Scoring multivariate forecasts
In scoringutils: Utilities for Scoring and Assessing Predictions

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

This Vignette provides an overview about how to score multivariate forecasts.

Univariate forecasts

Let's start with a simple univariate forecast: The number of cases of COVID-19 in Germany on 2021-05-15, forecasted by the EuroCOVIDhub-ensemble model on 2021-05-03. In our example, this forecast is represented by a set of 40 samples from the predictive distribution.

library(scoringutils)

example_univ_single <- example_sample_continuous[
  target_type == "Cases" &
    location == "DE" &
    forecast_date == "2021-05-03" &
    target_end_date == "2021-05-15" &
    horizon == 2 &
    model == "EuroCOVIDhub-ensemble"
]
example_univ_single

We can score this forecast and will receive a single score.

score(example_univ_single)

Now, of course, we can also score multiple similar forecasts at the same time. Let's say we're not only interested in Germany, but other countries as well.

example_univ_multi <- example_sample_continuous[
  target_type == "Cases" &
    forecast_date == "2021-05-03" &
    target_end_date == "2021-05-15" &
    horizon == 2 &
    model == "EuroCOVIDhub-ensemble"
]
example_univ_multi

Now, we have a set of 4 forecasts for 4 different countries, each of them represented by a set of 40 samples from the predictive distribution.

When we score these forecasts, we will get 4 scores, one for each forecast and observed value.

score(example_univ_multi)

Multivariate forecasts

Now, instead of treating the four observations as independent, we could also think of them as a single realisation of a draw from the multivariate distribution of COVID-19 cases across several countries.

The corresponding multivariate forecast would similarly specify a predictive distribution for the number of cases across all 4 countries. The samples are then not draws from four independent distributions, but instead samples from a joint multivariate predictive distribution.

In the following, let's assume that our samples were draws from a multivariate distribution all along (we just treated them as independent for the univariate case).

To tell scoringutils that we want to treat these as a multivariate forecast, we need to specify the columns that are pooled together to form a single multivariate forecast. We do this via the joint_across argument. For example, if we want to pool forecasts across locations and treat them as a single multivariate forecast, we could set joint_across = c("location", "location_name") (in our example, the two columns contain essentially the same information - we therefore have to include both in joint_across (or could alternatively delete one of them)).

example_multiv <- as_forecast_multivariate_sample(
  data = example_univ_multi,
  c("location", "location_name")
)
example_multiv

The column .mv_group_id is created automatically and represents an identifier for each multivariate forecast. .mv_group_id is 1 everywhere, because we only have a single multivariate forecast. When scoring this forecast using an appropriate multivariate scoring function, we will get a single score, even though we have 4 observations, one for each country. (Note that for the purposes of scoring, it doesn't matter that sample ids are still 1-40, repeated 4 times, instead of 1-160. scoringutils handles this appropriately.)

score(example_multiv)

By default, score() computes both the energy score and the variogram score for multivariate sample forecasts. The energy score is a multivariate generalisation of the CRPS that measures overall forecast accuracy. The variogram score (Scheuerer and Hamill, 2015) specifically targets the correlation structure between the targets being forecast jointly. For each pair of targets (e.g. two countries), it compares the observed absolute difference |y_i - y_j|^p against what the forecast distribution predicts for that difference. A forecast that gets the correlations between targets wrong will predict pairwise differences that do not match the observations, producing a higher score. This makes the variogram score more sensitive to misspecified correlations than the energy score.

You can customise parameters using purrr::partial(). The order parameter p controls how differences are scaled: p = 0.5 (the default) is more robust to outliers, while p = 1 gives a standard absolute difference. See ?variogram_score_multivariate for full parameter documentation. For example, to use p = 1:

score(
  example_multiv,
  metrics = list(
    energy_score = energy_score_multivariate,
    variogram_score = purrr::partial(
      variogram_score_multivariate, p = 1
    )
  )
)

Multivariate point forecasts

If you have point forecasts rather than samples, you can score them using the variogram score via as_forecast_multivariate_point(). This treats each point forecast as a single-sample ensemble.

example_point_multi <- example_point[
  target_type == "Cases" &
    forecast_date == "2021-05-03" &
    target_end_date == "2021-05-15" &
    horizon == 2 &
    model == "EuroCOVIDhub-ensemble"
]

example_mv_point <- as_forecast_multivariate_point(
  data = na.omit(example_point_multi),
  joint_across = c("location", "location_name")
)
score(example_mv_point)

If, at any point, you want to score the same forecast using different groupings, you'd have create a new separate forecast object with a different grouping and score that new forecast object.