EQRN is a framework for forecasting and extrapolating measures of conditional risk (e.g. of extreme or unprecedented events), including quantiles and exceedance probabilities, using extreme value statistics and flexible neural network architectures. It allows for capturing complex multivariate dependencies, including dependencies between observations, such as sequential (time) dependence. This implementation is based on the methodology introduced in Pasche and Engelke (2024) [doi, pdf, suppl.].
To install EQRN from CRAN, simply run from R:
install.packages("EQRN")Or, to install the development version of EQRN, run:
# install.packages("devtools")
devtools::install_github("opasche/EQRN")When the package is first loaded interactively after installation
(e.g. with library(EQRN) or with any
EQRN::fct()), the necessary backend software from the torch dependency is
automatically installed. Alternatively,
EQRN::install_backend() can be called to perform the
backend installation manually (necessary for non-interactive
environments). For more information about the torch backend and
troubleshooting, visit the torch
installation guide.
Risk assessment for extreme events requires accurate estimation of high quantiles that go beyond the range of historical observations. When the risk depends on the values of observed predictors, regression techniques are used to interpolate in the predictor space. In this package we propose the EQRN model that combines tools from neural networks and extreme value theory into a method capable of extrapolation in the presence of complex predictor dependence. Neural networks can naturally incorporate additional structure in the data. The recurrent version of EQRN is able to capture complex sequential dependence in time series.
In the corresponding article, EQRN is applied to forecasting of flood risk in the Swiss Aare catchment. It exploits information from multiple covariates in space and time to provide one-day-ahead predictions of return levels and exceedances probabilities. This output complements the static return level from a traditional extreme value analysis and the predictions are able to adapt to distributional shifts as experienced in a changing climate. Our model can help authorities to manage flooding more effectively and to minimize their disastrous impacts through early warning systems.
The minimal example below illustrates, in three simple steps, how to use the package functions to fit the EQRN model and predict extreme conditional quantiles and other metrics on new test data. In this example, a toy i.i.d. dataset is used.
scale_fct <- function(x1,x2){ 3 + cos(x1 + x2 + 0.5) }
set.seed(1)
X_train <- matrix(stats::runif(5120), ncol=2)
y_train <- scale_fct(X_train[,1], X_train[,2]) * stats::rt(2560, 4)
X_test <- matrix(stats::runif(2560), ncol=2)This can be achieved with any suitable quantile regression method. We
here use generalised random forests from the grf package, as
they are very easy to use and already quite flexible. One could for
example use a quantile regression neural network instead.
library(grf)
# Choose an intermediate probability level.
interm_lvl <- 0.8
# Fit a GRF for quantile regression with 500 trees (the more the better) on the training set (with a seed for reproducibility).
fit_grf <- grf::quantile_forest(X_train, y_train, num.trees=1000, seed=21)
# Construct out-of-bag intermediate quantiles on the training set.
intermediateq_train <- predict(fit_grf, newdata=NULL, quantiles=c(interm_lvl))$predictionsFit the EQRN network on the training set, with the intermediate quantiles as a varying threshold. Here:
shape_fixed=TRUE removes covariate
dependence from the shape output,net_structure=c(5,5) sets two hidden
layers of 5 neurons each as an architecture,library(EQRN)
fit_eqrn <- EQRN_fit(X_train, y_train, intermediateq_train, interm_lvl,
shape_fixed=TRUE, net_structure=c(5,5), n_epochs=100, seed=42)
#> Epoch: 1 out of 100 , average train loss: 2.371921
#> Epoch: 100 out of 100 , average train loss: 2.281698The arguments values are here arbitrarily chosen for illustration. As for any machine learning approach, hyperparameters should be tuned using set-aside validation data to obtain an accurate fit. Stopping criteria are also available for the number of fitting epochs. Refer to the documentation for a detailed description of the arguments, and to the article’s repository for more advanced examples.
# Desired probability levels at which to predict the conditional quantiles.
levels_predict <- c(0.999, 0.9999)
# Predict intermediate test quantiles using the intermediate model.
intermediateq_test <- predict(fit_grf, newdata=X_test, quantiles=c(interm_lvl))$predictions
# Predict the desired conditional extreme quantiles on the test set.
qpred_eqrn <- EQRN_predict(fit_eqrn, X_test, levels_predict, intermediateq_test)
# Forecast the probability that Y_test would exceed a certain large value.
large_value <- 10
ppred_eqrn <- EQRN_excess_probability(large_value, fit_eqrn, X_test, intermediateq_test)# Print some predictions:
hn <- 10
results <- data.frame(X1=X_test[1:hn,1], X2=X_test[1:hn,2], pred_Y_Q_80=intermediateq_test[1:hn],
pred_Y_Q_99.9=qpred_eqrn[1:hn,1], pred_Y_Q_99.99=qpred_eqrn[1:hn,2], Pr_Y_exceed_10=ppred_eqrn[1:hn])
print(results)
#> X1 X2 pred_Y_Q_80 pred_Y_Q_99.9 pred_Y_Q_99.99 Pr_Y_exceed_10
#> 1 0.5876351 0.83797214 2.763170 15.98885 34.31405 0.004118278
#> 2 0.9493471 0.74616973 2.123687 15.42404 33.85271 0.003459142
#> 3 0.7456916 0.24237508 2.335307 15.40873 33.52297 0.003537242
#> 4 0.2319869 0.70261432 3.041587 16.04881 34.07131 0.004325655
#> 5 0.7706744 0.19874048 4.725495 18.14131 36.72994 0.008187588
#> 6 0.7746018 0.03440777 5.414535 18.87249 37.51951 0.010778085
#> 7 0.7776956 0.33728896 2.796728 15.98395 34.25587 0.004133909
#> 8 0.2586140 0.49574342 3.775091 16.84983 34.96588 0.005546904
#> 9 0.7935616 0.60815766 2.608452 15.86056 34.22237 0.003949164
#> 10 0.1613134 0.52083200 1.770021 14.48766 32.10893 0.002831878Pasche, O. C. and Engelke, S. (2024). “Neural networks for extreme quantile regression with an application to forecasting of flood risk”. Annals of Applied Statistics 18(4), 2818–2839. https://doi.org/10.1214/24-AOAS1907
Published article: DOI:10.1214/24-AOAS1907
(PDF,
Supplement).
Article’s usage examples: https://github.com/opasche/EQRN_Results
Preprint (2022): ArXiv:2208.07590 (PDF).
Package created by Olivier C. PASCHE
Research Center for Statistics, University of Geneva (CH), 2022.
Supported by the Swiss National Science Foundation.