Evaluate forecasts — score • scoringutils

This function allows automatic scoring of forecasts using a range of metrics. For most users it will be the workhorse for scoring forecasts as it wraps the lower level functions package functions. However, these functions are also available if you wish to make use of them independently.

A range of forecasts formats are supported, including quantile-based, sample-based, binary forecasts. Prior to scoring, users may wish to make use of check_forecasts() to ensure that the input data is in a supported format though this will also be run internally by score(). Examples for each format are also provided (see the documentation for data below or in check_forecasts()).

Each format has a set of required columns (see below). Additional columns may be present to indicate a grouping of forecasts. For example, we could have forecasts made by different models in various locations at different time points, each for several weeks into the future. It is important, that there are only columns present which are relevant in order to group forecasts. A combination of different columns should uniquely define the unit of a single forecast, meaning that a single forecast is defined by the values in the other columns. Adding additional unrelated columns may alter results.

To obtain a quick overview of the currently supported evaluation metrics, have a look at the metrics data included in the package. The column metrics$Name gives an overview of all available metric names that can be computed. If interested in an unsupported metric please open a feature request or consider contributing a pull request.

For additional help and examples, check out the Getting Started Vignette as well as the paper Evaluating Forecasts with scoringutils in R.

Usage

score(data, metrics = NULL, ...)

Arguments

data

A data.frame or data.table with the predictions and observations. For scoring using score(), the following columns need to be present:

true_value - the true observed values
prediction - predictions or predictive samples for one true value. (You only don't need to provide a prediction column if you want to score quantile forecasts in a wide range format.)

For scoring integer and continuous forecasts a sample column is needed:

sample - an index to identify the predictive samples in the prediction column generated by one model for one true value. Only necessary for continuous and integer forecasts, not for binary predictions.

For scoring predictions in a quantile-format forecast you should provide a column called quantile:

quantile: quantile to which the prediction corresponds

In addition a model column is suggested and if not present this will be flagged and added to the input data with all forecasts assigned as an "unspecified model").

You can check the format of your data using check_forecasts() and there are examples for each format (example_quantile, example_continuous, example_integer, and example_binary).

metrics

the metrics you want to have in the output. If NULL (the default), all available metrics will be computed. For a list of available metrics see available_metrics(), or check the metrics data set.

...

additional parameters passed down to score_quantile() (internal function used for scoring forecasts in a quantile-based format).

Value

A data.table with unsummarised scores. There will be one score per quantile or sample, which is usually not desired, so you should almost always run summarise_scores() on the unsummarised scores.

References

Funk S, Camacho A, Kucharski AJ, Lowe R, Eggo RM, Edmunds WJ (2019) Assessing the performance of real-time epidemic forecasts: A case study of Ebola in the Western Area region of Sierra Leone, 2014-15. PLoS Comput Biol 15(2): e1006785. doi:10.1371/journal.pcbi.1006785

Author

Nikos Bosse nikosbosse@gmail.com

Examples

library(magrittr) # pipe operator
# \dontshow{
  data.table::setDTthreads(2) # restricts number of cores used on CRAN
# }

check_forecasts(example_quantile)
#> The following messages were produced when checking inputs:
#> 1.  144 values for `prediction` are NA in the data provided and the corresponding rows were removed. This may indicate a problem if unexpected.
#> Your forecasts seem to be for a target of the following type:
#> $target_type
#> [1] "integer"
#> 
#> and in the following format:
#> $prediction_type
#> [1] "quantile"
#> 
#> The unit of a single forecast is defined by:
#> $forecast_unit
#> [1] "location"        "target_end_date" "target_type"     "location_name"  
#> [5] "forecast_date"   "model"           "horizon"        
#> 
#> Cleaned data, rows with NA values in prediction or true_value removed:
#> $cleaned_data
#>        location target_end_date target_type true_value location_name
#>     1:       DE      2021-05-08       Cases     106987       Germany
#>     2:       DE      2021-05-08       Cases     106987       Germany
#>     3:       DE      2021-05-08       Cases     106987       Germany
#>     4:       DE      2021-05-08       Cases     106987       Germany
#>     5:       DE      2021-05-08       Cases     106987       Germany
#>    ---                                                              
#> 20397:       IT      2021-07-24      Deaths         78         Italy
#> 20398:       IT      2021-07-24      Deaths         78         Italy
#> 20399:       IT      2021-07-24      Deaths         78         Italy
#> 20400:       IT      2021-07-24      Deaths         78         Italy
#> 20401:       IT      2021-07-24      Deaths         78         Italy
#>        forecast_date quantile prediction                 model horizon
#>     1:    2021-05-03    0.010      82466 EuroCOVIDhub-ensemble       1
#>     2:    2021-05-03    0.025      86669 EuroCOVIDhub-ensemble       1
#>     3:    2021-05-03    0.050      90285 EuroCOVIDhub-ensemble       1
#>     4:    2021-05-03    0.100      95341 EuroCOVIDhub-ensemble       1
#>     5:    2021-05-03    0.150      99171 EuroCOVIDhub-ensemble       1
#>    ---                                                                
#> 20397:    2021-07-12    0.850        352  epiforecasts-EpiNow2       2
#> 20398:    2021-07-12    0.900        397  epiforecasts-EpiNow2       2
#> 20399:    2021-07-12    0.950        499  epiforecasts-EpiNow2       2
#> 20400:    2021-07-12    0.975        611  epiforecasts-EpiNow2       2
#> 20401:    2021-07-12    0.990        719  epiforecasts-EpiNow2       2
#> 
#> Number of unique values per column per model:
#> $unique_values
#>                    model location target_end_date target_type true_value
#> 1: EuroCOVIDhub-ensemble        4              12           2         96
#> 2: EuroCOVIDhub-baseline        4              12           2         96
#> 3:  epiforecasts-EpiNow2        4              12           2         95
#> 4:       UMass-MechBayes        4              12           1         48
#>    location_name forecast_date quantile prediction horizon
#> 1:             4            11       23       3969       3
#> 2:             4            11       23       3733       3
#> 3:             4            11       23       3903       3
#> 4:             4            11       23       1058       3
#> 
#> $messages
#> [1] "144 values for `prediction` are NA in the data provided and the corresponding rows were removed. This may indicate a problem if unexpected."
#> 
score(example_quantile) %>%
  add_coverage(by = c("model", "target_type")) %>%
  summarise_scores(by = c("model", "target_type"))
#> The following messages were produced when checking inputs:
#> 1.  144 values for `prediction` are NA in the data provided and the corresponding rows were removed. This may indicate a problem if unexpected.
#>                    model target_type interval_score dispersion underprediction
#> 1: EuroCOVIDhub-baseline       Cases    28483.57465 4102.50094    10284.972826
#> 2: EuroCOVIDhub-baseline      Deaths      159.40387   91.40625        2.098505
#> 3: EuroCOVIDhub-ensemble       Cases    17943.82383 3663.52458     4237.177310
#> 4: EuroCOVIDhub-ensemble      Deaths       41.42249   30.18099        4.103261
#> 5:       UMass-MechBayes      Deaths       52.65195   26.87239       16.800951
#> 6:  epiforecasts-EpiNow2       Cases    20831.55662 5664.37795     3260.355639
#> 7:  epiforecasts-EpiNow2      Deaths       66.64282   31.85692       15.893314
#>    overprediction coverage_deviation        bias   ae_median coverage_50
#> 1:   14096.100883        -0.11211957  0.09796875 38473.60156   0.3281250
#> 2:      65.899117         0.11614130  0.33906250   233.25781   0.6640625
#> 3:   10043.121943        -0.09785326 -0.05640625 24101.07031   0.3906250
#> 4:       7.138247         0.19528533  0.07265625    53.13281   0.8750000
#> 5:       8.978601        -0.02312500 -0.02234375    78.47656   0.4609375
#> 6:   11906.823030        -0.06660326 -0.07890625 27923.81250   0.4687500
#> 7:      18.892583        -0.04287176 -0.00512605   104.74790   0.4201681
#>    coverage_90
#> 1:   0.8203125
#> 2:   1.0000000
#> 3:   0.8046875
#> 4:   1.0000000
#> 5:   0.8750000
#> 6:   0.7890625
#> 7:   0.9075630

# set forecast unit manually (to avoid issues with scoringutils trying to
# determine the forecast unit automatically), check forecasts before scoring
example_quantile %>%
  set_forecast_unit(
    c("location", "target_end_date", "target_type", "horizon", "model")
  ) %>%
  check_forecasts() %>%
  score()
#> The following messages were produced when checking inputs:
#> 1.  144 values for `prediction` are NA in the data provided and the corresponding rows were removed. This may indicate a problem if unexpected.
#>        location target_end_date target_type horizon                 model range
#>     1:       DE      2021-05-08       Cases       1 EuroCOVIDhub-baseline     0
#>     2:       DE      2021-05-08       Cases       1 EuroCOVIDhub-baseline    10
#>     3:       DE      2021-05-08       Cases       1 EuroCOVIDhub-baseline    10
#>     4:       DE      2021-05-08       Cases       1 EuroCOVIDhub-baseline    20
#>     5:       DE      2021-05-08       Cases       1 EuroCOVIDhub-baseline    20
#>    ---                                                                         
#> 20397:       IT      2021-07-24      Deaths       3  epiforecasts-EpiNow2    90
#> 20398:       IT      2021-07-24      Deaths       3  epiforecasts-EpiNow2    95
#> 20399:       IT      2021-07-24      Deaths       3  epiforecasts-EpiNow2    95
#> 20400:       IT      2021-07-24      Deaths       3  epiforecasts-EpiNow2    98
#> 20401:       IT      2021-07-24      Deaths       3  epiforecasts-EpiNow2    98
#>        interval_score dispersion underprediction overprediction coverage
#>     1:       25620.00       0.00               0          25620        0
#>     2:       25599.50     184.50               0          25415        0
#>     3:       25599.50     184.50               0          25415        0
#>     4:       25481.00     556.00               0          24925        0
#>     5:       25481.00     556.00               0          24925        0
#>    ---                                                                  
#> 20397:          15.85      15.85               0              0        1
#> 20398:          11.75      11.75               0              0        1
#> 20399:          11.75      11.75               0              0        1
#> 20400:           6.52       6.52               0              0        1
#> 20401:           6.52       6.52               0              0        1
#>        coverage_deviation bias quantile ae_median quantile_coverage
#>     1:               0.00 0.95    0.500     25620              TRUE
#>     2:              -0.10 0.95    0.450     25620              TRUE
#>     3:              -0.10 0.95    0.550     25620              TRUE
#>     4:              -0.20 0.95    0.400     25620              TRUE
#>     5:              -0.20 0.95    0.600     25620              TRUE
#>    ---                                                             
#> 20397:               0.10 0.50    0.950        26              TRUE
#> 20398:               0.05 0.50    0.025        26             FALSE
#> 20399:               0.05 0.50    0.975        26              TRUE
#> 20400:               0.02 0.50    0.010        26             FALSE
#> 20401:               0.02 0.50    0.990        26              TRUE

# forecast formats with different metrics
if (FALSE) {
score(example_binary)
score(example_quantile)
score(example_integer)
score(example_continuous)
}

# score point forecasts (marked by 'NA' in the quantile column)
score(example_point) %>%
  summarise_scores(by = "model", na.rm = TRUE)
#> The following messages were produced when checking inputs:
#> 1.  144 values for `prediction` are NA in the data provided and the corresponding rows were removed. This may indicate a problem if unexpected.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
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#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#> Median not available, computing as mean of two innermost quantiles in order to compute bias.
#>                    model interval_score dispersion underprediction
#> 1: EuroCOVIDhub-baseline     14092.7647 2192.26967      4996.92898
#> 2: EuroCOVIDhub-ensemble      8852.4196 1930.80064      2005.56357
#> 3:  epiforecasts-EpiNow2     10659.5125 3084.85850      1604.96135
#> 4:       UMass-MechBayes        51.4781   28.09387        15.32315
#>    overprediction coverage_deviation        bias    ae_point     se_point
#> 1:     6903.56605        0.002102273  0.21851562 19353.42969 2.883446e+09
#> 2:     4916.05540        0.050752841  0.00812500 12077.10156 1.945118e+09
#> 3:     5969.69268       -0.057861612 -0.04376518 14521.10526 2.680928e+09
#> 4:        8.06108       -0.024886364 -0.02312500    78.47656 1.170976e+04
#>    ae_median
#> 1:       NaN
#> 2:       NaN
#> 3:       NaN
#> 4:       NaN