuous parameters (viz., $ψ$ , θ, ϕ, $σ_{m}$ , $σ_{f}$ ), except p₀, attained or improved upon the nominal coverage probability of 0.95 (see Table 4). We note that the somewhat lower coverage probability for the parameters p₀, N and N_Male is perhaps due to variation in the simulated data sets. However, coverage probabilities are expected to increase with a better detection and trap entry rates (i.e., higher ϕ and p₀) as we have discussed above.

Table 4.

Posterior Estimates of Parameters from the Nagarahole Tiger Analysis

Parameters	Mean	SD	2.5%	50%	97.5%	CI Width	Coverage Probability
N	133	8.89	117	133	152	35	0.880
$ψ$	0.333	0.032	0.272	0.332	0.397	0.125	0.980
N_Male	41	4.293	33	41	50	17	0.900
θ	0.312	0.050	0.220	0.310	0.413	0.193	1
ϕ	0.489	0.029	0.430	0.487	0.543	0.113	0.950
p ₀	0.041	0.004	0.033	0.040	0.049	0.015	0.910
$σ_{m}$	1.970	0.083	1.814	1.967	2.140	0.326	0.960
$σ_{f}$	1.209	0.056	1.103	1.207	1.323	0.220	0.940

Source: The authors.

Figure 1.

Source: The authors.

4 Discussion

In this article, we have developed a novel SECR estimator that successfully disentangles the ecological process of animal trap entry from the observation process of trap detection rates (see Section 2). Our simulation results highlight the relative importance of ensuring that trap stations are chosen based on good locations as compared to the importance of detector choice, especially, when there is more than one detector located at each station. When adequate spatial coverage is achieved by the array of detectors, as per the recommendation by Karanth and Nichols¹⁴, it is preferred that good spots are to be selected locally to maximize the probability of animal arrival.

Our SECR model is built upon an earlier Bayesian hierarchical model by Royle JA¹⁵ and makes full use of all data available (including information on partially identified individuals). We demonstrate how our model provides unbiased estimates of population size N when trap detection rate is less than one. We justify the importance of estimating trap detection rate ϕ by showing the bias in the estimate of N when we use the Royle JA¹⁵ model under certain simulation conditions.

We have developed the estimator using the special case of having only two detectors at each station, each detector capturing a set of unique traits about the identities of individuals. The assumption, however, is that each detector contains enough information on its own to ascertain individual identity. For example, as this study was motivated by the tiger example we have discussed in the article, we find a field situation where two profile flanks of an individual tiger are attempted to be caught at the same time at trap stations. When we do not have simultaneous captures it is not possible to tell if a right flank image of a tiger has an equivalent left flank image or not. We recognize that the situation will not directly apply if the same idea is extended to genotyping problems^{34, 35} because at each locus there is not enough information to convincingly identify individuals. We discuss more on this application later.

It is possible to extend our model to include three or more detectors per station based on the idea of how occupancy models³⁶ were constructed to include multiple sampling occasions. However, we envisage some complications with regard to explicitly defining the permutative arrangement of capture histories. For this, we need to understand how many detectors (implying how many sets of unique features) are necessary to establish full identity of an individual. For example, in genotyping problems³⁵, workers identify a panel of loci to achieve a desirably low level of probability of identity (PID). During field surveys^{34, 37}, workers often gather faecal samples for subsequent genotyping. However, not all faecal samples amplify in the laboratory. We envisage the application of our model to estimate this probability using the parameter ϕ.

As with most estimators, the utility of our SECR model is enhanced when meaningful covariates are applied on the specific model parameters. Ecologists interested in obtaining an understanding about fine scale movements of animals can now do so without the worry about the confounding problem of detector efficiency. We envisage that our estimator will find much use in optimal allocation problems¹⁶ in wildlife surveys. For example, many camera traps are available in the market at various costs. Since our model specifically estimates a parameter ϕ associated with trap efficiency, it would come of use to evaluate the relative gains in precision of estimates of abundance when, for example, cheap cameras are replaced by expensive cameras or to decide how many traps are needed at each station. Further, for defined monitoring budgets our model can be used to determine the most optimal allocation of the number of trap stations and the types of traps with available resources.

Beyond ecology, our SECR estimator lays the foundation for solving the statistical reconciliation problem in administrative lists.³⁸ In this problem, individuals do appear in different administrative lists at a region and the problem is to identify the population size from captures of individuals in the multiple lists. We find equivalence between multiple detectors discussed in our problem with the presence of different administrative lists in the problem described by Madigon and York.³⁸

An inherent problem in the application of a complex model for real world problems, and a larger problem in the statistical literature, is that selecting the appropriate model for prediction and characterization of populations is not straightforward. Some of us are currently working on evaluating and applying various model selection tools on this class of Bayesian SECR problems. We also encourage the extension of this estimator to include multiple detectors (more than two) as described above. With these developments, we envision wide application of the general approach presented here.

Supplemental Material

bilateral_supp_010119_highlighted_xyz13726f955f904 - Supplemental material for A Spatially Explicit Capture–Recapture Model for Partially Identified Individuals When Trap Detection Rate Is Less than One

Supplemental material, bilateral_supp_010119_highlighted_xyz13726f955f904 for A Spatially Explicit Capture–Recapture Model for Partially Identified Individuals When Trap Detection Rate Is Less than One by Soumen Dey, Mohan Delampady, K. Ullas Karanth, Arjun M. Gopalaswamy in Calcutta Statistical Association Bulletin

Footnotes

Acknowledgements

We thank the anonymous reviewers for some very useful comments and suggestions, which have brought improvements to our presentation. We thank the Indian Statistical Institute for financial and administrative support and the Centre for Wildlife Studies and Wildlife Conservation Society, New York for providing the data and analytical support. We also thank Ravishankar Parameshwaran and Devcharan Jathanna for help with the computer simulations and helpful advice. AMG thanks the Wildlife Conservation Society, New York for partial funding support.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

Supplementary Materials

Supplemental material for this article is available online.

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