Lecture 15

class: center, middle, inverse, title-slide

# Lecture 15
## Multi-state models
### <br/><br/><br/>WILD6900 (Spring 2020)

---

## Readings

> ### Kéry & Schaub 264-313

---
## What is a "state"

#### Alive/dead
- CJS model

--
#### Occupied/unoccupied
- Occupancy model

--
#### Geographic location (location `$1, 2, ...,S$`)

--
#### Un-infected/infected/dead

--
#### Age class

--
#### Any combination of above states*

.footnote[
*At least in theory
]

---
## Two states

#### State model

`$$\Large z_{i,t} \sim Bernoulli(z_{i,t-1}\phi)$$`

---
## Two states

#### Observation model

<br/>

`$$\Large y_{i,t} \sim Bernoulli(z_{i,t}p)$$`

---
## Three states

#### State model

---
## Three states

#### State model

---
## Three states - example

#### Movement between two sites

---
## Three states - example

#### Movement between two sites

- `$\large \phi_A$`: Probability alive at site A

- `$\large \phi_B$`: Probability alive at site B

- `$\large \psi_{AB}$`: Probability of moving from site A to site B

- `$\large \psi_{BA}$`: Probability of moving from site B to site A

---
## Three states - example

#### State model

.left-column[
<br/>
<br/>
<br/>
#### True state at time *t*
]

.right-column[
<center>
#### True state at time *t + 1*
</center>
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;">  </th>
   <th style="text-align:center;"> Site A </th>
   <th style="text-align:center;"> Site B </th>
   <th style="text-align:center;"> Dead </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> Site A </td>
   <td style="text-align:center;"> $\phi_A (1 - \psi_{AB})$ </td>
   <td style="text-align:center;"> $\phi_A \psi_{AB}$ </td>
   <td style="text-align:center;"> $1 - \phi_A$ </td>
  </tr>
  <tr>
   <td style="text-align:center;"> Site B </td>
   <td style="text-align:center;"> $\phi_B \psi_{BA}$ </td>
   <td style="text-align:center;"> $\phi_B (1 - \psi_{BA})$ </td>
   <td style="text-align:center;"> $1 - \phi_B$ </td>
  </tr>
  <tr>
   <td style="text-align:center;"> Dead </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
</tbody>
</table>

]

---
## Three states - example

#### Observation model

.left-column[
<br/>
<br/>
<br/>
#### True state at time *t*
]

.right-column[
<center>
#### Observation at time *t*
</center>
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;">  </th>
   <th style="text-align:center;"> Site A </th>
   <th style="text-align:center;"> Site B </th>
   <th style="text-align:center;"> Not Seen </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> Site A </td>
   <td style="text-align:center;"> $p_A$ </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> $1 - p_A$ </td>
  </tr>
  <tr>
   <td style="text-align:center;"> Site B </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> $p_B$ </td>
   <td style="text-align:center;"> $1 - p_B$ </td>
  </tr>
  <tr>
   <td style="text-align:center;"> Dead </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
</tbody>
</table>

]

---
## Three states - example

#### The data

- `$1 =$` Seen alive at site A

- `$2 =$` Seen alive at site B

- `$3 =$` Not seen

```r
head(rCH)
```

```
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    1    1    3    1    1    1
## [2,]    1    1    1    1    1    1
## [3,]    1    3    1    2    3    3
## [4,]    1    3    3    3    3    3
## [5,]    1    1    2    3    3    3
## [6,]    1    3    3    2    3    1
```

---
## Three states - example

#### The data

- `$f =$` First capture occasion

```r
tail(rCH)
```

```
##        [,1] [,2] [,3] [,4] [,5] [,6]
## [795,]    3    3    3    3    2    3
## [796,]    3    3    3    3    2    3
## [797,]    3    3    3    3    2    3
## [798,]    3    3    3    3    2    3
## [799,]    3    3    3    3    2    1
## [800,]    3    3    3    3    2    3
```

```r
tail(f)
```

```
## [1] 5 5 5 5 5 5
```

---
## Three states - example

#### The likelihood
- The Bernoulli distribution describes the probability of one (of two) outcomes

--
- The categorical distribution describes the probability of one (of `$N$`) outcomes
    + `$\Omega$` and `$\Theta$` are matrices defining the state/observation probabilities

--
`$$\large z_{i, t+1}|z_{i,t} \sim categorical(\Omega_{z_{i,t},1:S})$$`

`$$\large y_{i, t}|z_{i,t} \sim categorical(\Theta_{z_{i,t},1:S})$$`

---
## Analysis in JAGS

```r
sink("ms.jags")
cat("
model {
  # Priors 
   phiA ~ dunif(0, 1)
   phiB ~ dunif(0, 1)
   psiAB ~ dunif(0, 1)
   psiBA ~ dunif(0, 1)
   pA ~ dunif(0, 1)
   pB ~ dunif(0, 1)
```

---
## Analysis in JAGS

```r
  # Define state-transition and observation matrices
   # Define probabilities of state S(t+1) given S(t)
      ps[1,1] <- phiA * (1-psiAB)
      ps[1,2] <- phiA * psiAB
      ps[1,3] <- 1-phiA
      ps[2,1] <- phiB * psiBA
      ps[2,2] <- phiB * (1-psiBA)
      ps[2,3] <- 1-phiB
      ps[3,1] <- 0
      ps[3,2] <- 0
      ps[3,3] <- 1
```

---
## Analysis in JAGS

```r
      # Define probabilities of O(t) given S(t)
      po[1,1] <- pA
      po[1,2] <- 0
      po[1,3] <- 1-pA
      po[2,1] <- 0
      po[2,2] <- pB
      po[2,3] <- 1-pB
      po[3,1] <- 0
      po[3,2] <- 0
      po[3,3] <- 1
```

---
## Analysis in JAGS

```r
# Likelihood 
for (i in 1:nInd){
   # Define latent state at first capture
   z[i,f[i]] <- y[i,f[i]]
   for (t in (f[i]+1):nOcc){
      # State process: draw S(t) given S(t-1)
      z[i,t] ~ dcat(ps[z[i,t-1],])
      # Observation process: draw O(t) given S(t)
      y[i,t] ~ dcat(po[z[i,t],])
      } #t
   } #i
}
",fill = TRUE)
sink()
```

---
## Multi-state models

#### Very flexible
- Most capture-recapture models can be formulated as multi-state models

- Can accommodate many states

- Can include covariates and individual/temporal variation

#### But complexity comes at a cost
- Data hungry

- Often not fully identifiable

- Computationally intensive