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<!DOCTYPE html>
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<head>
<title>Ecological Time Series Analysis and Forecasting</title>
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class: inverse, middle, left, my-title-slide, title-slide
.title[
# Ecological Time Series Analysis and Forecasting
]
.author[
### Nicholas Clark
]
.institute[
### School of Veterinary Science, University of Queensland, Australia
]
.date[
### Friday 13th December, 2024
]
---
class: middle center
<style>.panelset{--panel-tab-foreground: #8F2727;--panel-tab-inactive-opacity: 0.8;}</style>
# Welcome
???
---
## Workflow
Press the "o" key on your keyboard to navigate among html slides
Access the [workshop materials here](https://github.com/nicholasjclark/ESA_2024_timeseries)
- View the sample <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> scripts in [live_code_examples](https://github.com/nicholasjclark/ESA_2024_timeseries/tree/main/live_code_examples)
- Use the [Google Doc](https://docs.google.com/document/d/1xd3icf1wxGxO3SVt2AmKO8CkeKv1QpsxgqK7rR15U08/edit?tab=t.0#heading=h.ds87nag4ykyb) to ask questions
Relevant open-source materials include:
- [Forecasting Principles and Practice](https://otexts.com/fpp3/)
- [Applied Time Series Analysis](https://atsa-es.github.io/atsa-labs/)
- [Ecological Forecasting & Dynamics Course](https://github.com/nicholasjclark/physalia-forecasting-course/tree/main)
- [A blog on how to use and interpret GAMs](https://ecogambler.netlify.app/blog/)
---
## This workshop's topics
Introductions
Why are time series difficult?
Common time series models
Why they fail in ecology
State-Space GAMs and the `mvgam` 📦
---
class: inverse middle center big-subsection
# Tell us about yourself
---
## Some challenges of time series
Temporal autocorrelation
Lagged effects
Non-Gaussian data and missing observations
Measurement error
Time-varying effects
Nonlinearities
Multi-series clustering
---
## A *positively* autocorrelated series
.panelset[
.panel[.panel-name[Code]
```r
# set seed for reproducibility
set.seed(1111)
# number of timepoints
T <- 100
# use arima.sim to simulate from an AR(1) model
series <- arima.sim(model = list(ar = 0.8), n = T, sd = 1)
# plot the time series as a line
plot(series, type = 'l', bty = 'l', lwd = 2,
col = 'darkred', ylab = 'Y', xlab = 'Time')
```
]
.panel[.panel-name[Model]
`$$\boldsymbol{Y}_{t}\sim \text{Normal}(\color{darkred}{0.8} * \boldsymbol{Y}_{t-1},\color{darkred}{1})$$`
]
.panel[.panel-name[Plot]
.center[]
]
]
---
## A *negatively* autocorrelated series
.panelset[
.panel[.panel-name[Code]
```r
# set seed for reproducibility
set.seed(1111)
# number of timepoints
T <- 100
# use arima.sim to simulate from an AR(1) model
series <- arima.sim(model = list(ar = -0.8), n = T, sd = 1)
# plot the time series as a line
plot(series, type = 'l', bty = 'l', lwd = 2,
col = 'darkred', ylab = 'Y', xlab = 'Time')
```
]
.panel[.panel-name[Model]
`$$\boldsymbol{Y}_{t}\sim \text{Normal}(\color{darkred}{-0.8} * \boldsymbol{Y}_{t-1},\color{darkred}{1})$$`
]
.panel[.panel-name[Plot]
.center[]
]
]
---
class: full-size
## Seasonality
.pull-right-bigger[]
Many time series show .emphasize[*repeated periodic cycles*]
- Breeding seasons
- Migration
- Green-ups / green-downs
- Lunar cycles
- Predator / prey dynamics
Often change slowly over time
---
## Example seasonal series
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-3-1.svg" style="display: block; margin: auto;" />
---
## Decompose: trend + seasonality
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-4-1.svg" style="display: block; margin: auto;" />
---
class: middle center
### Modelling these multiple components, either additively or multiplicatively, is a major goal of most time series analysis procedures
---
## Common time series models
Random Walk ([RW](https://atsa-es.github.io/atsa-labs/sec-tslab-random-walks-rw.html))
Autoregressive ([AR](https://atsa-es.github.io/atsa-labs/sec-tslab-autoregressive-ar-models.html))
Autoregressive Integrated Moving Average ([ARIMA](https://otexts.com/fpp3/arima.html); require [stationarity](https://otexts.com/fpp3/stationarity.html))
Exponential Smoothing ([ETS](https://otexts.com/fpp3/expsmooth.html))
[Regression with ARIMA errors](https://otexts.com/fpp3/regarima.html)
---
## *Very* easy to apply 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 src="resources/fc_logo.png" style="position:fixed; right:8%; top:4%; width:100px; height:117px; border:none;" />
Tools in the [`forecast` 📦](https://pkg.robjhyndman.com/forecast/) are hugely popular and accessible for time series analysis / forecasting
[ETS](https://pkg.robjhyndman.com/forecast/reference/ets.html) handles many types of seasonality and nonlinear trends
[Regression with ARIMA errors](https://pkg.robjhyndman.com/forecast/reference/auto.arima.html) includes additive fixed effects of predictors while capturing trends and seasonality
*Some* of these algorithms can handle missing data
*All* are extremely fast to fit and forecast
---
## Great! But what about these?
.grey[Temporal autocorrelation
Lagged effects]
.emphasize[*Non-Gaussian data and missing observations*
*Measurement error*
*Time-varying effects*
*Nonlinearities*
*Multi-series clustering*]
---
## Ecological time series include
Counts of multiple species over time
Presence-absence of species
Repeated captures in multiple plots
Censored measures (OTUs / pollutants with limits of detection)
Phenology records
Tree rings
etc...
---
## Example ecological time series
</br>
.pull-left[
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-5-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-6-1.svg" style="display: block; margin: auto;" />
]
---
## Another ecological time series
</br>
.pull-left[
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-7-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-8-1.svg" style="display: block; margin: auto;" />
]
---
## Collections of ecological series
<img src="ESA_slidedeck_files/figure-html/unnamed-chunk-9-1.svg" style="display: block; margin: auto;" />
---
class: inverse white-subsection
background-image: url('./resources/bwbirds.jpeg')
background-size: cover
## All can have measurement error
---
class: inverse middle center big-subsection
# How can we do better?
---
background-image: url('./resources/SS_model.svg')
## State-Space models
---
## State-Space models
</br>
<img align="center" width="1200" height="300" src="resources/auger.jpg">
.small[[Auger-Methe *et al* 2021](https://esajournals.onlinelibrary.wiley.com/doi/10.1002/ecm.1470)]
---
## State-Space linear models
`\begin{align*}
Y_t & \sim Normal(\alpha + \sum_{j=1}^{J}(\beta_j x_{jt}) + Zz_t, \sigma_y) \\
z_t & \sim Normal(f(z_{t-lag}) + \sum_{k=1}^{K}(\beta_k q_{kt}), \sigma_z)
\end{align*}`
<br/>
Where:
- `\(\beta_j\)` capture linear effects of the `\(J\)` observation model predictors
- `\(z_t\)` is a .emphasize[*latent process*], weighted by a loading matrix `\(Z\)`
- `\(\beta_k\)` capture linear effects of the `\(K\)` process model predictors
---
background-image: url('./resources/df_with_series.gif')
## *Z* &#8680; induced correlations
---
# Gaussian!?!
Gaussian observation models won't give sensible predictions for bounded / discrete / non-Normal data
We can do better by choosing observation distributions that obey the constraints on our outcome variables
.emphasize[*Generalizes*] the linear regression by replacing parameters from other probability distributions with linear models
`\(\alpha + \sum_{j=1}^{J}(\beta_j x_{jt}) + Zz_t\)` &#8680; `\(g^{-1}(\alpha + \sum_{j=1}^{J}(\beta_j x_{jt}) + Zz_t)\)`
`\(g^{-1}\)` is the inverse of a nonlinear .emphasize[*link function*]
---
## Many relevant distributions
[Many useful GLM probability distributions exist](https://cran.r-project.org/web/packages/brms/vignettes/brms_families.html). Some of these include:
- .emphasize[*Negative Binomial*] &mdash; overdispersed integers in `\((0,1,2,...)\)`
- .emphasize[*Bernoulli*] &mdash; presence-absence data in `\(\{0,1\}\)`
- .emphasize[*Student's T*] &mdash; heavy-tailed (skewed) real values in `\((-\infty, \infty)\)`
- .emphasize[*Lognormal*] &mdash; heavy-tailed (right skewed) real values in `\((0, \infty)\)`
- .emphasize[*Gamma*] &mdash; lighter-tailed (less skewed) real values in `\((0, \infty)\)`
- .emphasize[*Multinomial*] &mdash; integers representing `\(K\)` unordered categories in `\((0,1,..., K)\)`
- .emphasize[*Ordinal*] &mdash; integers representing `\(K\)` ordered categories in `\((0,1,..., K)\)`
---
class: middle center
### State-Space GLMs allow us to build models that respect the bounds and distributions of our observed data
<br>
### They traditionally assume the appropriately transformed mean response depends *linearly* on the predictors, as well as on the latent states
<br>
### But there are many other properties we'd like to model, including nonlinearities. Time to get .multicolor[W I G G L Y]
---
background-image: url('./resources/smooth_only.gif')
## GAMs use splines ...
---
background-image: url('./ESA_slidedeck_files/figure-html/basis-functions-1.svg')
## ... made of basis functions
---
background-image: url('./resources/basis-functions-weights-1.svg')
## Weighting basis functions ...
---
background-image: url('./resources/basis_weights.gif')
## ... gives a spline `\((f(x))\)`
---
background-image: url('./resources/smooth_to_data.gif')
## Penalize `\(f"(x)\)` to learn weights
---
background-image: url('./ESA_slidedeck_files/figure-html/complexity-1.svg')
---
## State-Space GAMs
`\begin{align*}
\mathbb{E}(\boldsymbol{Y_t}|\boldsymbol{X_t}, \boldsymbol{Q_t}) & = g^{-1}(\alpha + \sum_{j=1}^{J}f(x_{jt}) + Zz_t) \\
z_t & \sim Normal(f(z_{t-lag}) + \sum_{k=1}^{K}f(q_{kt}), \sigma_z)
\end{align*}`
<br/>
Where:
- `\(f(x)\)` are potentially nonlinear functions of the `\(J\)` predictors
- `\(z_t\)` is a .emphasize[*latent process*], weighted by a loading matrix `\(Z\)`
- `\(f(q)\)` are potentially nonlinear functions of the `\(K\)` predictors
---
class: inverse middle center big-subsection
# Questions?
---
<img src="resources/mvgam_logo.png" style="position:fixed; right:8%; top:4%; width:100px; height:117px; border:none;" />
## The [`mvgam` 📦](https://github.com/nicholasjclark/mvgam/tree/master)
Bayesian framework to fit State-Space GAMs
- Hierarchical intercepts, slopes, smooths and Gaussian Processes
- Learning `\(Z\)` &#8680; JSDMs, N-mixture models, Dynamic Factors
Built off [`mgcv`](https://cran.r-project.org/web/packages/mgcv/index.html), [`brms` ](https://paulbuerkner.com/brms/) and [`splines2`](https://cran.r-project.org/web/packages/splines2/index.html) 📦's for flexible effects
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
---
class: middle center big-subsection
# [Package overview](https://nicholasjclark.github.io/mvgam/articles/mvgam_overview.html)
---
## Example of the interface
```r
model <- mvgam(
formula = y ~
s(series, bs = 're') +
s(x0, series, bs = 're') +
x1,
trend_formula = ~ gp(x2, k = 20) +
te(x3, x4, bs = c('cr', 'tp')),
data = data,
family = poisson(),
trend_model = AR(p = 1, ma = TRUE, cor = TRUE),
burnin = 500,
samples = 500,
chains = 4)
```
---
## Produce all `Stan` code and objects
```r
stancode(model)
```
.small[
```
## // Stan model code generated by package mvgam
## functions {
## /* Spectral density of a squared exponential Gaussian process
## * Args:
## * x: array of numeric values of dimension NB x D
## * sdgp: marginal SD parameter
## * lscale: vector of length-scale parameters
## * Returns:
## * numeric vector of length NB of the SPD evaluated at 'x'
## */
## vector spd_gp_exp_quad(data array[] vector x, real sdgp, vector lscale) {
## int NB = dims(x)[1];
## int D = dims(x)[2];
## int Dls = rows(lscale);
## real constant = square(sdgp) * sqrt(2 * pi()) ^ D;
## vector[NB] out;
## if (Dls == 1) {
## // one dimensional or isotropic GP
## real neg_half_lscale2 = -0.5 * square(lscale[1]);
## constant = constant * lscale[1] ^ D;
## for (m in 1 : NB) {
## out[m] = constant * exp(neg_half_lscale2 * dot_self(x[m]));
## }
## } else {
## // multi-dimensional non-isotropic GP
## vector[Dls] neg_half_lscale2 = -0.5 * square(lscale);
## constant = constant * prod(lscale);
## for (m in 1 : NB) {
## out[m] = constant * exp(dot_product(neg_half_lscale2, square(x[m])));
## }
## }
## return out;
## }
## }
## data {
## int<lower=1> k_gp_trend_x2_; // basis functions for approximate gp
## array[k_gp_trend_x2_] vector[1] l_gp_trend_x2_; // approximate gp eigenvalues
## array[20] int b_trend_idx_gp_x2_; // gp basis coefficient indices
## int<lower=0> total_obs; // total number of observations
## int<lower=0> n; // number of timepoints per series
## int<lower=0> n_sp_trend; // number of trend smoothing parameters
## int<lower=0> n_lv; // number of dynamic factors
## int<lower=0> n_series; // number of series
## matrix[n_series, n_lv] Z; // matrix mapping series to latent states
## int<lower=0> num_basis; // total number of basis coefficients
## int<lower=0> num_basis_trend; // number of trend basis coefficients
## vector[num_basis_trend] zero_trend; // prior locations for trend basis coefficients
## matrix[total_obs, num_basis] X; // mgcv GAM design matrix
## matrix[n * n_lv, num_basis_trend] X_trend; // trend model design matrix
## array[n, n_series] int<lower=0> ytimes; // time-ordered matrix (which col in X belongs to each [time, series] observation?)
## array[n, n_lv] int ytimes_trend;
## int<lower=0> n_nonmissing; // number of nonmissing observations
## matrix[24, 72] S_trend2; // mgcv smooth penalty matrix S_trend2
## array[n_nonmissing] int<lower=0> flat_ys; // flattened nonmissing observations
## matrix[n_nonmissing, num_basis] flat_xs; // X values for nonmissing observations
## array[n_nonmissing] int<lower=0> obs_ind; // indices of nonmissing observations
## }
## transformed data {
## vector[n_lv] trend_zeros = rep_vector(0.0, n_lv);
## }
## parameters {
## // gp term sd parameters
## real<lower=0> alpha_gp_trend_x2_;
##
## // gp term length scale parameters
## array[1] vector<lower=0>[1] rho_gp_trend_x2_;
##
## // gp term latent variables
## vector[k_gp_trend_x2_] z_gp_trend_x2_;
##
## // raw basis coefficients
## vector[num_basis] b_raw;
## vector[num_basis_trend] b_raw_trend;
##
## // latent state SD terms
## vector<lower=0>[n_lv] sigma;
## cholesky_factor_corr[n_lv] L_Omega;
##
## // random effect variances
## vector<lower=0>[2] sigma_raw;
##
## // random effect means
## vector[2] mu_raw;
##
## // latent state AR1 terms
## vector<lower=-1, upper=1>[n_lv] ar1;
##
## // ma coefficients
## matrix<lower=-1, upper=1>[n_lv, n_lv] theta;
##
## // dynamic error parameters
## array[n] vector[n_lv] error;
##
## // smoothing parameters
## vector<lower=0>[n_sp_trend] lambda_trend;
## }
## transformed parameters {
## // latent states and loading matrix
## vector[n * n_lv] trend_mus;
## matrix[n, n_series] trend;
## array[n] vector[n_lv] LV;
## array[n] vector[n_lv] epsilon;
##
## // LKJ form of covariance matrix
## matrix[n_lv, n_lv] L_Sigma;
##
## // computed error covariance matrix
## cov_matrix[n_lv] Sigma;
## matrix[n_series, n_lv] lv_coefs;
##
## // basis coefficients
## vector[num_basis] b;
## vector[num_basis_trend] b_trend;
##
## // observation model basis coefficients
## b[1 : 2] = b_raw[1 : 2];
## b[3 : 6] = mu_raw[1] + b_raw[3 : 6] * sigma_raw[1];
## b[7 : 10] = mu_raw[2] + b_raw[7 : 10] * sigma_raw[2];
##
## // process model basis coefficients
## b_trend[1 : num_basis_trend] = b_raw_trend[1 : num_basis_trend];
## b_trend[b_trend_idx_gp_x2_] = sqrt(spd_gp_exp_quad(l_gp_trend_x2_,
## alpha_gp_trend_x2_,
## rho_gp_trend_x2_[1]))
## .* z_gp_trend_x2_;
##
## // latent process linear predictors
## trend_mus = X_trend * b_trend;
##
## // derived latent states
## LV[1] = trend_mus[ytimes_trend[1, 1 : n_lv]] + error[1];
## epsilon[1] = error[1];
## for (i in 2 : n) {
## // lagged error ma process
## epsilon[i] = theta * error[i - 1];
##
## // full ARMA process
## LV[i] = trend_mus[ytimes_trend[i, 1 : n_lv]]
## + ar1 .* (LV[i - 1] - trend_mus[ytimes_trend[i - 1, 1 : n_lv]])
## + epsilon[i] + error[i];
## }
## L_Sigma = diag_pre_multiply(sigma, L_Omega);
## Sigma = multiply_lower_tri_self_transpose(L_Sigma);
## lv_coefs = Z;
## for (i in 1 : n) {
## for (s in 1 : n_series) {
## trend[i, s] = dot_product(lv_coefs[s, : ], LV[i, : ]);
## }
## }
## }
## model {
## // prior for random effect population variances
## sigma_raw ~ student_t(3, 0, 2.5);
##
## // prior for random effect population means
## mu_raw ~ std_normal();
##
## // prior for (Intercept)...
## b_raw[1] ~ student_t(3, 0, 2.5);
##
## // prior for x1B...
## b_raw[2] ~ student_t(3, 0, 2);
##
## // prior (non-centred) for s(series)...
## b_raw[3 : 6] ~ std_normal();
##
## // prior (non-centred) for s(x0,series)...
## b_raw[7 : 10] ~ std_normal();
##
## // priors for AR parameters
## ar1 ~ std_normal();
##
## // priors for latent state SD parameters
## sigma ~ student_t(3, 0, 2.5);
##
## // dynamic process models
##
## // prior for (Intercept)_trend...
## b_raw_trend[1] ~ student_t(3, 0, 2);
##
## // prior for te(x3,x4)_trend...
## b_raw_trend[22 : 45] ~ multi_normal_prec(zero_trend[22 : 45],
## S_trend2[1 : 24, 1 : 24]
## * lambda_trend[3]
## + S_trend2[1 : 24, 25 : 48]
## * lambda_trend[4]
## + S_trend2[1 : 24, 49 : 72]
## * lambda_trend[5]);
##
## // prior for gp(x2)_trend...
## z_gp_trend_x2_ ~ std_normal();
## alpha_gp_trend_x2_ ~ student_t(3, 0, 2.5);
## rho_gp_trend_x2_[1] ~ inv_gamma(1.494197, 0.056607);
## b_raw_trend[b_trend_idx_gp_x2_] ~ std_normal();
## lambda_trend ~ normal(5, 30);
##
## // contemporaneous errors
## L_Omega ~ lkj_corr_cholesky(2);
## for (i in 1 : n) {
## error[i] ~ multi_normal_cholesky(trend_zeros, L_Sigma);
## }
##
## // ma coefficients
## for (i in 1 : n_lv) {
## for (j in 1 : n_lv) {
## if (i != j) {
## theta[i, j] ~ normal(0, 0.2);
## }
## }
## }
## {
## // likelihood functions
## vector[n_nonmissing] flat_trends;
## flat_trends = to_vector(trend)[obs_ind];
## flat_ys ~ poisson_log_glm(append_col(flat_xs, flat_trends), 0.0,
## append_row(b, 1.0));
## }
## }
## generated quantities {
## vector[total_obs] eta;
## matrix[n, n_series] mus;
## vector[n_sp_trend] rho_trend;
## vector[n_lv] penalty;
## array[n, n_series] int ypred;
## penalty = 1.0 / (sigma .* sigma);
## rho_trend = log(lambda_trend);
##
## // posterior predictions
## eta = X * b;
## for (s in 1 : n_series) {
## mus[1 : n, s] = eta[ytimes[1 : n, s]] + trend[1 : n, s];
## ypred[1 : n, s] = poisson_log_rng(mus[1 : n, s]);
## }
## }
```
]
---
## Workflow
Fit models that can include splines, GPs, and multivariate dynamic processes to sets of time series; use informative priors for effective regularization
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 forecast distributions using proper scoring rules
---
class: middle center big-subsection
# [`?mvgam`](https://nicholasjclark.github.io/mvgam/reference/mvgam.html#details)
---
background-image: url('./resources/mvgam_cheatsheet.png')
background-size: contain
---
class: inverse middle center big-subsection
# Live code examples
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