postforecasts

The goal of the postforecasts package is to unite a variety of post-processing methods into a single easy-to-use interface.

This repository was created in context of the Practical Statistical Training course offered by the Chair of Statistics at the University of Göttingen during Winter Term 2021/22. If you want to jump straight to the results of our project, please refer to the final course paper term_paper/Herp_Beck_Post-Processing.pdf which explains the theory as well as the empirical results of all implemented post-processing methods in great detail.

Installation

You can install the development version of postforecasts from GitHub with:

# install.packages("devtools")
devtools::install_github("nikosbosse/post-processing-forecasts")

Data

The postforecasts package is tailored towards post-processing Covid-19 forecasts which are provided by two separate data sets:

UK Covid-19 Crowd Forecasting Challenge

As part of an ongoing research project by the EpiForecasts group at the London School of Hygiene & Tropical Medicine, the UK Covid-19 Crowd Forecasting Challenge consisted of submitting weekly predictions of Covid-19 Cases and Deaths in the United Kingdom in 2021. The challenge was not restricted to experienced researchers in the field but rather intended to collect quantile predictions for the upcoming four weeks by non-expert individuals.

One of the main motivations was to gather evidence for or against the hypothesis that humans are highly capable of precise point forecasts. Yet, at the same time, they tend to be too confident in their beliefs such that prediction intervals are chosen too narrow. In fact, this tendency presents one motivation for post-processing: Extract valuable information from point forecasts and adjust the corresponding prediction intervals with a systematic correction procedure.

In case of individuals that are unfamiliar with statistical methodology, specifying forecasts for very specific quantiles of the predictive distribution might lead to inconsistencies. Therefore all participants could determine an uncertainty parameter around their median prediction via an interactive web application such that all quantile predictions could be concluded in an automatic fashion. Note that this procedure inherently leads to symmetric forecast intervals. The results of the 12-week challenge are publicly available.

European Covid-19 Forecast Hub

According to their webpage the European Covid-19 Forecast Hub collects “short-term forecasts of Covid-19 cases and deaths across Europe, created by a multitude of infectious disease modelling teams”.

In contrast to the compact UK data, the European Forecast Hub data contains almost two million observations for over 20 European countries. Further, the forecasters are knowledgeable research groups that submit their weekly predictions based on statistical models. Although the data collection continues in regular frequency up to this day, our data set is limited to a 32-week span from March 2021 until October 2021.

---

The overall structure of the two data sets above is very similar. To get a better understanding of the data the most important columns for our analysis are briefly explained here:

• location: The country for which the forecasts were submitted. Equals GB for the UK data. Our analysis for the European Forecast Hub data selects 18 different European countries.

• model: The forecaster (group). Mostly (non-expert) individuals for the UK data and international research groups for the European Forecast Hub.

• target_type: Either Covid-19 Cases or Covid-19 Deaths.

• horizon: The time horizon how far in advance the predictions were submitted. Ranges from 1 week-ahead to 4 weeks-ahead.

• forecast_date: The exact date when the forecasts were submitted.

• target_end_date: The exact date for which the forecasts were submitted.

• quantile: One of 23 different quantile values ranging from 0.01 to 0.99.

• prediction: The predicted value for one specific combination of the variables above.

• true_value: The actual, observed number of Covid-19 Cases or Deaths. This value is repeated 23 times, once for each quantile value.

Weighted Interval Score

In order to quantify if the post-processed prediction intervals improve the original forecasts we chose the Weighted Interval Score (WIS) as our evaluation metric. The WIS is a so-called Proper Scoring Rule: It incentivizes the forecaster to state their true best belief and cannot be manipulated in favour of own interests. It combines measures for interval sharpness as well as overprediction and underprediction and can thus be understood as a trade-off between interval coverage and precision. A lower WIS score indicates a better overall performance.

For a rigorous mathematical introduction of the WIS see Section 2.1.3 of our course paper.

Main Findings

The postforecasts package implements three versions for each of two post-processing frameworks: Conformalized Quantile Regression (CQR) and Quantile Spread Averaging (QSA) as well an as Ensemble method, where each quantile prediction is a convex combination of the individual methods, i.e. a linear combination where all weights are contained in the unit interval and sum up to one.

This section briefly presents the main findings of comparing all post-processing methods in terms of the Weighted Interval Score for the UK Covid-19 Crowd Forecasting Challenge data set. Note that only five of the six building block methods are included in the comparison since the multiplicative CQR version could not show competitive results early on and is thus omitted.

The data set which contains original and updated forecasts for all post-processing methods for the UK data can be conveniently loaded with the command

library(postforecasts)
uk_results <- readRDS("data_results/uk_complete.rds")

The required computations for obtaining these results can be reproduced with the following commands but may take a lot of computation time:

uk_data <- readRDS("data_modified/uk_data_incidences.csv")

uk_results <- uk_data |>
  update_predictions(
    methods = c(
      "cqr", "cqr_asymmetric", "qsa_uniform", "qsa_flexible", "qsa_flexible_symmetric"
    ),
    cv_init_training = 0.5
  ) |>
  collect_predictions() |>
  add_ensemble()

The following figure provides a visual illustration of original and adjusted prediction intervals of all post-processing methods including the ensemble for one specific combination of the variables model, target_type, horizon and quantile. The color legend displays the ensemble weights for each method. In this case only the asymmetric CQR and the flexible (not symmetric) QSA methods contribute to the ensemble. As a simple weighted average with weights close to 0.5 the lower (upper) bounds of the ensemble intervals are approximately halfway between the lower (upper) bounds of the cqr_asymmetric and qsa_flexible intervals:

Except for the last observations on the horizontal axis the forecasts of the two CQR versions are quite similar and significantly closer to the original predictions than the QSA intervals. Within the QSA family qsa_flexible and qsa_flexible_symmetric produce almost identical corrections whereas qsa_uniform behaves quite differently from all other methods and consistently causes the largest intervals.

The previous figure shows that different methods can have significantly different effects, yet it does not provide any hints which method improves the Weighted Interval Score most. Thus, the following table collects the WIS for each method on the training and validation set, aggregated over all models, target types, horizons and quantiles and sorted by increasing validation score:

method	validation score	training score	dispersion
ensemble	57.69	18.22	21.73
qsa_uniform	60.00	20.88	26.84
qsa_flexible	60.47	19.48	25.31
qsa_flexible_symmetric	60.92	20.49	33.22
cqr	62.15	20.82	24.10
cqr_asymmetric	63.97	14.46	17.99
original	65.74	23.62	12.00

Based on the table above, our comparison of the postforecasts post-processing methods yields a couple of interesting findings:

• All six custom methods improve out-of-sample performance compared to the original predictions on the UK data set.

• All three QSA versions lead to lower validation scores than any CQR variant. Thus, based on this first impression, the familiy of QSA post-processing methods clearly outperforms the CQR algorithm for the UK data.

• The ensemble model is the clear winner: Combining information from multiple QSA and CQR methods works better on new data than any individual method on its own. This suggests that the five building block methods are not redundant in the sense that they have different strengths and weaknesses depending on the location in feature space.

• The asymmetric CQR method suffers most from overfitting as it results in the lowest training but highest validation score for the small UK data set.

• In general, additional design restrictions such as identical weights in case of qsa_uniform and/or the symmetry assumption in case of cqr and qsa_flexible_symmetric have some kind of regularization effect which leads to better generalization to the validation set. Indeed, the least flexible versions of both method frameworks indicate the best validation performance and yet, unsurprisingly, the worst training score.

• All methods improve the original forecasts by expanding the prediction intervals which is indicated by the larger dispersion values. qsa_flexible_symmetric produces by far the widest intervals on average, yet we can not observe a correlation of better validation scores and either narrower or wider prediction intervals.

User Guide

This section provides an overview of the most important postforecasts functions as as a compact guide how to use our package effectively.

The postforecasts functions can be grouped into three categories:

1. Exploratory

   The plot_quantiles(), plot_intervals() and plot_intervals_grid() functions visualize the development of true Covid-19 Cases and Deaths over time as well as corresponding original and adjusted quantile predictions.

2. Model Fitting

   The update_predictions() function is the workhorse of the entire postforecasts package. It specifies both the raw data and the post-processing method(s) that should be applied to the data set. The function returns a list of k + 1 equally shaped data frames for k selected post-processing methods where the first element is given by the original, possibly filtered, data frame.

   All list elements can be analyzed separately or collectively by stacking them into one large data frame with the collect_predictions() function. The combined data frame is designed to work well with analysis functions of the scoringutils package. If multiple post-processing methods are applied, an ensemble model of all selected methods can be added via the add_ensemble() function, which lets the user access both the weighted ensemble predictions and a data frame with the corresponding weights.

3. Evaluation

   As noted above the Weighted Interval Score is our primary metric to evaluate the quality of prediction intervals. The score() function of the scoringutils package computes this quantity for each observation in the data set which can then be aggregated by the related summarise_scores() function.

   Depending on the granularity of the aggregation the output might contain many interval scores of vastly different magnitudes. To simplify interpretation the eval_methods() function computes relative or percentage changes in the Weighted Interval Score for each selected method compared to the original quantile predictions. Further, these relative changes can be visualized by the plot_eval() function.

To illustrate the application of the functions above to a practical example we use the Covid-19 data for Germany in 2021 that is provided by the European Forecast Hub.

hub_1 <- readr::read_csv(here::here("data_modified", "hub_data_1_incidences.csv"))
hub_2 <- readr::read_csv(here::here("data_modified", "hub_data_2_incidences.csv"))
hub_3 <- readr::read_csv(here::here("data_modified", "hub_data_3_incidences.csv"))

hub <- dplyr::bind_rows(hub_1, hub_2, hub_3)

hub_germany <- hub |>
  dplyr::filter(location == "DE") |>
  dplyr::distinct(
    model, target_end_date, target_type, quantile, horizon,
    .keep_all = TRUE
  )

We start with a visual overview of the original 5%, 20% 80% and 95% quantile predictions during the summer months of 2021 in Germany that were submitted by the EuroCOVIDhub-ensemble forecasting model:

plot_quantiles(
  hub_germany,
  model = "EuroCOVIDhub-ensemble", quantiles = c(0.05, 0.2, 0.8, 0.95)
)

The original predictions look quite noisy overall with the clear trend that uncertainty and, hence, the interval width increases with growing forecast horizons. We want to analyze if one particular post-processing method, Conformalized Quantile Regression, improves the predictive performance for this model on a validation set by computing the Weighted Interval Scores for Covid-19 Cases and Covid-19 Deaths separately:

df_updated <- update_predictions(
  hub_germany,
  methods = "cqr", models = "EuroCOVIDhub-ensemble", cv_init_training = 0.5
)
df_combined <- collect_predictions(df_updated)

df_combined |>
  extract_validation_set() |>
  scoringutils::score() |>
  scoringutils::summarise_scores(by = c("method", "target_type"))

method	target type	interval score	dispersion
cqr	Cases	13.40	5.07
original	Cases	13.78	3.81
cqr	Deaths	0.06	0.01
original	Deaths	0.05	0.03

We can observe that CQR improved the WIS for Covid-19 Cases, whereas the predictive performance for Covid-19 Deaths dropped slightly.

The update_predictions() and collect_predictions() combination immediately generalize to multiple post-processing methods. The only syntax change is a vector input of strings for the methods argument instead of a single string. Hence, if not desired, the user does not have to learn separate interfaces for each method nor be familar with the precise implementation. This design allows for maximum syntactic consistency by masking the internal functionality.

Further, the update_predictions() function automatically takes care of quantile crossing by reordering the output predictions in increasing quantile order. The cv_init_training parameter specifies the fraction of observations that is used for the pure training set before the Time Series Cross Validation process starts.

As seen in the previous table CQR increases the dispersion of the predictions for Cases significantly. We can visualize one example of these wider intervals with the plot_intervals() function:

plot_intervals(df_combined, target_type = "Cases", horizon = 2, quantile = 0.05)

Indeed, the 2 weeks-ahead 90% prediction intervals for Covid Cases in Germany are widened by CQR. The solid black line represents the true Case incidences whereas the grey dashed line indicates the end of the training set within the Cross Validation as specified by the cv_init_training parameter.

Recall that prediction uncertainty increases with larger forecast horizons. Similarly, CQR corrections also increase in size for forecasts that are submitted further in advance, which can be seen in the next plot along the horizontal dimension. Interestingly, CQR expands the intervals only for Cases whereas the forecasts for Deaths are narrowed!

plot_intervals_grid(df_combined, facet_by = "horizon", quantiles = 0.05)

Besides the target type (Cases or Deaths), it is also useful to compare CQR effects across forecast horizons or quantiles. Quite intuitively, CQR generally has a stronger relative benefit for large time horizons and extreme quantiles, where the original forecaster faced a greater uncertainty. The figure below illustrates how, in special cases like this one, the effect on the validation set can show rather mixed trends due to disadvantageous adjustments for the two and three weeks-ahead 98% prediction intervals:

df_eval <- eval_methods(df_combined, summarise_by = c("quantile", "horizon"))
plot_eval(df_eval)