mvgam

MultiVariate (Dynamic) Generalized Addivite Models

The goal of mvgam is to fit Bayesian (Dynamic) Generalized Additive Models. This package constructs State-Space models that can include highly flexible nonlinear predictor effects for both process and observation components by leveraging functionalities from the impressive brms and mgcv packages. This allows mvgam to fit a wide range of models, including hierarchical ecological models such as N-mixture or Joint Species Distribution models, as well as univariate and multivariate time series models with imperfect detection. The original motivation for the package is described in Clark & Wells 2022 (published in Methods in Ecology and Evolution), with additional inspiration on the use of Bayesian probabilistic modelling coming from Michael Betancourt, Michael Dietze and Sarah Heaps, among many others.

Resources

A series of vignettes cover data formatting, forecasting and several extended case studies of DGAMs. A number of other examples, including some step-by-step introductory webinars, have also been compiled:

• Time series in R and Stan using the mvgam package
• Ecological Forecasting with Dynamic Generalized Additive Models
• Distributed lags (and hierarchical distributed lags) using mgcv and mvgam
• State-Space Vector Autoregressions in mvgam
• Ecological Forecasting with Dynamic GAMs; a tutorial and detailed case study
• How to interpret and report nonlinear effects from Generalized Additive Models
• Phylogenetic smoothing using mgcv
• Incorporating time-varying seasonality in forecast models

Please also feel free to use the mvgam Discussion Board to hunt for or post other discussion topics related to the package, and do check out the mvgam changelog for any updates about recent upgrades that the package has incorporated.

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, Stan must be installed (along with either rstan and/or cmdstanr). Please refer to installation links for Stan with rstan here, or for Stan with cmdstandr here.

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.

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, please additionally cite:

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). DOI: https://doi.org/10.18637/jss.v076.i01

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 and Gaussian Processes, which rely on the mgcv, brms and splines2 packages. The rstan and cmdstanr packages together with Rcpp makes Stan conveniently accessible in R. If you use some of these features, please also consider citing the related packages.

Cheatsheet

Introducing mvgam for fitting Dynamic Generalized Additive Models

We can explore the package’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

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

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 most mvgam models, we need an indicator of the series name as a factor. A time column is also needed for most models to index time (but note that these variables are not necessarily needed for other models supported by mvgam, such as Joint Species Distribution Models)

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

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

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:

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

summary(lynx_mvgam)
#> GAM formula:
#> population ~ s(season, bs = "cc", k = 12)
#> 
#> Family:
#> poisson
#> 
#> Link function:
#> log
#> 
#> Trend model:
#> AR(p = 1)
#> 
#> 
#> N series:
#> 1 
#> 
#> N timepoints:
#> 60 
#> 
#> Status:
#> Fitted using Stan 
#> 4 chains, each with iter = 1000; warmup = 500; thin = 1 
#> Total post-warmup draws = 2000
#> 
#> 
#> GAM coefficient (beta) estimates:
#>                2.5%   50% 97.5% Rhat n_eff
#> (Intercept)   6.400  6.60  6.90    1   965
#> s(season).1  -0.640 -0.13  0.37    1  1069
#> s(season).2   0.720  1.30  2.00    1  1005
#> s(season).3   1.300  1.90  2.50    1   700
#> s(season).4  -0.078  0.53  1.10    1   999
#> s(season).5  -1.300 -0.69 -0.11    1  1052
#> s(season).6  -1.300 -0.54  0.19    1  1089
#> s(season).7   0.013  0.73  1.40    1  1102
#> s(season).8   0.610  1.40  2.10    1   758
#> s(season).9  -0.360  0.24  0.85    1   729
#> s(season).10 -1.400 -0.87 -0.35    1  1149
#> 
#> Approximate significance of GAM smooths:
#>            edf Ref.df Chi.sq p-value    
#> s(season) 9.98     10   46.7  <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Latent trend parameter AR estimates:
#>          2.5%  50% 97.5% Rhat n_eff
#> ar1[1]   0.60 0.83  0.97 1.00   609
#> sigma[1] 0.38 0.47  0.61 1.01   614
#> 
#> Stan MCMC diagnostics:
#> n_eff / iter looks reasonable for all parameters
#> Rhat looks reasonable for all parameters
#> 0 of 2000 iterations ended with a divergence (0%)
#> 2 of 2000 iterations saturated the maximum tree depth of 10 (0.1%)
#>  *Run with max_treedepth set to a larger value to avoid saturation
#> E-FMI indicated no pathological behavior
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Feb 18 9:28:11 PM 2025.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split MCMC chains
#> (at convergence, Rhat = 1)
#> 
#> Use how_to_cite(lynx_mvgam) to get started describing this model

As with any MCMC software, we can inspect traceplots. Here for the GAM smoothing parameters, using mvgam’s reliance on the excellent bayesplot library:

mcmc_plot(lynx_mvgam, variable = 'rho', regex = TRUE, type = 'trace')
#> No divergences to plot.

and for the latent trend parameters

mcmc_plot(lynx_mvgam, variable = 'trend_params', regex = TRUE, type = 'trace')
#> No divergences to plot.

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)

pp_check(lynx_mvgam, type = "hist", ndraws = 5)
#> `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Next examine simulated empirical Cumulative Distribution Functions (CDF) for posterior predictions

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. 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

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

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

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

require(gratia)
#> Loading required package: gratia
#> Warning: package 'gratia' was built under R version 4.2.3
#> 
#> Attaching package: 'gratia'
#> The following object is masked from 'package:mvgam':
#> 
#>     add_residuals
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:

require(ggplot2); require(marginaleffects)
#> Loading required package: 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)

plot(lynx_mvgam, type = 'forecast', newdata = lynx_test)

#> Out of sample CRPS:
#> 2383.745918

And the estimated latent trend component, again using the more flexible plot_mvgam_...() option to show first derivatives of the estimated trend

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. We can estimate relative contributions to forecast uncertainty for the GAM component and the latent trend component using mvgam

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 medians 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

plot(lynx_mvgam, type = 'residuals')

We can use the how_to_cite() function to generate a scaffold for describing the model and sampling details in scientific communications

description <- how_to_cite(lynx_mvgam)

description

#> Methods text skeleton
#> We used the R package mvgam (version 1.1.4; Clark & Wells, 2023) to construct, fit and int
#> errogate the model. mvgam fits Bayesian State-Space models that can include flexible predi
#> ctor effects in both the process and observation components by incorporating functionaliti
#> es from the brms (Burkner 2017), mgcv (Wood 2017) and splines2 (Wang & Yan, 2023) packages
#> . The mvgam-constructed model and observed data were passed to the probabilistic programmi
#> ng environment Stan (version 2.34.1; Carpenter et al. 2017, Stan Development Team 2025), s
#> pecifically through the cmdstanr interface (Gabry & Cesnovar, 2021). We ran 4 Hamiltonian 
#> Monte Carlo chains for 500 warmup iterations and 500 sampling iterations for joint posteri
#> or estimation. Rank normalized split Rhat (Vehtari et al. 2021) and effective sample sizes
#>  were used to monitor convergence.

#> 
#> Primary references
#> Clark, NJ and Wells K (2022). Dynamic Generalized Additive Models (DGAMs) for forecasting discrete ecological time series. Methods in Ecology and Evolution, 14, 771-784. doi.org/10.1111/2041-210X.13974
#> Burkner, PC (2017). brms: An R Package for Bayesian Multilevel Models Using Stan. Journal of Statistical Software, 80(1), 1-28. doi:10.18637/jss.v080.i01
#> Wood, SN (2017). Generalized Additive Models: An Introduction with R (2nd edition). Chapman and Hall/CRC.
#> Wang W and Yan J (2021). Shape-Restricted Regression Splines with R Package splines2. Journal of Data Science, 19(3), 498-517. doi:10.6339/21-JDS1020 https://doi.org/10.6339/21-JDS1020.
#> Carpenter, B, Gelman, A, Hoffman, MD, 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.
#> Gabry J, Cesnovar R, Johnson A, and Bronder S (2025). cmdstanr: R Interface to 'CmdStan'. https://mc-stan.org/cmdstanr/, https://discourse.mc-stan.org.
#> Vehtari A, Gelman A, Simpson D, Carpenter B, and Burkner P (2021). Rank-normalization, folding, and localization: An improved Rhat for assessing convergence of MCMC (with discussion). Bayesian Analysis 16(2) 667-718. https://doi.org/10.1214/20-BA1221.
#> 
#> Other useful references
#> Arel-Bundock, V, Greifer, N, and Heiss, A (2024). How to interpret statistical models using marginaleffects for R and Python. Journal of Statistical Software, 111(9), 1-32. https://doi.org/10.18637/jss.v111.i09
#> Gabry J, Simpson D, Vehtari A, Betancourt M, and Gelman A (2019). Visualization in Bayesian workflow. Journal of the Royal Statatistical Society A, 182, 389-402. doi:10.1111/rssa.12378.
#> Vehtari A, Gelman A, and Gabry J (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27, 1413-1432. doi:10.1007/s11222-016-9696-4.
#> Burkner, PC, Gabry, J, and Vehtari, A. (2020). Approximate leave-future-out cross-validation for Bayesian time series models. Journal of Statistical Computation and Simulation, 90(14), 2499-2523. https://doi.org/10.1080/00949655.2020.1783262

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)

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:

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')

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:

summary(mod, include_betas = FALSE)
#> GAM formula:
#> y ~ s(season, bs = "cc", k = 7) + s(season, by = series, m = 1, 
#>     k = 5)
#> 
#> Family:
#> beta
#> 
#> Link function:
#> logit
#> 
#> Trend model:
#> GP()
#> 
#> 
#> N series:
#> 3 
#> 
#> N timepoints:
#> 80 
#> 
#> Status:
#> Fitted using Stan 
#> 4 chains, each with iter = 1000; warmup = 500; thin = 1 
#> Total post-warmup draws = 2000
#> 
#> 
#> Observation precision parameter estimates:
#>        2.5%  50% 97.5% Rhat n_eff
#> phi[1]  7.6 12.0  18.0    1  1220
#> phi[2]  5.6  8.6  13.0    1  1685
#> phi[3]  4.0  6.0   8.8    1  1577
#> 
#> GAM coefficient (beta) estimates:
#>             2.5%  50% 97.5% Rhat n_eff
#> (Intercept) 0.11 0.46   0.7    1   978
#> 
#> Approximate significance of GAM smooths:
#>                            edf Ref.df Chi.sq p-value  
#> s(season)                3.757      5   8.68   0.049 *
#> s(season):seriesseries_1 0.906      4   8.38   0.268  
#> s(season):seriesseries_2 3.353      4   1.53   0.349  
#> s(season):seriesseries_3 3.235      4   2.27   0.139  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Latent trend marginal deviation (alpha) and length scale (rho) estimates:
#>              2.5%   50% 97.5% Rhat n_eff
#> alpha_gp[1] 0.110  0.40  0.82 1.00   819
#> alpha_gp[2] 0.550  0.93  1.40 1.00  1748
#> alpha_gp[3] 0.056  0.39  0.95 1.00   667
#> rho_gp[1]   1.200  3.90 12.00 1.01   457
#> rho_gp[2]   3.000 13.00 31.00 1.01   379
#> rho_gp[3]   1.300  5.00 23.00 1.00   443
#> 
#> Stan MCMC diagnostics:
#> n_eff / iter looks reasonable for all parameters
#> Rhat looks reasonable for all parameters
#> 15 of 2000 iterations ended with a divergence (0.75%)
#>  *Try running with larger adapt_delta to remove the divergences
#> 0 of 2000 iterations saturated the maximum tree depth of 10 (0%)
#> E-FMI indicated no pathological behavior
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Feb 18 9:30:17 PM 2025.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split MCMC chains
#> (at convergence, Rhat = 1)
#> 
#> Use how_to_cite(mod) to get started describing this model

Plot the hindcast and forecast distributions for each series

layout(matrix(1:4, nrow = 2, byrow = TRUE))
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 State-Space GAMs that include dynamic and spatially varying coefficient models, dynamic factors and Vector Autoregressive processes. See the package documentation 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. This can be helpful if users wish to make changes to the model to better suit their own bespoke research / analysis goals. The Stan Discourse is a helpful place to troubleshoot.

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). Other contributions are also very welcome, but please see The Contributor Instructions for general guidelines. Note that by participating in this project you agree to abide by the terms of its Contributor Code of Conduct.