Scoring multivariate forecasts
Source:vignettes/scoring-multivariate-forecasts.Rmd
scoring-multivariate-forecasts.RmdThis 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
#> Forecast type: sample
#> Forecast unit:
#> location, location_name, target_end_date, target_type, forecast_date, model,
#> and horizon
#>
#> location location_name target_end_date target_type forecast_date
#> <char> <char> <Date> <char> <Date>
#> 1: DE Germany 2021-05-15 Cases 2021-05-03
#> 2: DE Germany 2021-05-15 Cases 2021-05-03
#> 3: DE Germany 2021-05-15 Cases 2021-05-03
#> 4: DE Germany 2021-05-15 Cases 2021-05-03
#> 5: DE Germany 2021-05-15 Cases 2021-05-03
#> 6: DE Germany 2021-05-15 Cases 2021-05-03
#> 7: DE Germany 2021-05-15 Cases 2021-05-03
#> 8: DE Germany 2021-05-15 Cases 2021-05-03
#> 9: DE Germany 2021-05-15 Cases 2021-05-03
#> 10: DE Germany 2021-05-15 Cases 2021-05-03
#> 11: DE Germany 2021-05-15 Cases 2021-05-03
#> 12: DE Germany 2021-05-15 Cases 2021-05-03
#> 13: DE Germany 2021-05-15 Cases 2021-05-03
#> 14: DE Germany 2021-05-15 Cases 2021-05-03
#> 15: DE Germany 2021-05-15 Cases 2021-05-03
#> 16: DE Germany 2021-05-15 Cases 2021-05-03
#> 17: DE Germany 2021-05-15 Cases 2021-05-03
#> 18: DE Germany 2021-05-15 Cases 2021-05-03
#> 19: DE Germany 2021-05-15 Cases 2021-05-03
#> 20: DE Germany 2021-05-15 Cases 2021-05-03
#> 21: DE Germany 2021-05-15 Cases 2021-05-03
#> 22: DE Germany 2021-05-15 Cases 2021-05-03
#> 23: DE Germany 2021-05-15 Cases 2021-05-03
#> 24: DE Germany 2021-05-15 Cases 2021-05-03
#> 25: DE Germany 2021-05-15 Cases 2021-05-03
#> 26: DE Germany 2021-05-15 Cases 2021-05-03
#> 27: DE Germany 2021-05-15 Cases 2021-05-03
#> 28: DE Germany 2021-05-15 Cases 2021-05-03
#> 29: DE Germany 2021-05-15 Cases 2021-05-03
#> 30: DE Germany 2021-05-15 Cases 2021-05-03
#> 31: DE Germany 2021-05-15 Cases 2021-05-03
#> 32: DE Germany 2021-05-15 Cases 2021-05-03
#> 33: DE Germany 2021-05-15 Cases 2021-05-03
#> 34: DE Germany 2021-05-15 Cases 2021-05-03
#> 35: DE Germany 2021-05-15 Cases 2021-05-03
#> 36: DE Germany 2021-05-15 Cases 2021-05-03
#> 37: DE Germany 2021-05-15 Cases 2021-05-03
#> 38: DE Germany 2021-05-15 Cases 2021-05-03
#> 39: DE Germany 2021-05-15 Cases 2021-05-03
#> 40: DE Germany 2021-05-15 Cases 2021-05-03
#> location location_name target_end_date target_type forecast_date
#> model horizon predicted sample_id observed
#> <char> <num> <num> <int> <num>
#> 1: EuroCOVIDhub-ensemble 2 109365.73 1 64985
#> 2: EuroCOVIDhub-ensemble 2 63041.27 2 64985
#> 3: EuroCOVIDhub-ensemble 2 186364.05 3 64985
#> 4: EuroCOVIDhub-ensemble 2 127841.64 4 64985
#> 5: EuroCOVIDhub-ensemble 2 79550.56 5 64985
#> 6: EuroCOVIDhub-ensemble 2 193981.34 6 64985
#> 7: EuroCOVIDhub-ensemble 2 122953.97 7 64985
#> 8: EuroCOVIDhub-ensemble 2 148088.41 8 64985
#> 9: EuroCOVIDhub-ensemble 2 104570.23 9 64985
#> 10: EuroCOVIDhub-ensemble 2 130718.45 10 64985
#> 11: EuroCOVIDhub-ensemble 2 154126.24 11 64985
#> 12: EuroCOVIDhub-ensemble 2 164671.65 12 64985
#> 13: EuroCOVIDhub-ensemble 2 118330.18 13 64985
#> 14: EuroCOVIDhub-ensemble 2 107950.08 14 64985
#> 15: EuroCOVIDhub-ensemble 2 151033.84 15 64985
#> 16: EuroCOVIDhub-ensemble 2 120649.63 16 64985
#> 17: EuroCOVIDhub-ensemble 2 114380.55 17 64985
#> 18: EuroCOVIDhub-ensemble 2 104300.98 18 64985
#> 19: EuroCOVIDhub-ensemble 2 144538.28 19 64985
#> 20: EuroCOVIDhub-ensemble 2 66689.95 20 64985
#> 21: EuroCOVIDhub-ensemble 2 131096.85 21 64985
#> 22: EuroCOVIDhub-ensemble 2 120698.00 22 64985
#> 23: EuroCOVIDhub-ensemble 2 199890.08 23 64985
#> 24: EuroCOVIDhub-ensemble 2 132037.17 24 64985
#> 25: EuroCOVIDhub-ensemble 2 89928.75 25 64985
#> 26: EuroCOVIDhub-ensemble 2 144859.42 26 64985
#> 27: EuroCOVIDhub-ensemble 2 148745.59 27 64985
#> 28: EuroCOVIDhub-ensemble 2 97248.30 28 64985
#> 29: EuroCOVIDhub-ensemble 2 73744.04 29 64985
#> 30: EuroCOVIDhub-ensemble 2 117133.25 30 64985
#> 31: EuroCOVIDhub-ensemble 2 197014.73 31 64985
#> 32: EuroCOVIDhub-ensemble 2 137847.82 32 64985
#> 33: EuroCOVIDhub-ensemble 2 120085.18 33 64985
#> 34: EuroCOVIDhub-ensemble 2 91030.07 34 64985
#> 35: EuroCOVIDhub-ensemble 2 133265.23 35 64985
#> 36: EuroCOVIDhub-ensemble 2 161345.08 36 64985
#> 37: EuroCOVIDhub-ensemble 2 52633.20 37 64985
#> 38: EuroCOVIDhub-ensemble 2 104926.13 38 64985
#> 39: EuroCOVIDhub-ensemble 2 162582.41 39 64985
#> 40: EuroCOVIDhub-ensemble 2 143421.88 40 64985
#> model horizon predicted sample_id observedWe can score this forecast and will receive a single score.
score(example_univ_single)
#> location location_name target_end_date target_type forecast_date
#> <char> <char> <Date> <char> <Date>
#> 1: DE Germany 2021-05-15 Cases 2021-05-03
#> model horizon bias dss crps overprediction
#> <char> <num> <num> <num> <num> <num>
#> 1: EuroCOVIDhub-ensemble 2 0.9 24.00559 42655.41 34690.28
#> underprediction dispersion log_score mad ae_median se_mean
#> <num> <num> <num> <num> <num> <num>
#> 1: 0 7965.135 12.64899 31078.55 60412.8 3823196821Now, 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
#> Forecast type: sample
#> Forecast unit:
#> location, location_name, target_end_date, target_type, forecast_date, model,
#> and horizon
#>
#> location location_name target_end_date target_type forecast_date
#> <char> <char> <Date> <char> <Date>
#> 1: DE Germany 2021-05-15 Cases 2021-05-03
#> 2: DE Germany 2021-05-15 Cases 2021-05-03
#> 3: DE Germany 2021-05-15 Cases 2021-05-03
#> 4: DE Germany 2021-05-15 Cases 2021-05-03
#> 5: DE Germany 2021-05-15 Cases 2021-05-03
#> ---
#> 156: IT Italy 2021-05-15 Cases 2021-05-03
#> 157: IT Italy 2021-05-15 Cases 2021-05-03
#> 158: IT Italy 2021-05-15 Cases 2021-05-03
#> 159: IT Italy 2021-05-15 Cases 2021-05-03
#> 160: IT Italy 2021-05-15 Cases 2021-05-03
#> model horizon predicted sample_id observed
#> <char> <num> <num> <int> <num>
#> 1: EuroCOVIDhub-ensemble 2 109365.73 1 64985
#> 2: EuroCOVIDhub-ensemble 2 63041.27 2 64985
#> 3: EuroCOVIDhub-ensemble 2 186364.05 3 64985
#> 4: EuroCOVIDhub-ensemble 2 127841.64 4 64985
#> 5: EuroCOVIDhub-ensemble 2 79550.56 5 64985
#> ---
#> 156: EuroCOVIDhub-ensemble 2 72194.00 36 50453
#> 157: EuroCOVIDhub-ensemble 2 82507.14 37 50453
#> 158: EuroCOVIDhub-ensemble 2 102956.27 38 50453
#> 159: EuroCOVIDhub-ensemble 2 55985.84 39 50453
#> 160: EuroCOVIDhub-ensemble 2 65929.64 40 50453Now, 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)
#> location location_name target_end_date target_type forecast_date
#> <char> <char> <Date> <char> <Date>
#> 1: DE Germany 2021-05-15 Cases 2021-05-03
#> 2: FR France 2021-05-15 Cases 2021-05-03
#> 3: GB United Kingdom 2021-05-15 Cases 2021-05-03
#> 4: IT Italy 2021-05-15 Cases 2021-05-03
#> model horizon bias dss crps overprediction
#> <char> <num> <num> <num> <num> <num>
#> 1: EuroCOVIDhub-ensemble 2 0.90 24.00559 42655.413 34690.28
#> 2: EuroCOVIDhub-ensemble 2 0.50 22.37188 21960.030 7820.70
#> 3: EuroCOVIDhub-ensemble 2 -0.60 17.19740 2334.652 0.00
#> 4: EuroCOVIDhub-ensemble 2 0.95 21.73164 16262.604 12531.20
#> underprediction dispersion log_score mad ae_median se_mean
#> <num> <num> <num> <num> <num> <num>
#> 1: 0.000 7965.1347 12.64899 31078.550 60412.802 3823196821
#> 2: 0.000 14139.3296 11.99198 57243.099 38228.018 1763097632
#> 3: 1629.644 705.0089 10.03031 2680.797 3902.441 10157513
#> 4: 0.000 3731.3990 11.57288 16954.657 23892.603 626560551Multivariate 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
#> Forecast type: sample_multivariate
#> Forecast unit:
#> location, location_name, target_end_date, target_type, forecast_date, model,
#> and horizon
#>
#> location location_name target_end_date target_type forecast_date
#> <char> <char> <Date> <char> <Date>
#> 1: DE Germany 2021-05-15 Cases 2021-05-03
#> 2: DE Germany 2021-05-15 Cases 2021-05-03
#> 3: DE Germany 2021-05-15 Cases 2021-05-03
#> 4: DE Germany 2021-05-15 Cases 2021-05-03
#> 5: DE Germany 2021-05-15 Cases 2021-05-03
#> ---
#> 156: IT Italy 2021-05-15 Cases 2021-05-03
#> 157: IT Italy 2021-05-15 Cases 2021-05-03
#> 158: IT Italy 2021-05-15 Cases 2021-05-03
#> 159: IT Italy 2021-05-15 Cases 2021-05-03
#> 160: IT Italy 2021-05-15 Cases 2021-05-03
#> model horizon predicted sample_id observed .mv_group_id
#> <char> <num> <num> <int> <num> <int>
#> 1: EuroCOVIDhub-ensemble 2 109365.73 1 64985 1
#> 2: EuroCOVIDhub-ensemble 2 63041.27 2 64985 1
#> 3: EuroCOVIDhub-ensemble 2 186364.05 3 64985 1
#> 4: EuroCOVIDhub-ensemble 2 127841.64 4 64985 1
#> 5: EuroCOVIDhub-ensemble 2 79550.56 5 64985 1
#> ---
#> 156: EuroCOVIDhub-ensemble 2 72194.00 36 50453 1
#> 157: EuroCOVIDhub-ensemble 2 82507.14 37 50453 1
#> 158: EuroCOVIDhub-ensemble 2 102956.27 38 50453 1
#> 159: EuroCOVIDhub-ensemble 2 55985.84 39 50453 1
#> 160: EuroCOVIDhub-ensemble 2 65929.64 40 50453 1The 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)
#> target_end_date target_type forecast_date model horizon
#> <Date> <char> <Date> <char> <num>
#> 1: 2021-05-15 Cases 2021-05-03 EuroCOVIDhub-ensemble 2
#> energy_score .mv_group_id
#> <num> <int>
#> 1: 54795.73 1If, 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.
Univariate and multivariate scoring for matrices
Note: this section may only be relevant to you if you’re planning to score forecasts in matrix format.
Let’s construct a simple multivariate forecast:
# parameters for multivariate normal example
set.seed(123)
d <- 10 # number of dimensions
m <- 50 # number of samples from multivariate forecast distribution
mu0 <- rep(0, d)
mu <- rep(1, d)
S0 <- S <- diag(d)
S0[S0 == 0] <- 0.2
S[S == 0] <- 0.1
# generate samples from multivariate normal distributions
obs <- drop(mu0 + rnorm(d) %*% chol(S0))
fc_sample <- replicate(m, drop(mu + rnorm(d) %*% chol(S)))
obs2 <- drop(mu0 + rnorm(d) %*% chol(S0))
fc_sample2 <- replicate(m, drop(mu + rnorm(d) %*% chol(S)))Now, we can compute the Energy Score. Let’s compare the
scoringutils implementation with that of the
scoringRules package, on which the
scoringutils implementation is based. The only difference
is that scoringRules always expects a single multivariate
forecast, while the scoringutils
implementation can handle multiple multivariate forecasts together,
identified via a grouping vector (assuming they all have the same
dimension).
scoringRules::es_sample(y = obs, dat = fc_sample)
#> [1] 2.684649
# in the univariate case, Energy Score and CRPS are the same
# illustration: Evaluate forecast sample for the first variable
es_sr1 <- scoringRules::es_sample(y = obs, dat = fc_sample)
es_sr2 <- scoringRules::es_sample(y = obs2, dat = fc_sample2)
es_sr <- c(es_sr1, es_sr2)
es_su <- energy_score_multivariate(
observed = c(obs, obs2),
predicted = rbind(fc_sample, fc_sample2),
mv_group_id = c(rep(1, d), rep(2, d))
)
all.equal(es_sr, es_su, tolerance = 1e-6, check.attributes = FALSE)
#> [1] TRUEYou can provide observation weights when computing the Energy Score.
# illustration of observation weights for Energy Score
# example: equal weights for first half of draws; zero weights for other draws
w <- rep(c(1, 0), each = 0.5 * m) / (0.5 * m)
es_sr1 <- scoringRules::es_sample(y = obs, dat = fc_sample, w = w)
es_sr2 <- scoringRules::es_sample(y = obs2, dat = fc_sample2, w = w)
es_sr <- c(es_sr1, es_sr2)
es_su <- energy_score_multivariate(
observed = c(obs, obs2),
predicted = rbind(fc_sample, fc_sample2),
mv_group_id = c(rep(1, d), rep(2, d)),
w = w
)
all.equal(es_sr, es_su, tolerance = 1e-6, check.attributes = FALSE)
#> [1] TRUE