Lecture 9

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# Lecture 9
## State-space models
### WILD6900 (Spring 2021)

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## Readings

> Kéry & Schaub 115-131

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## Ecological state variables

#### **State variables** are the ecological quantities of interest in our model that change over space or time

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#### Abundance

> the number of individual organisms in a population at a particular point in time

#### Occurrence

> the spatial distribution of organisms with a particular region at a particular point in time

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#### Richness

> the number of co-occurring species at a given location and a particular point in time

???

`$^2$` These are not the only state variables, but are among the most common in ecological studies

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## Ecological parameters

#### **Parameters** determine how the state variables change over space and time

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- Survival

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- Reproduction

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- Movement

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- Population growth rate

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- Carrying capacity

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- Colonization/extinction rate

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## Process models

`$$\Huge [z|g(\theta_p, x), \sigma^2_p]$$`

- Mathematical description of our hypothesis about how the *state variables* we are interested in change over space and time

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- Represent the **true** value of our state variables at any given point in space or time

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- Deterministic

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- Abstraction

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## Observation models

- Rarely observe the true state perfectly

+ Animals are elusive and may hide from observers  
    + Even plants may be cryptic and hard to find  
    + Measurements may be taken with error  
    + May count the same individual > 1

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- *Observation uncertainty* `$\large (\sigma^2_o)$` can lead to biased estimates of model parameters, so generally requires its own model

`$$\Huge [y_i|d(\Theta_o,z_i), \sigma^2_o]$$`

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# State-space models

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## State-space models

- Hierarchical models

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- Decompose time series into:

+ process variation in state process  
    + observation error

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- Generally used for *Markovian* state process models

+ Population dynamics  
    + Survival  
    + Occupancy

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## Process models

### Population dynamics

`$$\Large N_{t+1} \sim Poisson(N_t \lambda)$$`

`$$\Large N_{t+1} \sim Normal(N_t e^{\bigg[r_0 \bigg(1-\frac{N_t}{K}\bigg)\bigg]}, \sigma^2)$$`

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### Survival

`$$\Large z_{t+1} \sim Bernoulli\big(z_t \phi_t\big)$$`

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### Occupancy

`$$\Large z_{t+1} \sim Bernoulli\bigg(z_t (1-\epsilon_t) + (1 - z_t)\gamma_t\bigg)$$`

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## Observation models

### Population dynamics

`$$\Large C_{t} \sim Normal(N_t, \sigma_o^2)$$`

`$$\Large C_{t} \sim Binomial(N_t, p)$$`

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### Survival

`$$\Large y_{t} \sim Bernoulli\big(z_t p\big)$$`

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### Occupancy

`$$\Large y_{t} \sim Bernoulli\big(z_t p_t\big)$$`

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## Including covariates

#### For any of these models, we can use GLMs to include covariates effects on parameters:

`$$\Large log(\lambda_t) = \alpha + \mathbf \beta \mathbf X + \varepsilon_t$$`

`$$\Large logit(\phi_t/\gamma_t/\epsilon_t/p_t) = \alpha + \mathbf \beta \mathbf X + \varepsilon_t$$`

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#### And random effect structure

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`$$\Large \varepsilon_t \sim normal\big(0, \tau_{\lambda}\big)$$`

`$$\Large \tau_{\lambda} \sim gamma\big(0.25, 0.25\big)$$`
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## Simple state-space population growth model

#### Process model
`$$\Large N_{t+1} = N_t \lambda_t$$`

`$$\Large \lambda_t \sim normal(\mu_{\lambda}, \tau_{\lambda})$$`

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#### Observation model
`$$\Large C_{t} = N_t + \epsilon_t$$`

`$$\Large \epsilon_t\sim Normal(0, \sigma_o^2)$$`

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## Simple state-space population growth model

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## Simple state-space population growth model

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## Simple state-space population growth model

#### What if instead of:

`$$\Large C_{t} = N_t + \epsilon_t$$`

`$$\Large \epsilon_t\sim Normal(0, \sigma_o^2)$$`

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#### The observation model is:

`$$\Large C_{t} \sim binomial(N_t, p)$$`

#### and

`$$\Large N_t = 50$$`

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## Systematic bias in state-space models

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## Systematic bias in state-space models

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## Systematic bias in state-space models

#### What if instead of:

`$$\Large C_{t} \sim binomial(N_t, p)$$`

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#### The observation model is:

`$$\Large C_{t} \sim binomial(N_t, p_t)$$`

#### and

`$$\Large logit(p_{t}) = \alpha + \beta \times year_t$$`

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## Systematic bias in state-space models

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## Systematic bias in state-space models

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## State-space models

#### Produce unbiased estimates of `$\Large N$` **only** when false-positives and false-negatives cancel each other out on average

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#### Produce unbiased estimates of population **indices** `$\Large (Np)$` if detection probability has no pattern over time

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#### Do **not** produce unbiased estimates of `$\Large N$` or `$\Large Np$` if their are temporal patterns in detection probability or false-positive rates

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## State-space models

#### Advantages

- explicit decomposition of process and observation models

- flexible

- mechanistic "smoothing" of process model

- latent state (and uncertainty) can be monitored

- possible to "integrate" data on state/observation parameters

#### Disadvantages

- computationally intensive

- usually produce biased estimates of `$N$`