The purpose of this vignette is to show how the `mvgam`

package can be used to produce probabilistic forecasts and to evaluate
those forecasts using a variety of proper scoring rules.

We begin by simulating some data to show how forecasts are computed
and evaluated in `mvgam`

. The `sim_mvgam()`

function can be used to simulate series that come from a variety of
response distributions as well as seasonal patterns and/or dynamic
temporal patterns. Here we simulate a collection of three time
count-valued series. These series all share the same seasonal pattern
but have different temporal dynamics. By setting
`trend_model = GP()`

and `prop_trend = 0.75`

, we
are generating time series that have smooth underlying temporal trends
(evolving as Gaussian Processes with squared exponential kernel) and
moderate seasonal patterns. The observations are Poisson-distributed and
we allow 10% of observations to be missing.

```
set.seed(2345)
simdat <- sim_mvgam(T = 100,
n_series = 3,
trend_model = GP(),
prop_trend = 0.75,
family = poisson(),
prop_missing = 0.10)
```

The returned object is a `list`

containing training and
testing data (`sim_mvgam()`

automatically splits the data
into these folds for us) together with some other information about the
data generating process that was used to simulate the data

```
str(simdat)
#> List of 6
#> $ data_train :'data.frame': 225 obs. of 5 variables:
#> ..$ y : int [1:225] 0 1 3 0 0 0 1 0 3 1 ...
#> ..$ season: int [1:225] 1 1 1 2 2 2 3 3 3 4 ...
#> ..$ year : int [1:225] 1 1 1 1 1 1 1 1 1 1 ...
#> ..$ series: Factor w/ 3 levels "series_1","series_2",..: 1 2 3 1 2 3 1 2 3 1 ...
#> ..$ time : int [1:225] 1 1 1 2 2 2 3 3 3 4 ...
#> $ data_test :'data.frame': 75 obs. of 5 variables:
#> ..$ y : int [1:75] 0 1 1 0 0 0 2 2 0 NA ...
#> ..$ season: int [1:75] 4 4 4 5 5 5 6 6 6 7 ...
#> ..$ year : int [1:75] 7 7 7 7 7 7 7 7 7 7 ...
#> ..$ series: Factor w/ 3 levels "series_1","series_2",..: 1 2 3 1 2 3 1 2 3 1 ...
#> ..$ time : int [1:75] 76 76 76 77 77 77 78 78 78 79 ...
#> $ true_corrs : num [1:3, 1:3] 1 0.465 -0.577 0.465 1 ...
#> $ true_trends : num [1:100, 1:3] -1.45 -1.54 -1.61 -1.67 -1.73 ...
#> $ global_seasonality: num [1:100] 0.0559 0.6249 1.3746 1.6805 0.5246 ...
#> $ trend_params :List of 2
#> ..$ alpha: num [1:3] 0.767 0.988 0.897
#> ..$ rho : num [1:3] 6.02 6.94 5.04
```

Each series in this case has a shared seasonal pattern. The resulting time series are similar to what we might encounter when dealing with count-valued data that can take small counts:

For individual series, we can plot the training and testing data, as well as some more specific features of the observed data:

The first model we will fit uses a shared cyclic spline to capture the repeated seasonality, as well as series-specific splines of time to capture the long-term dynamics. We allow the temporal splines to be fairly complex so they can capture as much of the temporal variation as possible:

```
mod1 <- mvgam(y ~ s(season, bs = 'cc', k = 8) +
s(time, by = series, bs = 'cr', k = 20),
knots = list(season = c(0.5, 12.5)),
trend_model = 'None',
data = simdat$data_train,
silent = 2)
```

The model fits without issue:

```
summary(mod1, include_betas = FALSE)
#> GAM formula:
#> y ~ s(season, bs = "cc", k = 8) + s(time, by = series, bs = "cr",
#> k = 20)
#> <environment: 0x000002c8194b2058>
#>
#> Family:
#> poisson
#>
#> Link function:
#> log
#>
#> Trend model:
#> None
#>
#> N series:
#> 3
#>
#> N timepoints:
#> 75
#>
#> 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) -0.4 -0.21 -0.045 1 1035
#>
#> Approximate significance of GAM smooths:
#> edf Ref.df Chi.sq p-value
#> s(season) 3.28 6 20.4 0.0215 *
#> s(time):seriesseries_1 6.95 19 14.0 0.8153
#> s(time):seriesseries_2 10.86 19 156.5 0.0044 **
#> s(time):seriesseries_3 6.79 19 18.1 0.5529
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> 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 Wed Sep 04 11:27:30 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)
```

And we can plot the conditional effects of the splines (on the link scale) to see that they are estimated to be highly nonlinear

Before showing how to produce and evaluate forecasts, we will fit a
second model to these data so the two models can be compared. This model
is equivalent to the above, except we now use Gaussian Processes to
model series-specific dynamics. This makes use of the `gp()`

function from `brms`

, which can fit Hilbert space approximate
GPs. See `?brms::gp`

for more details.

```
mod2 <- mvgam(y ~ s(season, bs = 'cc', k = 8) +
gp(time, by = series, c = 5/4, k = 20,
scale = FALSE),
knots = list(season = c(0.5, 12.5)),
trend_model = 'None',
data = simdat$data_train,
silent = 2)
```

The summary for this model now contains information on the GP parameters for each time series:

```
summary(mod2, include_betas = FALSE)
#> GAM formula:
#> y ~ s(season, bs = "cc", k = 8) + gp(time, by = series, c = 5/4,
#> k = 20, scale = FALSE)
#> <environment: 0x000002c8194b2058>
#>
#> Family:
#> poisson
#>
#> Link function:
#> log
#>
#> Trend model:
#> None
#>
#> N series:
#> 3
#>
#> N timepoints:
#> 75
#>
#> 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) -1.1 -0.51 0.25 1 731
#>
#> GAM gp term marginal deviation (alpha) and length scale (rho) estimates:
#> 2.5% 50% 97.5% Rhat n_eff
#> alpha_gp(time):seriesseries_1 0.19 0.75 2 1.00 932
#> alpha_gp(time):seriesseries_2 0.76 1.40 3 1.00 845
#> alpha_gp(time):seriesseries_3 0.48 1.10 3 1.00 968
#> rho_gp(time):seriesseries_1 1.10 4.90 25 1.01 678
#> rho_gp(time):seriesseries_2 2.20 10.00 17 1.00 645
#> rho_gp(time):seriesseries_3 1.50 9.20 24 1.00 908
#>
#> Approximate significance of GAM smooths:
#> edf Ref.df Chi.sq p-value
#> s(season) 3.4 6 21.1 0.011 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Stan MCMC diagnostics:
#> n_eff / iter looks reasonable for all parameters
#> Rhat looks reasonable for all parameters
#> 7 of 2000 iterations ended with a divergence (0.35%)
#> *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 Wed Sep 04 11:28:27 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)
```

We can plot the posteriors for these parameters, and for any other
parameter for that matter, using `bayesplot`

routines. First
the marginal deviation (\(\alpha\))
parameters:

And now the length scale (\(\rho\)) parameters:

We can again plot the nonlinear effects:

The estimates for the temporal trends are fairly similar for the two models, but below we will see if they produce similar forecasts

`forecast()`

functionProbabilistic forecasts can be computed in two main ways in
`mvgam`

. The first is to take a model that was fit only to
training data (as we did above in the two example models) and produce
temporal predictions from the posterior predictive distribution by
feeding `newdata`

to the `forecast()`

function. It
is crucial that any `newdata`

fed to the
`forecast()`

function follows on sequentially from the data
that was used to fit the model (this is not internally checked by the
package because it might be a headache to do so when data are not
supplied in a specific time-order). When calling the
`forecast()`

function, you have the option to generate
different kinds of predictions (i.e. predicting on the link scale,
response scale or to produce expectations; see
`?forecast.mvgam`

for details). We will use the default and
produce forecasts on the response scale, which is the most common way to
evaluate forecast distributions

```
fc_mod1 <- forecast(mod1, newdata = simdat$data_test)
fc_mod2 <- forecast(mod2, newdata = simdat$data_test)
```

The objects we have created are of class `mvgam_forecast`

,
which contain information on hindcast distributions, forecast
distributions and true observations for each series in the data:

```
str(fc_mod1)
#> List of 16
#> $ call :Class 'formula' language y ~ s(season, bs = "cc", k = 8) + s(time, by = series, bs = "cr", k = 20)
#> .. ..- attr(*, ".Environment")=<environment: 0x000002c8194b2058>
#> $ trend_call : NULL
#> $ family : chr "poisson"
#> $ family_pars : NULL
#> $ trend_model : chr "None"
#> $ drift : logi FALSE
#> $ use_lv : logi FALSE
#> $ fit_engine : chr "stan"
#> $ type : chr "response"
#> $ series_names : Factor w/ 3 levels "series_1","series_2",..: 1 2 3
#> $ train_observations:List of 3
#> ..$ series_1: int [1:75] 0 0 1 1 0 0 0 0 0 0 ...
#> ..$ series_2: int [1:75] 1 0 0 1 1 0 1 0 1 2 ...
#> ..$ series_3: int [1:75] 3 0 3 NA 2 1 1 1 1 3 ...
#> $ train_times : int [1:75] 1 2 3 4 5 6 7 8 9 10 ...
#> $ test_observations :List of 3
#> ..$ series_1: int [1:25] 0 0 2 NA 0 2 2 1 1 1 ...
#> ..$ series_2: int [1:25] 1 0 2 1 1 3 0 1 0 NA ...
#> ..$ series_3: int [1:25] 1 0 0 1 0 0 1 0 1 0 ...
#> $ test_times : int [1:25] 76 77 78 79 80 81 82 83 84 85 ...
#> $ hindcasts :List of 3
#> ..$ series_1: num [1:2000, 1:75] 0 0 0 1 0 1 0 0 0 0 ...
#> .. ..- attr(*, "dimnames")=List of 2
#> .. .. ..$ : NULL
#> .. .. ..$ : chr [1:75] "ypred[1,1]" "ypred[2,1]" "ypred[3,1]" "ypred[4,1]" ...
#> ..$ series_2: num [1:2000, 1:75] 0 0 0 0 0 1 0 0 0 0 ...
#> .. ..- attr(*, "dimnames")=List of 2
#> .. .. ..$ : NULL
#> .. .. ..$ : chr [1:75] "ypred[1,2]" "ypred[2,2]" "ypred[3,2]" "ypred[4,2]" ...
#> ..$ series_3: num [1:2000, 1:75] 3 2 2 0 1 3 2 3 1 3 ...
#> .. ..- attr(*, "dimnames")=List of 2
#> .. .. ..$ : NULL
#> .. .. ..$ : chr [1:75] "ypred[1,3]" "ypred[2,3]" "ypred[3,3]" "ypred[4,3]" ...
#> $ forecasts :List of 3
#> ..$ series_1: num [1:2000, 1:25] 0 1 0 0 0 1 1 1 1 1 ...
#> ..$ series_2: num [1:2000, 1:25] 0 0 1 1 0 0 0 1 0 0 ...
#> ..$ series_3: num [1:2000, 1:25] 0 0 1 1 0 3 1 0 3 2 ...
#> - attr(*, "class")= chr "mvgam_forecast"
```

We can plot the forecasts for some series from each model using the
`S3 plot`

method for objects of this class:

Clearly the two models do not produce equivalent forecasts. We will come back to scoring these forecasts in a moment.

`newdata`

in `mvgam()`

The second way we can produce forecasts in `mvgam`

is to
feed the testing data directly to the `mvgam()`

function as
`newdata`

. This will include the testing data as missing
observations so that they are automatically predicted from the posterior
predictive distribution using the `generated quantities`

block in `Stan`

. As an example, we can refit
`mod2`

but include the testing data for automatic
forecasts:

```
mod2 <- mvgam(y ~ s(season, bs = 'cc', k = 8) +
gp(time, by = series, c = 5/4, k = 20,
scale = FALSE),
knots = list(season = c(0.5, 12.5)),
trend_model = 'None',
data = simdat$data_train,
newdata = simdat$data_test,
silent = 2)
```

Because the model already contains a forecast distribution, we do not
need to feed `newdata`

to the `forecast()`

function:

The forecasts will be nearly identical to those calculated previously:

A primary purpose of the `mvgam_forecast`

class is to
readily allow forecast evaluations for each series in the data, using a
variety of possible scoring functions. See
`?mvgam::score.mvgam_forecast`

to view the types of scores
that are available. A useful scoring metric is the Continuous Rank Probability Score (CRPS). A CRPS
value is similar to what we might get if we calculated a weighted
absolute error using the full forecast distribution.

```
crps_mod1 <- score(fc_mod1, score = 'crps')
str(crps_mod1)
#> List of 4
#> $ series_1 :'data.frame': 25 obs. of 5 variables:
#> ..$ score : num [1:25] 0.1797 0.1341 1.3675 NA 0.0386 ...
#> ..$ in_interval : num [1:25] 1 1 1 NA 1 1 1 1 1 1 ...
#> ..$ interval_width: num [1:25] 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
#> ..$ eval_horizon : int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> ..$ score_type : chr [1:25] "crps" "crps" "crps" "crps" ...
#> $ series_2 :'data.frame': 25 obs. of 5 variables:
#> ..$ score : num [1:25] 0.375 0.283 1.003 0.516 0.649 ...
#> ..$ in_interval : num [1:25] 1 1 1 1 1 1 1 1 1 NA ...
#> ..$ interval_width: num [1:25] 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
#> ..$ eval_horizon : int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> ..$ score_type : chr [1:25] "crps" "crps" "crps" "crps" ...
#> $ series_3 :'data.frame': 25 obs. of 5 variables:
#> ..$ score : num [1:25] 0.318 0.604 0.4 0.353 0.212 ...
#> ..$ in_interval : num [1:25] 1 1 1 1 1 1 1 1 1 1 ...
#> ..$ interval_width: num [1:25] 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
#> ..$ eval_horizon : int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> ..$ score_type : chr [1:25] "crps" "crps" "crps" "crps" ...
#> $ all_series:'data.frame': 25 obs. of 3 variables:
#> ..$ score : num [1:25] 0.873 1.021 2.77 NA 0.9 ...
#> ..$ eval_horizon: int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> ..$ score_type : chr [1:25] "sum_crps" "sum_crps" "sum_crps" "sum_crps" ...
crps_mod1$series_1
#> score in_interval interval_width eval_horizon score_type
#> 1 0.17965150 1 0.9 1 crps
#> 2 0.13411725 1 0.9 2 crps
#> 3 1.36749550 1 0.9 3 crps
#> 4 NA NA 0.9 4 crps
#> 5 0.03860225 1 0.9 5 crps
#> 6 1.55334350 1 0.9 6 crps
#> 7 1.49198325 1 0.9 7 crps
#> 8 0.64088650 1 0.9 8 crps
#> 9 0.61613650 1 0.9 9 crps
#> 10 0.60528025 1 0.9 10 crps
#> 11 1.30624075 1 0.9 11 crps
#> 12 2.06809025 1 0.9 12 crps
#> 13 0.61887550 1 0.9 13 crps
#> 14 0.13920225 1 0.9 14 crps
#> 15 0.67819725 1 0.9 15 crps
#> 16 0.07817500 1 0.9 16 crps
#> 17 0.07567500 1 0.9 17 crps
#> 18 0.09510025 1 0.9 18 crps
#> 19 0.12604375 1 0.9 19 crps
#> 20 NA NA 0.9 20 crps
#> 21 0.20347825 1 0.9 21 crps
#> 22 0.82202975 1 0.9 22 crps
#> 23 NA NA 0.9 23 crps
#> 24 1.06660275 1 0.9 24 crps
#> 25 0.67787450 1 0.9 25 crps
```

The returned list contains a `data.frame`

for each series
in the data that shows the CRPS score for each evaluation in the testing
data, along with some other useful information about the fit of the
forecast distribution. In particular, we are given a logical value (1s
and 0s) telling us whether the true value was within a pre-specified
credible interval (i.e. the coverage of the forecast distribution). The
default interval width is 0.9, so we would hope that the values in the
`in_interval`

column take a 1 approximately 90% of the time.
This value can be changed if you wish to compute different coverages,
say using a 60% interval:

```
crps_mod1 <- score(fc_mod1, score = 'crps', interval_width = 0.6)
crps_mod1$series_1
#> score in_interval interval_width eval_horizon score_type
#> 1 0.17965150 1 0.6 1 crps
#> 2 0.13411725 1 0.6 2 crps
#> 3 1.36749550 0 0.6 3 crps
#> 4 NA NA 0.6 4 crps
#> 5 0.03860225 1 0.6 5 crps
#> 6 1.55334350 0 0.6 6 crps
#> 7 1.49198325 0 0.6 7 crps
#> 8 0.64088650 1 0.6 8 crps
#> 9 0.61613650 1 0.6 9 crps
#> 10 0.60528025 1 0.6 10 crps
#> 11 1.30624075 0 0.6 11 crps
#> 12 2.06809025 0 0.6 12 crps
#> 13 0.61887550 1 0.6 13 crps
#> 14 0.13920225 1 0.6 14 crps
#> 15 0.67819725 1 0.6 15 crps
#> 16 0.07817500 1 0.6 16 crps
#> 17 0.07567500 1 0.6 17 crps
#> 18 0.09510025 1 0.6 18 crps
#> 19 0.12604375 1 0.6 19 crps
#> 20 NA NA 0.6 20 crps
#> 21 0.20347825 1 0.6 21 crps
#> 22 0.82202975 1 0.6 22 crps
#> 23 NA NA 0.6 23 crps
#> 24 1.06660275 1 0.6 24 crps
#> 25 0.67787450 1 0.6 25 crps
```

We can also compare forecasts against out of sample observations
using the Expected Log Predictive Density (ELPD; also known as the
log score). The ELPD is a strictly proper scoring rule that can be
applied to any distributional forecast, but to compute it we need
predictions on the link scale rather than on the outcome scale. This is
where it is advantageous to change the type of prediction we can get
using the `forecast()`

function:

```
link_mod1 <- forecast(mod1, newdata = simdat$data_test, type = 'link')
score(link_mod1, score = 'elpd')$series_1
#> score eval_horizon score_type
#> 1 -0.5285206 1 elpd
#> 2 -0.4286994 2 elpd
#> 3 -2.9660940 3 elpd
#> 4 NA 4 elpd
#> 5 -0.1988847 5 elpd
#> 6 -3.3821055 6 elpd
#> 7 -3.2797236 7 elpd
#> 8 -2.0571076 8 elpd
#> 9 -2.0794559 9 elpd
#> 10 -2.0882202 10 elpd
#> 11 -3.0870256 11 elpd
#> 12 -3.7065927 12 elpd
#> 13 -2.1601960 13 elpd
#> 14 -0.2931143 14 elpd
#> 15 -2.3694878 15 elpd
#> 16 -0.2110566 16 elpd
#> 17 -0.1986209 17 elpd
#> 18 -0.2069720 18 elpd
#> 19 -0.2193413 19 elpd
#> 20 NA 20 elpd
#> 21 -0.2379762 21 elpd
#> 22 -2.6261457 22 elpd
#> 23 NA 23 elpd
#> 24 -2.6309438 24 elpd
#> 25 -0.2817444 25 elpd
```

Finally, when we have multiple time series it may also make sense to
use a multivariate proper scoring rule. `mvgam`

offers two
such options: the Energy score and the Variogram score. The first
penalizes forecast distributions that are less well calibrated against
the truth, while the second penalizes forecasts that do not capture the
observed true correlation structure. Which score to use depends on your
goals, but both are very easy to compute:

```
energy_mod2 <- score(fc_mod2, score = 'energy')
str(energy_mod2)
#> List of 4
#> $ series_1 :'data.frame': 25 obs. of 3 variables:
#> ..$ in_interval : num [1:25] 1 1 1 NA 1 1 1 1 1 1 ...
#> ..$ interval_width: num [1:25] 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
#> ..$ eval_horizon : int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> $ series_2 :'data.frame': 25 obs. of 3 variables:
#> ..$ in_interval : num [1:25] 1 1 1 1 1 1 1 1 1 NA ...
#> ..$ interval_width: num [1:25] 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
#> ..$ eval_horizon : int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> $ series_3 :'data.frame': 25 obs. of 3 variables:
#> ..$ in_interval : num [1:25] 1 1 1 1 1 1 1 1 1 1 ...
#> ..$ interval_width: num [1:25] 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 ...
#> ..$ eval_horizon : int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> $ all_series:'data.frame': 25 obs. of 3 variables:
#> ..$ score : num [1:25] 0.755 1.12 1.245 NA 0.447 ...
#> ..$ eval_horizon: int [1:25] 1 2 3 4 5 6 7 8 9 10 ...
#> ..$ score_type : chr [1:25] "energy" "energy" "energy" "energy" ...
```

The returned object still provides information on interval coverage
for each individual series, but there is only a single score per horizon
now (which is provided in the `all_series`

slot):

```
energy_mod2$all_series
#> score eval_horizon score_type
#> 1 0.7546579 1 energy
#> 2 1.1200630 2 energy
#> 3 1.2447843 3 energy
#> 4 NA 4 energy
#> 5 0.4465348 5 energy
#> 6 1.8231460 6 energy
#> 7 1.4418019 7 energy
#> 8 0.7172890 8 energy
#> 9 1.0762943 9 energy
#> 10 NA 10 energy
#> 11 1.4112423 11 energy
#> 12 3.2385416 12 energy
#> 13 1.5836460 13 energy
#> 14 1.1953349 14 energy
#> 15 1.0412578 15 energy
#> 16 1.8348615 16 energy
#> 17 NA 17 energy
#> 18 0.7142977 18 energy
#> 19 0.9059773 19 energy
#> 20 NA 20 energy
#> 21 1.1043397 21 energy
#> 22 1.3292391 22 energy
#> 23 NA 23 energy
#> 24 2.1419570 24 energy
#> 25 1.2610880 25 energy
```

You can use your score(s) of choice to compare different models. For
example, we can compute and plot the difference in CRPS scores for each
series in data. Here, a negative value means the Gaussian Process model
(`mod2`

) is better, while a positive value means the spline
model (`mod1`

) is better.

```
crps_mod1 <- score(fc_mod1, score = 'crps')
crps_mod2 <- score(fc_mod2, score = 'crps')
diff_scores <- crps_mod2$series_1$score -
crps_mod1$series_1$score
plot(diff_scores, pch = 16, cex = 1.25, col = 'darkred',
ylim = c(-1*max(abs(diff_scores), na.rm = TRUE),
max(abs(diff_scores), na.rm = TRUE)),
bty = 'l',
xlab = 'Forecast horizon',
ylab = expression(CRPS[GP]~-~CRPS[spline]))
abline(h = 0, lty = 'dashed', lwd = 2)
gp_better <- length(which(diff_scores < 0))
title(main = paste0('GP better in ', gp_better, ' of 25 evaluations',
'\nMean difference = ',
round(mean(diff_scores, na.rm = TRUE), 2)))
```

```
diff_scores <- crps_mod2$series_2$score -
crps_mod1$series_2$score
plot(diff_scores, pch = 16, cex = 1.25, col = 'darkred',
ylim = c(-1*max(abs(diff_scores), na.rm = TRUE),
max(abs(diff_scores), na.rm = TRUE)),
bty = 'l',
xlab = 'Forecast horizon',
ylab = expression(CRPS[GP]~-~CRPS[spline]))
abline(h = 0, lty = 'dashed', lwd = 2)
gp_better <- length(which(diff_scores < 0))
title(main = paste0('GP better in ', gp_better, ' of 25 evaluations',
'\nMean difference = ',
round(mean(diff_scores, na.rm = TRUE), 2)))
```

```
diff_scores <- crps_mod2$series_3$score -
crps_mod1$series_3$score
plot(diff_scores, pch = 16, cex = 1.25, col = 'darkred',
ylim = c(-1*max(abs(diff_scores), na.rm = TRUE),
max(abs(diff_scores), na.rm = TRUE)),
bty = 'l',
xlab = 'Forecast horizon',
ylab = expression(CRPS[GP]~-~CRPS[spline]))
abline(h = 0, lty = 'dashed', lwd = 2)
gp_better <- length(which(diff_scores < 0))
title(main = paste0('GP better in ', gp_better, ' of 25 evaluations',
'\nMean difference = ',
round(mean(diff_scores, na.rm = TRUE), 2)))
```

The GP model consistently gives better forecasts, and the difference between scores grows quickly as the forecast horizon increases. This is not unexpected given the way that splines linearly extrapolate outside the range of training data

The following papers and resources offer useful material about Bayesian forecasting and proper scoring rules:

Hyndman, Rob J., and George Athanasopoulos. Forecasting: principles
and practice. *OTexts*, 2018.

Gneiting, Tilmann, and Adrian E. Raftery. Strictly
proper scoring rules, prediction, and estimation *Journal of the
American statistical Association* 102.477 (2007) 359-378.

Simonis, Juniper L., Ethan P. White, and SK Morgan Ernest. Evaluating
probabilistic ecological forecasts *Ecology* 102.8 (2021)
e03431.

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)