A nuanced quantile random forest approach for fast prediction of a stochastic marine flooding simulator applied to a macrotidal coastal site

Abstract

Integrating full-process high resolution hydrodynamic simulations within early warning system (EWS) for marine flooding is hindered by the large computation time cost of such numerical models. This problem can be alleviated through the statistical analysis of pre-calculated simulation results to build a fast (low computation time cost) statistical predictive model (named metamodel). Despite the success of this approach, a direct application of such techniques for EWS is not straightforward in all settings, more particularly in environments where the stochastic character of waves has a significant effect on the induced flood, i.e., where overtopping is on a duration smaller than 500 times the offshore wave period. In such environments, the numerical simulator is not deterministic and provides statistical quantities of the flooding indicators. By focusing on the estimates of quantiles, the objective of the present study is to explore the applicability of random forest (RF) models for marine flooding prediction by providing two levels of information: (1) the quantile of interest via a quantile random forest regression model (qRF); (2) the flooding probability via a classification random forest (cRF). We use the macrotidal site of Gâvres (French Atlantic coast) as an application case for which ~ 2000 numerical simulations were performed (i.e. stochastic simulations given 100 different extreme-but-realistic offshore meteo-oceanic input conditions were repeated 20 times) to compute local and global flooding indicators (respectively the maximum water depth at the coast and the total volume of water entering the territory). Through an extensive repeated cross-validation procedure, we tune the qRF parameters leading to high coefficient of determination of ~ 90% for the quantiles at 25–50–75%, and we show that the qRF models outperform the commonly used Tobit regression model. The comparison with the numerical results on historical events shows very satisfactory prediction for events both leading to major flooding and to absence of impact. For low quantile level and minor-to-moderate flooding events, the second level provided by the cRF-derived flooding probability shows its added value by enabling the EWS user to nuance the qRF prediction and to tag some situations where the prediction remains unsure.

Appendices

Appendix A: Sources of data for the coastal flooding application case

See Table 4.

Table 4 Original datasets for the database of offshore meteo-oceanic conditions (after Idier et al. 2020)

Appendix B: Extreme value analysis

The procedure holds as follows. First, two classes of offshore conditions are considered: ‘amplitude’ variables X = (tswl, hs, u), which can take very large values, and covariates X_c = (tp, dp, du), which are dependent on the values of the ‘amplitude’ variables. Considering ‘amplitude’ variables, a multivariate extreme value analysis is conducted to extrapolate their joint probability density to extreme values by taking into account the dependence structure.

This step holds as follows:

• The marginals of each ‘amplitude’ variable are first analysed and modelled by the combination of the empirical distribution of the X_i values, below a suitable high threshold, with the Generalised Pareto distribution (GPD), above the threshold (Coles and Tawn 1991). The threshold is selected by a combination of methods, namely visual inspection of quantile–quantile graphs, “mean residual life plots”, “modified scale and shape parameters plots” (Coles et al. 2001);

• The dependence structure of the ‘amplitude’ variables (transformed into common standard Gumbel margins) is modelled by following the approach by Heffernan and Tawn (2004) as described below.

• Once fitted, a Monte Carlo simulation procedure is used to randomly generate realizations of the ‘amplitude’ variables.

• Based on the generated dataset, the covariates are generated by using the empirical distribution conditional on the values of the ‘amplitude’ variables.

Consider X_i=1,2,3 the different ‘amplitude’ variables (tswl, hs, u). The variables X_i are first transformed into common standard Gumbel margins. Let use denote Y_-i the vector of all transformed variables except for the ith variable Y_i, when Y_i is large. A non-linear regression model is set up as follows:

$${\varvec{Y}}_{{ - {i}}} |Y_{i} = {\mathbf{a}} \cdot Y_{i} + { }Y_{i}^{{\text{b}}} \cdot {\varvec{W}}$$

(6)

with $Y_{i} > \upsilon$ (i.e. Y_i having large values), a and b are parameters vectors (one value per parameter for each pair of variables), ν a threshold to be defined and W a vector of residuals. The model is adjusted using the maximum likelihood method based on the assumption that the residuals W are Gaussian with a mean and variance to be calculated. For our case study, the threshold selected for ν in Eq. (6) was set up at 0.85 (expressed as a probability of non-exceedance) using the diagnostic tools described in Heffernan and Tawn (2004).

Appendix C: Historical events

Table 5 presents the offshore meteo-oceanic conditions associated to 21 historical events that were selected with respect to past damage data via historical data, past hydro-meteorological data and numerical hydrodynamic simulations (Idier et al. 2020).

Table 5 Description of the historical events used for validation; the three quantiles of interest for the local (H_max) and global (Volume) indicators are provided in the following order: at 25, 50 and 75%

Appendix D: Censored regression (Tobit)

The Tobit model (Tobin 1958) is based on the concept of censoring, which means that that the response y itself is limited to a certain threshold (or a range), and cannot be observed outside these limits. We assume that there is a latent unobservable process driving the response, which can be described by suitable covariates. In the present study, we focus on zero left-censored normal distribution that is defined as follows:

$$y = \max \left( {0,y^{*} } \right), y^{*} \sim N\left( {\mu ,\sigma } \right)$$

(7)

where y^* is the unobservable ‘latent’ response that is assumed to follow a normal distribution (denoted N), given the parameters location μ and scale σ. The ‘observable’ response y is simply the maximum of the latent response and the censoring point.

The density probability function $f_{{{\text{cens}}}}$ of y is:

$$f_{{{\text{cens}}}} \left( {y{|}\mu ,\sigma ,0} \right) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{if}}\;\;y < 0} \hfill \\ {F{(}y{|}\mu ,\sigma ) } \hfill & {{\text{if }}\;\;{\text{y}} = 0} \hfill \\ {f{(}y{|}\mu ,\sigma )} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$

(8)

where f and F are respectively the probability density function and the cumulative density function of the Normal distribution.

The parameters location μ and scale σ are assumed to follow a regression model as follows:

$$\mu = {\mathbf{x}}^{{\text{T}}} \cdot {\varvec{\beta}},\user2{ }\quad {\log}\left( {\upsigma } \right) = {\mathbf{x}}^{{\text{T}}} \cdot {\varvec{\gamma}}$$

(9)

where x are the covariates (the offshore meteo-oceanic conditions in the case of our study), and $\beta ,\gamma$ are the regression coefficients that can be estimated by maximizing the sum over the dataset of the log-likelihood function log($f_{{{\text{cens}}}} \left( {y{|}\mu ,\sigma ,0} \right)$).