Lecture 20

class: center, middle, inverse, title-slide

.title[
# Lecture 20
]
.subtitle[
## Open Population Mark Recapture Part 2
]
.author[
### <br/><br/><br/>Spring 2025
]

---

## Readings

> Analysis of Capture-Recapture Data By Rachel S. McCrea, Byron J. T. Morgan

> Kéry & Schaub Chapter 10

> http://www.phidot.org/software/mark/docs/book/pdf/chap12.pdf

---
## Review

Lots of parameters of potential interest in these models

- `$\phi$` = probability of survival

- `$\gamma$` = biologically meaningless parameter indicating if you are 'removed' from the available population

- `$N_t$` = realized abundance in time `$t$`

- `$N_s$` = total number of individuals ever alive (superpopulation)

- `$B_t$` = realized recruits in time `$t$`

- `$b_t$` = probability of being recruited in time `$t$` given that you ARE recruited at some point from time `$t = 1$` to time `$t = T$` (semi-meaningless biologically)

- `$\psi$` = proportion of the total augmented population (real + imaginary) that are real

- `$f = \frac{B_t}{N_t}$` = per capita realized recruitment

---
## JS Open Population Model

In the classic Jolly-Seber formulation, the population followed this structure:

Here, what the authors call `$M_t$` is just marked individuals, so `$M_t$` + `$U_t$` = `$N_t$`.
---
## POPAN Style Superpopulation Model

First published by Schwarz et al. (1993) and then Schwarz and Arnason (1996). The model was run using a software package called "POPAN", hence we now call it the "POPAN model".

This formulation was much more ground-breaking before we had data augmentation techniques.

---
## POPAN Style Superpopulation Model

This formulation has very little impact when we actually go to code the model in Nimble.

<br/>

In our Bayesian framework, `$N_s$` = `$\psi*M$` (proportion of total augmented population ever alive).

<br/>

Then at each time set, animals can enter the population with some probability `$b$`, such that the expected value of recruits in each time step is just `$N_s*b$` = `$\psi*M*b$`.

<br/>

Assuming you are using data augmentation and the robust design, the only difference is that animals can be alive (z = 1) but not real. Thus we hope to avoid some of the 'recruitment rate' issues by allowing animals to 'become real' at a different rate than they 'are born'.

---
## POPAN Model in Nimble
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.smaller[

``` r
for (i in 1:M){
    w[i] ~ dbern(psi)   # Draw latent inclusion
    z[i,1] ~ dbern(nu[1])
    # Observation process
    mu1[i] <- p[i,1] * z[i,1] * w[i]
    y[i,1] ~ dbern(mu1[i])
    # Subsequent occasions
    for (t in 2:n.occasions){
       # State process
       q[i,t-1] <- 1-z[i,t-1]
       mu2[i,t] <- phi[i,t-1] * z[i,t-1] + nu[t] * prod(q[i,1:(t-1)])
       z[i,t] ~ dbern(mu2[i,t])
       # Observation process
       mu3[i,t] <- z[i,t] * p[i,t] * w[i]
       y[i,t] ~ dbern(mu3[i,t])
    } #t
 } #i
for(t in 1:n.occasions){
  nu[t] ~ dbeta(1,1)
}
```
]

---
## Pros and Cons - General

####Regardless of parameterization, fully time-dependent models are hard

In a fully time-dependent model (both `$p$` and `$\phi$` vary with time) 
<br/>

+ Cannot separately estimate first entry probability and first capture probability 
    
<br/>
    
    + Cannot separately estimate last surival and last capture probability without robust design

---
## Other Models Exist

+ Pradel Model (reverse-time formulation)
  
  <br/>
  
  + Link-Barker Model (allows for direct prior on per-capita fecundity)
  
  <br/>
  
  + Multi-state Formulation
  
  <br/>
  
--
<br/>

All these model requires a lot of data on marked individuals - we need a lot of data to understand complex processes

---
## Example open population - Dipper!

As promised, this would not be a wildlife Bayesian statistics class without learning about the Dipper dataset.

European Dipper (*Cinclus cinclus*) were captured annually from 1981--1987. This dataset was originally analyzed in a CJS format and later in POPAN. It has be analyzed excessively since its original publication.

Consider this a history lesson :)

We will use the version of this data contained in the `bayess` package

``` r
library(bayess)
data("eurodip")
```

---
## Dippers

In eurodip, each row corresponds to a capture-recapture story for a given adult dipper, 0 indicating an absence of capture that year and, in the case of a capture, 1, 2, or 3 representing the zone where the dipper is captured.

We will first analyze this data using the per-capita restricted dynamic occupancy form of the JS model, then in the POPAN style, and then as a hidden markov model. Note that we *do not* have a robust design here.

To save computation time, we'll only analyze the first 100 individuals in the dataset.

``` r
head(eurodip)
```

```
##   t1 t2 t3 t4 t5 t6 t7
## 1  2  3  2  3  0  0  0
## 2  2  2  0  0  0  0  0
## 3  2  2  1  2  3  3  0
## 4  1  1  0  0  0  0  0
## 5  2  0  0  0  0  0  0
## 6  1  0  3  0  0  0  0
```