Toward the probabilistic forecasting of cyclone-induced marine flooding by overtopping at Reunion Island aided by a time-varying random-forest classification approach

Abstract

In 2017, Irma and Maria highlighted the vulnerability of small islands to cyclonic events and the necessity of advancing the forecast techniques for cyclone-induced marine flooding. In this context, this paper presents a generic approach to deriving time-varying inundation forecasts from ensemble track and intensity forecasts applied to the case of Reunion Island in the Indian Ocean. The challenge for volcanic islands is to account for the full complexity of wave overtopping processes while also ensuring a robustness and timeliness that are compatible with emergency requirements. The challenge is addressed by following a hybrid approach relying on the combination of process-based models with a statistical model (herein, a random-forest classifier) trained with a precalculated database. The latter enables one to translate any time series of coastal marine conditions into the time-varying probability of inundation for different sectors. The application detailed for the case of Cyclone Dumile at Sainte-Suzanne city shows that the proposed approach enables quick discrimination, in both space and time, thereby identifying safe and exposed areas and demonstrating that probabilistic forecasting of marine flooding by overtopping is feasible. The whole method can be easily adapted to other territories and scales provided that validated process-based models are available. Beyond early warning applications, the developed database and statistical models may also be used for informing risk prevention and adaptation strategies.

Appendix: Random forest method

Based on the set of n training data D = (X_i; y_i)_i=1,…,n, the objective of setting up a classification model (classifier) is to identify the rules for assigning a new “yet-unseen” input scenario into class 1 or 0 without resorting to a long-running numerical simulation. A basic classification model can rely on the concept of a decision tree (Breiman et al. 1984), which is recursively built using binary partitioning of the domain space into regions that are increasingly homogeneous with respect to the class variable. As an illustration, Fig. 

Fig. 10

Schematic overview of a tree model for binary classification (0 and 1). X_1–4 correspond to the input variables; C_1–6 correspond to the cut-off values for partitioning

10 schematically depicts a tree model for binary classification (class label 0 or 1). The homogeneous regions correspond to the nodes. The splitting process aims to subdivide one parent node into two subnodes. Different splitting rules exist, the most popular of which relies on the maximization of node homogeneity, which is measured by the Gini impurity index denoted G (Breiman et al. 1984) defined as follows:

$$G = 1 - p_{1}^{2} - p_{0}^{2}$$

(3)

where $p_{k} = \frac{{n_{k} }}{n}$ is the fraction of the n_k samples from class k = {0,1}. The decrease of G is calculated as G_parent − G_split1 − G_split2 after splitting the parent node into two subnodes that depends on the cut-off value C. At each step of the construction, the algorithm selects the cut-off value C that results in the largest decrease of G. This process is performed until subdivision no longer decreases the Gini index, or until a minimum node size is reached (typically 1 for classification).

The random forest (RF) builds on the same principles as classification decision tree models but extends them by adding a random character to the construction process at the following two levels: (1) each tree is constructed using a different bootstrap sample of the data (Breiman 2001), which enables overcoming the instability issue of classical decision tree (i.e., the tree structure and resulting predicted classes might not be stable when new samples are provided); (2) each node is split using the best among a subset of input parameters randomly chosen at that node, which enables handling situations which have highly correlated input parameters (Cutler et al. 2007). Figure 

Fig. 11

Schematic overview of the algorithm for RF setup (0 and 1). The original training dataset D = {X_i; y_i} is randomly split into n_tree bootstrap samples, which are used to set up the subsequent classification tree model. The out-of-bag samples D^C are used to measure the prediction error Err. The ensemble of classification tree models constitutes the RF model (i.e., the aggregation of the different tree models)

11 schematically depicts the principles of the RF algorithm, as follows:

• Step 1: Select n_tree bootstrap samples D_j (using random sampling with replacement) from the original training data D = {X_i; y_i}_i=1…n;

• Step 2: For each of the bootstrap samples D_j, grow a classification tree by splitting until all the observations in each terminal node come from the same class (“pure”). The split can rely, for instance, on the Gini index (an original formulation of Breiman 2001) or can be fully random (Geurts et al. 2006). At each node, rather than choosing the best split among all predictors (input parameters), choose it among m_try randomly selected predictors. For each tree j, the classification error is measured by the misclassification rate denoted Err_j as follows:

  $${\text{Err}}_{j} = \frac{1}{{\left| {{\varvec{D}}_{j}^{C} } \right|}}\sum\limits_{{i \in {\varvec{D}}_{j}^{C} }}^{{}} {I(y_{i} \ne \hat{y}_{i} )}$$

  (4)

  where D^C is the set of samples which are not used for the construction of the considered classification decision tree; this set constitutes the out-of-bag (OOB) samples so that ${\mathbf{D}}_{j}^{C} \cup {\mathbf{D}}_{j} = {\mathbf{D}};\,|{\varvec{D}}_{j}^{C} |$ is the size of the jth OOB set; y_i is the ith class indicator within the jth OOB set; $\hat{y}_{i}$ is the corresponding class prediction using the jth classification tree;

• Step 3: Predict new data by aggregating the predictions of the n_tree trees. The class of the “yet-unseen” input parameters is estimated as the one with the maximum of occurrence among the different trees (votes). A class probability (probability of belonging to class 1 or 0 otherwise) can be estimated either by relying on a voting-based approach (by counting the proportion of the considered class estimated using the n_tree trees) or by using a regression-based RF model as described by Malley et al. (2012).