1. Spatially Explicit Capture-recapture Models for Density Estimation 5.11 UF-2015

2. References Efford (2004): initial idea, ad hoc simulation based approach Efford (2004): initial idea, ad hoc simulation based approach Likelihood-based approaches: Borchers and Efford 2008), Efford et al. (2008, 2009) Likelihood-based approaches: Borchers and Efford 2008), Efford et al. (2008, 2009) Hierarchical MCMC approach: Royle and Young (2008), Royle et al. (2009ab), Royle and Gardner (2010), Gardner et al. (2009, 2010) Hierarchical MCMC approach: Royle and Young (2008), Royle et al. (2009ab), Royle and Gardner (2010), Gardner et al. (2009, 2010)

3. Spatially Explicit Capture- recapture (SECR) Models Developed to deal explicitly with 2 main problems associated with density estimation in trapping studies Developed to deal explicitly with 2 main problems associated with density estimation in trapping studies Unknown sample area varies with trap layout – home range size Unknown sample area varies with trap layout – home range size Heterogeneity in capture probability associated with animal location relative to traps Heterogeneity in capture probability associated with animal location relative to traps Edge effects

4. Known sample area? Previous models: Geographic closure Previous models: Geographic closure

5. Unknown sample area Boundary strip, nested grids… Trapping web – distance sampling

6. Unknown sample area Open-field perspective Open-field perspective Perhaps more realistic and useful Perhaps more realistic and useful Animal density is defined as the local intensity of a spatial point process Animal density is defined as the local intensity of a spatial point process Model activity centers of animal with a 2D spatial point process (Poisson)

7. Context SECR is a method for modelling data collected with an array of detectors SECR is a method for modelling data collected with an array of detectors Goal = estimate population density (D) (or population size N) Goal = estimate population density (D) (or population size N) Model includes population parameters (D) and parameters for the detection process (p) Model includes population parameters (D) and parameters for the detection process (p)

8. Sampling context Fixed traps (cameras, live-traps, snags) Fixed traps (cameras, live-traps, snags) Individual ID’s available (mark, DNA, …) Individual ID’s available (mark, DNA, …) 2D sampling grid (array of detectors) 2D sampling grid (array of detectors)

9. (SECR) Models: Conceptual Framework I Animals have activity centers (s i ), viewed as random variables Animals have activity centers (s i ), viewed as random variables s i = (s 1i, s 2i ) for i = 1, 2, …, N animals s i = (s 1i, s 2i ) for i = 1, 2, …, N animals Traps in array have known locations: Traps in array have known locations: x j = (x 1j, x 2j ) for j = 1, 2, …, J x j = (x 1j, x 2j ) for j = 1, 2, …, J

10. (SECR) Models: Conceptual Framework II Capture probability at any trap, j, is a function of distance between trap and animal activity center: Capture probability at any trap, j, is a function of distance between trap and animal activity center: d ij = || s i – x j || =  (s 1i – x 1i ) 2 + (s 2i – x 2i ) 2 d ij = || s i – x j || =  (s 1i – x 1i ) 2 + (s 2i – x 2i ) 2 Euclidian distance

12. SECR: Data y ijk = y ijk = 1 if individual i is caught in trap j at occasion k 1 if individual i is caught in trap j at occasion k 0 if individual i is not caught in trap j at interval occasion k 0 if individual i is not caught in trap j at interval occasion k rows = animals ID’s (i) values (e.g., A9) = trap ID (j) where animal (i) was ‘captured’ in interval k

13. SECR: Data Data for a single animal i j k

14. SECR: Model State Process (spatial point process) State Process (spatial point process) Distribution of individual activity centers = function of animal density D (Homogeneous or Inhomogeneous Poisson) Observation process Observation process Detection/non-detection = function of distance: activity center to trap

15. SECR: 3 Observation Models: Binomial: (e.g., hair snags) Binomial: (e.g., hair snags) Animal can be caught at most 1 time in any trap, but in any number of traps within an interval Multinomial: (e.g., real traps) Multinomial: (e.g., real traps) Animal can be caught at most one time in at most 1 trap within an interval Animal can be caught at most one time in at most 1 trap within an interval (not released from trap) Poisson: (Camera traps) Poisson: (Camera traps) Animal can be caught several times at a trap within an interval (example shown below)

16. SECR: Poisson Observation Model y ijk ~ Poisson (λ 0.g ij ) y ijk is a count λ 0 = baseline capture intensity at distance d ij = 0 (if trap was at center of activity) g ij = exp (- d 2 ij / σ 2 ) ; half-normal function specifying decrease in capture intensity with distance between activity center and trap σ 2 = parameter to be estimated (spatial scale over which p decreases)

17. Distance (m) Pr(detection) g ij = exp (- d 2 ij / σ 2 ) half-normal dist.

18. SECR: Poisson Observation Model Analysis: Log transformation of data Linear function: log (E [y ijk ]) = α + β d 2 ij α = log (λ 0 ) α = log (λ 0 ) β = - (1 / σ 2 ) β = - (1 / σ 2 ) d ij is a random effect, because activity centers are unknown d ij is a random effect, because activity centers are unknown E [y ijk ] is the expected number of captures at trap j in interval k

19. Density process S unknown and N unknown S unknown and N unknown Region S Sample unit S = region that contains the activity centers

20. Parameters to be estimated Detection process: λ 0 and σ 2 N = Number of activity centers Other Unknown: Location of activity centers (s i ) (so sample area S is unknown)

21. SECR Poisson Observation Model: Implementation I Assume special case of Assume special case of activity centers (s i ) known activity centers (s i ) known number of animals (N) known number of animals (N) known Specify priors for: Specify priors for: σ 2 ~ Uniform σ 2 ~ Uniform λ 0 ~ Gamma λ 0 ~ Gamma Implement using MCMC Implement using MCMC

22. SECR Poisson Observation Model: Implementation II Assume special case of Assume special case of number of animals known number of animals known but but activity centers unknown => specify distribution Specify priors: Specify priors: σ 2 ~ Uniform σ 2 ~ Uniform λ 0 ~ Gamma λ 0 ~ Gamma ; where S is region of spatial point process s ~ Uniform (S); where S is region of spatial point process Implement using MCMC Implement using MCMC

23. SECR Poisson Observation Model: Implementation III & activity centers Number of animals & activity centers unknown s ~ Uniform (S) s ~ Uniform (S) N ~ Binomial (M, ψ) ψ ~ Uniform (0, 1)

24. Data Augmentation; Unknown N Royle et al. (2007), Royle and Dorazio (2008) Royle et al. (2007), Royle and Dorazio (2008) Select number M that is larger than largest possible N; i.e., M > N Select number M that is larger than largest possible N; i.e., M > N Discrete uniform prior for N based on: Discrete uniform prior for N based on: N ~ Binomial (M, ψ) N ~ Binomial (M, ψ) ψ ~ Uniform (0, 1) ψ ~ Uniform (0, 1)

25. Data Augmentation; Unknown N Latent indicator variables, z i = Latent indicator variables, z i = 1 if individual i is member of population 1 if individual i is member of population 0 if individual i is not in population 0 if individual i is not in population z i ~ Bernoulli (ψ) z i ~ Bernoulli (ψ) Observation model: Observation model: y ijk ~ Poisson (λ 0 g ij ) y ijk ~ Poisson (λ 0 g ij ) if z i = 1 y ijk = 0 y ijk = 0 if z i = 0 Provides way to estimate number of all-0 capture histories Provides way to estimate number of all-0 capture histories

26. Data Augmentation Indiv.Capt. Hist.zizi 11 1 0 3 0 21 …0 1 2 0 1 41 …0 0 1 2 2 01 …0 3 1 0 4 21 n3 0 1 2 1 01 n+10 0 0 1 … 1 … 1 N 1 N+10 0 0 0 … 0 … 0 … 0 M 0 (real animals) ψ (not in pop) 1 - ψ

27. SECR Poisson Observation Model: Implementation III Number of animals & activity centers unknown Number of animals & activity centers unknown Specify priors: Specify priors: σ 2 ~ Uniform σ 2 ~ Uniform λ 0 ~ Gamma λ 0 ~ Gamma s ~ Uniform (S) s ~ Uniform (S) ψ ~ Uniform (0, 1) z i ~ Bernoulli (ψ) Implement using MCMC Implement using MCMC

28. Other SECR models Binomial and multinomial observation models differ from Poisson in details, but with same basic ideas Binomial and multinomial observation models differ from Poisson in details, but with same basic ideas Programs: DENSITY (Efford) Programs: DENSITY (Efford) SPACECAP (WCS-India) R package ‘secr’

29. SECR Summary Deals formally and explicitly with 2 main problems associated with density estimation based on closed CR models Deals formally and explicitly with 2 main problems associated with density estimation based on closed CR models Unknown sample area Unknown sample area Heterogeneity in capture probability associated with animal location relative to traps Heterogeneity in capture probability associated with animal location relative to traps Models are flexible (e.g., they permit trap response, etc.) Models are flexible (e.g., they permit trap response, etc.) New method of choice for density estimation New method of choice for density estimation

30. Some references Efford (2004): initial idea, ad hoc simulation based approach Efford (2004): initial idea, ad hoc simulation based approach Likelihood-based approaches: Borchers and Efford 2008), Efford et al. (2008, 2009) Likelihood-based approaches: Borchers and Efford 2008), Efford et al. (2008, 2009) Hierarchical MCMC approach: Royle and Young (2008), Royle et al. (2009ab), Royle and Gardner (2010), Gardner et al. (2009, 2010) Hierarchical MCMC approach: Royle and Young (2008), Royle et al. (2009ab), Royle and Gardner (2010), Gardner et al. (2009, 2010)