README.Rmd

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*mvgam*
================
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The goal of `mvgam` is to fit Bayesian Dynamic Generalized Additive Models to time series data. The motivation for the package is described in [Clark & Wells 2022](https://besjournals.onlinelibrary.wiley.com/doi/10.1111/2041-210X.13974){target="_blank"} (published in *Methods in Ecology and Evolution*), with additional inspiration on the use of Bayesian probabilistic modelling coming from [Michael Betancourt](https://betanalpha.github.io/writing/){target="_blank"}, [Michael Dietze](https://www.bu.edu/earth/profiles/michael-dietze/){target="_blank"} and [Emily Fox](https://emilybfox.su.domains/){target="_blank"}, among many others. 

## Resources
A series of [vignettes cover data formatting, forecasting and several extended case studies of DGAMs](https://nicholasjclark.github.io/mvgam/){target="_blank"}. A number of other examples have also been compiled:
  
* [Ecological Forecasting with Dynamic Generalized Additive Models](https://www.youtube.com/watch?v=0zZopLlomsQ){target="_blank"}
* [Distributed lags (and hierarchical distributed lags) using `mgcv` and `mvgam`](https://ecogambler.netlify.app/blog/distributed-lags-mgcv/){target="_blank"}
* [Ecological Forecasting with Dynamic GAMs; a tutorial and detailed case study](https://www.youtube.com/watch?v=RwllLjgPUmM){target="_blank"}
* [How to interpret and report nonlinear effects from Generalized Additive Models](https://ecogambler.netlify.app/blog/interpreting-gams/){target="_blank"}
* [Phylogenetic smoothing using `mgcv`](https://ecogambler.netlify.app/blog/phylogenetic-smooths-mgcv/){target="_blank"}
* [Incorporating time-varying seasonality in forecast models](https://ecogambler.netlify.app/blog/time-varying-seasonality/){target="_blank"}
  
## Installation
Install the stable version from `CRAN` using: `install.packages('mvgam')`, or install the development version from `GitHub` using:
`devtools::install_github("nicholasjclark/mvgam")`. Note that to condition models on observed data, either `JAGS` (along with packages `rjags` and `runjags`) or `Stan` must be installed (along with either `rstan` and/or `cmdstanr`). Please refer to installation links for `JAGS` [here](https://sourceforge.net/projects/mcmc-jags/files/){target="_blank"}, for `Stan` with `rstan` [here](https://mc-stan.org/users/interfaces/rstan){target="_blank"}, or for `Stan` with `cmdstandr` [here](https://mc-stan.org/cmdstanr/){target="_blank"}. You will need a fairly recent version of `Stan` to ensure all syntax is recognized. If you see warnings such as `variable "array" does not exist`, this is usually a sign that you need to update `Stan`. 

We highly recommend you use `Cmdstan` through the `cmdstanr` interface. This is because `Cmdstan` is easier to install, is more up to date with new features, and uses less memory than `rstan`. See [this documentation from the `Cmdstan` team for more information](http://mc-stan.org/cmdstanr/articles/cmdstanr.html#comparison-with-rstan){target="_blank"}. 

## Citing `mvgam` and related software
When using any software please make sure to appropriately acknowledge the hard work that developers and maintainers put into making these packages available. Citations are currently the best way to formally acknowledge this work, so we highly encourage you to cite any packages that you rely on for your research.

When using `mvgam`, please cite the following:

- Clark, N.J. and Wells, K. (2022). Dynamic Generalized Additive Models (DGAMs) for forecasting discrete ecological time series. *Methods in Ecology and Evolution*. DOI: https://doi.org/10.1111/2041-210X.13974

As `mvgam` acts as an interface to `Stan` and `JAGS`, please additionally cite whichever software you use for parameter estimation:

- Carpenter B., Gelman A., Hoffman M. D., Lee D., Goodrich B., Betancourt M.,
  Brubaker M., Guo J., Li P., and Riddell A. (2017). Stan: A probabilistic
  programming language. *Journal of Statistical Software*. 76(1).
  10.18637/jss.v076.i01
- Plummer, M. (2013). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.          *Proceedings of the 3rd International Workshop on Distributed Statistical Computing*. 124(125.10).

`mvgam` relies on several other `R` packages and, of course, on `R` itself. To
find out how to cite R and its packages, use the `citation` function. There are
some features of `mvgam` which specifically rely on certain packages. The most important of these is the generation of data necessary to estimate smoothing splines, which entirely rely on `mgcv`. The `rstan` and `cmdstanr` packages together with `Rcpp` makes `Stan` conveniently accessible in `R`, while the `rjags` and `runjags` packages together with the `coda` package make `JAGS` accessible in `R`. If you use some of these features, please also consider citing the related packages.

## Cheatsheet
[![`mvgam` usage cheatsheet](https://github.com/nicholasjclark/mvgam/raw/master/misc/mvgam_cheatsheet.png)](https://github.com/nicholasjclark/mvgam/raw/master/misc/mvgam_cheatsheet.pdf)

## Introducing `mvgam` for fitting Generalized Additive Models to time series
We can explore the model’s primary functions using a dataset that is available with all `R` installations. Load the `lynx` data and plot the series as well as its autocorrelation function
```{r include = FALSE}
library(mvgam)
library(ggplot2); theme_set(theme_bw())
```

```{r echo = FALSE}
library(mvgam)
```

```{r, fig.alt = "Visualizing the lynx time series in R"}
data(lynx)
lynx_full <- data.frame(year = 1821:1934, 
                        population = as.numeric(lynx))
plot(lynx_full$population, type = 'l', ylab = 'Lynx trappings',
     xlab = 'Time', bty = 'l', lwd = 2)
box(bty = 'l', lwd  = 2)
acf(lynx_full$population, main = '', bty = 'l', lwd = 2,
    ci.col = 'darkred')
box(bty = 'l', lwd  = 2)
```

Along with serial autocorrelation, there is a clear ~19-year cyclic pattern. Create a `season` term that can be used to model this effect and give a better representation of the data generating process than we would likely get with a linear model
```{r, fig.alt = "Visualizing and decomposing the lynx time series in R"}
plot(stl(ts(lynx_full$population, frequency = 19), s.window = 'periodic'),
     lwd = 2, col.range = 'darkred')
lynx_full$season <- (lynx_full$year%%19) + 1
```

For `mvgam` models, we need an indicator of the series name as a `factor` (if the column `series` is missing, this will be added automatically by assuming that all observations are from a single time series). A `time` column is needed to index time
```{r}
lynx_full$time <- 1:NROW(lynx_full)
lynx_full$series <- factor('series1')
```

Split the data into training (first 50 years) and testing (next 10 years of data) to evaluate forecasts
```{r}
lynx_train = lynx_full[1:50, ]
lynx_test = lynx_full[51:60, ]
```

Inspect the series in a bit more detail using `mvgam`'s plotting utility
```{r, message=FALSE, warning=FALSE, fig.width=6.5, fig.height=6.5, dpi=160, fig.alt = "Plotting time series features with the mvgam R package"}
plot_mvgam_series(data = lynx_train, y = 'population')
```

Formulate an `mvgam` model; this model fits a GAM in which a cyclic smooth function for `season` is estimated jointly with a full time series model for the temporal process (in this case an `AR1` process). We assume the outcome follows a Poisson distribution and will condition the model in `Stan` using MCMC sampling with the `Cmdstan` interface:
```{r, include=FALSE}
lynx_mvgam <- mvgam(data = lynx_train,
               newdata = lynx_test,
               formula = population ~ s(season, bs = 'cc', k = 12),
               knots = list(season = c(0.5, 19.5)),
               family = poisson(),
               trend_model = AR(p = 1),
               noncentred = TRUE,
               use_stan = TRUE)
```

```{r, eval=FALSE}
lynx_mvgam <- mvgam(population ~ s(season, bs = 'cc', k = 12),
                    knots = list(season = c(0.5, 19.5)),
                    data = lynx_train,
                    newdata = lynx_test,
                    family = poisson(),
                    trend_model = AR(p = 1),
                    backend = 'cmdstanr')
```

Have a look at this model's summary to see what is being estimated. Note that no pathological behaviours have been detected and we achieve good effective sample sizes / mixing for all parameters
```{r}
summary(lynx_mvgam)
```

As with any MCMC software, we can inspect traceplots. Here for the GAM smoothing parameters, using `mvgam`'s reliance on the excellent `bayesplot` library:
```{r, fig.alt = "Smoothing parameter posterior distributions estimated with Stan in mvgam"}
mcmc_plot(lynx_mvgam, variable = 'rho', regex = TRUE, type = 'trace')
```

and for the latent trend parameters
```{r, fig.alt = "Dynamic temporal autocorrelation parameters estimated with Stan in mvgam"}
mcmc_plot(lynx_mvgam, variable = 'trend_params', regex = TRUE, type = 'trace')
```

Use posterior predictive checks, which capitalize on the extensive functionality of the `bayesplot` package, to see if the model can simulate data that looks realistic and unbiased. First, examine histograms for posterior retrodictions (`yhat`) and compare to the histogram of the observations (`y`)
```{r, fig.alt = "Posterior predictive checks for discrete time series in R"}
pp_check(lynx_mvgam, type = "hist", ndraws = 5)
```

Next examine simulated empirical Cumulative Distribution Functions (CDF) for posterior predictions
```{r, fig.alt = "Posterior predictive checks for discrete time series in R"}
pp_check(lynx_mvgam, type = "ecdf_overlay", ndraws = 25)
```

Rootograms are [popular graphical tools for checking a discrete model's ability to capture dispersion properties of the response variable](http://arxiv.org/pdf/1605.01311){target="_blank"}. Posterior predictive hanging rootograms can be displayed using the `ppc()` function. In the plot below, we bin the unique observed values into `25` bins to prevent overplotting and help with interpretation. This plot compares the frequencies of observed vs predicted values for each bin. For example, if the gray bars (representing observed frequencies) tend to stretch below zero, this suggests the model's simulations predict the values in that particular bin less frequently than they are observed in the data. A well-fitting model that can generate realistic simulated data will provide a rootogram in which the lower boundaries of the grey bars are generally near zero. For this plot we use the `S3` function `ppc.mvgam()`, which is not as versatile as `pp_check()` but allows us to bin rootograms to avoid overplotting
```{r, fig.alt = "Posterior predictive rootograms for discrete time series in R"}
ppc(lynx_mvgam, type = "rootogram", n_bins = 25)
```

All plots indicate the model is well calibrated against the training data. Inspect the estimated cyclic smooth, which is shown as a ribbon plot of posterior empirical quantiles. We can also overlay posterior quantiles of partial residuals (shown in red), which represent the leftover variation that the model expects would remain if this smooth term was dropped but all other parameters remained unchanged. A strong pattern in the partial residuals suggests there would be strong patterns left unexplained in the model *if* we were to drop this term, giving us further confidence that this function is important in the model
```{r, fig.width=5, fig.height=4, fig.align='center', dpi=150, fig.alt = "Plotting GAM smooth functions in mvgam and R"}
plot(lynx_mvgam, type = 'smooths', residuals = TRUE)
```

First derivatives of smooths can be plotted to inspect how the slope of the function changes. To plot these we use the more flexible `plot_mvgam_smooth()` function
```{r, fig.alt = "Plotting GAM smooth functions in mvgam and R"}
plot_mvgam_smooth(lynx_mvgam, series = 1, 
                  smooth = 'season', 
                  derivatives = TRUE)
```

If you have the `gratia` package installed, it can also be used to plot partial effects of smooths on the link scale
```{r, fig.alt = "Plotting GAM smooth functions in mvgam using gratia"}
require(gratia)
draw(lynx_mvgam)
```

As for many types of regression models, it is often more useful to plot model effects on the outcome scale. `mvgam` has support for the wonderful `marginaleffects` package, allowing a wide variety of posterior contrasts, averages, conditional and marginal predictions to be calculated and plotted. Below is the conditional effect of season plotted on the outcome scale, for example:
```{r, fig.alt = "Using marginaleffects and mvgam to plot GAM smooth functions in R"}
require(ggplot2); require(marginaleffects)
plot_predictions(lynx_mvgam, condition = 'season', points = 0.5) +
  theme_classic()
```

We can also view the `mvgam`'s posterior predictions for the entire series (testing and training)
```{r, fig.alt = "Plotting forecast distributions using mvgam in R"}
plot(lynx_mvgam, type = 'forecast', newdata = lynx_test)
```

And the estimated latent trend component, again using the more flexible `plot_mvgam_...()` option to show first derivatives of the estimated trend
```{r, fig.alt = "Plotting dynamic trend components using mvgam in R"}
plot_mvgam_trend(lynx_mvgam, newdata = lynx_test, derivatives = TRUE)
```

A key aspect of ecological forecasting is to understand [how different components of a model contribute to forecast uncertainty](https://esajournals.onlinelibrary.wiley.com/doi/full/10.1002/eap.1589){target="_blank"}. We can estimate relative contributions to forecast uncertainty for the GAM component and the latent trend component using `mvgam`
```{r, fig.alt = "Decomposing uncertainty contributions to forecasts in mvgam in R"}
plot_mvgam_uncertainty(lynx_mvgam, newdata = lynx_test, legend_position = 'none')
text(1, 0.2, cex = 1.5, label="GAM component", 
     pos = 4, col="white", family = 'serif')
text(1, 0.8, cex = 1.5, label="Trend component", 
     pos = 4, col="#7C0000", family = 'serif')
```

Both components contribute to forecast uncertainty. Diagnostics of the model can also be performed using `mvgam`. Have a look at the model's residuals, which are posterior empirical quantiles of Dunn-Smyth randomised quantile residuals so should follow approximate normality. We are primarily looking for a lack of autocorrelation, which would suggest our AR1 model is appropriate for the latent trend
```{r, fig.width=6.5, fig.height=6.5, dpi=160, fig.alt = "Plotting Dunn-Smyth residuals for time series analysis in mvgam and R"}
plot(lynx_mvgam, type = 'residuals')
```

## Extended observation families
`mvgam` was originally designed to analyse and forecast non-negative integer-valued data. These data are traditionally challenging to analyse with existing time-series analysis packages. But further development of `mvgam` has resulted in support for a growing number of observation families. Currently, the package can handle data for the following:  
  
* `gaussian()` for real-valued data 
* `student_t()` for heavy-tailed real-valued data
* `lognormal()` for non-negative real-valued data
* `Gamma()` for non-negative real-valued data
* `betar()` for proportional data on `(0,1)`
* `bernoulli()` for binary data
* `poisson()` for count data
* `nb()` for overdispersed count data
* `binomial()` for count data with known number of trials
* `beta_binomial()` for overdispersed count data with known number of trials
* `nmix()` for count data with imperfect detection (unknown number of trials)
* `tweedie()` for overdispersed count data
  
Note that only `poisson()`, `nb()`, and `tweedie()` are available if using `JAGS`. All families, apart from `tweedie()`, are supported if using `Stan`. See `??mvgam_families` for more information. Below is a simple example for simulating and modelling proportional data with `Beta` observations over a set of seasonal series with independent Gaussian Process dynamic trends:
```{r beta_sim, message=FALSE, warning=FALSE, fig.width=6.5, fig.height=6.5, dpi=160}
set.seed(100)
data <- sim_mvgam(family = betar(),
                  T = 80,
                  trend_model = 'GP',
                  prop_trend = 0.5, 
                  seasonality = 'shared')
plot_mvgam_series(data = data$data_train, series = 'all')
```

```{r, include=FALSE}
mod <- mvgam(y ~ s(season, bs = 'cc', k = 7) +
               s(season, by = series, m = 1, k = 5),
             trend_model = 'GP',
             data = data$data_train,
             newdata = data$data_test,
             family = betar())
```

```{r, eval=FALSE}
mod <- mvgam(y ~ s(season, bs = 'cc', k = 7) +
               s(season, by = series, m = 1, k = 5),
             trend_model = 'GP',
             data = data$data_train,
             newdata = data$data_test,
             family = betar())
```

Inspect the summary to see that the posterior now also contains estimates for the `Beta` precision parameters $\phi$. We can suppress a summary of the $\beta$ coefficients, which is useful when there are many spline coefficients to report:
```{r}
summary(mod, include_betas = FALSE)
```

Plot the hindcast and forecast distributions for each series
```{r eval=FALSE}
layout(matrix(1:4, nrow = 2, byrow = TRUE))
for(i in 1:3){
  plot(mod, type = 'forecast', series = i)
}
```

```{r beta_fc, echo=FALSE, message=FALSE}
layout(matrix(1:4, nrow = 2, byrow = TRUE))
par(mar=c(3, 3, 2, 2),
    oma = c(2, 2, 0, 0),
    mgp = c(2, 0.5, 0))
for(i in 1:3){
  plot(mod, type = 'forecast', series = i)
}
```

There are many more extended uses of `mvgam`, including the ability to fit hierarchical GAMs that include dynamic coefficient models, dynamic factor and Vector Autoregressive processes. See the [package documentation](https://nicholasjclark.github.io/mvgam/){target="_blank"} for more details. The package can also be used to generate all necessary data structures, initial value functions and modelling code necessary to fit DGAMs using `Stan` or `JAGS`. This can be helpful if users wish to make changes to the model to better suit their own bespoke research / analysis goals. The following resources can be helpful to troubleshoot:
  
* [Stan Discourse](https://discourse.mc-stan.org/){target="_blank"}
* [JAGS Discourse](https://sourceforge.net/projects/mcmc-jags/){target="_blank"}

## License
This project is licensed under an `MIT` open source license

## Interested in contributing?
I'm actively seeking PhD students and other researchers to work in the areas of ecological forecasting, multivariate model evaluation and development of `mvgam`. Please reach out if you are interested (n.clark'at'uq.edu.au)