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.
A series of vignettes cover data formatting, forecasting and several extended case studies of DGAMs. A number of other examples have also been compiled:
- Ecological Forecasting with Dynamic Generalized Additive Models
- Distributed lags (and hierarchical distributed lags)
using
mgcvandmvgam - 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
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.
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, which rely on mgcv and
splines2. 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.
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
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$year%%19) + 1For most mvgam models, we need an indicator of the series name as a
factor. A time column is also needed for most models to index time
(but note that these variables are not necessarily needed for other
models supported by mvgam, such as Joint Species Distribution
Models)
lynx_full$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 6.900 1 660
#> s(season).1 -0.640 -0.12 0.390 1 1035
#> s(season).2 0.690 1.30 1.900 1 893
#> s(season).3 1.200 1.90 2.500 1 842
#> s(season).4 -0.079 0.53 1.100 1 830
#> s(season).5 -1.300 -0.71 -0.058 1 849
#> s(season).6 -1.200 -0.56 0.110 1 1168
#> s(season).7 0.058 0.72 1.400 1 997
#> s(season).8 0.660 1.40 2.100 1 1019
#> s(season).9 -0.350 0.24 0.840 1 906
#> s(season).10 -1.400 -0.85 -0.370 1 1124
#>
#> Approximate significance of GAM smooths:
#> edf Ref.df Chi.sq p-value
#> s(season) 9.99 10 45.6 <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] 0.59 0.82 0.98 1.00 660
#> sigma[1] 0.39 0.48 0.61 1.01 628
#>
#> 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%)
#> 0 of 2000 iterations saturated the maximum tree depth of 12 (0%)
#> E-FMI indicated no pathological behavior
#>
#> Samples were drawn using NUTS(diag_e) at Thu Oct 31 9:16:35 AM 2024.
#> 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)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
#>
#> 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 DRPS:
#> 2380.3470865
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 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
plot(lynx_mvgam, type = 'residuals')
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 datastudent_t()for heavy-tailed real-valued datalognormal()for non-negative real-valued dataGamma()for non-negative real-valued databetar()for proportional data on(0,1)bernoulli()for binary datapoisson()for count datanb()for overdispersed count databinomial()for count data with known number of trialsbeta_binomial()for overdispersed count data with known number of trialsnmix()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
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] 5.4 8.3 12 1 1248
#> phi[2] 5.7 8.6 13 1 1312
#> phi[3] 5.6 8.5 12 1 1724
#>
#> GAM coefficient (beta) estimates:
#> 2.5% 50% 97.5% Rhat n_eff
#> (Intercept) -0.2 0.19 0.46 1.01 566
#>
#> Approximate significance of GAM smooths:
#> edf Ref.df Chi.sq p-value
#> s(season) 3.872 5 29.63 1.6e-05 ***
#> s(season):seriesseries_1 0.615 4 0.77 0.98
#> s(season):seriesseries_2 1.012 4 0.30 0.99
#> s(season):seriesseries_3 1.106 4 1.54 0.81
#> ---
#> 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] 0.051 0.41 0.92 1.01 525
#> alpha_gp[2] 0.360 0.72 1.20 1.00 946
#> alpha_gp[3] 0.150 0.46 1.00 1.00 659
#> rho_gp[1] 1.100 3.80 15.00 1.01 370
#> rho_gp[2] 1.900 7.80 37.00 1.01 365
#> rho_gp[3] 1.400 5.10 21.00 1.00 645
#>
#> Stan MCMC diagnostics:
#> n_eff / iter looks reasonable for all parameters
#> Rhat looks reasonable for all parameters
#> 12 of 2000 iterations ended with a divergence (0.6%)
#> *Try running with larger adapt_delta to remove the divergences
#> 0 of 2000 iterations saturated the maximum tree depth of 12 (0%)
#> E-FMI indicated no pathological behavior
#>
#> Samples were drawn using NUTS(diag_e) at Thu Oct 31 9:17:20 AM 2024.
#> 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)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 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.
This project is licensed under an MIT open source license
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)