An Enhanced Spatial Capture Model for Population Analysis Using Unidentified Counts through Camera Encounters
Abstract
:1. Introduction
 (1)
 Employs a prior distribution for the essential parameter of the zeroinflated population;
 (2)
 Regularizes the Markov chain Monte Carlo (MCMC) by controlling the effective sample size;
 (3)
2. Methods
2.1. Hierarchical Spatial Capture–Recapture Model
2.2. Proposed Method
2.3. Sensitivity of the Model to $\psi $
2.4. Autocorrelation Plot
2.5. Effective Sample Size
3. Results
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
MCMC  Markov chain Monte Carlo 
HSCR  Hierarchical spatial capture–recapture 
$N$  Population size 
$\widehat{N}$  Estimated population size 
$K$  Sampling occasion 
${z}_{ijk}$  Camera encounter history for individual $i$, at camera $j$, on occasion $k$ 
${\lambda}_{ij}$  The encounter rate for individual $i$ at camera $j$ 
${\lambda}_{0}$  The baseline encounter rate 
$\sigma $  Home range radius. 
${d}_{ij}$  The Euclidean distance between activity center ${s}_{i}$ and the camera location ${x}_{j}$ 
${n}_{jk}$  The number of camera encounters at camera $j$ on occasion $k$ 
$M$  The augmented parameter (the total number of hypothetical individuals) 
$\psi $  Probability of success, i.e., the probability that an individual in the occupancy model of size $M$ is a member of the original model of size $N$ 
$\alpha $ and $\beta $  Parameters of Beta probability distribution 
ESS  The effective sample size 
$L$  The number of zeros added to the model $L=MN$ (data augmentation size) 
$Lagk$  The autocorrelation between the current sample and kth preceding sample 
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$$\mathit{M}$$

$$\mathit{\psi}$$

$$\widehat{\mathit{\sigma}}$$

$${\widehat{\mathit{\lambda}}}_{0}$$

$$\widehat{\mathit{\psi}}$$

$${\mathit{S}\mathit{d}}_{\widehat{\mathit{\psi}}}$$

$$\left\left.\mathbf{\%}{\mathit{e}}_{\widehat{\mathit{\psi}}}\right\right.$$

$$\widehat{\mathit{N}}$$

$${\mathit{S}\mathit{d}}_{\widehat{\mathit{N}}}$$

$${\mathit{E}\mathit{S}\mathit{S}}_{\mathit{N}}$$

$${\mathit{E}\mathit{S}\mathit{S}}_{\mathit{\psi}}$$

$${\mathit{L}\mathit{a}\mathit{g}10}_{\mathit{N}}$$

$${\mathit{L}\mathit{a}\mathit{g}10}_{\mathit{\psi}}$$


100  0.00–1.00  0.437  0.589  0.313  0.046  25.200  30.887  2.577  1113.2  1297.6  0.668  0.625 
0.00–0.50  0.451  0.606  0.274  0.045  9.600  27.107  2.322  1850.8  2180.9  0.482  0.412  
0.10–0.40  0.465  0.585  0.253  0.043  1.200  25.255  2.185  2679.4  3232.7  0.403  0.324  
0.05–0.35  0.479  0.584  0.231  0.042  7.600  23.326  2.036  2505.5  3145.0  0.395  0.309  
0.10–0.35  0.471  0.589  0.237  0.043  5.200  23.862  2.077  2738.4  3681.9  0.352  0.260  
200  0.00–1.00  0.416  0.596  0.199  0.028  59.200  39.231  2.501  250.6  263.3  0.909  0.894 
0.00–0.50  0.443  0.579  0.158  0.026  26.400  30.979  2.030  1066.8  1140.4  0.674  0.623  
0.10–0.40  0.415  0.579  0.174  0.027  39.200  33.433  2.192  2340.8  2727.2  0.496  0.431  
0.05–0.35  0.443  0.579  0.153  0.025  22.400  29.922  1.969  2082.8  2203.2  0.547  0.486  
0.10–0.35  0.411  0.592  0.173  0.027  38.400  33.340  2.184  3017.5  3701.7  0.446  0.375 
$$\mathit{M}$$

$$\mathit{\psi}$$

$$\widehat{\mathit{\sigma}}$$

$${\widehat{\mathit{\lambda}}}_{0}$$

$$\widehat{\mathit{\psi}}$$

$$\left\left.\mathbf{\%}{\mathit{e}}_{\widehat{\mathit{\psi}}}\right\right.$$

$$\widehat{\mathit{N}}$$

$${\mathit{E}\mathit{S}\mathit{S}}_{\mathit{N}}$$

$${\mathit{E}\mathit{S}\mathit{S}}_{\mathit{\psi}}$$

$${\mathit{L}\mathit{a}\mathit{g}10}_{\mathit{N}}$$

$${\mathit{L}\mathit{a}\mathit{g}10}_{\mathit{\psi}}$$


100  0.00–1.00  0.560  0.497  0.323  29.200  31.943  1674.3  1862.4  0.635  0.593 
0.00–0.50  0.563  0.546  0.273  9.200  27.173  3661.7  4368.0  0.393  0.334  
0.10–0.40  0.516  0.557  0.261  4.400  26.251  5987.0  7677.7  0.270  0.205  
200  0.00–1.00  0.546  0.527  0.170  36.000  33.321  1490.3  1778.8  0.670  0.620 
0.00–0.50  0.527  0.564  0.158  26.400  31.026  2647.5  3152.3  0.528  0.473  
0.10–0.40  0.496  0.545  0.165  32.00  31.168  3880.0  4606.6  0.412  0.355 
95% CI  95% CI  

$$\widehat{\mathit{N}}$$

$$\widehat{\mathit{N}}2\ast {\mathit{S}\mathit{d}}_{\widehat{\mathit{N}}}$$

$$\widehat{\mathit{N}}+2\ast {\mathit{S}\mathit{d}}_{\widehat{\mathit{N}}}$$

$$\widehat{\mathit{\psi}}$$

$$\widehat{\mathit{\psi}}2\ast {\mathit{S}\mathit{d}}_{\widehat{\mathit{\psi}}}$$

$$\widehat{\mathit{\psi}}+2\ast {\mathit{S}\mathit{d}}_{\widehat{\mathit{\psi}}}$$
 
$M$ = 100  30.887  25.733  36.041  0.313  0.220  0.406 
27.107  22.463  31.751  0.274  0.185  0.363  
25.255  20.886  29.624  0.253  0.166  0.340  
23.326  19.255  27.397  0.231  0.147  0.315  
23.862  19.707  28.017  0.237  0.152  0.322  
$M$ = 200  39.231  34.230  44.232  0.199  0.143  0.255 
30.979  26.919  35.039  0.158  0.106  0.210  
33.433  29.049  37.817  0.174  0.120  0.228  
29.922  25.984  33.860  0.153  0.102  0.204  
33.340  28.972  37.708  0.173  0.120  0.226 
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Jaber, M.; Hamad, F.; Breininger, R.D.; Kachouie, N.N. An Enhanced Spatial Capture Model for Population Analysis Using Unidentified Counts through Camera Encounters. Axioms 2023, 12, 1094. https://doi.org/10.3390/axioms12121094
Jaber M, Hamad F, Breininger RD, Kachouie NN. An Enhanced Spatial Capture Model for Population Analysis Using Unidentified Counts through Camera Encounters. Axioms. 2023; 12(12):1094. https://doi.org/10.3390/axioms12121094
Chicago/Turabian StyleJaber, Mohamed, Farag Hamad, Robert D. Breininger, and Nezamoddin N. Kachouie. 2023. "An Enhanced Spatial Capture Model for Population Analysis Using Unidentified Counts through Camera Encounters" Axioms 12, no. 12: 1094. https://doi.org/10.3390/axioms12121094