2.1 FRamework for Index-based Drought Analysis

Figure 2FRamework for Index-based Drought Analysis (FRIDA): 1. Identification of basin characteristics, 2. feature extraction, and 3. drought index modeling.

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FRIDA designs drought indexes in three steps as reported in Fig. 2.

The identification of basin characteristics is a preliminary empirical process, which consists of the selection of a target variable and the collection of candidate predictors. The target variable is an appropriately chosen water deficit, representative of the actual drought conditions in the basin (e.g., water supply deficit, soil moisture deficit). The dataset of predictors contains the candidate features to reproduce the target variable and consists of observed hydro-meteorological variables and composite drought indicators over different spatiotemporal scales.

Target variable and candidate predictors constitute the input to the “feature extraction” step, the second building block of the framework. This block employs an input variable selection (IVS) algorithm that explores the space of candidate predictors to select Pareto-efficient subsets of predictors with respect to multiple assessment metrics. Most commonly, these metrics quantify the subset accuracy in reproducing the target and the parsimony (i.e., the cardinality of the subset), crucial characteristics for an operational index expected to balance precision and ease of use. In some cases, relevance and redundancy can also be considered in order to explore the input space more effectively. In particular, the metric of relevance favors highly informative subsets (i.e., constituted by predictors that are highly correlated with the target), while the redundancy metric ensures low intra-subset similarity. The objectives of relevance and redundancy are essential to stimulate the search process towards the identification of a diversified and comprehensive set of solutions, which would not be achieved by optimizing cardinality and accuracy only.

In this work, we use an advanced IVS algorithm called the Wrapper for Quasi-Equally Informative Subset Selection. W-QEISS provides as output a number of efficient subsets that are collected in a selection matrix.

In the “drought index” modeling block, the preferred efficient solution is selected by the user, balancing the trade-off between competing objectives and, possibly, considering additional operative needs neglected in the IVS search (e.g., cost and reliability of the variable monitoring). Lastly, an appropriate regressor is fit to the sample dataset of Pareto-efficient inputs and the target variable. The choice of model class is determined by the application of interest. In general, highly non-linear learning machines like artificial neural networks (ANNs) provide a good balance between accuracy and flexibility. On the other hand, such black-box models lack intuitive interpretability and might be unsuitable for applications that affect several stakeholders and require a wide acceptance of the tool to be employed (Estrela and Vargas, 2012). In these cases, a simpler model (e.g., a linear model) might be preferred, as it grants an immediate understanding of the physical meaning, though at the price of poorer approximation skills.

2.2 Feature extraction via the Wrapper for Quasi-Equally Informative Subset Selection

Feature extraction techniques, employed in the second block of FRIDA, are an ensemble of data pre-processing algorithms that transform the original input dataset into a more compact, while still highly informative, subset (Cunningham, 2008). Among the feature extraction algorithms, IVS techniques specifically address the problem of the reduction of the input space by identifying the relevant predictors to be used to calibrate a model of the target variable (Bowden et al., 2005). There are two main classes of IVS techniques: “filters” and “wrappers”. Filters evaluate the relevance of each variable separately, computing an error metric on the features (Yang and Pedersen, 1997; Sharma, 2000; Galelli and Castelletti, 2013). Wrappers, on the other hand, assess the relevance of a variables ensemble, evaluating the prediction performance of a given learning machine calibrated on the input set, and thus considering the interactions and dependencies between variables (Guyon, 2003). In terms of performance, wrappers are often more accurate than filters, although computationally more intensive (Galelli et al., 2014).

In this study, we used W-QEISS as a wrapper (Karakaya et al., 2015; Taormina et al., 2016). The W-QEISS algorithm receives as input the set X of candidate predictors, i.e., $\mathbf{X}=\mathit{\{}{x}_i,\mathrm{\dots },x_{n_X}\mathit{\}}$ and the trajectory y of the target variable. The algorithm is composed of three main steps (Karakaya et al., 2015), as illustrated in Fig. 3.

Figure 3The W-QEISS flowchart. Step 1: generate Pareto-efficient solutions with respect to the four objectives of relevance, redundancy, cardinality, and accuracy; step 2: select high accuracy subsets; step 3: eliminate inferior subsets.

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• Step 1: a set A⊆X of Pareto-efficient solutions is built according to the four-objective functions of relevance f₁(⋅), redundancy f₂(⋅), cardinality f₃(⋅), and accuracy f₄(⋅). A global multi-objective optimization algorithm is employed to explore the space of possible subsets. In this study, we use the self-adaptive Borg Multi-Objective Evolutionary Algorithm (MOEA; Hadka and Reed, 2013), which has shown to outperform other benchmark evolutionary algorithms in terms of number of solutions returned, ability to handle many-objective problems, ease of use, and overall consistency across a suite of challenging multi-objective problems (Reed et al., 2013). A learning machine is used to compute the predictive accuracy f₄ of each set. In this study, we employ extreme learning machines (ELMs; Huang et al., 2006), belonging to the family of ANNs, which were shown to provide a good performance in terms of accuracy and flexibility in a variety of problems, while being up to a thousand times faster than benchmark feedforward ANNs (Huang et al., 2012). ELMs, in fact, bypass the time consuming gradient-based search of optimal neuron parameters required by traditional ANN techniques, by defining randomly parameterized hidden nodes, and subsequently optimizing their output weights. Such optimization is solved through a one-step matrix product and essentially amounts to learning a linear model.

  However, we do not expect the choice of the learning machine or MOEA to be crucial for the attainment of the result. A different benchmark MOEA (e.g., NGSAII, MOEAD, eps-MOEA) is likely to achieve a comparable result, although requiring a possibly significant effort in the manual calibration of the evolution parameters, which is automated in Borg MOEA. Similarly, ELM could in principle be substituted by other ANN techniques, although incrementing the computational time to possibly unbearable levels, given the multiple calibration and validation processes reiterated in W-QEISS.

• Step 2: among the Pareto-efficient subsets, the maximum value of accuracy $f_{\mathrm{4}}^{\ast }$ is identified, associated with subset ${\mathbf{S}}_{f_{\mathrm{4}}^{*}}\subseteq \mathbf{A}$. Then, solutions with significantly lower accuracy are discarded from ensemble A, obtaining A_δ. The ensemble A_δ contains quasi-equally informative subsets with respect to ${\mathbf{S}}_{f_{\mathrm{4}}^{*}}\subseteq {\mathbf{A}}_{\mathit{\delta }}\subseteq \mathbf{A}$, i.e., subsets that have (almost) the same predictive accuracy with respect to a given model class. When the dataset of candidate variables presents a significant correlation among features, numerous subsets characterized by a wide range of cardinalities are generally available to achieve a relative small range of accuracies. This is often the case in environmental problems, where spatial and temporal correlation of hydro-meteorological variables and associated indicators is significant. Therefore, at this stage, the accuracy metric is used to retain accurate solutions only, provided that they feature different cardinalities and predictor combinations.

  Formally, on the basis of a predefined small value of δ, S_i is δ-quasi-equally informative to subset ${\mathbf{S}}_{f_{\mathrm{4}}^{*}}$ if

  $$\begin{array}{}\text{(1)}& f_{\mathrm{4}}\left({\mathbf{S}}_i\right)\ge (\mathrm{1}-\mathit{\delta })f_{\mathrm{4}}^{*}\mathrm{for}\mathrm{0}\le \mathit{\delta }\le \mathrm{1}\end{array}$$

• Step 3: the final ensemble ${\mathbf{A}}_{\mathit{\delta }}^{\ast }$ is computed after the elimination of the inferior subsets. The subset S_j is considered inferior to S_i, if it is a superset of S_i, and does not score higher accuracy. Formally
  S_i⊂S_j and f₄(S_i)≥f₄(S_j).
  In this step, all subsets contained in A_δ are compared in order to find possible inferior subsets and eliminate them. By doing this, the final ensemble of δ-quasi-equally informative subsets ${\mathbf{A}}_{\mathit{\delta }}^{\ast }$ is provided as output of the procedure and reported in a selection matrix.

The W-QEISS algorithm differs from a traditional IVS approach as it introduces the consideration that, for a given cardinality, multiple subsets of variables can have almost indistinguishable accuracy performance. The outcome of W-QEISS variable selection is thus not a single most accurate subset for each cardinality, but a pool of δ-quasi-equally accurate solutions among which the preference can be determined by other metrics not directly considered in the optimization (e.g., cost and reliability of the variable observation).

Another innovative feature of the W-QEISS approach relies on the formulation of a four-objective optimization problem. Besides the two traditional objectives of accuracy ad complexity commonly employed in wrappers, W-QEISS includes two other metrics of relevance and redundancy (Sharma and Mehrotra, 2014). The maximization of accuracy ensures a precise reproduction of the data, while the minimization of cardinality aims at simplifying the final models. These characteristics are key for an operational index, expected to balance precision and ease of use. Relevance and redundancy optimization is instead an asset for an effective subset search process, as it fosters the diversification of the solutions explored within the MOEA algorithm, guaranteeing low intra-subset similarity and high information content of the solutions. A two-objective search based on cardinality and accuracy would only, in fact, identify optimal solutions, but at the same time disregard a number of quasi-equally informative subsets with an almost identical operational behavior. The identification of such alternative solutions, nevertheless, grants flexibility and a multiplicity of options for the expert-based choice of the preferred subset, where certain combinations of predictors can be favored according to case-specific operative purposes, e.g., more robust or less costly data gathering process, enhanced acceptability, or immediacy of the index.

Three of the four objectives formulations make use of the symmetric uncertainty (SU), a measure of the dependence and similarity between two variables (Witten and Frank, 2005). SU assumes values between 0 (independent variables) and 1 (complete dependence) and is computed for two features A and B as

where H(⋅) is the entropy of variable (⋅) (see for instance Scott, 2012 for the definition).

W-QEISS bases its objectives formulation on information theory, as discussed in Karakaya et al. (2015). Information theoretic criteria (e.g., SU, mutual information, and partial mutual information) do not assume any functional relationship between the variables and thus are well suited to quantify the dependence between two variables in any modeling context (MacKay, 2003). Other objectives formulations could in principle be explored, for instance substituting the use of symmetric uncertainty with more traditional correlation coefficients, although with the risk of losing generality by assuming linear dependence between variables.

The four assessment metrics are formulated as follows:

1. Relevance f₁(S): to be maximized, is formulated as

   $$\begin{array}{}\text{(3)}& f_{\mathrm{1}}\left(\mathbf{S}\right)=\sum _{x_i\in \mathbf{S}\subseteq \mathbf{X}}\mathrm{SU}(x_i,y),\end{array}$$

   where the term SU(x_i,y) represents the symmetric uncertainty between the feature x_i and the output y. The relevance is therefore a measure of the explanatory power of the features with respect to the output.

2. Redundancy f₂(S): to be minimized, is formulated as

   $$\begin{array}{}\text{(4)}& f_{\mathrm{2}}\left(\mathbf{S}\right)=\sum _{x_i\in \mathbf{S}\subseteq \mathbf{X}}\mathrm{SU}(x_i,x_j),\end{array}$$

   where SU(x_i,x_j) represents the SU between two features x_i and x_j. High redundancy thus means high similarity between the features. By minimizing the redundancy, the algorithm ensures that the search will be oriented towards the selection of subsets with mutually dissimilar features.

3. Cardinality f₃(S): to be minimized, is formulated as

   $$\begin{array}{}\text{(5)}& f_{\mathrm{3}}\left(\mathbf{S}\right)=\left|\mathbf{S}\right|,\end{array}$$

   where |S| is the number of predictors within the subset. Its minimization guarantees that the resulting model will not be unnecessarily complex.

4. Accuracy f₄(S): to be maximized, is formulated as

   $$\begin{array}{}\text{(6)}& f_{\mathrm{4}}\left(\mathbf{S}\right)=\mathrm{SU}(y,\widehat{y}(\mathbf{S}\left)\right),\end{array}$$

   where SU$(y,\widehat{y}(\mathbf{S}\left)\right)$ is the correlation, measured in SU, between the observed output y and the prediction $\widehat{y}\left(\mathbf{S}\right)$ obtained from the model.

4.1 Identification of the basin's characteristics

In the first report concerning the I_e development (CHJ, 2007b), the index was validated for the time span from January 1986 to June 2000 against the supply deficit recorded in the basin with respect to agricultural and urban water demand, and the procedure for the state index computation was approved. To ensure comparability between the I_e and the FRIDA constructed index, we decided to employ the same supply deficit as target variable for the application of the FRIDA approach to the Jucar case study. The Jucar supply deficit employed in this work was simulated via the AQUATOOL model (Andreu et al., 1996). The model can run in simulation mode with a monthly time step, and it is conceived in the form of a flow network with oriented connections reproducing water losses, hydraulic relations between nodes, reservoirs and aquifers, and flow limitations based on elevation. Within AQUATOOL, complex processes such as evaporation and infiltration are effectively reproduced. The modeled supply deficit, employed as target variable, represents the monthly nominal shortage of water conveyed to the irrigation districts, and is only quantifiable a posteriori, when the water shortage has already jeopardized the fields. On the other hand, a drought index can be constantly monitored, and thus represents a valuable management tool for restraining drought impacts and identifying effective drought management strategies.

The database of candidate input variables was assembled by retrieving the available observed variables in the basin and computing traditional drought indicators at multiple time aggregations. The resulting candidate predictors, listed in Table 1, are the following:

• two temporal features: date of the measurement and month of the year (Moy);

• 12 monthly observed variables, current inputs to the I_e, reported in Fig. 4: average monthly storage and groundwater levels, average 3-month river runoff, and cumulated areal precipitation over 12 months;

• eight additional observed variables in the basin: outflows from, and inflows to, the main reservoirs, and mean monthly areal temperatures;

• six traditional drought indicators: the Standardized Precipitation Index (SPI) and the Standardized Precipitation and Evapotranspiration Index (SPEI). SPI and SPEI indicators are computed on mean monthly data over the entire basin for 3-, 6-, and 12-month time aggregations. SPI requires as input the precipitation and SPEI requires precipitation and temperature, as it uses the difference between precipitation and potential evapotranspiration as reference variable.

  Their values express the water availability conditions of a basin in terms of units of standard deviation from the mean: negative (positive) values indicate drier (wetter) conditions than average (see McKee et al., 1993; Vicente-Serrano et al., 2010 for details on definition and calculation of these indicators).

Table 1Set of candidate input features for the feature extraction step via W-QEISS.

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4.2 Feature extraction via W-QEISS

The result of the W-QEISS algorithm is not a single most-accurate set of variables for a given cardinality, but several quasi-equally informative subsets, whose accuracy is lower than the best one by a small percentage (δ⋅100 %). Figure 5 represents a “selection matrix”, which reports the composition of each alternative subset of predictors within 15 % of accuracy with respect to the highest one. The value δ=0.15 was chosen since it provides a reasonable trade-off between the number of solutions and their accuracy. The accuracy is measured in symmetric uncertainty between the target variable and the ELM calibrated using the reported subset.

The alternative subsets are sorted in ascending order of cardinality (from top to bottom), and accuracy (within each cardinality level). A rectangular marker is placed at the intersection between the row identifying a given subset and the columns corresponding to the selected predictors. The marker color varies with the cardinality of the subset, with lighter shades of gray indicating smaller subsets. In this case the cardinality spans from 3 to 9 features. The highest accuracy is reported in red and recorded for subset number 14. The five corresponding selected predictors, marked on the horizontal axis with a blue background, are the following:

• Moy: month of the year;

• S1: total storage aggregated for the reservoirs Alarcón, Contreras, and Tous;

• F3: river flow measured in the Jucar middle basin, after the confluence with smaller rivers Jardín and Lezuza coming from the southwest;

• Pz2: groundwater level measured at the piezometer situated in the central area of the basin, in correspondence with a rainfed agricultural area;

• SPEI₆: SPEI at 6 months time aggregation computed with precipitation and temperature data averaged for the whole basin.

Figure 5Selection matrix: the left vertical axis represents the subset number (id) and the right vertical axis the corresponding accuracy measured in SU. A colored marker is put in correspondence with the variables, listed on the horizontal axis, selected by each subset. The shade of gray is an indication of the cardinality of the subset, lighter shades for lower cardinality. The highest accuracy is reported in red and the corresponding variables, constituting the most accurate subset, have a blue background.

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From the analysis of the selection matrix, several insights can be gained from modeling and decision-making viewpoints. To begin with, insights into the relevance of a predictor can be obtained from the detection of the vertical bars traced by joining markers across multiple rows. Uninterrupted bars indicate strongly relevant predictors that cannot be substituted by other input combinations without incurring a substantial drop of predictive accuracy. This is the case for the cumulated storage of the three main reservoirs Alarcón, Contreras, and Tous (S1). This information is essential to the final model, as the exclusion of such predictors highly affects the model performance.

Increasing gaps in the vertical bars are found when considering predictors with progressively weaker relevance, while irrelevant inputs are recognizable by isolated markers or their total absence. The variables Moy, F3, and Pz2 are considered relevant variables, as they are selected quite frequently, although high accuracy solutions exist that do not make use of all of them. Finally, the variable SPEI₆, while included in the most accurate subset, is overall present in four subsets only, whereas in other solutions with comparable accuracy it is replaced by different predictors, mainly carrying a similar precipitation-based information, such as pluviometer measures, or SPI/SPEI indicators at different time aggregations.

The presence of alternative subsets helps explore the trade-off between multiple measures of predictive accuracy with respect to other metrics not directly considered in the optimization routine, and the choice of the preferred subset is determined by the index application. Given the cardinality, one can decide to sacrifice a small amount of predictive accuracy for an easier-to-yield (or more reliable) combination of predictors. For example, with a loss smaller than 1 % in accuracy, subset 13 selects SPI₆ instead of SPEI₆. This possible replacement is interesting from an operational point of view as SPI is easier to compute than SPEI. In fact, SPI only requires the precipitation for its computation whereas precipitation and temperature or evapotranspiration are needed for the computation of SPEI. In addition, even after the preferred subset is chosen and the system is operating, knowing that one specific predictor can be replaced by another (or multiple) predictor(s) can aid management in case of monitoring network maintenance or instrument failure. When the main predictor is not observable, one can temporarily resort to alternative predictors incurring a minimum loss of accuracy.

An additional consideration is related to the possibility to effectively address the uncertainty deriving from the choice of model inputs (Taormina et al., 2016). When multiple alternative subsets are provided, it is possible to explore the uncertainty related to the selection of predictors yielding similar accuracy. For instance, in this case study, we can observe that almost all subsets carry groundwater and rain information, but while the piezometric level is consistently provided by Pz2, the source of the precipitation information highly varies among the precipitation-based features (pluviometers or other SPI/SPEI indicators).

Finally, through the selection matrix analysis we can compare the features selected by W-QEISS and the variables that constitute the state index input set. Apart from sporadic single selections, all the observed variables not included in the state index are consistently discarded by the W-QEISS as well, suggesting that the algorithm comes to the same conclusion as the Spanish experts considering inflows, outflows, and temperatures as non-relevant for the description of the state of water resources in the Jucar river basin. Note that this result is a consequence of the use of the nominal agricultural demand to compute the target deficit. Temperature information is likely to become relevant if a real, weather-influenced, agricultural demand is employed instead. The feature month of the year is not explicitly an input to the state index, nevertheless, analogous information is implicitly included in the I_e through the normalization of the indicators described in Eq. (7). On the other hand, several features are considered in the I_e, but generally neglected by the W-QEISS selection. Among them, two out of three piezometers, the river flows upstream from the reservoirs, one pluviometer and the storage at Forata. These inputs are probably redundant due to their spatial correlation. Spatial variability is considered in the computation of I_e by including several spatially distributed observations of the main information categories: two measures of reservoir storages, four of river flows, three groundwater levels, and three precipitation measures. Conversely, the selection matrix supports the gain of a deeper understanding of the spatial interdependence of variables by identifying the best location for measuring the variables, sparing the need for several distributed measures. The highest accuracy-subset, in fact, selects only one variable out of each category: one storage, one river flows measure, one piezometer, and a spatially distributed precipitation information, i.e., SPEI₆ which replaces three areal pluviometers.

4.3 Drought index modeling

Among the pool of solutions, the choice of the preferred subsets is driven by the index application. For instance, an online use of the index that requires its frequent computation may benefit from an agile, easy-to-observe subset. With respect to the highest accuracy solution (subset 14), for instance, subset number 7 neglects predictor F3 thus presenting lower cardinality with an accuracy loss of only 3 %. Similarly, the already mentioned subset 13 contains an easier-to-compute indicator (SPI instead of SPEI) with a negligible performance degradation. Nevertheless, for our methodological purpose we will employ the most accurate subset 14, as we are interested in discussing the potential of the method.

Concerning the model class choice, a highly flexible non-linear model is likely to yield the highest accuracy in reproducing the target. However, strong non-linearity and black-box behavior typically result in poor interpretability, a feature that is detrimental to the use of the index for management purposes as in the Jucar system, where restrictive measures in water use are activated when certain threshold values of the state index are reached. As a consequence, the index outcome exerts a direct influence on many water-related activities requiring an easily interpretable and widely acceptable tool.

The calibration of a linear model on the chosen 5-dimensional subset seems to be a good compromise between accuracy and transparency. As mentioned above, the feature Moy represents the succession of the months in the year, and is an expression of the seasonality of hydro-meteorological processes. Moy is constructed as the repetition of an array of numbers from 1 to 12 for the length of the considered time horizon, and thus presents a discontinuous shape: a slow and steady increase followed by a steep decrease in correspondence to the onset of a new year. While the non-linear models employed in the feature selection can effortlessly handle such an intermittent vector, linear models struggle with similar shapes. We therefore decided to account for the seasonality in the linear model indirectly, i.e., excluding Moy from the predictors set, but, consistently, considering seasonality by subtracting the annual cyclo-stationary mean from the predictors time series.

The calibrated linear model representing the supply deficit is reported in Fig. 6 and provides a very satisfying result, with an accuracy measured with the coefficient of determination in cross validation of $R_{\mathrm{FRIDA}-\mathrm{linear}}^{\mathrm{2}}=\mathrm{0.904}$, significantly higher than the $R_{I_{\mathrm{e}}}^{\mathrm{2}}=\mathrm{0.739}$ scored by the state index, and a set of weights of immediate physical interpretability reported in Table 2. By inspecting the weights, one can notice that those assigned to the predictors “flow” and SPEI₆ are very low, although not null, and the index trajectory is mainly determined by the “storage” and “piezometer” values. S1 and Pz2, in fact, describe the trajectories of the main water reservoirs of the region, lakes and groundwater, whose fluctuations are the result of natural variability as well as human regulation, mainly for irrigation purposes.

Table 2Weights of the linear model calibrated for the optimal subset of predictors. The predictor Moy (month of the year), providing seasonal information, is not directly included in the weights optimization but it is accounted for by subtracting the annual cyclo-stationary mean from the predictors time series.

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As a further analysis, we reiterated the model calibration and cross-validation steps with a more complex, highly flexible model class, the ELM architecture, which scored an accuracy of $R_{\mathrm{FRIDA}-\mathrm{ELM}}^{\mathrm{2}}=\mathrm{0.907}$. On the one hand, the arguably insignificant 0.005 % improvement in accuracy of ELM with respect to the linear class probably does not justify the loss of immediacy and transparency induced by the transition to a black-box model. On the other hand, this experiment proves the robustness of the linear model in constituting the model class of choice for this drought index. In Table 3 we report a more detailed comparison between state index, FRIDA-linear and FRIDA-ELM indexes with several accuracy metrics. The analysis of other metrics seem to reinforce the conclusions drawn by considering R² only: both FRIDA indexes (linear and ELM) outperform the state index quite significantly, while the difference among them is negligible, although the non-linear index is always the top performing.

Table 3Accuracy of the state index, FRIDA-linear, and FRIDA-ELM in reproducing the supply deficit, quantified in terms of coefficient of determination R², the Pearson correlation coefficient, the root mean square error (RMSE), and the fourth grade root mean square error (R4MS4E).

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The reported metrics do not distinguish between errors above and below the target deficit. Indeed, we consider these two error types to be of comparable importance. On the one hand, the underestimation of a deficit value may find the water users unprepared to face a serious drought. On the other hand, the overestimation of drought conditions may ignite repeated false alarms that will compromise the index trustworthiness and its efficacy in triggering an alert state. Therefore, rather than penalizing an error above or below the target trajectory, we find it more compelling to focus on errors in the most crucial drought situations i.e., at the maximum level of deficit recorded. One way of doing so is considering R4MS4E, as in Table 3, which penalizes errors in the deficit peaks. Another specific assessment tool for analyzing the indexes performance during critical droughts is the confusion matrix, reporting the classification performance of critical droughts, here arbitrarily defined as months reporting deficit values above the 85th percentile (Tables 4, 5, 6). The rows of the confusion matrix represent the instances in a predicted class, while the columns represent the instances in an actual class. Consequently, the main diagonal reports the number of correctly classified points. Cells outside the main diagonal specify the errors: the value in the bottom-left cell (first column, second row) indicates a situation in which the index does not recognize an ongoing drought, while the value in the top-right cell (first row and second column) indicates the number of false alarms. The FRIDA-ELM confusion matrix seems to significantly exceed the competitors performances by having an error of only 0.57 % of the times, as opposed to the 10.91 % of I_e, and the 6.3 % of FRIDA-linear.

Table 4State index confusion matrix.

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Table 5FRIDA-linear confusion matrix.

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Table 6FRIDA-ELM confusion matrix.

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Figure 6Comparison between the FRIDA-linear index (red plot, blue scale on the left) and the state index (green plot, green scale on the right) in reproducing the monthly aggregated supply deficit (blue dashed plot, blue scale on the left). The FRIDA index presents a higher similarity with the deficit and only requires 5 inputs instead of the 12 required by the state index.

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