Artificial Neural Networks for Drought Forecasting in the Central Region of the State of Zacatecas, Mexico

Abstract

Drought is, among natural hazards, one of the most harmful to humanity. The forecasting of droughts is essential to reduce their impact on the economy, agriculture, tourism and water resource systems. In this study, drought forecast in the central region of the state of Zacatecas, a semi-arid region of Mexico, is explored by means of artificial neural networks (ANNs), forecasting numerical values of three drought indices—the standardized precipitation index (SPI), the standardized precipitation and evapotranspiration index (SPEI) and the reconnaissance drought index (RDI)—in an effort to establish the most suitable index for drought forecasting with ANNs in semi-arid regions. Records of 52 years of monthly precipitation and temperature were used. The indices were calculated in three different time scales: 3, 6 and 12 months. The analyzed models showed great capacity to forecast the values of the three drought indices, and it was found that for the trial set, the RDI was the drought index that was best fitted by the models, with the evaluation metrics R² (determination coefficient), RMSE (root mean square error), MAE (mean absolute error) and MBE (Mean Bias Error) showing ranges of 0.834–0.988, 0.099–0.402, 0.072–0.343 and 0.017–0.095, respectively. For the validation set, the evaluation metrics were slightly better.

1. Introduction

Drought is a natural phenomenon that occurs in all climates and is one of the most serious natural hazards, causing significant economic, social and environmental losses [1]. The phenomenon may be considered to be a water deficit that persists over a prolonged period of time as a result of a deficiency in the input of atmospheric, surface or underground water [2,3]. Drought has a devastating effect on agriculture, water supply, ecosystems, public health, energy generation and the economy, making it a challenging issue in water resource management [4,5]. It is estimated that yearly global economic losses caused by drought are over USD 6–8 billion greater than those produced by other weather phenomena, such as floods, storms and hurricanes [6]. In recent years, droughts in the United States of America have caused annual losses of almost USD 9 billion [7], while in Latin America, the drought that occurred during the period from 2005 to 2015 is regarded as the costliest natural disaster due to the economic impact on the agricultural and livestock sectors, which amounted to losses of USD 13 billion [8]. In Mexico, it is estimated that the drought of 2011 was among the most severe in the last 70 years, with a cost of MXN 7750 million (approximately USD 700 million) to the agricultural and livestock sectors and covered approximately 80% of Mexico’s territory [9]. In Mexico, the National Water Commission (Spanish acronym CONAGUA) is the official institution that declares drought conditions (through the Mexico Drought Monitor), which are based on the calculations of drought parameters (SPI, percent of normal rainfall anomaly and vegetation health index, among others).

Since human activity is the main driver of climate change [10,11], droughts are expected to increase in frequency and intensity in the future [12], mostly as a result of decreases in regional precipitation and increases in evapotranspiration due to global warming [13,14].

Drought monitoring systems are essential tools for managing the risks associated with this natural hazard. These systems allow tracking of the onset, progress, severity and extent of drought, and can be used to trigger contingency plans, if available [15]. An important aspect of drought monitoring and the development of an early warning system is the ability to effectively forecast future drought episodes. Forecasting future drought events in a region is important for finding sustainable solutions for water management and drought risk assessment [16,17]. This information may be obtained by means of accurate drought forecasting models, which are usually generated using drought indices [18]. Today, there are many indices for drought quantification. Among the most used worldwide are the SPI [19], the RDI [20] and the SPEI [21], among others, and the most used in Mexico are the SPI and SPEI. See, for example, [9,22,23,24]. Because of the complexity of the drought phenomenon, none of the drought indices can be applied in all regions or natural systems. Therefore, for an analysis of drought, more than one index should be considered, and it is important to examine the sensitivity and precision of each index and to explore their behavior in specific conditions and aims [25].

Drought forecasting may be conducted by means of physical-, conceptual- and data-based models. However, the physical and conceptual models require information from the study area regarding a large number of parameters, such as soil humidity, soil type, slope, topography, temperature, and evapotranspiration, resulting in very complex models [26,27]. Data-based models are the most widely used for drought forecasting due to faster development times and minimal data requirements [28,29].

In recent decades, different machine learning (ML) models have been used for forecasting with hydrological models. Specifically, drought forecasting models such as ANN (e.g., Maca and Pech [30]; Poornima and Pushpalatha [31]; Yan et al. [32]; Shrivastava et al. [33]), adaptive neuro-fuzzy inference systems (ANFIS) (e.g., Choubin et al. [34]; Achite et al. [35]), support vector regression (SVR) (e.g., Das et al. [36]; Fung et al. [37]; Shamshirband et al. [38]), external learning machines (ELM) (e.g., Mouatadid et al. [39]; Yaseen et al. [40]) and ML models with wavelet processing (e.g., Karbasi et al. [41]; Khan et al. [42]) have been used. ML models have also been successfully applied in other areas of water resources, as well as in other fields of science (e.g., Truong et al. [43]; Kumar et al. [44], Shams et al. [45]; Khan et al. [46]; Onyango et al. [47]; Wang et al. [48]).

Several studies have shown that artificial intelligence models could provide better results for semi-arid regions than conventional drought modeling techniques; for example, Markov chains, the regression method and linear stochastic models, including the autoregressive integrated moving average (ARIMA) and seasonal autoregressive integrated moving average (SARIMA) [49]. In Mexico, studies on the drought phenomenon have focused on the characterization and analysis of drought events, while few have been developed in the field of forecasting using ANN, such as in the work of [23,50].

The objectives of this study are as follows: (1) to develop an ANN model with multi-layer perceptron architecture to forecast drought in the central region of the state of Zacatecas, within Mexico´s semi-arid region, and to evaluate the forecasting ability of this model using the SPI, SPEI and RDI indices at the 3-, 6- and 12-month time scales; and (2) to test which of these drought indices is best modeled by ANNs, in an effort to establish which index is most suitable for ANN-assisted drought forecasting in semi-arid regions with characteristics similar to those of the study area. As a result, the joint implementation of ANN and the drought index that best models drought conditions could significantly improve the ability to forecast periods of drought and inform the development of water management strategies for these conditions.

2. Materials and Methods

2.1. Description of the Study Area

The state of Zacatecas is located between 25°07′31″ and 21°02′31″ N latitude and 100°44′32″ and 104°21′13″ W longitude, with a mean elevation of 2040 m above sea level (masl) [51]. It has an area of 75,284 km², which represents 3.8% of Mexico’s territory. Most of the state has an arid or semi-arid climate (73%), followed by a temperate subhumid climate (17%); its mean annual temperature is 17 °C, and its maximum (30 °C) and minimum (3 °C) monthly temperatures occur, respectively, in the months of May and January. The mean annual precipitation is 510 mm, most of which falls during the summer (75%), from June to September [52]. Specifically, at the two analysis sites, the climatic conditions are as follows: At the Agua Nueva (32001) weather station, the average rainfall is 365.9 mm/year. The month with the highest rainfall is August, 72.7 mm, and the month with the lowest rainfall is March, with 6.3 mm. The average annual temperature recorded at the station is 17.7 °C. The coldest month is January, with an average temperature of 12.1 °C, and the hottest month is June, with an average temperature of 22.1 °C. On the other hand, at the Calera (32003) weather station, located south of Agua Nueva weather station, the average rainfall is 439.9 mm/year. The months with the highest and lowest average monthly rainfall are July and April, respectively, with 94.0 and 6.7 mm. At this climatological station, the average annual temperature is 15.6 °C, and the coldest and hottest months are January and July, with 10.7 and 19.8 °C, respectively.

2.2. Description of Data

The weather data used in this study were obtained from the CLICOM (CLImate COMputing project) database [53], which is the official weather database in Mexico, and the data are public and free of charge. The weather data corresponds to two weather stations located in central Zacatecas: Agua Nueva and Calera (Figure 1). The data series spans over a period of 52 years, from 1961 to 2012.

Data selection was based on three criteria, in accordance with [25]: (1) the weather station had to be operational; (2) the weather data records needed to span at least 50 years; and (3) each weather station could have at most 15% of missing data. Prior to drought analysis, missing data were imputed, and the homogeneity of the weather data series was tested. The normal ratio method was used to fill in the missing data on a daily basis. The homogeneity of precipitation and temperature data was tested using the student’s t-test, Cramer’s test [54], the standard normal homogeneity test (SNHT) and Buishand’s test [55].

2.3. Potential Evapotranspiration

Calculation of the SPEI and RDI indices requires previous calculations of monthly potential evapotranspiration (PET), which was estimated by using Thornthwaite´s method [56] since it is one of the simplest methods for calculating the PET.

2.4. Drought Indices

Three drought indices were used in this study: the standardized precipitation index, the standardized precipitation and evapotranspiration index, and the drought reconnaissance index.

2.4.1. Standardized Precipitation Index

The SPI was developed by McKee et al. [19] for the identification and monitoring of drought events using only precipitation as an input. The SPI is calculated by fitting an appropriate probability density function (PDF) (usually the gamma distribution function) to the cumulative monthly precipitation data for each time scale of interest k and for each site of interest. The SPI may be calculated for several time scales, with 1, 2, 3, 6, 9, 12, 18, 24, 36 and 48 months being the most commonly used. In this study, scales of 3, 6 and 12 months were selected to monitor drought in the short-, medium-, and long-term, respectively. The SPI values were obtained by transforming values of the gamma distribution into values of the normal standard variable [57]. Positive SPI values indicate wet periods, while negative values indicate dry periods compared to the normal conditions in the study area. A drought event is considered to occur when the SPI is consecutively negative, reaches a value of −1 or less, and ends when the SPI becomes positive [58]. According to the SPI values, drought may be classified from slight to extreme (

Table 1. Classifications of SPI, SPEI and RDI.

Classification	SPI/SPEI/RDI Value
Extremely wet	≥2
Severely wet	1.50 to 1.99
Moderately wet	1.00 to 1.49
Slightly humid (close to normal)	0 to 0.99
Slight drought (close to normal)	0 to −0.99
Moderately drought	−1.00 to −1.49
Severely drought	−1.50 to −1.99
Extremely drought	≤−2

). A detailed description of the SPI calculation may be found in [25,59,60], among others.

2.4.2. Standardized Precipitation and Evapotranspiration Index

The SPEI was developed by Vicente-Serrano et al. [21] as an extension of the SPI. In comparison with the SPI, which only uses precipitation as an input, the SPEI is based on a monthly climatic water balance, that is, the difference between precipitation and potential evapotranspiration [37], which is calculated at different time scales. Its calculation follows a similar approach to the SPI calculation but uses the three-parameter log-logistic PDF instead of the two-parameter gamma PDF. A detailed calculation of the SPEI may be found in [21,25], among others. Different types of droughts may be classified according to the SPEI, as seen in Table 1.

2.4.3. Drought Reconnaissance Index

The RDI was developed by Tsakiris and Vangelis [20] to provide a more realistic representation of drought conditions, considering a balance between the inputs and outputs in a water system. Since the RDI incorporates potential evapotranspiration as well as precipitation, it can be used to compare drought conditions between areas with different climatic characteristics on different time scales [58]. The RDI is expressed in three ways: the initial value $\left({\alpha }_k\right)$, the normalized RDI ($RD{I}_n$) and the standardized RDI ($RD{I}_{st}$). In this study, the RDI was obtained by fitting the gamma distribution to the values of ${\alpha }_k$, in a similar way as was conducted in the calculation of the SPI. A detailed description of the RDI calculation is given by [25,61,62], among others. As with the SPI and SPEI, drought severity may be classified according to the RDI values, as shown in Table 1.

2.5. Artificial Neural Networks

ANNs are nonlinear black-box mathematical models. In practice, this approach is used to define a deterministic relationship between the process variables when the physical basis for their generation is unknown beforehand [49]. An ANN is a processing system that has certain performance characteristics resembling biological neural networks of the human brain and has the ability to identify a relationship from given patterns to solve large-scale complex problems such as pattern recognition, nonlinear modeling, classification, association and control [63]. ANN models are particularly useful for modeling nonlinear relationships in data, as often occur in hydrological studies [64].

The ANN models used in this study have a feedforward multi-layer perceptron (MLP) architecture that was trained with the Levenberg–Marquardt (LM) back-propagation algorithm. MLPs consist of an input layer, one or more hidden layers and an output layer (Figure 2).

Each neuron in a layer is connected to every neuron in the following layer at varying weights. The activation function of individual neurons can be either linear or nonlinear, and it maps weighted inputs to their output. As stated in [65], the value of the output is provided by:

$$y_k^{\prime }\left(t\right)=f_o\left[\sum _{j=1}^M{w}_{kj}.f_n\left(\sum _{i=1}^N{w}_{ji}{x}_i\left(t\right)+w_{jo}\right)+w_{ko}\right]$$

(1)

where $N$ is the number of input variables; $M$ is the number of hidden neurons; $x_i\left(t\right)$ is the $i$-th input variable at time step $t$; $w_{ji}$ is the weight that connects the $i$-th neuron in the input layer and the $j$-th neuron in the hidden layer; $w_{j0}$ is the bias for the $j$-th hidden neuron; $f_n$ is the activation function of the hidden neuron; $w_{kj}$ is the weight that connects the $j$-th neuron in the hidden layer and $k$-th neuron in the output layer; $w_{k0}$ is the bias for the $k$-th output neuron; $f_0$ is the activation function for the output neuron; $y_k^{\prime }\left(t\right)$ is the forecasted $k$-th output at time step $t$.

The LM algorithm is used for training because it is considered one of the fastest methods for training ANNs [66].

The ANN model with MLP architecture is among the most studied [30] and is frequently used in hydrological forecasting due to its design simplicity [28,67]. The MLP model has been used in different studies involving climatological variables, e.g., Dimitriadou et al. [68] and Vafaeipour et al. [69], among others, and in drought prediction, e.g., Jalalkamali et al. [70], Belayneh et al. [71], Choubin et al. [34] and Achour et al. [49], among others.

2.6. Development of the MLP Models

The development of the MLP models for drought forecasting was conducted in three stages: (1) data preprocessing, (2) model calibration and (3) the evaluation of models and selection of the best model.

2.6.1. Data Preprocessing

ANN models frequently integrate the historical and current values of one or more input variables to forecast future values. The sliding window method is usually used to accomplish this [69,72]. This leads to the development of several models based on various combinations of the possible input variables. To prevent differences in data variability between the inputs and outputs, data must be handled through a normalization process prior to being used in the building of MLP models. As a result, the model´s parameter range is simplified. Similarly, data must be divided into separate portions for various purposes, such as training, model validation and model evaluation [73].

2.6.2. Model Calibration

MLP model development implies the optimization of different hyperparameters, such as the number of neurons and hidden layers, the number of training cycles, the training optimizer and the loss function. Typically, this is achieved through the calibration process, which is composed of stages of model training and validation. MLP model training fits the weights and biases that minimize the selected error function. This is usually achieved by using an optimization algorithm based on gradient descent [70]. However, there is a possibility of overfitting the network by continuously minimizing the error function. Thus, the model will end up memorizing individual examples rather than trends in the training dataset, resulting in a model that exhibits poor forecasting capabilities when it is supplied with data that was not used in training [74]. To prevent overfitting, the early stopping technique was used in this study. The purpose of this procedure is to stop training when the network starts becoming overfitted, which is achieved by using the training and validation data simultaneously during network training.

2.6.3. Model Evaluation

The last stage of model development is based on evaluating the forecasting capacity of the calibrated models using the trial dataset, which is not used in the calibration stage. To measure model performance, four statistical metrics were used in this study: the determination coefficient ($R^2$), the root mean squared error (RMSE), the mean absolute error (MAE) and the mean bias error (MBE). The ANN model, which performs better according to the four statistical metrics, was selected as the final drought forecasting model. Figure 3 shows a schematic representation of the methodology followed in this research for drought forecasting.

• Determination Coefficient

The determination coefficient ($R^2$) shows the percentage of variability in $y$ explained by the $x$ variables, where $x$ and $y$ represent a set of data. $R^2$ is calculated using Equation (2):

$$R^2={\left({\displaystyle \frac{{\sum }_{i=1}^n\left(y_{o,i}-{\overline{y}}_o\right)\left(y_{p,i}-{\overline{y}}_p\right)}{\sqrt{{\sum }_{i=1}^n{\left(y_{o,i}-{\overline{y}}_o\right)}^2{\sum }_{i=1}^n{\left(y_{p,i}-{\overline{y}}_p\right)}^2}}}\right)}^2$$

(2)

where n is the total number of data points in the drought index series; $y_{o,i}$ and $y_{p,i}$ represent, respectively, the observed (calculated) and forecasted monthly values of the SPI, SPEI and RDI indices; and ${\overline{y}}_o$ and ${\overline{y}}_p$ represent the mean values of said indices.

• Root Mean Squared Error

The root mean squared error (RMSE) measures the square root of the mean standard deviation between the observed and forecasted values. It is used when an error is highly nonlinear. The RMSE is calculated using Equation (3):

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_{p,i}-y_{o,i}\right)}^2}$$

(3)

• Mean Absolute Error

The mean absolute error (MAE) measures the sum of the absolute variation between the forecasted and the observed values. The MAE is calculated using Equation (4):

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_{p,i}-y_{o,i}\right|$$

(4)

• Mean Bias Error

The mean bias error (MBE) measures the average difference between the forecasted and actual values. If MBE < 0, the model underestimates the actual values, and if MBE > 0, it overestimates these values. The MBE is calculated using Equation (5):

$$MBE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left(y_{p,i}-y_{o,i}\right)$$

(5)

2.7. Programming Tools

The filling in of missing data, the evaluation of homogeneity of precipitation and temperature data series, the calculation of ETP and the calculation of the three drought indices were performed in MATLAB R2020b. On the other hand, the forecast of the drought indices SPI, SPEI and RDI with ANN (with MLP architecture) was carried out using Python 3.8 under the Tensorflow 2.9.2, Keras 2.9.0 and Scikit-Learn 1.0.2 libraries on the Google Colab platform.

3. Results and Discussion

Once the daily data series for the precipitation and minimum and maximum temperatures were completed, the homogeneity of these series was tested at a 5% significance level using the statistical tests: Student’s t-test, Cramer, Buishand and SNHT. Regarding the data from the Agua Nueva weather station, the precipitation data were homogeneous according to two of the statistical tests, and the minimum and maximum temperature series were homogeneous according to three of the analysis tests. On the other hand, for the data from the Calera weather station, the precipitation data were found to be homogeneous according to four statistical tests, while the minimum and maximum temperature series were considered homogeneous according to two and four statistical tests, respectively. Based on the results obtained, the data series under study are considered homogeneous.

In this study, different ANN models with a three-layer MLP (ANN MLP) architecture were developed. The models developed are based on a combination of past values for the SPI, SPEI and RDI indices for the 3-, 6- and 12-month analysis scales. From 3 to 6 past values were used in this study to make one month in advance forecasts $\left(t+1\right)$ (

Table 2. Inputs and outputs for all tested neural network models.

Model Output	Model Input	Window Size
3	${DI}_t,{DI}_{t-1},{DI}_{t-2},$	${DI}_{\left(t+1\right)}$
4	${DI}_t,{DI}_{t-1},{DI}_{t-2},{DI}_{t-3}$	${DI}_{\left(t+1\right)}$
5	${DI}_t,{DI}_{t-1},{DI}_{t-2},{DI}_{t-3},{DI}_{t-4}$	${DI}_{\left(t+1\right)}$
6	${DI}_t,{DI}_{t-1},{DI}_{t-2},{DI}_{t-3},{DI}_{t-4},{DI}_{t-5}$	${DI}_{\left(t+1\right)}$

where DI represents the drought index used as model input and output.

). This very number of observed values is referred to as the number of neurons in the input layer [75]. All inputs and outputs were normalized using the Min–Max scaling method, which consists of scaling the data within the 0 to 1 range and implies estimating the available minimum and maximum values in the dataset [76]. Based on some research, such as [17,28,29], the dataset was divided into three subsets: the training set for the 1961–1996 period (~70% of the data), the validation set for the 1997–2004 period (~15% of the data) and the trial set, which spans from 2005 to 2012 (~15% of the data). Models were trained with the stochastic gradient descent with the momentum algorithm, using the mean squared error (MSE) as the loss function.

3.1. Calibration Stage

For the series corresponding to each drought index, the optimal model structure was identified by a trial-and-error process. The optimal structure was selected according to the lowest MSE value in the validation set. Different combinations of input and hidden neurons were tested. For each window size, between 3 and 13 neurons were tested in the hidden layer. Only one neuron was used in the output layer.

Table 3. Statistical metrics of the drought forecasting models for the SPI, SPEI and RDI indices in the 3-, 6-, and 12-month time scales for the validation and trial datasets.

Station	Index	Scale	MLP Architecture	Validation					Trial
				MSE	R2	RMSE	MAE	MBE	R2	RMSE	MAE	MBE
Agua Nueva (32001)	SPI	3	4-3-1	0.105	0.873	0.339	0.277	−0.049	0.882	0.336	0.267	−0.014
		6	4-3-1	0.064	0.923	0.266	0.211	−0.023	0.912	0.282	0.230	−0.036
		12	6-9-1	0.003	0.996	0.059	0.046	0.010	0.985	0.081	0.065	−0.027
	SPEI	3	3-3-1	0.129	0.856	0.363	0.305	−0.022	0.866	0.346	0.297	−0.043
		6	5-4-1	0.061	0.932	0.253	0.210	−0.020	0.908	0.268	0.226	−0.080
		12	6-9-1	0.005	0.995	0.069	0.060	0.026	0.983	0.090	0.064	−0.036
	RDI	3	3-8-1	0.089	0.873	0.314	0.256	−0.044	0.881	0.341	0.282	0.028
		6	3-5-1	0.039	0.950	0.204	0.163	−0.014	0.936	0.240	0.189	−0.017
		12	6-11-1	0.003	0.996	0.057	0.043	0.006	0.979	0.099	0.072	−0.019
Calera (32003)	SPI	3	3-10-1	0.191	0.817	0.443	0.370	0.068	0.822	0.412	0.352	0.076
		6	3-10-1	0.198	0.853	0.450	0.362	0.077	0.804	0.452	0.359	0.066
		12	4-10-1	0.010	0.993	0.102	0.082	0.019	0.987	0.111	0.091	0.005
	SPEI	3	3-5-1	0.215	0.765	0.466	0.364	0.051	0.713	0.571	0.435	0.154
		6	3-5-1	0.072	0.929	0.268	0.227	0.049	0.914	0.313	0.268	0.075
		12	3-11-1	0.009	0.993	0.094	0.082	0.033	0.988	0.109	0.089	0.024
	RDI	3	3-10-1	0.163	0.831	0.409	0.336	0.070	0.834	0.402	0.343	0.095
		6	6-5-1	0.146	0.894	0.388	0.294	0.075	0.867	0.385	0.289	0.084
		12	3-9-1	0.010	0.992	0.102	0.083	0.017	0.988	0.107	0.083	0.014

The bold values indicate the best results for each station, in each of the analysis scales, and in the trial dataset for each analyzed station.

shows the ANN MLP models with the best MSE result for each series, as well as the values of the performance metrics or efficiency indices for the validation and trial sets of these models.

According to the MSE results, the models experienced a lower loss for the validation set during the training stage at the 12-month scale, with values ranging between 0.003 and 0.010. The highest MSE values were obtained in the models for the 3-month timeframe, with values ranging from 0.089 to 0.215. The ANN MLP models best fit the data from the Agua Nueva (32001) station. The structure of the models in Table 3 shows that most have three neurons in the input layer, indicating that the best results are obtained using three past values for one month in an advance forecast of the drought indices. Scatter diagrams for the validation and trial datasets for the models described in Table 3 are displayed in Figure 4 and Figure 5, in which the calculated and predicted values are contrasted. These figures show that forecasts of the SPI, SPEI and RDI drought indices with the developed models improve as the scale of analysis increases. Likewise, in Figure 4d and Figure 5d, it can be seen that the models developed for the SPEI at the 3-month scale exhibit the worst fit relative to the series of calculated drought indices, both for the validation and trial datasets.

According to the results of the MBE performance metric, for the data from the Agua Nueva weather station, of the nine proposed ANN forecast models, six underestimate the values of the drought indices (MBE < 0), while for the Calera weather station, all models overestimate the values of said indices (MBE > 0).

3.2. Trial Stage

At the trial stage, the results from the Agua Nueva station show that, for the same scale of analysis, the models had a similar performance with the three drought indices. At the 3-month scale, with a 4-3-1 architecture model, the SPI performed slightly better than the SPEI and RDI, yielding values of R², RMSE, MAE and MBE of 0.882, 0.336, 0.267 and −0.014, respectively. At the 6-month time scale, the RDI with a 3-5-1 architecture model had the best performance, with the following statistical metrics: 0.936, 0.240, 0.189 and −0.017 for R², RMSE, MAE and MBE, respectively. And at the 12-month time scale, the SPI, with a 6-9-1 architecture model, produced the best results, yielding values of 0.985 for the R², 0.081 for the RMSE, 0.065 for the MAE and −0.027 for the MBE.

For Calera station, results of the efficiency indices show that, at the 3-month scale, the RDI forecast with a 3-10-1 architecture model had the best results, with values of 0.834 for R², 0.402 for RMSE, 0.343 for MAE and 0.095 for MBE. At the 6-month scale, the SPEI forecast with a 3-5-1 model yielded the best results, with efficiency index values of 0.914, 0.313, 0.268 and 0.075 for R², RMSE, MAE and MBE, respectively. Finally, at the 12-month scale, the RDI with a 3-9-1 model had the best performance, with values of 0.988 for R², 0.107 for RMSE, 0.083 for MAE and 0.014 for MBE.

At the 12-month scale, the models exhibited similar performances, with few variations in the results of the efficiency indices for the forecasts of the three drought indices at both weather stations. Additionally, compared to the other two scales of analysis, the results from the 12-month scale were the best. It was discovered that as the scale of analysis becomes smaller, the models’ performance decreases, with the 3-month scale showing the lowest model performance.

As previously stated, of all three drought indices and all three scales of analysis, the worst ANN MLP model fit was for the SPEI at the 3-month scale (Figure 4d and Figure 5d). Figure 6 and Figure 7 show the behavior of the calculated and forecasted SPEIs for the trial dataset at the three scales of analysis for the Agua Nueva and Calera stations, respectively. In these figures, it can be seen that the forecast of this index is very good at the 12-month scale (Figure 6c and Figure 7c) and even at the 6-month scale (Figure 6b and Figure 7b), confirming what the statistical metrics (Table 3) and the results of Figure 4 and Figure 5 indicate. However, at the 3-month time scale, although the forecasts have the same behavior (Figure 6a) and produce similar values, differences between the maximum and minimum calculated and forecasted values are, in some cases, notorious for changing the category of wet and dry events (Figure 7a).

At this stage, the results of the MBE performance metric show that for the Agua Nueva weather station, with the exception of the model for the RDI on the 3-month scale, all the proposed ANN forecast models underestimate the values of the three drought indices analyzed (MBE < 0), while for the data from the Calera weather station, all the proposed models for the three drought indices and the three analysis scales overestimate the values of said indices (MBE > 0).

In Mexico, studies on drought forecasting using drought indices and artificial neural networks are scarce. Hernández-Vásquez et al. [23] used a multi-layer perceptron neural network with feedforward connections to forecast the drought in northwest Mexico using the SPI and SPEI drought indices at the 3-, 6-, 12- and 24-month time scales. Magallanes-Quintanar et al. [77] used a nonlinear autoregressive artificial neural network with exogenous inputs (NARX) to forecast the drought in central Mexico using the SPI drought index at the 12-month time scale. On the other hand, Magallanes-Quintanar et al. [24] used a multi-layer perceptron neural network model to forecast the drought in central Mexico (Zacatecas) using the SPI drought index at the 12-month time scale. Magallanes-Quintanar et al. [78] also evaluated the drought forecast in the same study region using the SPI on a 12-month time scale with auto-machine-learning models. None of these drought forecasting investigations used the RDI drought index.

The results found in this investigation of the R² values for the validation stage for the SPI-3 (0.817–0.873), SPI-6 (0.853–0.923), SPI-12 (0.993–0.996), SPEI-3 (0.765–0.856), SPEI-6 (0.929–0.932) and SPEI-12 (0.993–0.995) drought indices are higher than those obtained by Hernández-Vásquez et al. [23] for the same indices, SPI-3 (0.54–0.74), SPI-6 (0.58–0.83), SPI-12 (0.68–0.89), SPEI-3 (0.51–0.69), SPEI-6 (0.63–0.90) and SPEI-12 (0.62–0.94). For their part, Magallanes-Quintanar et al. [24] in the validation stage, report R² values for the SPI-12 drought index in the range of 0.843–0.879, which are also lower than those found in this research (0.993–0.996); furthermore, these authors report MAE values in the range of 0.071–0.083, which are slightly higher than the values found here for this performance metric (0.046–0.082). On the other hand, for the test stage, Magallanes-Quintanar et al. [77] report R² values for the SPI-12 drought index in the range of 0.799–0.928, which are lower than those found in this research (0.985–0.987). In this regard, Magallanes-Quintanar et al. [78], for this same stage, obtained R² values for the SPI-12 in the range of 0.871–0.930, which are also lower than those found in this study.

Likewise, the results of this research are similar to those presented by Tuğrul and Hinis [79] in Türkiye, by Choubin et al. [34] in southwestern Iran, by Zhang et al. [29] in northern China and by Achour et al. [49] in northwestern Algeria, who successfully used ANN models to forecast the SPI drought index at different scales of analysis, and comparable to those presented by Ghasemi et al. [80], Karbasi et al. [41] and Karbasi et al. [81] in Iran to forecast drought using the SPEI drought index at different scales of analysis. Although it is true that the conditions of this study are different from the conditions of the reference studies cited (regional characteristics, length of the data series, analysis period, etc.), its statistical metrics are used only to compare the performance of the ANN MLP models developed here.

In general, according to the results of the four statistical metrics at the three scales of analysis, the RDI was the drought index best fitted by the developed ANN MLP models at Calera station. On the other hand, at Agua Nueva station, the results from the ANN MLP models show little variation among the three drought indices at the three scales of analysis, with the SPI being the drought index best fitted by the ANN MLP models, winning over the RDI index by a very small margin.

The ANN models developed present satisfactory results; however, they correspond to the analyses of only two data series, so it is advisable to employ a larger number of data series to extend the analysis of the drought forecast in the analysis region. Similarly, it is convenient to explore other types of machine learning models that have produced satisfactory results in other studies, and that may allow for improving the results obtained here.

4. Conclusions

In this study, drought forecasting was analyzed by applying an ANN model with an MLP architecture to two climatological stations in the state of Zacatecas and predicting the SPI, SPEI and RDI drought indices using past values as the predictors. The dataset used in this research covers the period from 1961 to 2012.

A comparison of validation and trial set results shows that, in most cases, the results of the performance metrics are slightly better for the validation set. Furthermore, the difference between these sets indicates that the models have the ability to generalize forecasts to data other than the training and validation sets used in the calibration process. Evaluation of the drought index results shows that for both the validation and the trial datasets, model performance increases for greater scales of analysis; that is, the drought index forecasts are better at the 12-month scales than at the 6- and 3-month scales. This is because as the scale of analysis increases, drought events become less frequent but last longer, which makes learning easier for the ANN MLP models. In general, the analyzed models showed great capacity to forecast the values of the three drought indices at the three time scales of analysis, and it was found that for the trial set, the RDI was the drought index that was best fitted by the models, followed by SPI and SPEI. Its ability to integrate both precipitation and potential evapotranspiration positions it as the best alternative for forecasting drought using an ANN model.

Based on the obtained results, it appears that the developed ANN MLP models may prove to be an effective instrument for predicting droughts in the central region of the Mexican state of Zacatecas. To ensure that these methods are effective across the state’s various climate zones, it is best to apply them to a greater number of data series of drought indices. In order to better understand the characteristics of droughts in the study region and, consequently, be able to establish strategies for the mitigation of drought in different watersheds, it is also advised that future studies apply ANN models with architectures different from those used in this study and explore the forecast of droughts for two or more months in advance.