To investigate the predictability of drought data from the single-model initial-condition large ensemble (SMILE) consisting of 50 members, the Canadian Regional Climate Model 5 Large Ensemble (CRCM5-LE) is used. The data were produced within the scope of the ClimEx project (, http://www.climex-project.org, last access: 28 November 2021). The CRCM5-LE was generated by dynamical downscaling of the data provided by the 50-member initial-condition Canadian Earth System Model 2 using the Canadian Regional Climate Model 5 . The data have a resolution of 0.11∘ (12 km) and are produced for the years 1950–2099 for a European and an eastern North America domain. For the years 1950–2005 the historical greenhouse gas concentrations and aerosol emissions are being used. Starting from 2005, the model introduces the RCP8.5 forcing scenario. A total of 42 atmospheric variables are available at a temporal resolution of 1 to 3 h. They are used on a monthly basis as input to the machine learning algorithms. The list of variables is provided in Table .

In the study we use monthly sea level pressure (psl) from the driving model CanESM2-LE for the calculation of North Atlantic Oscillation (NAO), Scandinavian Oscillation (SCA), East Atlantic Oscillation (EA) and East Atlantic–Western Russia Oscillation (EAWR) over the whole Atlantic basin (20–80∘ N, 90∘ W–40∘ E). The Atlantic Multidecadal Oscillation (AMO) is calculated using the sea surface temperature (SST) over 0–60∘ N, 0–80∘ W from the CanESM2. Only the period 1955–2005 is considered in order to stay within the scope of historical climate. The CRCM5 domain is displayed in Fig. . For the machine learning training a grid point situated at 48.11∘ N, 11.91∘ E is referenced as Munich, and 38.67∘ N, 9.17∘ W is referenced as Lisbon.

In order to calculate NAO, SCA, EA and EAWR, the method introduced by is used: a principal component analysis (PCA) of the monthly psl is performed over the 20–80∘ N, 90∘ W–40∘ E domain. The leading eigenvectors, scaled by the amount of variance they explain, represent the leading circulation patterns of the atmospheric system. The first eigenvector corresponds to NAO, the second one to SCA, the third one to EA and the fourth one to EAWR. To calculate the teleconnection indices (NAO, SCA, EA, EAWR) the eofs package described in is used. It is an implementation of the technique of empirical orthogonal functions (EOFs) . The leading modes of the PCA corresponding to NAO, SCA, EA and EAWR derived from the CanEsm2 dataset are shown in Fig. .

AMO is calculated by spatial averaging over the 0–60∘ N, 0–80∘ W area of the anomaly of sea surface temperature . Additionally the 10-year running mean of AMO is calculated as an input variable as it is widely used in various studies and was shown to be correlated with precipitation .

Variable subset selection helps to limit the computational time and to improve predictive accuracy . In order to eliminate redundant variables, Pearson's R between all the CRCM5 variables for the chosen domains is calculated. Pearson's R (ρX,Y) is a measure of linear correlation between two variables X and Y; ρX,Y equals 1 if the correlation is totally positive, 0 if there is no linear correlation and -1 if the correlation is total negative . For two samples x and y, Pearson's R is defined in the following way: 3ρx,y=∑i=1n(xi-x¯)(yi-y¯)∑i=1n(xi-x¯)2∑i=1n(yi-y¯)2.

The bar refers to the average over the index i . Pearson's R is a popular and easy method for feature selection of continuous variables as introduced in ; ρ is calculated for all possible permutations of the 41 input variables. The ones correlating to a high degree are examined, and a threshold of 0.95 is chosen. In Table  a list of sorted-out variables and the corresponding values of Pearson's R is given. The high correlation values can be explained by a physical relationship between the variables: e.g., the total evaporation (evspsbl) is almost the same as evaporation from land (evspsblland) as there are no relevant water bodies in the chosen domains. Out of the full list of 42 variables, 14 are sorted out as being redundant.

The standardized precipitation index (SPI) is a precipitation-based index introduced by . For the calculation of SPI a continuous monthly precipitation dataset is used. The index can be calculated on different timescales: typically, it is 1, 3, 6, 12 or 24 months. As a first step the precipitation values are accumulated for the needed timescale. The resulting dataset is fitted to a gamma distribution for each month separately and then transformed to a normal distribution such that the mean SPI is zero. The SPI value for a given precipitation is then the number of standard deviations from normal. Because of the normalization, the SPI is especially useful to represent wetter and drier climates as well as to account for differences among seasons. As the two study sites are having different meteorological conditions, the SPI provides a convenient and comparable measure . As noted in the accumulation period of the SPI value needs to be chosen equal to or less than the prediction lead time as otherwise the precipitation values needed for the mathematical calculation of the SPI would be given as input to the machine learning algorithm. Therefore the accumulation period of 1 month is chosen. SPI1 is calculated for Lisbon and Munich each using the data from 1955–2005 from all members as reference.

This study investigates drought predictability applying the technique of supervised machine learning for this purpose. Machine learning is a promising tool for the analysis of complex and data-rich phenomena as droughts . The Python package Keras, a high-level neural network package, is used for the design of the machine learning models as it allows the design of neural networks in an easy way by adding layers. Three crucial elements are needed to perform drought prediction by supervised machine learning: input data; a target variable to be predicted; and a computation pipeline, which includes the machine learning algorithm.

The data from the years 1957–1999 are used as training data; the years 2000–2005 are used for the testing purpose. Each of the time periods is available 50 times as we are dealing with an ensemble of 50 members. This results in 2150 model years for training and 250 years for testing. A small fraction of the training data are used for the validation of the machine learning algorithms. The target variable chosen for the prediction of droughts is SPI1. Two classes for the prediction are identified in the following way: SPI1<-1 is defined as an event and is initialized with 1; SPI1>-1 is initialized with 0 and corresponds to a non-drought event. The lead time of 1 month is chosen for the prediction as it has been used in previous studies by and . Moreover shorter prediction lead times usually obtain better results when compared to longer periods, as seen in . After the feature selection 28 variables originating directly from the CRCM5-LE dataset are used as input. In addition to those the teleconnection indices NAO, SCA, EA, EAWR, AMO and AMO10 are used as input.

To predict, for example, a drought or non-drought in April of 1980, the data for 12 months before the event are used as input. This is NAO and other teleconnection and atmospheric variables for the period April 1989–March 1980; for a prediction of an event in May 1980, May 1989–April 1980 is used as input. The 12 months before the event are chosen in accordance with the study by , who found that the best-performing drought prediction model was the one including the value up to 12 months before the predicted one. We perform a time series prediction with no limitation on special months or seasons to be inspected.

For this analysis we use a supervised machine learning algorithm, an artificial neural network (ANN). ANNs are algorithms whose design is inspired by the architecture of the human brain with its neurons ; they both consists of connected nodes. A link between the node i and the node j serves to propagate the activation ai from i to j. To each connection a numeric weight wi,j is assigned. The output of the node is computed by 4ai=g(inj)=g∑i=0nwi,jai p. 728. The activation function defines the output of the node. In order to have stable learners with confident predictions a function with a soft threshold is recommended . In this study the following three activation functions are used: sigmoid, rectified linear unit (ReLU), exponential linear unit (ELU). Sigmoid activation is especially useful for the output layer , while ReLU and ELU both have the property of allowing very fast optimization .

The sigmoid function, also called the logistic function, is defined in the following way: 5Logistic(x)=11+e-x . This function has an output between 0 and 1. This can be interpreted as a probability of belonging to the class 1. One of the main disadvantages of the sigmoid activation function is the vanishing gradient problem: at higher, almost saturated layers with values of 1 or -1, the gradients become nearly 0, resulting in a slow optimization convergence p. 726.

ReLU refers to rectified linear unit and shows better performance when dealing with the vanishing gradient problem . ReLU is defined in the following way: 6f(x)=max⁡(0,x).

ELU refers to the exponential linear unit and was introduced by . claim that in experiments the ELU activation led to faster learning and significantly better generalization performance than ReLU and sigmoid activation. The function is defined as 7f(x)=x if x>0α(exp⁡(x)-1) if x≤0; α controls the value to which an ELU saturates for negative inputs. Per default the value is set to 1 such that the function saturates at -1.

Two kinds of layers are used in this study: dense and dropout. Dense refers to a regular fully connected neural network layer. Dropout refers to a layer which is randomly setting a fraction of inputs to zero at each update. This technique is used to prevent overfitting and therefore to improve the performance of the algorithm . The first part of the study concentrates on the methodological search for the best-performing algorithms. A pipeline to search for the best-performing architecture, value for L2 regularization and loss function is built up.

The model performance is evaluated using accuracy and F1 score . The latter one is especially useful when training on datasets with an imbalanced class distribution as it is in the case of our dataset. Accuracy is defined in the following way: 8Accuracy=Number of right predictionsTotal number of samples.

F1 score is a harmonic measure between precision and recall. Precision is the amount of true positives with respect to the amount of positively classified data. Recall is the amount of true positives with respect to the total number of positives in the data. The F1 score is defined in the following way: 9F1 score=2Precision⋅RecallPrecision+Recall.

Due to the class imbalance within the dataset we require that the accuracy of each class is at least 50 %. In that case given the distribution of the test dataset of 1803 non-drought events to 387 droughts for Lisbon and 1848 non-drought events to 352 drought events for Munich a marginal F1 score of 0.26 for Lisbon and 0.24 for Munich is given.

The best-performing models are additionally evaluated using the Heidke skill score (HSS). The range of the HSS is -∞ to 1. Values below zero indicate that the random forecast (a forecast which randomly assigns the labels) has a better performance than the trained model. HSS of 1 indicates a perfect forecast. HSS is defined in the following way: 10HSS=2ad-bc(a+c)(c+d)+(a+b)(b+d), where a is the number of true positives, b the number of false positives, c number of false negatives and d number of true negatives.

The second part of the study analyzes the best-performing algorithms (one for the Lisbon domain, one for the Munich domain) by applying explainable AI methods. SHAP (SHapley Additive exPlanations) is a state-of-the-art method for interpretation of machine learning models, which was inspired by game theory . It estimates for each input feature an average marginal contribution to the prediction of the result and therefore allows a comparison of the contributions among different features. In addition to that the difference in predictability among the seasons is calculated and compared to gain a better understanding on the influence of seasonal weather patterns.

An overview of the proposed methodology can be found in Fig. .

For the design of the ANN it is crucial to perform fine-tuning of the model parameters to find the optimal setup. An architecture has to have enough layers and neurons to capture the complexity of the dataset . In order to find the best architecture the learning curve of the algorithm is inspected. The learning curve shows the loss of the training and validation datasets on the weights during the training . Two examples are shown in Fig. . The plot shown at the top refers to an architecture which is not able to capture the complexity of the dataset: the loss is hardly decreasing in the training or validation data. The bottom figure refers to an architecture which overfits: in the last epochs the loss of the validation dataset is rising, while it decreases in the training dataset.

In this way a network is searched which captures the given complexity of the dataset. This is reached with an algorithm consisting of at least five layers. Additionally two dropout layers, which set a specified number of nodes to zero in a random way, are introduced in order to fight overfitting.