Introduction to Integrated Populations Models

.title[
# Introduction to Integrated Populations Models
]
.author[
### Mark Scheuerell<br><br><br>(use the arrows keys to navigate the slide deck)
]

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# RUN RECONSTRUCTION:

# Classical approach

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# Build a brood table

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# Build a brood table

`$$\Large ~~~~~~~~~~~~~ N_{y,a} = p_a S_y$$`

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# Build a brood table

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# Build a brood table

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# Build a brood table

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# Build a brood table

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# Problems with this approach

### .blue[1) Spawner counts are not exhaustive & contain errors]

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# Covariate measured with error

### .center.blue[Ricker model]

`$$\Large R_t = \alpha S_t \exp (- \beta S_t + \epsilon_t)$$`

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# Covariate measured with error

### .center.blue[Ricker model]

`$$\Large R_t = \alpha S_t \exp (- \beta S_t + \epsilon_t) \\ 
\Large \Downarrow \\
\Large \log \left( \frac{R_t}{S_t} \right) = \log (\alpha) - \beta \underline{S_t} + \epsilon_t \\
\Large \Downarrow \\
\Large y_i = \alpha + \beta \underline{x_i} + \epsilon_i$$`

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# Problems with this approach

### .blue[1) Spawner counts are not exhaustive & contain errors]

### .green[2) Age-composition data are typically non-exhaustive & imprecise]

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# Typical age expansion

### .blue[&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Fish sampled for age comp << total return!]

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# Problems with this approach

### .blue[1) Spawner counts are not exhaustive & contain errors]

### .green[2) Age-composition data are typically non-exhaustive & imprecise]

### .orange[3) Missing data cause problems]

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# Problems with missing data

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# Problems with missing data

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# Problems with missing data

### .red[We just lost 4 pairs of missing data!]

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# Problems with this approach

### .blue[1) Spawner counts are not exhaustive & contain errors]

### .green[2) Age-composition data are typically non-exhaustive & imprecise]

### .orange[3) Missing data cause problems]

### .purple[4) Stock-Recruit models are process models, not observation models]

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# Observation model

### .red[Time-ordering is irrelevant; estimated] `$\color{#F34213}{R_1}$` .red[has no effect on any later] `$\color{#F34213}{S_t}$`

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# Integrated population models

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# Integrated population models

### &nbsp; &nbsp; &nbsp; 😃 &nbsp; .blue[Do make the model match the data]

### &nbsp; &nbsp; &nbsp; 😱 &nbsp; .red[Don't pre-process the data to match the model]

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# 😃

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# 😱

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# Creating recruits from spawners

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# Forms for stock-recruit model

### .center.blue[Ricker model with autocorrelated errors]

`$$\Large R_t = \alpha S_t \exp (- \beta S_t + \epsilon_t) \\ 
\Large \Downarrow \\
\Large \log (R_t) = \log (\alpha) + \log(S_t) - \beta S_t + \epsilon_t$$`

`$$\Large \epsilon_t \sim \text{N}(\phi \epsilon_{t-1}, q)$$`

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# Forms for stock-recruit model

### .center.blue[Ricker model with covariates]

`$$\Large R_t = \alpha S_t \exp (- \beta S_t + \gamma X_{t-h} + \epsilon_t) \\ 
\Large \Downarrow \\
\Large \log (R_t) = \log (\alpha) + \log(S_t) - \beta S_t + \gamma X_{t-h} + \epsilon_t$$`

`$$\Large \epsilon_t \sim \text{N}(\phi \epsilon_{t-1}, q)$$`

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# Projecting recruits forward

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# Projecting recruits forward

<br>

### .center.blue[recruits-by-age = (proportion-by-age) &times; (total recruits)]

<br>

`$$\LARGE N_{a,t} = p_{a, t-a} R_{t-a}$$`

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# Observation models for spawners

### .pull-left.center[True spawners]

### .pull-right[&nbsp;]

### .pull-right.blue[True spawners are difference<br>between returns and catch]

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# Observation models for spawners

### .pull-left.center[True spawners]

### .pull-right[&nbsp;]

### .pull-right.blue[True spawners are difference<br>between returns and catch]

### .pull-left.center[<br><br><br>Observed spawners]

### .pull-right[<br><br><br>&nbsp;]

### .pull-right.green[Measured escapement is<br>estimate of true spawners]

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# Observation models for age-comp

### .pull-left.center[Total returns]

### .pull-right[&nbsp;]

<div style="float: left; width: 47%;">$$\Large D_t = \sum_a O_{a,t}$$</div>

### .pull-right.blue[Number of fish observed<br>in age class _a_ in year _t_]

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# Observation models for age-comp

### .pull-left.center[Total returns]

### .pull-right[&nbsp;]

<div style="float: left; width: 47%;">$$\Large D_t = \sum_a O_{a,t}$$</div>

### .pull-right.blue[Number of fish observed<br>in age class _a_ in year _t_]

### .pull-left.center[Proportion by age]

### .pull-right[&nbsp;<br>]

<div style="float: left; width: 47%;">$$\Large \pi_{a,t} = \frac{N_{a,t}}{\sum_a N_{a,t}}$$</div>

### <div style="float: right; padding: 15px; color: green; width: 46%;">Number of fish predicted<br>in age class _a_ in year _t_</div>

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# Observation models for age-comp

### .pull-left.center[Total returns]

### .pull-right[&nbsp;]

<div style="float: left; width: 47%;">$$\Large D_t = \sum_a O_{a,t}$$</div>

### .pull-right.blue[Number of fish observed<br>in age class _a_ in year _t_]

### .pull-left.center[Proportion by age]

### .pull-right[&nbsp;<br>]

<div style="float: left; width: 47%;">$$\Large \pi_{a,t} = \frac{N_{a,t}}{\sum_a N_{a,t}}$$</div>

### <div style="float: right; padding: 15px; color: green; width: 46%;">Number of fish predicted<br>in age class _a_ in year _t_</div>

`$$\Large \mathbf{O}_t \sim \text{Multinomial}(\boldsymbol{\pi}_t, D_t)$$`

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# Feedback between R & S

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# EXAMPLE: Washington Coho

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# Can we use a hierarchical approach to improve forecast performance?

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.citation.gray[DeFilippo et al. (2021) Fish Res]

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# WA natural-origin coho

### .blue[&#x2022; Eggs laid in brood year _y_]

### .blue[&#x2022; Fry emerge in spring of year _y_ + 1]

### .blue[&#x2022; Most juveniles spend ~1 year in freshwater]

### .blue[&#x2022; Outmigrate as smolts in spring of year _y_ + 2]

### .blue[&#x2022; Most spend ~1.5 years at sea]

### .blue[&#x2022; Return to spawn in fall of year _y_ + 3]

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# Data types (1986-2017)

### .blue[&#x2022; Quantity of habitat for each population (stream length)]

### .blue[&#x2022; Adult escapement counts]

### .blue[&#x2022; Adult harvest counts]

### .blue[&#x2022; Smolt outmigration counts]

### .blue[&#x2022; Coded wire tag marine survival estimates]

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# Forecast evaluation

### .blue[&#x2022; Initially fit to 16 years (1986-2001)]

### .blue[&#x2022; Skill evaluated for next 16 years (2002-2017) with model refit each year]

### .blue[&#x2022; Skill based on mean absolute scaled error (MASE)]

`$$\Large ASE_{i, y}=\frac{\left|P_{i, y}-A_{i, y}\right|}{\frac{1}{n-1} \sum_1^n\left|A_{i, t}-A_{i, t-1}\right|}$$`

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# Correlation in<br>marine survival

.citation.gray[DeFilippo et al. (2021) Fish Res]

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# Theoretical parent-offspring relationship

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background-image: url(figs/wa_coho_b-h.png)
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.citation.gray[DeFilippo et al. (2021) Fish Res]

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background-image: url(figs/wa_coho_fore.png)
background-position: 45% 65%
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.citation.gray[DeFilippo et al. (2021) Fish Res]

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background-image: url(figs/wa_coho_error.png)
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# IPM offers<br>improvement

.citation.gray[DeFilippo et al. (2021) Fish Res]

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# Learn more about IPMs

### https://github.com/lukasdefilippo/ST-IPM

### https://github.com/mdscheuerell/steelhead_IPM

### https://github.com/marksorel8/Sorel_etal_Wenatchee_IPM