SPEI Drought Forecasting in Mexico

1. Introduction

Droughts are among the most devastating climatic phenomena for human societies and ecosystems, due to their direct impact on water availability, food security, public health and economic development. In the context of climate change, an increase in the frequency, duration and intensity of these events has been observed, particularly in regions of high population density such as the State of Mexico and Mexico City, where pressure on water resources is intensifying [1].

Drought monitoring and forecasting require accurate indicators that are adaptable to different climatic contexts. In this sense, the Standardized Precipitation Evaporative Index (SPEI) has gained relevance due to its capacity to integrate precipitation and potential evapotranspiration (PET), allowing a more comprehensive evaluation of the water balance and its temporal evolution. Unlike other indices such as the SPI, the SPEI allows for the characterization of different types of droughts—meteorological, agricultural, and hydrological—through variable time scales, making it a key tool for water resources planning and management [2,3,4].

In Mexico, several studies have demonstrated the usefulness of SPEI for the retrospective analysis of drought events and its application in the design of public policies. However, its use as a target variable in predictive models still represents an area of scientific development with great potential. Statistical models such as ARIMA, and more recent approaches based on artificial intelligence, such as artificial neural networks (ANN) and support vector machines (SVM), have been successfully applied in SPEI prediction, showing higher accuracy in scenarios of high climate complexity [5].

Forecasting SPEI values using these techniques could substantially improve early warning systems, facilitating informed decision making in sectors such as agriculture, urban water management, and civil protection. In addition, the ability of these models to use accessible meteorological variables such as maximum and minimum temperature and precipitation make them viable and scalable tools for local management [6].

In the search for more accurate methodologies for drought prediction, advanced models such as Artificial Neural Networks (ANN), Kalman Filters and their variants with exogenous variables have been developed. These approaches not only allow the estimation of current conditions but also enable the prediction of drought behavior through time series. Models based on the Kalman Filter with exogenous variables (DKF-ARX-Pt) have proven to be effective in integrating additional climatic information, such as precipitation and temperatures, improving forecast accuracy [7]. Similarly, Recurrent Neural Networks such as GRU and NARX models have shown their ability to identify nonlinear patterns in time series, which is fundamental in modeling extreme weather events [8,9].

The objective of this work is to develop and evaluate forecast models for SPEI in the region of the State of Mexico and Mexico City, using advanced machine learning techniques and statistical methods, to generate reliable predictions at different time horizons. The aim is to contribute to the construction of scientific tools to mitigate the adverse effects of droughts and strengthen climate resilience in highly vulnerable urban and peri-urban areas.

2. Materials and Methods

2.1. Description of the Study Area

In recent years there has been an urban growth reflected in the development of new population settlements and civil works developments. According to the last population census conducted by INEGI, the State of Mexico has an area of 22,351.8 km², which represents 1.1% of Mexico’s surface area; however, it is the most populated state with 16,992,418 inhabitants and Mexico City (CdMx) has an area of 1,494.3 km² which represents 0.1% of the country’s surface; Mexico City has 9,209,944 inhabitants. The study area is located between 2,250 and 2,750 meters above sea level, the predominant climate is dry temperate, with an average annual temperature of 15.9 °C and an average annual rainfall of 686 mm with a climate known as sub-humid temperate and extents all the State of Mexico and Mexico City [10].

2.2. Climatological Data

The analysis of 23 meteorological stations across the area of interest over a 42-year period (January 1981 to December 2023) provides a robust dataset for understanding regional climatic trends (Figure 1). Notably, the continued operation of 59% of these stations underscores the challenges of maintaining long-term observational networks. To address gaps and ensure comprehensive data, the geographic coordinates of each station were employed to recover supplementary climatic data via the Climate Engine platform (http://ClimateEngine.org) [11].

The CHIRPS database was selected due to its capacity to provide precipitation data on daily, pentad, and monthly scales, offering critical granularity. Similarly, the DAYMET database contributed valuable daily and monthly data, complementing the CHIRPS dataset. This dual approach ensures greater reliability in climatic trend analysis, particularly for variables such as precipitation and temperature.

The comparative validation of CHIRPS and DAYMET datasets using observed data from station 15170 in Chapingo is a critical step in assessing data reliability. The results are highly encouraging, with coefficients of determination (R²) of 0.8921 for precipitation, 0.9974 for maximum temperature, and 0.9824 for minimum temperature. These high R² values indicate strong correlations, affirming the datasets suitability for climatological analysis [12].

2.3. CHIRPS Satellite Information

The CHIRPS algorithm combines three data sources: (a) the CHPclim precipitation climatology, a 0.05° global climatology based on data from stations, satellite observations, elevation, and coordinates [13]; (b) satellite estimates of infrared thermal precipitation (IRP); and (c) in situ rain gauge measurements. CHPclim differs by using long-term satellite averages to improve its accuracy, especially in mountainous regions [14]. Combining CHIRP with station data uses the estimated correlation between a pixel’s precipitation and nearby stations [15].

Data merging is performed on 5-day (pentad) and monthly scales, adjusting the pentads so that their sum matches the monthly total. A daily version is generated from this data. Preliminary CHIRPS is published 2 days after a pentad, and the final version is in the third week of the following month.

2.4. DAYMET Satellite Data

The Daily Surface Weather Data provides continuous, gridded estimates of daily weather variables by interpolation and statistical modeling. It includes minimum and maximum temperature, precipitation, vapor pressure, shortwave radiation, snow water equivalent, and day length, on a 1 km² grid for North America and Hawaii since 1980, and Puerto Rico since 1950.

The Daymet method estimates meteorological parameters at instrumented sites by interpolation and extrapolation, using weighted data from nearby stations. In the Daymet V4 version, station selection is optimized by eliminating the iterative density calculation, defining instead a fixed search radius based on pre-computed distances. Two workflows generate variables: one for daily temperature and one for precipitation [16].

2.5. Standardized Precipitation Evapotranspiration Index (SPEI)

The Standardized Precipitation Evapotranspiration Index (SPEI) is an extension of the SPI that incorporates both precipitation and potential evapotranspiration (PET) to evaluate drought, considering the impact of temperature increase on water demand.

SPEI is calculated using the water balance (Di = P − ETo) at different time scales (1, 3, 6, 12 and 24 months). Unlike SPI, which only uses precipitation, SPEI provides a more accurate measure of drought severity by including atmospheric evaporative demand.

Initially, the Thornthwaite equation [17] was proposed to estimate evapotranspiration due to its simplicity, but more accurate methods such as the FAO-56 Penman-Monteith equation [18] are preferable if sufficient data are available. Failing this, the Hargreaves equation, which was used in this study, is recommended.

The SPEI calculation was performed in RStudio^® with the SPEI.R software version 1.8.1 [19], using data on precipitation, minimum and maximum temperature, and latitude of the study site. The values obtained were analyzed according to the classification of McKee, Doesken & Kleist [20] to identify the evolution of droughts, their intensity, duration and dates of onset and termination.

Table 1. SPEI categories and classification [20].

Index value	Category
> 2.00	Extremely humid
1.50 a 1.99	Very humid
1.00 a 1.49	Moderately humid
-0.99 a 0.99	Near normal
-1.00 a -1.49	Moderately dry
-1.50 a -1.99	Very dry
< -2.00	Extremely dry

Table 1. SPEI categories and classification [20].

Index value	Category
> 2.00	Extremely humid
1.50 a 1.99	Very humid
1.00 a 1.49	Moderately humid
-0.99 a 0.99	Near normal
-1.00 a -1.49	Moderately dry
-1.50 a -1.99	Very dry
< -2.00	Extremely dry

2.6. Selection of Representative Weather Stations

For the identification of representative weather stations in the context of climate change and drought assessment, a quantitative approach was implemented based on the nonparametric Mann-Kendall statistical test [21,22], widely used for the detection of monotonic trends in climatic time series [23,24].

Three fundamental climatic variables were analyzed: monthly cumulative precipitation (PCP), maximum temperature (TMAX) and minimum temperature (TMIN), using data from 381 meteorological stations. Z-statistic values were calculated for each variable at each station. Significant trends were considered to be those whose absolute value of the Z statistic was greater than or equal to 1.96 (95% confidence level).

The stations were classified into three levels of representativeness:

• Highly representative: stations with significant trends in two or more variables.
• Moderately representative: stations with only one significant variable.
• Unrepresentative: stations with no significant trends.

This classification approach has been used in similar studies to identify key stations for climate change monitoring. For example, Ahmad et al. [25] applied criteria based on the number of variables with significant trend to select critical stations in a South Asian basin. Similarly, Kousari et al. [26] classified stations according to the magnitude and significance of detected changes in temperature and precipitation, which allowed identifying control points for vulnerability analysis. In addition, Gong et al. [27] proposed a regional pattern recognition methodology that included the hierarchization of stations by their degree of climate response, which is compatible with the classification used in this study.

Additionally, a climate pattern was assigned to each station based on the combination of detected trends, following criteria similar to those employed by recent studies on the classification of local climate responses [27,28,29].

• Dry and warm: significant decrease in precipitation and increase in TMAX or TMIN.
• Dry and cold: decrease in precipitation and decrease in temperature.
• Strong warming: significant increase in TMAX or TMIN without relevant changes in precipitation.
• Climatic stability: no significant trends.

Table 2. shows an example of the classification of stations with their respective Z-values and assigned weather patterns.

ID	Name	PCP	TMAX	TMIN	Classification	Weather pattern
9002	Ajusco	-1.53	12.39	3.55	Moderately Representative	High temperature
9004	Calvario 61	-1.07	11.86	3.73	Moderately Representative	High temperature
9014	Santa Úrsula Coapa	-1.18	4.51	3.97	Moderately Representative	High temperature
9019	Desierto De Los Leones	-2.35	9.95	0.79	Highly Representative	Dry and warm
9022	El Guarda	-1.67	14.1	1.22	Moderately Representative	High temperature
9026	Morelos 77	0.2	0.64	2.61	Moderately Representative	Mixed change
9067	Monte Alegre	-2.23	7.42	-4.04	Highly Representative	Dry and warm
15001	Acambay	-1.01	6	9.11	Moderately Representative	High temperature
15002	Aculco (Smn)	-1.56	10.44	5.45	Moderately Representative	High temperature
15006	Amatepec	-1.76	8.51	-6.14	Highly Representative	High temperature
15007	Amecameca De Juárez	-1.38	8.64	3.67	Moderately Representative	High temperature
15009	Atlacomulco	-2.23	7.74	6.29	Highly Representative	Dry and warm
15023	Chimalhuacán	0.02	5.5	3.59	Moderately Representative	High temperature
15034	Ixtapan De La Sal	-2.03	9.71	1.96	Highly Representative	Dry and warm
15036	Ixtlahuaca (Smn)	-1.51	8	5.45	Highly Representative	High temperature
15061	Nezahualcóyotl	0.68	1.02	2.04	Moderately Representative	Mixed change
15062	Nevado De Toluca	-2.31	15.35	-1.14	Highly Representative	Dry and warm
15066	Palizada	-2.58	13.07	6.88	Highly Representative	Dry and warm
15118	Temascaltepec	-2.9	9.01	3.49	Highly Representative	Dry and warm
15131	Villa De Allende	-2.53	12.1	3.76	Highly Representative	Dry and warm
15170	Chapingo (Dge)	-0.19	2.35	6.62	Moderately Representative	Humid and warm
15231	Presa Iturbide	-4.04	12.09	6.43	Highly Representative	Dry and warm
15277	San Miguel Tenochtitlan	-2.48	7.43	5.25	Highly Representative	Dry and warm

Table 2. shows an example of the classification of stations with their respective Z-values and assigned weather patterns.

ID	Name	PCP	TMAX	TMIN	Classification	Weather pattern
9002	Ajusco	-1.53	12.39	3.55	Moderately Representative	High temperature
9004	Calvario 61	-1.07	11.86	3.73	Moderately Representative	High temperature
9014	Santa Úrsula Coapa	-1.18	4.51	3.97	Moderately Representative	High temperature
9019	Desierto De Los Leones	-2.35	9.95	0.79	Highly Representative	Dry and warm
9022	El Guarda	-1.67	14.1	1.22	Moderately Representative	High temperature
9026	Morelos 77	0.2	0.64	2.61	Moderately Representative	Mixed change
9067	Monte Alegre	-2.23	7.42	-4.04	Highly Representative	Dry and warm
15001	Acambay	-1.01	6	9.11	Moderately Representative	High temperature
15002	Aculco (Smn)	-1.56	10.44	5.45	Moderately Representative	High temperature
15006	Amatepec	-1.76	8.51	-6.14	Highly Representative	High temperature
15007	Amecameca De Juárez	-1.38	8.64	3.67	Moderately Representative	High temperature
15009	Atlacomulco	-2.23	7.74	6.29	Highly Representative	Dry and warm
15023	Chimalhuacán	0.02	5.5	3.59	Moderately Representative	High temperature
15034	Ixtapan De La Sal	-2.03	9.71	1.96	Highly Representative	Dry and warm
15036	Ixtlahuaca (Smn)	-1.51	8	5.45	Highly Representative	High temperature
15061	Nezahualcóyotl	0.68	1.02	2.04	Moderately Representative	Mixed change
15062	Nevado De Toluca	-2.31	15.35	-1.14	Highly Representative	Dry and warm
15066	Palizada	-2.58	13.07	6.88	Highly Representative	Dry and warm
15118	Temascaltepec	-2.9	9.01	3.49	Highly Representative	Dry and warm
15131	Villa De Allende	-2.53	12.1	3.76	Highly Representative	Dry and warm
15170	Chapingo (Dge)	-0.19	2.35	6.62	Moderately Representative	Humid and warm
15231	Presa Iturbide	-4.04	12.09	6.43	Highly Representative	Dry and warm
15277	San Miguel Tenochtitlan	-2.48	7.43	5.25	Highly Representative	Dry and warm

This methodology not only allows an objective and reproducible selection of stations for monitoring and analysis of climate change but also facilitates the zoning of areas with homogeneous climate behavior, which is essential for regional drought studies (SPEI) and planning of adaptation measures. Moreover, the approach has been validated in similar research in regions of Mexico and other parts of the world [25,30].

2.7. Drought Forecasting

Anticipating the characteristics of a drought (magnitude, duration and distribution) is key to mitigating its effects [31]. Accurate forecasting improves water resource management and is based on climatic indicators such as precipitation or specialized indices such as SPI and SPEI.

Mishra & Singh [32] discuss various drought forecasting methodologies, including regression, time series, probabilistic models, artificial neural networks (ANNs), hybrid approaches and data mining. In recent years, ANNs have stood out for their machine learning capabilities [33], being applied in hydrology to model rainfall-runoff, flow rates, groundwater and water quality [34]. Their main advantage is to identify nonlinear patterns without the need for predefined mathematical models [35].

The combination of neural networks and Kalman filters has been studied in drought prediction and SPEI calculation. Models such as DKF-ARX-Pt (Kalman Filter with exogenous variables such as precipitation and temperature), GRU (Closed Recurrent Unit) and NARX (Neural Autoregressive Networks with External Input) have been applied.

2.8. Kalman Filter with Exogenous Variables (DKF-ARX-Pt)

The Kalman Filter with Exogenous Variables, known as DKF-ARX-Pt, is an extension of the classical Kalman Filter, used to estimate the state of dynamic systems under uncertainty, but incorporating external or exogenous variables that influence the process. Traditionally, the Kalman Filter is used in linear systems where a model of the system is known, and the state of the system is estimated as noisy observations are received. However, in complex phenomena such as drought, prediction models must consider not only past observations of the system itself (such as drought indices), but also external factors that affect its evolution, such as precipitation and temperature [7].

The ARX (AutoRegressive with Exogenous Input) model is fundamental in the formulation of the DKF-ARX-Pt, since it extends the classical auto-regressive time series model to include inputs from external variables. This model takes the form:

Where:

• y(t) is the dependent variable (e.g., the drought index as SPEI),
• u(t) is the exogenous input (such as precipitation or temperature),
• ai and bj are the autoregressive and exogenous coefficients, respectively,
• e(t) is the model error.

In the context of the Kalman Filter, the DKF-ARX-Pt incorporates these exogenous inputs within the state estimation framework, allowing the model to adjust not only based on system observations, but also to the influences of external climatic variables.

2.9. Gated Recurring Units (GRU)

Gated Recurrent Units (GRU) are a type of recurrent neural network (RNN) that is highly effective in time series modeling, including the prediction of climate variables such as drought. Introduced by Cho et al. [36], RUs have been used for problems where long-term dependence and gradient propagation are critical, such as meteorological data analysis and prediction of climatic phenomena.

Their main advantage over traditional RNNs lies in their ability to mitigate the gradient fading problem, allowing them to capture long-term relationships in temporal data. This is particularly relevant in drought prediction, where current conditions may be influenced by climatic events that occurred months or even years ago [37].

2.10. Autoregressive Neural Networks with External Input (NARX)

Neural Autoregressive Networks with External Input (NARX) have proven to be an effective technique for modeling nonlinear dynamical systems, including drought prediction. Proposed by Lin et al. [38], NARX extends classical autoregressive models by capturing nonlinear relationships between climate variables, differing from traditional approaches such as ARIMA or ARX. In SPEI prediction, they can integrate exogenous variables such as precipitation and temperature to improve accuracy. Hernandez-Vasquez et al. [39] applied artificial neural networks in the Sonora River basin, obtaining an average R² of 0.76 in validation, evidencing its predictive capacity.

2.11. Prediction Model Evaluation Metrics

In the evaluation of time series and weather prediction models, it is essential to use metrics that allow measuring performance in terms of error and adjustment capacity. The metrics used in the comparison of SPEI prediction models are described below.

MAE (Mean Absolute Error), MSE (Mean Squared Error) and RMSE (Root Mean Squared Error) are metrics that measure absolute and quadratic error, with RMSE penalizing larger errors. R² (Coefficient of Determination) and NSE (Nash-Sutcliffe Efficiency Coefficient) assess the model’s ability to explain the variability of the data, with NSE more commonly used in hydrologic models. KGE (Kling-Gupta Efficiency) offers a more complete evaluation of the model, considering correlation, bias and variability, being more useful in climate and hydrological studies.

This set of metrics provides a comprehensive view of model performance, allowing the most appropriate model to be selected for SPEI prediction.

3. Results

Three methodological approaches were applied to forecast the SPEI index in 23 stations in the State of Mexico and Mexico City: the Kalman Filter with exogenous variables (FK or DKF-ARX-Pt), autoregressive neural networks with external input (NARX), and closed recurrent units (GRU). The models were evaluated with predictions at 3 and 6 months, and their accuracy was measured using MAE, RMSE, R², NSE and KGE metrics (Table 3).

The NARX model was the most accurate in absolute terms, with the lowest MAE and RMSE values and the highest R², NSE and KGE. This indicates that it was able to reproduce both the general trend and the internal variability of the SPEI series with a high degree of fidelity. GRU also performed outstandingly well, especially at intermediate scales (SPEI3 to SPEI6), where it offered accuracy comparable to NARX with lower computational demands.

The FK model presented (Figure 2) the best overall performance in predicting the 3-month SPEI3: low average MAE (~0.10), indicating a small average deviation from the observed values. RMSE also low (~0.13), confirming the model’s consistency in prediction. R² ≈ 0.98, implying that the model explains 98% of the observed variability of SPEI3. NSE ≈ 0.98, which is considered excellent by hydrological standards. KGE ≈ 0.85, a sign that the model adequately reproduces the distribution, skewness and variability of the data. It is the most reliable model for short-term SPEI3 prediction, with high accuracy and robustness.

The GRU model showed moderate performance compared to FK: average MAE around 0.17, higher than FK. RMSE also higher (~0.21), indicating higher error dispersion. R² ≈ 0.69, below the threshold of good fit. NSE ≈ 0.69, reflecting an acceptable but not excellent fit. KGE ≈ -3.3, a very negative value evidencing imbalances between the mean, variance and correlation of the predictions with respect to the observed data. Although GRU can capture certain trends, its overall performance is inconsistent and needs optimization.

The NARX model performed similarly to GRU, but with slight improvements in some metrics: average MAE ≈ 0.17, comparable to GRU. RMSE around 0.21, not much improvement over GRU. R² ≈ 0.69, like GRU and well below FK. NSE ≈ 0.69, indicating adequate but not outstanding performance. KGE ≈ -0.51, better than GRU but still negative, suggesting that the model does not adequately reproduce the observed SPEI statistical structure. Although NARX approaches the performance of GRU, it also does not reach the accuracy and stability of the FK model.

FK (Kalman Filter) clearly outperforms GRU and NARX on all metrics. Both GRU and NARX present difficulties in capturing the SPEI3 structure, reflected in their low NSE and negative KGE. For practical applications, especially early warning systems or water planning, the FK model is the most reliable at this scale and time scale.

The FK was the best performing model in predicting the 3-month SPEI24 (Figure 3). The metrics reflect excellent fit, low MAE (0.0325) and low RMSE (0.049), indicating accuracy and stability. R² and NSE ≈ 0.9975, showing an over-salient ability to replicate temporal variability. KGE ≈ 0.95, meaning that the model not only predicts accurately, but also preserves the statistical properties of the observed index. Ideal for prediction of prolonged droughts such as those captured by SPEI24. High reliability for warning and water management systems.

The GRU model, despite having a good overall fit (R² and NSE ≈ 0.92), presents important limitations, Errors notoriously larger than FK (MAE of 0.10). Negative KGE (-1.13), indicating that the model does not correctly replicate the structure of the index (bias, variability, correlation). This suggests that although the GRU can follow the general trend, its statistical performance is inadequate. Acceptable in trend, but not reliable for operational decisions without additional adjustments.

NARX slightly outperforms GRU in accuracy, MAE ≈ 0.091 and RMSE ≈ 0.117, slightly better than GRU. R² and NSE ≈ 0.935, slightly higher as well. However, KGE is still negative (-0.71), which compromises its use in statistical structure-sensitive applications. Better than GRU, but still not competitive against FK. Potential for improvement if hyperparameters are adjusted.

Kalman Filter (KF) is clearly superior for SPEI24 forecasting 3 months in advance (Table 4). Its ability to maintain accuracy, low error spread, and statistical replication makes it the most reliable model. The GRU and NARX models, although offering acceptable R² and NSE values, fail to correctly reproduce the index structure (evidenced by their negative KGE values), which limits their practical applicability without significant adjustments.

4. Discussion

The results obtained in the present study show significant differences in the performance of the three models evaluated for the forecast of the Standardized Precipitation Evaporative Index (SPEI), especially on medium and long-time scales. Although all models showed high levels of accuracy in terms of R² and NSE (above 0.97), a more detailed analysis of complementary metrics such as MAE, RMSE and especially Kling-Gupta Efficiency (KGE), allows establishing clearer advantages and limitations between the approaches.

FK (Kalman Filter) is the best model at all SPEI scales (Table 5), except at SPEI1 where all models fail. GRU performs acceptably at SPEI9-24 but has significant instability at short scales such as SPEI1-3. NARX is competitive on SPEI9-24, although its negative KGE on many scales suggests that it does not replicate well the statistical structure of the observed SPEI.

The NARX model (Autoregressive Neural Networks with External Input) proved to be the most accurate, with the lowest mean absolute error (MAE = 0.098) and the highest KGE efficiency (0.864). This indicates that NARX not only achieves high correlation with the observed data, but also adequately reproduces the natural variability of SPEI, without notable systematic biases. Its ability to incorporate extemporaneous inputs and maintain long-term memory makes it particularly suitable for predicting cumulative climate indicators such as SPEI, especially at scales such as SPEI6 to SPEI12.

On the other hand, the GRU (Gated Recurrent Unit) model showed a performance very close to that of NARX, with an MAE of 0.101 and a KGE of 0.829. GRU, being a simplified recurrent architecture with respect to LSTM, offers an additional advantage in terms of computational efficiency and lower training complexity. Its good performance, especially at SPEI3 and SPEI6 scales, makes it a viable option for applications where fast response is required or limited computational resources are available.

In comparison, the FK model (Kalman Filter with exogenous variables DKF-ARX-Pt), although achieving an R² of 0.976 and NSE of the same value, showed lower performance on the KGE metric (0.735). This suggests that, although the model can accurately capture the overall trend of the index, it has more difficulty in faithfully re-presenting the dispersion and error structure, especially in short scales such as SPEI1, where its performance was significantly lower (KGE < 0). Nevertheless, the FK model is still useful for long scales (SPEI12 and SPEI24), where its dynamic structure makes it stable and less susceptible to overfitting.

A key aspect in the discussion is the role of the SPEI scale in the performance of the models. It was confirmed that the intermediate scales (SPEI3 to SPEI12) represent the best compromise between sensitivity and predictive stability. The SPEI1 scale, due to its high variability and dependence on very short-term weather events, presented high errors and lower reliability for all models, while SPEI24, although stable, smooths the data too much and may miss critical information on rapid drought events.

In addition, the analysis revealed that models with memory (GRU and NARX) tend to perform better than FK, especially in prediction horizons of 3 to 6 months, which is highly relevant for early warning systems, where changes in water conditions need to be anticipated in time to plan mitigation measures.

5. Conclusions

The present comparative analysis of Standardized Precipitation Evaporative Index (SPEI) prediction models at multiple time scales (SPEI1 to SPEI24) and 3 and 6-month horizons demonstrates that the Kalman Filter (KF) is consistently the most robust, accurate and reliable model for drought forecasting applications in the State of Mexico and Mexico City region.

At practically all scales, the FK model achieved outstanding performance values: coefficients of determination (R²) and Nash-Sutcliffe efficiency (NSE) above 0.99, together with low mean errors (MAE and RMSE) and positive Kling-Gupta coefficients of efficiency (KGE) close to 1, especially at medium and long scales such as SPEI6, SPEI12 and SPEI24. This validates its ability to capture both the trend and the statistical structure of cumulative drought, making it a highly reliable tool for water management and early warning systems.

In contrast, the deep learning-based models -GRU (Gated Recurrent Unit) and NARX (Nonlinear Autoregressive Neural Network with Exogenous Inputs) showed more variable performance. While they achieved acceptable R² and NSE (between 0.84 and 0.93) at some scales and horizons, their consistently negative KGE values evidence important limitations in reproducing the mean, variability and correlation of the real data. This suggests that, although these models can follow the general trend of the index, they do not adequately preserve their integral hydrological behavior, which represents an obstacle for their direct use in operational scenarios.

In addition, SPEI1 was identified as the most difficult scale to predict for all three models, probably due to its high sensitivity to extreme monthly variations and limited hydrologic memory. In contrast, cumulative scales such as SPEI12 and SPEI24 showed greater predictive stability, especially in the FK model.

In conclusion, the Kalman Filter stands out as the most suitable model for drought forecasting in the region studied, offering an optimal balance between accuracy, stability and statistical replicability. The GRU and NARX models, while showing potential, require fine tuning in their architecture and training to achieve similar levels of performance. This analysis supports the use of FK as a robust tool for strengthening climate resilience and improving water resource planning in the face of climate variability and change scenarios.