Lecture 11

class: center, middle, inverse, title-slide

# Lecture 11
## Estimating abundance: Occupancy modeling
### <br/><br/><br/>WILD6900 (Spring 2020)

---

## Readings

> Kéry & Schaub 383-409

---
## Estimating abundance?

#### For the past few weeks, we've been modeling abundance:

`$$\Large N_t \sim Poisson(\lambda)$$`

--
#### Occupancy is the probability a site is occupied

- reduced-information form of abundance
    + If `$\large N_i > 0$`, `$\large z_i = 1$`

+ If `$\large N_i = 0$`, `$\large z_i = 0$`

--
#### So even when occupancy is the state-variable, we are still modeling something related to abundance

---
## Estimating abundance?

#### Typical state model for occupancy

`$$\Large z_i \sim Bernoulli(\psi)$$`

--
#### If the expected abundance is `$\large \lambda$`, what is the probability `$\large N = 0$`?

`$$\Large Pr(N=0|\lambda)=\frac{\lambda^0e^-\lambda}{0!}=e^{-\lambda}$$`

--
#### If the expected abundance is `$\large \lambda$`, what is the probability `$\large N > 0$`?

`$$\Large 1 - Pr(N=0|\lambda)=1 -e^{-\lambda}$$`

--
#### So (if our assumptions are valid):

`$$\Large \psi = 1 - e^{-\lambda}$$`

---
## Why estimate occupancy?

#### Less effort

--
#### Historical data sets

--
#### More reliable when `$\large N$` very small

--
#### Occupancy = abundance (e.g., breeding territory)

--
#### Metapopulation dynamics

--
#### Distribution/range size

--
#### Disease dynamics

---
## Why not just use observed presence/absence?

#### As in all ecological studies, we rarely (if ever) observe the state process perfectly

--
#### In the case of occupancy, some sites will be occupied but we will fail to detect the species

- i.e., `$\large p < 1$`

--
#### Also possible (though generally more rare) that we record the species when it's not present (false positive)

- see [Royle & Link 2006](https://www.uvm.edu/rsenr/vtcfwru/spreadsheets/occupancy/Occupancy%20Exercises/Exercise12/Royle_Link_2006.pdf)

--
#### Similar to N-mixture models, estimating `$\large p$` requires temporal replication

---
## Estimating `$\LARGE p$`

#### Imagine a single site surveyed 3 times:

- Assume site is closed across samples

- Assume constant `$\large p$`

`$$\LARGE y_i = [111]$$`

--
#### What is the likelihood of this observation?

--
`$$\LARGE \psi p^3$$`

---
## Estimating `$\LARGE p$`

#### What about?

`$$\LARGE y_i = [011]$$`

--
`$$\LARGE \psi (1-p)p^2$$`

---
## Estimating `$\LARGE p$`

#### What about?

`$$\LARGE y_i = [000]$$`

--
`$$\LARGE (1 - \psi) + \psi (1-p)^3$$`

---
## Single-season (static) occupancy model

#### State-space formulation

- State-model

`$$\Large z_i \sim Bernoulli(\psi_i)$$`

`$$\Large logit(\psi_i) = \alpha_0 + \alpha_1x_i$$`

- Observation-model

`$$\Large y_{i,k} \sim Bernoulli(z_ip_{i,k})$$`

`$$\Large logit(p_{ik}) = \beta_0 + \beta_1x_{i,k}$$`

---
## Single-season (static) occupancy model

```r
model{
  # Priors 
  psi ~ dbeta(1, 1)
  p ~ dbeta(1, 1)
  
  # Likelihood
  for(i in 1:M){
    # State model
    z[i] ~ dbern(psi)
    
    # Observation model
    for(k in 1:K){
      y[i, k] ~ dbern(p * z[i])
    }
  }
}
```

---
## Single-season (static) occupancy model

```r
model{
  alpha0 ~ dnorm(0, 0.1)
  alpha1 ~ dnorm(0, 0.1)
  mu.lp ~ dnorm(0, 0.1)
  tau.p ~ dunif(0, 10)

for(i in 1:M){
    z[i] ~ dbern(psi[i])
    logit(psi[i]) <- alpha0 + alpha1 * x1[i]
    for(k in 1:K){
      y[i, k] ~ dbern(p[i,k] * z[i])
      logit(p[i,k]) <- lp[i,k]
      lp[i,k] ~ dnorm(mu.lp, tau)
    }
  }

N.occ <- sum(z[1:M])
}
```

---
## Multi-season (dynamic) occupancy model

#### What if occupancy can change over time?

- Data collection using the *robust design*
    + Population open between primary periods (e.g., years)
    + Population closed within secondary periods (e.g., occasions)

`$$\Large y_i = [\underbrace{101}_{Year\;1}\;\;\;\;\; \underbrace{000}_{Year\;2}\;\;\;\;\;\underbrace{110}_{Year\;3}\;\;\;\;\;\underbrace{100}_{Year\;4}]$$`

--
- In year 1:
`$$\Large z_{i,1} \sim Bernoulli(\psi)$$`

- In years 2+:
`$$\Large z_{i,t} \sim Bernoulli(z_{i,t-1}(1-\epsilon)+(1-z_{i,t-1}\gamma))$$`