# Time-varying effects in mvgam

#### Nicholas J Clark

#### 2025-03-14

The purpose of this vignette is to show how the `mvgam` package can be used to estimate and forecast regression coefficients that vary through time.

## Time-varying effects

Dynamic fixed-effect coefficients (often referred to as dynamic linear models) can be readily incorporated into GAMs / DGAMs. In `mvgam`, the `dynamic()` formula wrapper offers a convenient interface to set these up. The plan is to incorporate a range of dynamic options (such as random walk, AR1 etc…) but for the moment only low-rank Gaussian Process (GP) smooths are allowed (making use either of the `gp` basis in `mgcv` of of Hilbert space approximate GPs). These are advantageous over splines or random walk effects for several reasons. First, GPs will force the time-varying effect to be smooth. This often makes sense in reality, where we would not expect a regression coefficient to change rapidly from one time point to the next. Second, GPs provide information on the ‘global’ dynamics of a time-varying effect through their length-scale parameters. This means we can use them to provide accurate forecasts of how an effect is expected to change in the future, something that we couldn’t do well if we used splines to estimate the effect. An example below illustrates.

### Simulating time-varying effects

Simulate a time-varying coefficient using a squared exponential Gaussian Process function with length scale \(\rho\)=10. We will do this using an internal function from `mvgam` (the `sim_gp` function):

    set.seed(1111)
    N <- 200
    beta_temp <- mvgam:::sim_gp(rnorm(1),
      alpha_gp = 0.75,
      rho_gp = 10,
      h = N
    ) + 0.5

A plot of the time-varying coefficient shows that it changes smoothly through time:

    plot(beta_temp,
      type = "l", lwd = 3,
      bty = "l", xlab = "Time", ylab = "Coefficient",
      col = "darkred"
    )
    box(bty = "l", lwd = 2)

Next we need to simulate the values of the covariate, which we will call `temp` (to represent \(temperature\)). In this case we just use a standard normal distribution to simulate this covariate:

    temp <- rnorm(N, sd = 1)

Finally, simulate the outcome variable, which is a Gaussian observation process (with observation error) over the time-varying effect of \(temperature\)

    out <- rnorm(N,
      mean = 4 + beta_temp * temp,
      sd = 0.25
    )
    time <- seq_along(temp)
    plot(out,
      type = "l", lwd = 3,
      bty = "l", xlab = "Time", ylab = "Outcome",
      col = "darkred"
    )
    box(bty = "l", lwd = 2)

Gather the data into a `data.frame` for fitting models, and split the data into training and testing folds.

    data <- data.frame(out, temp, time)
    data_train <- data[1:190, ]
    data_test <- data[191:200, ]

### The `dynamic()` function

Time-varying coefficients can be fairly easily set up using the `s()` or `gp()` wrapper functions in `mvgam` formulae by fitting a nonlinear effect of `time` and using the covariate of interest as the numeric `by` variable (see `?mgcv::s` or `?brms::gp` for more details). The `dynamic()` formula wrapper offers a way to automate this process, and will eventually allow for a broader variety of time-varying effects (such as random walk or AR processes). Depending on the arguments that are specified to `dynamic`, it will either set up a low-rank GP smooth function using `s()` with `bs = 'gp'` and a fixed value of the length scale parameter \(\rho\), or it will set up a Hilbert space approximate GP using the `gp()` function with `c=5/4` so that \(\rho\) is estimated (see `?dynamic` for more details). In this first example we will use the `s()` option, and will mis-specify the \(\rho\) parameter here as, in practice, it is never known. This call to `dynamic()` will set up the following smooth: `s(time, by = temp, bs = "gp", m = c(-2, 8, 2), k = 40)`

    mod <- mvgam(out ~ dynamic(temp, rho = 8, stationary = TRUE, k = 40),
      family = gaussian(),
      data = data_train,
      silent = 2
    )

Inspect the model summary, which shows how the `dynamic()` wrapper was used to construct a low-rank Gaussian Process smooth function:

    summary(mod, include_betas = FALSE)
    #> GAM formula:
    #> out ~ s(time, by = temp, bs = "gp", m = c(-2, 8, 2), k = 40)
    #> <environment: 0x000001e003956610>
    #>
    #> Family:
    #> gaussian
    #>
    #> Link function:
    #> identity
    #>
    #> Trend model:
    #> None
    #>
    #> N series:
    #> 1
    #>
    #> N timepoints:
    #> 190
    #>
    #> Status:
    #> Fitted using Stan
    #> 4 chains, each with iter = 1000; warmup = 500; thin = 1
    #> Total post-warmup draws = 2000
    #>
    #>
    #> Observation error parameter estimates:
    #>              2.5%  50% 97.5% Rhat n_eff
    #> sigma_obs[1] 0.23 0.25  0.28    1  2708
    #>
    #> GAM coefficient (beta) estimates:
    #>             2.5% 50% 97.5% Rhat n_eff
    #> (Intercept)    4   4   4.1    1  3517
    #>
    #> Approximate significance of GAM smooths:
    #>               edf Ref.df Chi.sq p-value
    #> s(time):temp 17.4     40    165  <2e-16 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Stan MCMC diagnostics:
    #> ✔ No issues with effective samples per iteration
    #> ✔ Rhat looks good for all parameters
    #> ✔ No issues with divergences
    #> ✔ No issues with maximum tree depth
    #>
    #> Samples were drawn using sampling(hmc). 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() to get started describing this model

Because this model used a spline with a `gp` basis, it’s smooths can be visualised just like any other `gam`. We can plot the estimates for the in-sample and out-of-sample periods to see how the Gaussian Process function produces sensible smooth forecasts. Here we supply the full dataset to the `newdata` argument in `plot_mvgam_smooth()` to inspect posterior forecasts of the time-varying smooth function. Overlay the true simulated function to see that the model has adequately estimated it’s dynamics in both the training and testing data partitions

    plot_mvgam_smooth(mod, smooth = 1, newdata = data)
    abline(v = 190, lty = "dashed", lwd = 2)
    lines(beta_temp, lwd = 2.5, col = "white")
    lines(beta_temp, lwd = 2)

We can also use `plot_predictions()` from the `marginaleffects` package to visualise the time-varying coefficient for what the effect would be estimated to be at different values of \(temperature\):

    require(marginaleffects)
    range_round <- function(x) {
      round(range(x, na.rm = TRUE), 2)
    }
    plot_predictions(mod,
      newdata = datagrid(
        time = unique,
        temp = range_round
      ),
      by = c("time", "temp", "temp"),
      type = "link"
    )

This results in sensible forecasts of the observations as well

    fc <- forecast(mod, newdata = data_test)
    plot(fc)

The syntax is very similar if we wish to estimate the parameters of the underlying Gaussian Process, this time using a Hilbert space approximation. We simply omit the `rho` argument in `dynamic()` to make this happen. This will set up a call similar to `gp(time, by = 'temp', c = 5/4, k = 40)`.

    mod <- mvgam(out ~ dynamic(temp, k = 40),
      family = gaussian(),
      data = data_train,
      silent = 2
    )

This model summary now contains estimates for the marginal deviation and length scale parameters of the underlying Gaussian Process function:

    summary(mod, include_betas = FALSE)
    #> GAM formula:
    #> out ~ gp(time, by = temp, c = 5/4, k = 40, scale = TRUE)
    #> <environment: 0x000001e003956610>
    #>
    #> Family:
    #> gaussian
    #>
    #> Link function:
    #> identity
    #>
    #> Trend model:
    #> None
    #>
    #> N series:
    #> 1
    #>
    #> N timepoints:
    #> 190
    #>
    #> Status:
    #> Fitted using Stan
    #> 4 chains, each with iter = 1000; warmup = 500; thin = 1
    #> Total post-warmup draws = 2000
    #>
    #>
    #> Observation error parameter estimates:
    #>              2.5%  50% 97.5% Rhat n_eff
    #> sigma_obs[1] 0.24 0.26   0.3    1  1965
    #>
    #> GAM coefficient (beta) estimates:
    #>             2.5% 50% 97.5% Rhat n_eff
    #> (Intercept)    4   4   4.1    1  2617
    #>
    #> GAM gp term marginal deviation (alpha) and length scale (rho) estimates:
    #>                      2.5%   50% 97.5% Rhat n_eff
    #> alpha_gp(time):temp 0.630 0.890 1.400    1   776
    #> rho_gp(time):temp   0.028 0.052 0.067    1   942
    #>
    #> Stan MCMC diagnostics:
    #> ✔ No issues with effective samples per iteration
    #> ✔ Rhat looks good for all parameters
    #> ✔ No issues with divergences
    #> ✔ No issues with maximum tree depth
    #>
    #> Samples were drawn using sampling(hmc). 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() to get started describing this model

Effects for `gp()` terms can also be plotted as smooths:

    plot_mvgam_smooth(mod, smooth = 1, newdata = data)
    abline(v = 190, lty = "dashed", lwd = 2)
    lines(beta_temp, lwd = 2.5, col = "white")
    lines(beta_temp, lwd = 2)

## Salmon survival example

Here we will use openly available data on marine survival of Chinook salmon to illustrate how time-varying effects can be used to improve ecological time series models. Scheuerell and Williams (2005) used a dynamic linear model to examine the relationship between marine survival of Chinook salmon and an index of ocean upwelling strength along the west coast of the USA. The authors hypothesized that stronger upwelling in April should create better growing conditions for phytoplankton, which would then translate into more zooplankton and provide better foraging opportunities for juvenile salmon entering the ocean. The data on survival is measured as a proportional variable over 42 years (1964–2005) and is available in the `MARSS` package:

    load(url("https://github.com/atsa-es/MARSS/raw/master/data/SalmonSurvCUI.rda"))
    dplyr::glimpse(SalmonSurvCUI)
    #> Rows: 42
    #> Columns: 3
    #> $ year    <int> 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 19…
    #> $ logit.s <dbl> -3.46, -3.32, -3.58, -3.03, -3.61, -3.35, -3.93, -4.19, -4.82,…
    #> $ CUI.apr <int> 57, 5, 43, 11, 47, -21, 25, -2, -1, 43, 2, 35, 0, 1, -1, 6, -7…

First we need to prepare the data for modelling. The variable `CUI.apr` will be standardized to make it easier for the sampler to estimate underlying GP parameters for the time-varying effect. We also need to convert the survival back to a proportion, as in its current form it has been logit-transformed (this is because most time series packages cannot handle proportional data). As usual, we also need to create a `time` indicator and a `series` indicator for working in `mvgam`:

    SalmonSurvCUI %>%
      # create a time variable
      dplyr::mutate(time = dplyr::row_number()) %>%
      # create a series variable
      dplyr::mutate(series = as.factor("salmon")) %>%
      # z-score the covariate CUI.apr
      dplyr::mutate(CUI.apr = as.vector(scale(CUI.apr))) %>%
      # convert logit-transformed survival back to proportional
      dplyr::mutate(survival = plogis(logit.s)) -> model_data

Inspect the data

    dplyr::glimpse(model_data)
    #> Rows: 42
    #> Columns: 6
    #> $ year     <int> 1964, 1965, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1…
    #> $ logit.s  <dbl> -3.46, -3.32, -3.58, -3.03, -3.61, -3.35, -3.93, -4.19, -4.82…
    #> $ CUI.apr  <dbl> 2.37949804, 0.03330223, 1.74782994, 0.30401713, 1.92830654, -…
    #> $ time     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18…
    #> $ series   <fct> salmon, salmon, salmon, salmon, salmon, salmon, salmon, salmo…
    #> $ survival <dbl> 0.030472033, 0.034891409, 0.027119717, 0.046088827, 0.0263393…

Plot features of the outcome variable, which shows that it is a proportional variable with particular restrictions that we want to model:

    plot_mvgam_series(data = model_data, y = "survival")

### A State-Space Beta regression

`mvgam` can easily handle data that are bounded at 0 and 1 with a Beta observation model (using the `mgcv` function `betar()`, see `?mgcv::betar` for details). First we will fit a simple State-Space model that uses an AR1 dynamic process model with no predictors and a Beta observation model:

    mod0 <- mvgam(
      formula = survival ~ 1,
      trend_model = AR(),
      noncentred = TRUE,
      priors = prior(normal(-3.5, 0.5), class = Intercept),
      family = betar(),
      data = model_data,
      silent = 2
    )

The summary of this model shows good behaviour of the Hamiltonian Monte Carlo sampler and provides useful summaries on the Beta observation model parameters:

    summary(mod0)
    #> GAM formula:
    #> survival ~ 1
    #> <environment: 0x000001e003956610>
    #>
    #> Family:
    #> beta
    #>
    #> Link function:
    #> logit
    #>
    #> Trend model:
    #> AR()
    #>
    #>
    #> N series:
    #> 1
    #>
    #> N timepoints:
    #> 42
    #>
    #> 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]   83 230   600 1.01   285
    #>
    #> GAM coefficient (beta) estimates:
    #>             2.5%  50% 97.5% Rhat n_eff
    #> (Intercept) -4.6 -4.3    -4    1   802
    #>
    #> Latent trend parameter AR estimates:
    #>           2.5%  50% 97.5% Rhat n_eff
    #> ar1[1]   -0.28 0.69  0.99 1.00   442
    #> sigma[1]  0.12 0.40  0.67 1.01   264
    #>
    #> Stan MCMC diagnostics:
    #> ✔ No issues with effective samples per iteration
    #> ✔ Rhat looks good for all parameters
    #> ✔ No issues with divergences
    #> ✔ No issues with maximum tree depth
    #>
    #> Samples were drawn using sampling(hmc). 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() to get started describing this model

A plot of the underlying dynamic component shows how it has easily handled the temporal evolution of the time series:

    plot(mod0, type = "trend")

### Including time-varying upwelling effects

Now we can increase the complexity of our model by constructing and fitting a State-Space model with a time-varying effect of the coastal upwelling index in addition to the autoregressive dynamics. We again use a Beta observation model to capture the restrictions of our proportional observations, but this time will include a `dynamic()` effect of `CUI.apr` in the latent process model. We do not specify the \(\rho\) parameter, instead opting to estimate it using a Hilbert space approximate GP:

    mod1 <- mvgam(
      formula = survival ~ 1,
      trend_formula = ~ dynamic(CUI.apr, k = 25, scale = FALSE) - 1,
      trend_model = AR(),
      noncentred = TRUE,
      priors = prior(normal(-3.5, 0.5), class = Intercept),
      family = betar(),
      data = model_data,
      silent = 2
    )

The summary for this model now includes estimates for the time-varying GP parameters:

    summary(mod1, include_betas = FALSE)
    #> GAM observation formula:
    #> survival ~ 1
    #> <environment: 0x000001e003956610>
    #>
    #> GAM process formula:
    #> ~dynamic(CUI.apr, k = 25, scale = FALSE) - 1
    #> <environment: 0x000001e003956610>
    #>
    #> Family:
    #> beta
    #>
    #> Link function:
    #> logit
    #>
    #> Trend model:
    #> AR()
    #>
    #>
    #> N process models:
    #> 1
    #>
    #> N series:
    #> 1
    #>
    #> N timepoints:
    #> 42
    #>
    #> 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]  170 350   660    1   802
    #>
    #> GAM observation model coefficient (beta) estimates:
    #>             2.5%  50% 97.5% Rhat n_eff
    #> (Intercept) -4.4 -3.8  -2.9    1   687
    #>
    #> Process model AR parameter estimates:
    #>        2.5%  50% 97.5% Rhat n_eff
    #> ar1[1] 0.57 0.92     1    1   547
    #>
    #> Process error parameter estimates:
    #>          2.5%  50% 97.5% Rhat n_eff
    #> sigma[1] 0.18 0.32  0.52    1   822
    #>
    #> GAM process model gp term marginal deviation (alpha) and length scale (rho) estimates:
    #>                         2.5%  50% 97.5% Rhat n_eff
    #> alpha_gp(time):CUI.apr 0.027 0.32   1.2 1.00   695
    #> rho_gp(time):CUI.apr   1.300 5.50  32.0 1.01   578
    #>
    #> Stan MCMC diagnostics:
    #> ✔ No issues with effective samples per iteration
    #> ✔ Rhat looks good for all parameters
    #> ✖ 66 of 2000 iterations ended with a divergence (3.3%)
    #>     Try a larger adapt_delta to remove divergences
    #> ✔ No issues with maximum tree depth
    #>
    #> Samples were drawn using sampling(hmc). 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() to get started describing this model

The estimates for the underlying dynamic process, and for the hindcasts, haven’t changed much:

    plot(mod1, type = "trend")

    plot(mod1, type = "forecast")

But the process error parameter \(\sigma\) is slightly smaller for this model than for the first model:

    # Extract estimates of the process error 'sigma' for each model
    mod0_sigma <- as.data.frame(mod0, variable = "sigma", regex = TRUE) %>%
      dplyr::mutate(model = "Mod0")
    mod1_sigma <- as.data.frame(mod1, variable = "sigma", regex = TRUE) %>%
      dplyr::mutate(model = "Mod1")
    sigmas <- rbind(mod0_sigma, mod1_sigma)

    # Plot using ggplot2
    require(ggplot2)
    ggplot(sigmas, aes(y = `sigma[1]`, fill = model)) +
      geom_density(alpha = 0.3, colour = NA) +
      coord_flip()

Why does the process error not need to be as flexible in the second model? Because the estimates of this dynamic process are now informed partly by the time-varying effect of upwelling, which we can visualise on the link scale using `plot()`:

    plot(mod1, type = "smooths", trend_effects = TRUE)

### Comparing model predictive performances

A key question when fitting multiple time series models is whether one of them provides better predictions than the other. There are several options in `mvgam` for exploring this quantitatively. First, we can compare models based on in-sample approximate leave-one-out cross-validation as implemented in the popular `loo` package:

    loo_compare(mod0, mod1)
    #>      elpd_diff se_diff
    #> mod1  0.0       0.0
    #> mod0 -8.6       3.4

The second model has the larger Expected Log Predictive Density (ELPD), meaning that it is slightly favoured over the simpler model that did not include the time-varying upwelling effect. However, the two models certainly do not differ by much. But this metric only compares in-sample performance, and we are hoping to use our models to produce reasonable forecasts. Luckily, `mvgam` also has routines for comparing models using approximate leave-future-out cross-validation. Here we refit both models to a reduced training set (starting at time point 30) and produce approximate 1-step ahead forecasts. These forecasts are used to estimate forecast ELPD before expanding the training set one time point at a time. We use Pareto-smoothed importance sampling to reweight posterior predictions, acting as a kind of particle filter so that we don’t need to refit the model too often (you can read more about how this process works in Bürkner et al. 2020).

    lfo_mod0 <- lfo_cv(mod0, min_t = 30)
    lfo_mod1 <- lfo_cv(mod1, min_t = 30)

The model with the time-varying upwelling effect tends to provides better 1-step ahead forecasts, with a higher total forecast ELPD

    sum(lfo_mod0$elpds)
    #> [1] 35.6104
    sum(lfo_mod1$elpds)
    #> [1] 37.63404

We can also plot the ELPDs for each model as a contrast. Here, values less than zero suggest the time-varying predictor model (Mod1) gives better 1-step ahead forecasts:

    plot(
      x = 1:length(lfo_mod0$elpds) + 30,
      y = lfo_mod0$elpds - lfo_mod1$elpds,
      ylab = "ELPDmod0 - ELPDmod1",
      xlab = "Evaluation time point",
      pch = 16,
      col = "darkred",
      bty = "l"
    )
    abline(h = 0, lty = "dashed")

A useful exercise to further expand this model would be to think about what kinds of predictors might impact measurement error, which could easily be implemented into the observation formula in `mvgam()`. But for now, we will leave the model as-is.

## Further reading

The following papers and resources offer a lot of useful material about dynamic linear models and how they can be applied / evaluated in practice:

Bürkner, PC, Gabry, J and Vehtari, A Approximate leave-future-out cross-validation for Bayesian time series models. Journal of Statistical Computation and Simulation. 90:14 (2020) 2499-2523.

Herrero, Asier, et al. From the individual to the landscape and back: time‐varying effects of climate and herbivory on tree sapling growth at distribution limits. Journal of Ecology 104.2 (2016): 430-442.

Holmes, Elizabeth E., Eric J. Ward, and Wills Kellie. “MARSS: multivariate autoregressive state-space models for analyzing time-series data.” R Journal. 4.1 (2012): 11.

Scheuerell, Mark D., and John G. Williams. Forecasting climate induced changes in the survival of Snake River Spring/Summer Chinook Salmon (Oncorhynchus Tshawytscha) Fisheries Oceanography 14 (2005): 448–57.

## 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 see this small list of opportunities on my website and do reach out if you are interested (n.clark’at’uq.edu.au)