Ecological forecasting with dynamic GAMs

.title[
# Ecological forecasting with dynamic GAMs
]
.author[
### Nicholas Clark
]
.institute[
### School of Veterinary Science, University of Queensland, Australia
]
.date[
### November, 2023
]

---

###“Because all decision making is based on what will happen in the future, either under the status quo or different decision alternatives, decision making ultimately depends on forecasts”
  
[Dietze et al. 2018](https://ecoforecast.org/about/)

---

background-image: url('./resources/big_data.gif')
background-size: contain
background-color: #F2F2F2

---

## Properties of ecological series
Temporal autocorrelation

Lagged effects

Non-Gaussian data and missing observations

Measurement error

Time-varying effects

Nonlinearities

Multi-series clustering

---

## Properties of ecological series
.grey[Temporal autocorrelation

Lagged effects

Non-Gaussian data and missing observations

Measurement error

Time-varying effects]

---

---

---

background-image: url('./resources/smooth_to_data.gif')
## ...penalized to fit data

---

## Easy to fit in <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:steelblue;overflow:visible;position:relative;"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg>
<img align="center" width="306" height="464" src="resources/gam_book.jpg" style="float: right; 
  margin: 10px;">
`$$\mathbb{E}(\boldsymbol{Y_t}|\boldsymbol{X_t})=g^{-1}(\alpha + \sum_{j=1}^{J}f(x_{jt}))$$`

<br/>
Where: 
- `$g^{-1}$` is the inverse of the link function
- `${\alpha}$` is the intercept
- `$f(x)$` are potentially nonlinear functions of the `$J$` predictors

---

---
## Need more on GAMs?
---
class: inverse
background-image: url('./resources/simpson_screenshot.png')
background-size: cover

## Gavin has you covered <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 0 48 256a208 208 0 1 0 416 0zM0 256a256 256 0 1 1 512 0A256 256 0 1 1 0 256zm177.6 62.1C192.8 334.5 218.8 352 256 352s63.2-17.5 78.4-33.9c9-9.7 24.2-10.4 33.9-1.4s10.4 24.2 1.4 33.9c-22 23.8-60 49.4-113.6 49.4s-91.7-25.5-113.6-49.4c-9-9.7-8.4-24.9 1.4-33.9s24.9-8.4 33.9 1.4zM144.4 208a32 32 0 1 1 64 0 32 32 0 1 1 -64 0zm192-32a32 32 0 1 1 0 64 32 32 0 1 1 0-64z"/></svg>

<img width="230" height="230" src="resources/simpson_qr_code.svg" style="position:absolute; bottom:0px; right:0px">
  
---

class: middle center
### GAMs are just fancy GLMs, where some (or all) of the predictor effects are estimated as (possibly nonlinear) smooth functions
<br>
### But the complexity these smooth functions can handle is *enormous*

---

## What's the catch?

---

## A spline of `time`

```r
library(mgcv)
*model <- gam(y ~ s(time, k = 20, bs = 'bs', m = 2),
                data = data,
                family = gaussian())
```

A B-spline (`bs = 'bs'`) with `m = 2` sets the penalty on the second derivative

---

## Hindcasts <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 0 48 256a208 208 0 1 0 416 0zM0 256a256 256 0 1 1 512 0A256 256 0 1 1 0 256zm177.6 62.1C192.8 334.5 218.8 352 256 352s63.2-17.5 78.4-33.9c9-9.7 24.2-10.4 33.9-1.4s10.4 24.2 1.4 33.9c-22 23.8-60 49.4-113.6 49.4s-91.7-25.5-113.6-49.4c-9-9.7-8.4-24.9 1.4-33.9s24.9-8.4 33.9 1.4zm40-89.3l0 0 0 0-.2-.2c-.2-.2-.4-.5-.7-.9c-.6-.8-1.6-2-2.8-3.4c-2.5-2.8-6-6.6-10.2-10.3c-8.8-7.8-18.8-14-27.7-14s-18.9 6.2-27.7 14c-4.2 3.7-7.7 7.5-10.2 10.3c-1.2 1.4-2.2 2.6-2.8 3.4c-.3 .4-.6 .7-.7 .9l-.2 .2 0 0 0 0 0 0c-2.1 2.8-5.7 3.9-8.9 2.8s-5.5-4.1-5.5-7.6c0-17.9 6.7-35.6 16.6-48.8c9.8-13 23.9-23.2 39.4-23.2s29.6 10.2 39.4 23.2c9.9 13.2 16.6 30.9 16.6 48.8c0 3.4-2.2 6.5-5.5 7.6s-6.9 0-8.9-2.8l0 0 0 0zm160 0l0 0-.2-.2c-.2-.2-.4-.5-.7-.9c-.6-.8-1.6-2-2.8-3.4c-2.5-2.8-6-6.6-10.2-10.3c-8.8-7.8-18.8-14-27.7-14s-18.9 6.2-27.7 14c-4.2 3.7-7.7 7.5-10.2 10.3c-1.2 1.4-2.2 2.6-2.8 3.4c-.3 .4-.6 .7-.7 .9l-.2 .2 0 0 0 0 0 0c-2.1 2.8-5.7 3.9-8.9 2.8s-5.5-4.1-5.5-7.6c0-17.9 6.7-35.6 16.6-48.8c9.8-13 23.9-23.2 39.4-23.2s29.6 10.2 39.4 23.2c9.9 13.2 16.6 30.9 16.6 48.8c0 3.4-2.2 6.5-5.5 7.6s-6.9 0-8.9-2.8l0 0 0 0 0 0z"/></svg>

```
## No non-missing values in test_observations; cannot calculate forecast score
```

---

## Extrapolate 2-steps ahead <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 0 48 256a208 208 0 1 0 416 0zM0 256a256 256 0 1 1 512 0A256 256 0 1 1 0 256zm177.6 62.1C192.8 334.5 218.8 352 256 352s63.2-17.5 78.4-33.9c9-9.7 24.2-10.4 33.9-1.4s10.4 24.2 1.4 33.9c-22 23.8-60 49.4-113.6 49.4s-91.7-25.5-113.6-49.4c-9-9.7-8.4-24.9 1.4-33.9s24.9-8.4 33.9 1.4zM144.4 208a32 32 0 1 1 64 0 32 32 0 1 1 -64 0zm192-32a32 32 0 1 1 0 64 32 32 0 1 1 0-64z"/></svg>

---

## 5-steps ahead <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M464 256A208 208 0 1 0 48 256a208 208 0 1 0 416 0zM0 256a256 256 0 1 1 512 0A256 256 0 1 1 0 256zM174.6 384.1c-4.5 12.5-18.2 18.9-30.7 14.4s-18.9-18.2-14.4-30.7C146.9 319.4 198.9 288 256 288s109.1 31.4 126.6 79.9c4.5 12.5-2 26.2-14.4 30.7s-26.2-2-30.7-14.4C328.2 358.5 297.2 336 256 336s-72.2 22.5-81.4 48.1zM144.4 208a32 32 0 1 1 64 0 32 32 0 1 1 -64 0zm192-32a32 32 0 1 1 0 64 32 32 0 1 1 0-64z"/></svg>

---

## 20-steps ahead <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M175.9 448c-35-.1-65.5-22.6-76-54.6C67.6 356.8 48 308.7 48 256C48 141.1 141.1 48 256 48s208 93.1 208 208s-93.1 208-208 208c-28.4 0-55.5-5.7-80.1-16zM0 256a256 256 0 1 0 512 0A256 256 0 1 0 0 256zM128 369c0 26 21.5 47 48 47s48-21 48-47c0-20-28.4-60.4-41.6-77.7c-3.2-4.4-9.6-4.4-12.8 0C156.6 308.6 128 349 128 369zm128-65c-13.3 0-24 10.7-24 24s10.7 24 24 24c30.7 0 58.7 11.5 80 30.6c9.9 8.8 25 8 33.9-1.9s8-25-1.9-33.9C338.3 320.2 299 304 256 304zm47.6-96a32 32 0 1 0 64 0 32 32 0 1 0 -64 0zm-128 32a32 32 0 1 0 0-64 32 32 0 1 0 0 64z"/></svg>

---

background-image: url('./resources/basis_weights.gif')
## Basis functions &#8680; local knowledge

---
## Basis functions &#8680; local knowledge

---

## Basis functions &#8680; local knowledge

---

## Basis functions &#8680; local knowledge

---
## Forecasts <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M400 406.1V288c0-13.3-10.7-24-24-24s-24 10.7-24 24V440.6c-28.7 15-61.4 23.4-96 23.4s-67.3-8.5-96-23.4V288c0-13.3-10.7-24-24-24s-24 10.7-24 24V406.1C72.6 368.2 48 315 48 256C48 141.1 141.1 48 256 48s208 93.1 208 208c0 59-24.6 112.2-64 150.1zM256 512A256 256 0 1 0 256 0a256 256 0 1 0 0 512zM159.6 220c10.6 0 19.9 3.8 25.4 9.7c7.6 8.1 20.2 8.5 28.3 .9s8.5-20.2 .9-28.3C199.7 186.8 179 180 159.6 180s-40.1 6.8-54.6 22.3c-7.6 8.1-7.1 20.7 .9 28.3s20.7 7.1 28.3-.9c5.5-5.8 14.8-9.7 25.4-9.7zm166.6 9.7c5.5-5.8 14.8-9.7 25.4-9.7s19.9 3.8 25.4 9.7c7.6 8.1 20.2 8.5 28.3 .9s8.5-20.2 .9-28.3C391.7 186.8 371 180 351.6 180s-40.1 6.8-54.6 22.3c-7.6 8.1-7.1 20.7 .9 28.3s20.7 7.1 28.3-.9zM208 320v32c0 26.5 21.5 48 48 48s48-21.5 48-48V320c0-26.5-21.5-48-48-48s-48 21.5-48 48z"/></svg>

---
## We need *global knowledge*
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-11-1.svg" style="display: block; margin: auto;" />

---

## We need *global knowledge*
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-12-1.svg" style="display: block; margin: auto;" />

---

## We need *global knowledge*
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-13-1.svg" style="display: block; margin: auto;" />

---
## We need *global knowledge*
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-14-1.svg" style="display: block; margin: auto;" />

---

## Dynamic GAMs

`$$\mathbb{E}(\boldsymbol{Y_t}|\boldsymbol{X_t})=g^{-1}(\alpha + \sum_{j=1}^{J}f(x_{jt}) + z_t)$$`

<br/>
Where: 
- `$g^{-1}$` is the inverse of the link function
- `${\alpha}$` is the intercept
- `$f(x)$` are potentially nonlinear functions of the `$J$` predictors
- `$z_t$` is a .emphasize[*latent dynamic process*]

---

## Modelling with the [`mvgam` 📦](https://github.com/nicholasjclark/mvgam/tree/master)

Bayesian framework to fit Dynamic GLMs and Dynamic GAMs
- Hierarchical intercepts, slopes and smooths
- Latent dynamic processes
- State-Space models with measurement error

Built off the [`mgcv` 📦](https://cran.r-project.org/web/packages/mgcv/index.html) to construct penalized smoothing splines

Convenient and familiar <svg aria-hidden="true" role="img" viewBox="0 0 581 512" style="height:1em;width:1.13em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:steelblue;overflow:visible;position:relative;"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> formula interface

Uni- or multivariate series from a range of response distributions

Uses [Stan](https://mc-stan.org/) for efficient Hamiltonian Monte Carlo sampling

---

## Observation families

`gaussian()`, `student-t()` &#8680; real values in `$(-\infty, \infty)$`   
 
`lognormal()`, `Gamma()` &#8680; positive real values in `$[0, \infty)$`

`betar()`  &#8680; real values (proportional) in `$[0,1]$`

`poisson()`, `nb()` &#8680; non-negative integers in `$(0,1,2,...)$`

---

## Extended predictor effects

`s()` &#8680; Smoothing spline of one or more covariates

`s(bs = 're')` &#8680; Hierarchical slopes or intercepts

`te(), ti(), t2()` &#8680; Tensor product smoothing spline of two or more covariates

`gp()` &#8680; Gaussian Process function (with squared exponential kernel) of one covariate

`dynamic()` &#8680; Time-varying effect of one covariate

---

class: middle center
### We can fit models that include random effects, nonlinear effects and complex multidimensional smooth functions. All these effects can operate .emphasize[*on both process and observation models*]
<br>
### Can incorporate unobserved temporal dynamics; do not need to regress the outcome on its own past values
<br>
### Very advantageous for ecological time series. But what kinds of dynamic processes are available in the `mvgam` 📦?

---

## Random Walk or AR(1-3)
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-15-1.svg" style="display: block; margin: auto;" />

---

## Gaussian Process...
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-16-1.svg" style="display: block; margin: auto;" />

---
## ...where length scale  &#8680; *memory*
<img src="EFI_talk_slidedeck_files/figure-html/unnamed-chunk-17-1.svg" style="display: block; margin: auto;" />

---

background-image: url('./resources/VAR.svg')
background-size: contain
## VAR1 &#8680; Granger causality

---

background-image: url('./resources/df_with_series.gif')
## Factors &#8680; induced correlations
---

## Example of the interface

```r
model <- mvgam(
  formula = y ~ 
    s(series, bs = 're') + 
    s(x0, series, bs = 're') +
    x1 +
    gp() +
    te(x3, x4, bs = c('cr', 'tp')),
  data = data,
  family = poisson(),
  trend_model = 'AR1',
  burnin = 500,
  samples = 500,
  chains = 4,
  parallel = TRUE
  )
```

---

## Example data (long format)

---

## Response (`NA`s allowed)

---

## Series indicator (as `factor`)

---

## Time indicator

---

## Any other predictors

<table class=" lightable-minimal" style='font-family: "Trebuchet MS", verdana, sans-serif; width: auto !important; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:right;"> y </th>
   <th style="text-align:left;"> series </th>
   <th style="text-align:right;"> time </th>
   <th style="text-align:right;"> x0 </th>
   <th style="text-align:left;"> x1 </th>
   <th style="text-align:right;"> x2 </th>
   <th style="text-align:right;"> x3 </th>
   <th style="text-align:right;"> x4 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:left;"> species_1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.38 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> A </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.20 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.18 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.72 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:left;"> species_2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.71 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> A </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -2.67 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.02 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.67 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:left;"> species_3 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.05 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> B </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.33 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.12 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.50 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:left;"> species_4 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.77 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> B </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.65 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.86 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.49 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:left;"> species_1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.29 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> A </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.25 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.18 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.82 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:left;"> species_2 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.34 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> A </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.15 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 2.12 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.20 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> species_3 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.38 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> B </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.81 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.33 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -1.15 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> species_4 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.32 </td>
   <td style="text-align:left;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> B </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 0.22 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> -0.72 </td>
   <td style="text-align:right;font-weight: bold;background-color: rgba(81, 36, 122, 0.13) !important;"> 1.36 </td>
  </tr>
</tbody>
</table>

---

## Types of `mvgam` predictions
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.small[modified from [Heiss 2022](https://www.andrewheiss.com/blog/2022/09/26/guide-visualizing-types-posteriors/)]

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## Workflow in `mvgam` 📦

Fit models that can include nonlinear splines, GPs, and multivariate dynamic processes to ecological time series

Use posterior predictive checks and Randomized Quantile (Dunn-Smyth) residuals to assess model failures

Use `marginaleffects` 📦 to generate interpretable (and reportable) model predictions

Produce probabilistic forecasts

Evaluate forecasts from competing models with proper scoring rules                           
---

## More resources
Vignette &#8680; [Overview of the package](https://nicholasjclark.github.io/mvgam/articles/mvgam_overview.html)

Vignette &#8680;  [Formatting data for use in `mvgam`](https://nicholasjclark.github.io/mvgam/articles/data_in_mvgam.html)

Vignette &#8680; [Shared latent process models](https://nicholasjclark.github.io/mvgam/articles/shared_states.html)

Vignette &#8680; [Time-varying effects](https://nicholasjclark.github.io/mvgam/articles/time_varying_effects.html)

Vignette &#8680; [Multivariate State-Space models](https://nicholasjclark.github.io/mvgam/articles/trend_formulas.html)

Motivating publication &#8680; Clark & Wells 2023 [*Methods in Ecology and Evolution*](https://besjournals.onlinelibrary.wiley.com/doi/full/10.1111/2041-210X.13974)

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# Example
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class: animated fadeIn
background-image: url('./resources/pp_image.jpg')
background-size: cover

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class: animated fadeIn
background-image: url('./resources/dm_image.png')
background-size: cover

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class: animated fadeIn
background-image: url('./resources/mouse_image.jpg')
background-size: cover

---
class: full-size
## Long-term rodent monitoring
<img align="center" width="410" height="500" src="resources/portal_sampling.png" style="float: right; 
  margin: 10px;">

Monthly trapping in Portal, AZ, from 1977

20 rodent species recorded

Experimental evidence for interactions

Both gradual and sudden regime transitions

---
background-image: url('./resources/raw_series.jpg')
background-size: contain

---
background-image: url('./resources/model_definition.png')
background-size: contain

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class: full-size
## Model components
<img align="center" width="530" height="370" src="resources/crank.jpg" style="float: right; 
  margin: 10px;">

---
background-image: url('./resources/VAR.svg')
background-size: contain
## VAR1 &#8680; Granger causality

---
class: full-size
## Model components
<img align="center" width="530" height="370" src="resources/crank.jpg" style="float: right; 
  margin: 10px;">

1. Multispecies temporal dynamics

---
background-image: url('./resources/raw_acfs.jpg')
background-size: contain

---
background-image: url('./resources/dist_lag1.svg')
background-size: contain

---
background-image: url('./resources/dist_lag2.svg')
background-size: contain

---
class: full-size
## Model components
<img align="center" width="530" height="370" src="resources/crank.jpg" style="float: right; 
  margin: 10px;">

1. Multispecies temporal dynamics

2. Seasonal variation

---

---

---

background-image: url('./resources/krat_sketch.svg')
background-size: contain
## What did we find?

---

background-image: url('./resources/NDVI_contrasts.jpeg')
background-size: contain

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background-image: url('./resources/krat_sketch.svg')
background-size: contain
## What did we find?
  
.emphasize[*1. Higher NDVI = more captures*]

---

background-image: url('./resources/mintemp_conditionals.jpeg')
background-size: contain

---

background-image: url('./resources/krat_sketch.svg')
background-size: contain
## What did we find?

1. Higher NDVI = more captures

---
background-image: url('./resources/VAR_cors.jpeg')
background-size: contain

---
background-image: url('./resources/DM_imp_response.jpg')
background-size: contain

---

background-image: url('./resources/krat_sketch.svg')
background-size: contain
## What did we find?
  
1. Higher NDVI = more captures

2. Varying seasonal effects

---

---

---
background-image: url('./resources/krat_sketch.svg')
background-size: contain
## What did we find?
  
1. Higher NDVI = more captures

2. Varying seasonal effects

3. Complex dependencies

---

# Thank you. Questions?