Optimizing Time Series Models for Forecasting Environmental Variables: A Rainfall Case Study

Abstract

The application of time series models for forecasting environmental variables such as precipitation is essential for understanding climatic patterns and supporting sustainable urban planning in environments characterized by high or moderate levels of risk. This study aims to evaluate and optimize time series forecasting models for rainfall prediction in Barranquilla, Colombia. To this end, five models were applied, namely, Simple Moving Average (SMA), Weighted Moving Average (WMA), Exponential Smoothing (ES), and multiplicative and additive Holt–Winters models, using 139 monthly precipitation records from the IDEAM database covering the period 2013–2025. Model accuracy was evaluated using Mean Absolute Error (MAE) and Mean Squared Error (MSE), and nonlinear optimization techniques were applied to estimate smoothing and weighting parameters for improved accuracy. The results showed that optimization significantly enhances model performance, particularly in the multiplicative Holt–Winters model, which achieved the lowest errors, with a minimum MAE of 75.33 mm and an MSE of 9647.07. The comparative analysis with previous studies demonstrated that even simple models can yield substantial improvements when properly optimized. Furthermore, forecasts optimized using MAE were more stable and consistent, whereas those optimized with MSE were more sensitive to extreme variations. Overall, the findings confirm that seasonal models with optimized parameters offer superior predictive capacity, making them valuable tools for hydrological risk management.

1. Introduction

Currently, concern about the state of the environment has reached unprecedented levels, driven by growing evidence of the adverse effects that human activities have on natural resources and ecosystems [1,2]. The overexploitation of natural assets, habitat degradation, biodiversity loss, and the increase in pollutant emissions have intensified ecological challenges on a global scale [3,4]. In response, the scientific community, international organizations, companies, and governments emphasize the urgent need to adopt comprehensive strategies to mitigate environmental impacts and promote sustainable development [5,6,7,8]. One key element of these strategies is the development of reliable forecasting tools that anticipate environmental dynamics and support effective decision-making in policy and planning [9,10,11].

Environmental variables, such as air quality, temperature, humidity, and precipitation, are fundamental indicators for evaluating ecosystem health and understanding the interactions between natural and human systems [12,13,14]. Accurate forecasting of these variables is among the most pressing challenges in climate science and sustainability planning. Climate change has intensified the frequency and severity of extreme weather events, increasing the demand for models capable of predicting environmental risks at local and regional levels [15,16,17].

Forecasting methods applicable to environmental analysis enable the modeling of historical data and observed patterns to anticipate future scenarios [18]. These techniques simplify complex systems into interpretable structures, supporting informed decisions in environmental management, and public policy [19,20]. Their application is particularly valuable in anticipating adverse events such as heatwaves, droughts, or floods, which increasingly threaten human well-being and infrastructure [21]. In this context, precipitation forecasting is especially critical in regions vulnerable to hydrometeorological hazards.

Barranquilla, Colombia, is a coastal city located in a tropical region that experiences significant rainfall variability. Precipitation, defined as the amount of water that falls from the atmosphere in the form of rain, snow, or hail [22], is a fundamental component of the hydrological cycle and plays a key role in water availability for human consumption, agriculture, and energy generation [13]. In Barranquilla, intense rainfall events frequently lead to urban flooding, a problem exacerbated by topographic features and deficiencies in the stormwater drainage infrastructure [23]. These floods affect urban mobility, damage public infrastructure, and pose health and safety risks to communities [24,25,26]. In recent years, the city has experienced recurrent urban flooding during the rainy season, particularly between August and November, with daily precipitation often exceeding 80 mm [27]. According to local reports, over 108 people have died from being swept away by flash floods in the city and metropolitan area, with three deaths recorded in 2023 alone [28,29]. Understanding the seasonal behavior of rainfall in the city is therefore essential for building urban resilience in the face of growing climate variability.

Despite its strategic importance, rainfall forecasting in Barranquilla remains underexplored, with few locally adapted studies that rigorously evaluate or optimize classical time series models. Some recent approaches rely on fixed parameters or use a limited set of error metrics, which may reduce forecasting precision in highly variable tropical environments [30]. Addressing this gap requires a methodological approach that combines classical statistical models with robust parameter optimization techniques, enabling more accurate predictions.

This study aims to evaluate and optimize classical time series forecasting models for precipitation prediction in Barranquilla, using historical data from 2013 to 2025. The models analyzed include Simple Moving Average (SMA), Weighted Moving Average (WMA), Exponential Smoothing (ES), and the Holt–Winters models in both their multiplicative and additive forms. The novelty of this work lies in the application of nonlinear optimization techniques to adjust model parameters and enhance forecasting accuracy. Additionally, this study includes a comparative analysis with a previous work conducted in the same city [30], highlighting the performance obtained through parameter optimization. These contributions offer valuable insights for hydrological risk management and sustainable urban planning in vulnerable regions.

The structure of this article is as follows: Section 2 presents a literature review of relevant studies on the use of mathematical and computational techniques for analyzing and forecasting environmental variables. Section 3 describes the methodology, including the forecasting models, the optimization procedures, and the dataset. Section 4 presents the results and analysis of the numerical experiments. Section 5 provides a critical discussion on the practical implications of the findings and future directions. Finally, Section 6 outlines the main conclusions, highlighting contributions to climate resilience and recommendations for further research.

2. Literature Review

In recent years, predictive and forecasting models have been developed and applied to study the behavior of environmental variables in general. For example, [31] addressed the growing need to store and process data from multiple sensors in precision agriculture, particularly in remote areas. The study proposed the integration of Internet of Things (IoT) technologies to optimize cloud-based data processing and support informed decision-making. The research employed simple and multiple linear regression models to analyze relationships among variables such as temperature, light, pH, and dissolved oxygen, aiming to identify the key factors influencing microalgae growth. Studies such as [32] employed time series models to forecast precipitation in Manizales, Colombia, using ARIMA, SARIMA, SARIMAX, Prophet, and Neural Prophet, with data from the Integrated Environmental Monitoring System of Caldas (SIMAC). The results indicated a good model fit, with Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) values close to 19. The Prophet model stood out, achieving an RMSE of 19.06313 and an MAE of 16.24064, outperforming the stochastic models. These forecasts contribute to improved landslide risk management and enhance the effectiveness of early warning systems. Other studies, such as [33], proposed a deep learning approach based on Long Short-Term Memory (LSTM) networks to enhance daily streamflow prediction in watersheds with limited monitoring data. The model was evaluated in both near-natural and human-impacted basins and proved effective in predicting daily flows using aggregated monthly data. It demonstrated robustness across various stations and hydrological regimes, with a mean percentage bias within ±5%, except in arid regions. The analysis emphasized the importance of wet season data and showed that weekly data can produce results comparable to those obtained from daily observations, highlighting the model’s potential for water resource management in data-scarce areas. In [34], the authors explored the application of diffusion models, particularly appealing for their capacity to represent the probabilistic nature of climate, for short-term precipitation forecasting (1 to 3 h) in Central Europe, using data from 2016 to 2021. The study introduced a Generative Ensemble Diffusion (GED) approach, which generates multiple plausible climate scenarios and aggregates them into a final forecast, mirroring traditional ensemble simulation techniques. Compared to U-Net models and the CERRA system, GED achieved a 25% reduction in Mean Squared Error (MSE), demonstrating its effectiveness in simulating complex climate dynamics through inverse diffusion processes. In [35], machine learning and deep learning algorithms were evaluated to model changes in groundwater levels in the Coastal Aquifer System of South Africa’s West Coast. The study compared Random Forest (RF), Support Vector Machine (SVM), Simple Recurrent Neural Network (SimpleRNN), and Long Short-Term Memory (LSTM) models. The results showed that SVM outperformed the other models in terms of Root Mean Squared Error (RMSE) and overall accuracy, achieving a Mean Absolute Error (MAE) of 0.356 m and an RMSE of 0.372 m, while RF achieved the lowest MAE. The study concluded that these techniques are effective tools for improving groundwater level forecasting and recommended incorporating detailed geological information in future research to enhance model interpretability. In [36], a hybrid model combining CNN-BiLSTM and Random Forest was proposed to improve temperature forecasting by addressing the limitations of individual models in capturing nonlinear and spatiotemporal relationships. Relevant meteorological indicators were selected using Pearson correlation and mutual information analysis. Within a sequence-to-sequence framework, the CNN-BiLSTM component extracted spatiotemporal features and generated abstract data representations, which were subsequently processed by the Random Forest algorithm. This hybrid approach enhanced both the accuracy and stability of the forecasts. Experiments using meteorological data from Changsha, Hunan (2017–2021), demonstrated that the proposed model outperformed state-of-the-art methods, reducing the Mean Absolute Error (MAE) and Mean Squared Error (MSE) by 35.6% and 57.5%, respectively. In [30], the authors analyzed precipitation forecasting in Barranquilla, Colombia, highlighting its importance as a key variable for environmental management. Forecasting methods such as Simple Moving Average, Weighted Moving Average, and Exponential Smoothing (ES) were applied to monthly data from 2014 to 2024. The results indicated a significant increase in both the intensity and frequency of rainfall, particularly during the second half of the year. Among the evaluated methods, Exponential Smoothing achieved the highest accuracy, as reflected by the lowest Mean Squared Error (MSE). The study underscored the influence of climate change on rainfall variability and emphasized the need for environmental management strategies to support sustainable urban planning. The authors of [18] presented an innovative model called ST-iTransformer for air pollution forecasting, integrating spatiotemporal embeddings and inverted attention mechanisms. It used three-dimensional data from multiple monitoring stations and stages, along with meteorological variables. The model enhanced the extraction of spatiotemporal features and achieved more accurate multi-step forecasts. Validated in Beijing, China, it significantly outperformed existing models. Ablation experiments confirmed that spatiotemporal integration and the inclusion of meteorological data improved prediction accuracy. The model was identified as a promising tool for air quality forecasting and environmental management. In [19], an air pollution forecasting model was developed for Shanghai using machine learning with stacked generalization, based on a small but long-term dataset (2011–2021) of meteorological variables and pollutant concentrations (PM₁₀ and PM_2.5). The analysis incorporated Principal Component Analysis (PCA) to identify synoptic patterns and advection sources. The model, trained on 4240 samples, achieved a Root Mean Squared Error (RMSE) of 11.93 µg/m³ and an R² of 0.72, outperforming alternative methods despite the limited number of observations. It accurately predicted pollution peaks and was proposed as a valuable tool for environmental policy and public health alerts, emphasizing the need for integrated strategies to address climate change and air quality challenges. Finally, [20] evaluated the prediction of air pollutants (CO, O₃, NO₂, SO₂, PM₁₀, PM_2.5) in Tehran from 2013 to 2023 using deep learning (DL) models, compared to conventional machine learning (ML) methods. Meteorological variables such as temperature, humidity, pressure, and wind speed were included as inputs. The DL models—Gated Recurrent Unit (GRU), Fully Connected Neural Network (FCNN), and Convolutional Neural Network (CNN)—outperformed the ML models in terms of predictive accuracy, achieving high coefficients of determination (R² up to 0.9276) and a low Mean Squared Error (MSE). Among them, the FCNN model stood out for its operational efficiency. Temperature and humidity were identified as the most influential variables. The results confirmed the potential of DL models to support air quality control policies.

In general, there is limited scientific research on the application of forecasting methods for predicting precipitation in the city of Barranquilla. To address this gap, the present study applies SMA, WMA, ES, and Holt–Winters models (both additive and multiplicative), combined with nonlinear optimization techniques to determine the optimal smoothing and weighting parameters aimed at minimizing MAE and MSE. In addition, the results of this study are used to conduct a comparative analysis with a previous study [30], aiming to highlight the advantages of using optimization techniques in parametric forecasting methods.

Although various modern approaches such as deep learning and hybrid AI models (e.g., LSTM, CNN-BiLSTM, GED) have shown excellent predictive capabilities in different contexts [33,34,35,36], they often require extensive computational resources, large training datasets, and specialized tuning, which may limit their applicability in settings with data scarcity or constrained technical capacity. In contrast, classical methods like Holt–Winters, WMA, and ES offer greater interpretability, faster implementation, and reduced computational demands, advantages that are especially relevant in many tropical urban environments, where operational simplicity and adaptability are critical. These limitations are particularly considered in developing countries, where access to high-quality data, computing infrastructure, and technical expertise may be limited. Therefore, the use of optimized classical models presents a practical and accessible solution for improving environmental forecasting in resource-constrained contexts. This study builds on these classical models but enhances them using nonlinear optimization, providing a methodological middle ground between simplicity and precision.

Furthermore, the reviewed literature provides limited emphasis on cities located in tropical climates similar to Barranquilla. While some studies address precipitation forecasting in broader geographical settings, few explicitly focus on the specific challenges of tropical coastal regions, where rainfall variability is influenced by complex interactions between local and global climatic factors. By applying and optimizing classical forecasting techniques in such a setting, the present study contributes to filling this gap and demonstrates the value of adapting well-established models to environmentally sensitive urban areas.

3. Materials and Methods

As previously mentioned, this study applies five forecasting models to predict precipitation using historical data for the city of Barranquilla from 2013 to 2025. The accuracy of the models is evaluated using statistical indicators such as Mean Absolute Error (MAE) and Mean Squared Error (MSE). Additionally, a nonlinear optimization approach is employed to determine the optimal smoothing and weighting parameters for the Holt–Winters and WMA models, aiming to minimize the MAE and MSE individually. Time series and exponential smoothing models are essential tools for forecasting environmental variables such as precipitation. These models enable the identification of historical patterns, trends, and seasonality, thereby improving forecast accuracy. Moreover, exponential smoothing assigns greater weight to more recent observations, allowing models to adapt to dynamic environmental changes. The forecasting models are presented below.

3.1. Simple Moving Average (SMA)

The moving average method is a widely used statistical technique for smoothing random fluctuations in a time series to reveal underlying patterns or trends. This approach involves calculating the average of the most recent k + 1 observations within a defined time window and using that value as an estimate for the next point in the series (see Equation (1)) [37,38]. As time advances, the observation window shifts forward sequentially, generating a series of moving averages that produce a smoothed trajectory of the analyzed variable. This technique is particularly valuable when the objective is to reduce the impact of episodic variations or statistical noise while preserving the overall dynamics of the phenomenon under study.

$${\mathrm{Y}\prime }_{t+1}=(Y_t+Y_{t-1}+\cdots +Y_{t-k})/(k+1)$$

(1)

where ${\mathrm{Y}\prime }_{t+1}$ represents the forecasted value for period t + 1 in the time series, $Y_t$ corresponds to the actual observed value of the variable in period t, and k + 1 indicates the total number of periods considered in the moving average calculation. This parameter determines the size of the observation window used to smooth the series, directly influencing the model’s sensitivity to short-term variations.

3.2. Weighted Moving Average (WMA)

This method is an extension of the Simple Moving Average (SMA) model, in which differentiated weights are assigned to each observation within a defined time window, giving greater importance to more recent data. Unlike the simple approach, where all values carry equal weight, the Weighted Moving Average (WMA) allows for increased sensitivity to recent changes in the series trend. By applying decreasing weights to older observations, this method enhances responsiveness to abrupt variations, particularly valuable in scenarios characterized by high volatility or rapidly changing conditions, such as environmental variables influenced by extreme events. The corresponding mathematical formulation is presented in Equation (2) [39,40], where the weighting coefficients can be determined empirically or through optimization techniques.

$${\mathrm{Y}\prime }_{t+1}=Y_t\times {\mathrm{P}}_1+Y_{t-1}\times {\mathrm{P}}_2+\dots +Y_{t-k}\times {\mathrm{P}}_j$$

(2)

Here, ${\mathrm{Y}\prime }_{t+1}$ represents the forecasted value for period t + 1 in the time series, $Y_t$ corresponds to the actual observed value of the variable in period t, and k + 1 indicates the total number of periods considered in the analysis window. The ${\mathrm{P}}_j$ terms represent the weighting coefficients assigned to each observation, reflecting the relative importance of more recent data compared to older values. The total number of weights applied depends on the selected value of k + 1, and this choice directly influences the model’s sensitivity to recent changes in the series.

The SMA and WMA models assume that the data exhibit linear and relatively stationary behavior over time. These models perform adequately when fluctuations are moderate, and noise is limited. However, they may fail to capture nonlinear trends or high-frequency variations present in non-stationary or noisy time series. In such cases, their forecasts may lack responsiveness, especially during abrupt transitions in the data. The WMA model partially addresses this by giving greater weight to recent observations, which improves performance in short-term forecasting scenarios. Nevertheless, both models are best suited to environments with stable seasonal patterns and minimal structural breaks.

3.3. Exponential Smoothing (ES)

Exponential Smoothing is a univariate forecasting method that updates future estimates by applying a smoothing coefficient to the difference between the most recent observed value and the previous forecast (see Equation (3)) [41]. This operation assigns greater weight to recent observations, allowing the model to adapt quickly to changes in the time series without sacrificing stability. As such, it is particularly well suited for series that exhibit weak seasonality or smooth trends [42]. The core principle of this method is that the most recent data contain more relevant information about the future behavior of the variable and should therefore exert a stronger influence on the forecasting process. The smoothing parameter, typically denoted by θ, controls the model’s sensitivity.

$${\mathrm{Y}\prime }_{t+1}={\mathrm{Y}\prime }_t+\mathsf{\theta }(Y_t-{\mathrm{Y}\prime }_t)$$

(3)

where ${\mathrm{Y}\prime }_{t+1}$ represents the forecasted value for period t + 1, $Y_t$ is the actual observed value at time t, and ${\mathrm{Y}\prime }_t$ is the forecast generated for that same period t. The parameter θ denotes the smoothing coefficient, which must lie within the interval 0 ≤ θ ≤ 1. Typically, θ is estimated empirically or through numerical optimization methods, such as the Simplex algorithm, with the aim of minimizing error metrics such as the Mean Absolute Error (MAE) or Mean Squared Error (MSE) (see [42] for a more detailed discussion).

3.4. Holt–Winters Multiplicative and Additive Methods

The Holt–Winters method, also known as triple exponential smoothing, is an advanced forecasting technique for time series data. It extends simple exponential smoothing by incorporating both trend and seasonality components, making it particularly effective for series with regular and recurring seasonal patterns [43]. There are two main variants of the method (additive and multiplicative), which differ in how they model the seasonal component [44]. The Holt–Winters method is based on three main components:

• $A_t$ (Level): Smoothed value at period t or average of the series at time t.
• $T_t$ (Trend): Estimated trend at period t.
• $S_t$ (Seasonality): Estimated seasonality at period t.

These components are modeled through three fundamental smoothing equations that represent the level, trend, and seasonality of the time series. The combination of these equations enables the model to capture the structural dynamics of the observed phenomenon and to generate forecasts for one or more future periods as required [45]. Depending on the nature of the chosen model (additive or multiplicative) the interaction among these components is adjusted to appropriately reflect the characteristics of the series, particularly when seasonal variations exhibit changes in intensity (see Equations (4)–(11)).

Equations for Multiplicative Method:

$$A_t= \alpha \left({\displaystyle \frac{Y_t}{S_{t-L}}}\right)+ \left(1 - \alpha \right)\left(A_{t-1} + T_{t-1}\right)$$

(4)

$$T_t=\mathsf{\beta } \left(A_t-A_{t-1}\right)+\left(1-\beta \right)T_{t-1}$$

(5)

$$S_t=\mathsf{\gamma } \left({\displaystyle \frac{Y_t}{A_t}}\right)+\left(1-\gamma \right)S_{t-L}$$

(6)

$${\mathrm{Y}\prime }_{t+\mathrm{p}}=\left(A_t+p{T}_t\right)S_{t-L+p}$$

(7)

Equations for Additive Method:

$$A_t= \alpha \left(Y_t- S_{t-L}\right)+ \left(1 - \alpha \right)\left(A_{t-1} + T_{t-1}\right)$$

(8)

$$T_t=\mathsf{\beta } \left(A_t-A_{t-1}\right)+\left(1-\beta \right)T_{t-1}$$

(9)

$$S_t=\mathsf{\gamma } \left(Y_t-A_t\right)+\left(1-\gamma \right)S_{t-L}$$

(10)

$${\mathrm{Y}\prime }_{t+\mathrm{p}}=A_t+p{T}_t+S_{t-L+p}$$

(11)

Here, α is the smoothing constant for the level component (0 < α < 1), β is the smoothing constant for the trend component (0 < β < 1), and γ is the smoothing constant for the seasonal component (0 < γ < 1). L denotes the number of periods in a seasonal cycle (i.e., the seasonal length), and p represents the number of periods to forecast into the future.

3.5. Forecasting Accuracy Metrics and Optimization Approach

A key consideration when selecting a forecasting method is its predictive accuracy. In this study, two statistical metrics were used to evaluate model performance: Mean Absolute Error (MAE) and Mean Squared Error (MSE). MAE measures the average magnitude of the errors in a set of n forecasts, without considering their direction. It is calculated as the average of the absolute differences between the observed values and the predicted values. A lower MAE indicates better predictive accuracy. It is defined as

$$MAE={\displaystyle \frac{\sum _{t=1}^n\left|Y_t-{\mathrm{Y}}_t^{\prime }\right|}{n}}$$

(12)

For its part, MSE measures the average of the squares of the n errors, giving more weight to larger errors due to the squaring operation. This makes MSE more sensitive to outliers than MAE. It is calculated as

$$MSE={\displaystyle \frac{\sum _{t=1}^n{\left(Y_t-{\mathrm{Y}}_t^{\prime }\right)}^2}{n}}$$

(13)

In the case of the Weighted Moving Average (WMA), Exponential Smoothing (ES), and Holt–Winters models (both additive and multiplicative), the values of MAE and MSE can be minimized by identifying the optimal values of the parameters involved in the forecasting calculations. Given the nonlinear nature of the equations used in these models, nonlinear optimization techniques are required to efficiently and accurately adjust the parameters, ensuring improved forecasting performance. For example, the optimization of the optimal Pj values in the WMA model can be achieved by solving the following optimization problem:

Objective (to minimize forecasting error):

$$\min \mathit{Eror}\left(P_j\right)= \left\{or\begin{array}{c}\mathit{MAE}\left(P_j\right)={\displaystyle \frac{\sum _{t=1}^n\left|Y_t-Y_t^{\prime }\left(P_j\right)\right|}{n}} \\ \mathit{MSE}\left(P_j\right)=\left({\displaystyle \frac{\sum _{t=1}^n{\left(Y_t-Y_t^{\prime }\left(P_j\right)\right)}^2}{n}}\right)\end{array}\right.$$

(14)

subject to

$$\sum _{j=1}^{k+1}{P}_j=1 and P_j\ge 0 for all j=1,\dots ,k+1$$

(15)

For the Exponential Smoothing (ES) model, the corresponding optimization model can be expressed as follows:

Objective (to minimize forecasting error):

$$\min \mathit{Eror}\left(\theta \right)= \left\{or\begin{array}{c}\mathit{MAE}\left(\theta \right)={\displaystyle \frac{\sum _{t=1}^n\left|Y_t-Y_t^{\prime }\left(\theta \right)\right|}{n}} \\ \mathit{MSE}\left(\theta \right)=\left({\displaystyle \frac{\sum _{t=1}^n{\left(Y_t-Y_t^{\prime }\left(\theta \right)\right)}^2}{n}}\right)\end{array}\right.$$

(16)

subject to

$$0 \le \theta \le 1$$

(17)

Finally, for the Holt–Winters models the forecasting accuracy can be improved by optimizing the smoothing parameters for level (α), trend (β), and seasonality (γ). The optimization problem is formulated as follows:

Objective (to minimize forecasting error):

$$\min \mathit{Eror}\left(\alpha ,\beta ,\mathit{\gamma }\right)= \left\{or\begin{array}{c}\mathit{MAE}\left(\alpha ,\beta ,\gamma \right)={\displaystyle \frac{\sum _{t=1}^n\left|Y_t-Y_t^{\prime }(\alpha ,\beta ,\gamma )\right|}{n}} \\ \mathit{MSE}\left(\alpha ,\beta ,\gamma \right)=\left({\displaystyle \frac{\sum _{t=1}^n{\left(Y_t-Y_t^{\prime }(\alpha ,\beta ,\gamma )\right)}^2}{n}}\right)\end{array}\right.$$

(18)

subject to

$$0<\mathsf{\alpha }<1, 0<\mathsf{\beta }<1, 0<\mathsf{\gamma }<1$$

(19)

The Simplex method was selected as the optimization technique due to its effectiveness in handling nonlinear objective functions without requiring derivatives, making it especially suitable for smoothing models like Holt–Winters and Exponential Smoothing. This method is computationally efficient and robust when optimizing multiple interdependent parameters under bounded constraints. In the context of this study, the Simplex algorithm allowed for fast convergence when minimizing error metrics (MAE and MSE), even in models with up to three or more parameters. Its practical implementation also aligns with this study’s objective of maintaining methodological accessibility.

3.6. Data Description and Modeling Pipeline

A total of 139 monthly precipitation records were analyzed, corresponding to the period from December 2013 to June 2025. The data, expressed in millimeters (mm), represent the total rainfall recorded each month and were obtained from the open-access portal of Colombia’s Institute of Hydrology, Meteorology and Environmental Studies (IDEAM). These values were collected from the official meteorological station for the city of Barranquilla. Prior to analysis, the dataset was checked for completeness and consistency. No missing values were found during the period analyzed. Outliers were evaluated using descriptive statistics methods, but no values were excluded given their alignment with documented rainfall variability in the region.

The modeling pipeline used in this study consists of five main stages (see Figure 1): (1) Data Input, involving the collection of monthly rainfall data from the IDEAM for Barranquilla; (2) Model Selection, where classical time series models such as SMA, WMA, ES, and Holt–Winters (additive and multiplicative) are chosen based on their relevance to seasonal patterns; (3) Parameter Optimization, applying nonlinear optimization via the Simplex method to minimize MAE and MSE in the WMA, ES, and Holt–Winters models; (4) Performance Evaluation, assessing model accuracy through MAE, MSE, and forecast stability; and (5) Forecast Generation, producing rainfall predictions for comparative analysis. This pipeline ensures a systematic and transparent framework for selecting and validating forecasting models.

4. Results

This section presents the results of applying the proposed forecasting methods to analyze the precipitation variable in the city of Barranquilla. The performance of each forecasting model was evaluated using the dataset described in the previous section. The evaluation focused on forecasting accuracy across all models, highlighting improvements achieved through parameter optimization.

Table 1. Results for MAE, MSE, and ${\mathrm{Y}\prime }_{t+1}$ using the proposed forecasting methods.

$\mathit{k}+1$	SMA Model	$\mathit{L}$	Holt–Winters Multiplicative Model
3	MAE = 87.74, MSE = 13,908.29, ${\mathrm{Y}\prime }_{t+1}$ = 104.17 mm	4	Optimal values for MAE minimization $\alpha$ = 4.45 × 10−6, $\beta$ = 0.1183, $\gamma$= 0.0116
4	MAE = 93.49, MSE = 14,129.13, ${\mathrm{Y}\prime }_{t+1}$ = 78.13 mm		MAE* = 79.11, MSE = 12,901.06, ${\mathrm{Y}\prime }_{t+1}$ = 58.64 mm
5	MAE = 96.12, MSE = 14,429.65, ${\mathrm{Y}\prime }_{t+1}$ = 62.50 mm	4	Optimal values for MSE minimization $\alpha$ = 0.04591, $\beta$ = 0.00777, $\gamma$ = 0.00603
6	MAE = 99.21, MSE = 14,542.07, ${\mathrm{Y}\prime }_{t+1}$ = 52.20 mm		MAE = 86.20, MSE* = 11,557.56, ${\mathrm{Y}\prime }_{t+1}$ = 101.10 mm
$\mathit{k}+1$	WMA model	5	Optimal values for MAE minimization $\alpha$ = 0.9232, $\beta$ = 1 × 10−7, $\gamma$= 1 × 10−7
3	Optimal $P_j$ values for MAE minimization $P_1$ = 0.93, $P_2$ = 0.07, $P_3$ = 0.00		MAE* = 75.33, MSE = 12,968.22, ${\mathrm{Y}\prime }_{t+1}$ = 22.04 mm
	MAE* = 76.44, MSE = 13,207.28, ${\mathrm{Y}\prime }_{t+1}$ = 21.90 mm	5	Optimal values for MSE minimization $\alpha$ = 0.05958, $\beta$ = 0.003628, $\gamma$= 1 × 10−7
3	Optimal $P_j$ values for MSE minimization $P_1$ = 0.685, $P_2$ = 0.143, $P_3$ = 0.172		MAE = 86.65, MSE* = 11,600.84, ${\mathrm{Y}\prime }_{t+1}$ = 108.01 mm
	MAE = 77.51, MSE* = 12,312.44, ${\mathrm{Y}\prime }_{t+1}$ = 47.23 mm	6	Optimal values for MAE minimization $\alpha$ = 0.0007, $\beta$ = 0.01461, $\gamma$= 0.02327
4	Optimal $P_j$ values for MAE minimization $P_1$ = 0.907, $P_2$ = 0.071, $P_3$ = 0.016, $P_4$ = 0.006		MAE* = 76.68, MSE = 12,542.91, ${\mathrm{Y}\prime }_{t+1}$ = 22.49 mm
	MAE* = 76.99, MSE = 13,107.32, ${\mathrm{Y}\prime }_{t+1}$ = 22.40 mm	6	Optimal values for MSE minimization $\alpha$ = 0.161118, $\beta$ = 1 × 10−7, $\gamma$ = 0.08227
4	Optimal $P_j$ values for MSE minimization $P_1$ = 0.65, $P_2$ = 0.11, $P_3$ = 0.02, $P_4$ = 0.22		MAE = 80.10, MSE* = 9647.07, ${\mathrm{Y}\prime }_{t+1}$ = 34.86 mm
	MAE = 81.06, MSE* = 11,845.28, ${\mathrm{Y}\prime }_{t+1}$ = 33.57 mm	$\mathit{L}$	Holt–Winters additive model
5	Optimal $P_j$ values for MAE minimization $P_1$ = 0.913, $P_2$ = 0.072, $P_3$ = 0.00, $P_4$ = 0.00, $P_5$ = 0.015	4	Optimal values for MAE minimization $\alpha$ = 0.003163, $\beta$ = 0.002364, $\gamma$= 0.023723
	MAE* = 77.50, MSE = 13,238.17, ${\mathrm{Y}\prime }_{t+1}$ = 22.16 mm		MAE* = 79.72, MSE = 13455.09, ${\mathrm{Y}\prime }_{t+1}$ = 64.91 mm
5	Optimal $P_j$ values for MSE minimization $P_1$ = 0.628, $P_2$ = 0.112, $P_3$ = 0.005, $P_4$ = 0.142, $P_5$ = 0113	4	Optimal values for MSE minimization $\alpha$ = 0.054176, $\beta$ = 0.00333, $\gamma$ = 0.033139
	MAE = 81.29, MSE* = 11,790.73, ${\mathrm{Y}\prime }_{t+1}$ = 32.61 mm		MAE = 85.70, MSE* = 11,391.63, ${\mathrm{Y}\prime }_{t+1}$ = 103.05 mm
6	Optimal $P_j$ values for MAE minimization $P_1$ = 0.793, $P_2$ = 0.072, $P_3$ = 0.011, $P_4$ = 0.00, $P_5$ = 0.00, $P_6$= 0.124	5	Optimal values for MAE minimization $\alpha$ = 0.000618, $\beta$ = 0.26748, $\gamma$= 1 × 10−7
	MAE* = 77.11, MSE = 12,235.30, ${\mathrm{Y}\prime }_{t+1}$ = 22.30 mm		MAE* = 80.39, MSE = 13,874.74, ${\mathrm{Y}\prime }_{t+1}$ = 96.04 mm
6	Optimal $P_j$ values for MSE minimization $P_1$ = 0.612, $P_2$ = 0.09, $P_3$ = 0.004, $P_4$ = 0.127, $P_5$ = 0.01, $P_6$= 0.157	5	Optimal values for MSE minimization $\alpha$ = 0.05838, $\beta$ = 0.003793, $\gamma$ = 1 × 10−7
	MAE = 81.05, MSE* = 11,602.88, ${\mathrm{Y}\prime }_{t+1}$ = 26.44 mm		MAE = 86.64, MSE* = 11,595.23, ${\mathrm{Y}\prime }_{t+1}$ = 108.21 mm
Optimal  $\mathit{\theta }$  value	ES model	6	Optimal values for MAE minimization $\alpha$ = 1 × 10-07, $\beta$ = 0.004155, $\gamma$= 0.058804
$\theta$ = 0.92	MAE* = 76.44, MSE = 13,158.93, ${\mathrm{Y}\prime }_{t+1}$ = 22.04 mm		MAE* = 77.82, MSE = 12764.91, ${\mathrm{Y}\prime }_{t+1}$ = 26.58 mm
$\theta$ = 0.07	MAE = 87.56, MSE* = 11,779.45, ${\mathrm{Y}\prime }_{t+1}$ = 103.17 mm	6	Optimal values for MSE minimization $\alpha$ = 0.06650, $\beta$ = 4.17 × 10−5, $\gamma$ = 0.080602
			MAE = 83.53, MSE* = 10,704.47, ${\mathrm{Y}\prime }_{t+1}$ = 51.84 mm

summarizes the results, presenting the values of Mean Absolute Error (MAE), Mean Squared Error (MSE), and the forecasted value (${\mathrm{Y}\prime }_{t+1}$) under the various scenarios evaluated. It also highlights the optimal MAE* and MSE* values achieved through parameter optimization for the WMA, ES, and Holt–Winters models (both additive and multiplicative). Given the seasonal characteristics of rainfall in Barranquilla, the analysis considered values of k + 1 ranging from 3 to 6 for the SMA and WMA models, and seasonal lengths L between 4 and 6 for the Holt–Winters models. The table illustrates how the incorporation of a nonlinear optimization model enabled the optimization of key parameters in each forecasting method, resulting in a substantial improvement in error metrics and overall predictive accuracy. In this regard, the best performance in terms of MAE* was achieved by the Holt–Winters multiplicative model, which reached a minimum value of 75.33 mm when L = 5, followed by the ES model (MAE* = 76.44 mm with θ = 0.92) and WMA model (MAE* = 76.44 mm when k + 1 = 3). Regarding MSE, the lowest values were obtained with the Holt–Winters multiplicative (MSE* = 9647.07 when L = 6) and Holt–Winters additive (MSE* = 10,704.47 when L = 6), indicating a strong ability of these models to reduce large forecasting errors. This performance can be attributed to the model’s suitability for time series data exhibiting multiplicative seasonality and stable trend components, which characterize the monthly rainfall patterns in Barranquilla. The city’s precipitation displays strong annual seasonality and variable peaks during the rainy periods, which makes the multiplicative structure particularly well suited for this type of data. Additionally, the trend stability of the data favors exponential smoothing techniques that dynamically adapt to level and seasonal shifts. In contrast, while the additive Holt–Winters model yielded competitive results, it performed slightly worse due to its inability to model proportional seasonal effects. Moreover, the SMA method yielded the highest error values (MAE = 99.21 and MSE = 14,542.07 when k + 1 = 6), highlighting its limited capacity to capture the underlying complexity and variability of the precipitation series.

Figure 2 shows the graphs of observed precipitation values versus forecasts for the best-performing models in minimizing MAE* and MSE*. As shown in Figure 2, the graphical comparison between observed and forecasted rainfall provides visual confirmation of the model’s predictive capabilities. The Holt–Winters multiplicative model, in particular, demonstrates a strong alignment between observed peaks and predicted values, accurately capturing both the magnitude and timing of rainfall fluctuations. The stability of the forecasts generated under MAE optimization and the greater responsiveness of those under MSE optimization are also visually evident. These visual patterns support the numerical findings, highlighting the importance of model selection and optimization criteria in achieving reliable predictions.

The results also reveal a systematic pattern in the forecasted values, depending on the model used and the error metric selected for optimization. Specifically, it was observed that the forecasted values tend to be more moderate and stable when the MAE is minimized, compared to the forecasts obtained by minimizing the MSE. This suggests that MAE-based optimization yields less extreme predictions, likely due to its linear penalization of errors, whereas MSE places greater emphasis on larger deviations, which can lead to more variable forecast outputs.

It is important to highlight that optimization not only improves model fit but also reduces the dispersion of forecasted values, particularly in the WMA model, where the predicted values are more stable and display low variability. Moreover, it is evident that in all cases, the optimized versions of the models yield significantly lower MAE and MSE values, while the forecasted values become more consistent. In general, the results provide strong evidence of the advantages of applying nonlinear optimization techniques and selecting models with seasonal sensitivity to enhance the accuracy of precipitation forecasting.

5. Discussion

The results obtained in this study were compared with those from a previous analysis of the precipitation variable in the city of Barranquilla.

Table 2. Results for MSE and ${\mathrm{Y}\prime }_{t+1}$ using the forecasting methods proposed by [30].

$\mathit{k}+1$	SMA Model
3	MSE = 12,485, ${\mathrm{Y}\prime }_{t+1}$ = 8.60 mm
4	MSE = 12,784, ${\mathrm{Y}\prime }_{t+1}$ = 6.45 mm
5	MSE = 13,188 ${\mathrm{Y}\prime }_{t+1}$ = 5.16 mm
$\mathit{k}+1$	WMA model
	$P_1$ = 0.5	$P_2$ = 0.3	$P_3$ = 0.2
3	MSE = 11,322, ${\mathrm{Y}\prime }_{t+1}$ = 10.62 mm
	$P_1$ = 0.65	$P_2$ = 0.20	$P_3$ = 0.15
3	MSE = 10,878, ${\mathrm{Y}\prime }_{t+1}$ = 12.97 mm
	$P_1$ = 0.45	$P_2$ = 0.25	$P_3$ = 0.20	$P_4$ = 0.10
4	MSE = 11,281, ${\mathrm{Y}\prime }_{t+1}$ = 9.71 mm
	$P_1$ = 0.50	$P_2$ = 0.30	$P_3$ = 0.10	$P_4$ = 0.10
4	MSE = 11,016, ${\mathrm{Y}\prime }_{t+1}$ = 9.86 mm
	$P_1$ = 0.40	$P_2$ = 0.25	$P_3$ = 0.20	$P_4$ = 0.10	$P_5$= 0.05
5	MSE = 11,473, ${\mathrm{Y}\prime }_{t+1}$ = 8.80 mm
	$P_1$ = 0.45	$P_2$ = 0.25	$P_3$ = 0.15	$P_4$ = 0.10	$P_5$ = 0.05
5	MSE = 11,137, ${\mathrm{Y}\prime }_{t+1}$ = 9.33 mm
θ	ES model
0.5	MSE* = 10,487, ${\mathrm{Y}\prime }_{t+1}$ = 13.82 mm

summarizes the findings of that earlier study, which analyzed data from 2014 to 2024 and applied the SMA, WMA, and ES models for forecasting purposes [30]. Moreover, values of k + 1 ranging from 3 to 5 were used in the SMA and WMA models. Notably, that study used only the Mean Squared Error (MSE) as the performance metric for evaluating forecasting accuracy. Furthermore, only the smoothing parameter θ for the ES model was optimized, while the weighting coefficients in the WMA model were assigned empirically, without the use of a formal optimization procedure.

The comparison between Table 1 and Table 2 allows for the identification of key methodological differences and systematic trends in the behavior of the forecasting models. For example, in Table 1, the inclusion of optimization procedures for the evaluated methods resulted in minimum MSE values. In contrast, in Table 2, where only the smoothing parameter θ was optimized for the ES model, a minimum MSE* of 10,487 for θ = 0.5 is reported. Although the value of θ = 0.5 was not the result of a comprehensive optimization process, it still performed relatively well, outperforming most of the MSE values reported in Table 1, and being surpassed only by the Holt–Winters multiplicative method (MSE* = 9647.07). This finding suggests that the optimization of individual parameters can yield significant improvements, but its effect is more pronounced when applied to methods that incorporate multiple smoothing parameters. Moreover, the optimized methods presented in Table 1 tend to maintain internal consistency between low error metrics and realistic forecast values. In this sense, most of the model results in Table 2 exhibit moderate MSE values but tend to produce forecasts that underestimate precipitation, such as the WMA model, which yielded a forecast of ${\mathrm{Y}\prime }_{t+1}$ = 8.80 mm when k + 1 = 5, or the SMA model, with ${\mathrm{Y}\prime }_{t+1}$ = 6.45 mm when k + 1 = 4. This underscores the substantial impact that parameter optimization has on model performance and forecasting accuracy.

Overall, the comparative analysis supports the methodological superiority of the models presented in Table 1, not only due to the implementation of more advanced techniques but also because of the comprehensive optimization approach applied. At the same time, Table 2 demonstrates that even simpler models, when properly optimized, can achieve performance levels comparable to those of more complex methods. Nevertheless, the overall trend suggests that greater parametric tuning and higher structural complexity generally lead to improved predictive capacity and reduced forecasting errors.

Beyond the quantitative comparison of model performance, the results have broader practical and methodological implications. The findings of this study have important implications for real-world applications, particularly in tropical urban areas like Barranquilla, where seasonal rainfall variability poses significant challenges for flood management, drainage infrastructure, and urban planning. Accurate forecasting models can support the development of early warning systems and help municipal authorities allocate resources more efficiently during high-risk periods. For instance, the Holt–Winters multiplicative model, which demonstrated strong predictive performance under both MAE and MSE optimization, is well suited for capturing the nonlinear and seasonal behavior typical of precipitation in the region. This makes it a valuable tool for integrating rainfall forecasts into urban resilience and adaptation strategies.

Furthermore, while classical time series models offer interpretability and computational simplicity, there remains potential for improvement through the use of more advanced approaches. Techniques such as artificial neural networks (ANNs), support vector regression (SVR), and deep learning could be explored in future studies to capture more complex temporal dependencies and interactions in the data. However, the deployment of such methods must also consider limitations related to data availability, model transparency, and implementation costs, especially in the context of developing countries. Overall, this study establishes a solid foundation for data-driven rainfall forecasting in Barranquilla and highlights the potential for integrating hybrid approaches and AI-based models into future climate adaptation strategies.

6. Conclusions

Key findings: The present study highlights the effectiveness of time series models, particularly when combined with nonlinear optimization techniques, for forecasting precipitation in the city of Barranquilla, Colombia. Among the five models evaluated, the multiplicative Holt–Winters model demonstrated the best performance, with a minimum MAE of 75.33 mm (for L = 5) and an MSE of 9647.07 (for L = 6). This superior performance is attributed to its capacity to effectively model both trend and seasonality, two key characteristics of the analyzed data. It was also observed that models with a larger number of parameters, such as the Holt–Winters models, enable a more precise adjustment of the seasonal pattern, although they require more complex optimization procedures. The Exponential Smoothing (ES) and Weighted Moving Average (WMA) models also demonstrated competitive results when optimized, each achieving an MAE of 76.44 mm. These findings underscore the importance of parameter calibration, as models that initially exhibited moderate or low performance improved significantly through the application of nonlinear optimization techniques, such as the Simplex algorithm. The comparative analysis with previous studies further revealed that even simpler models can attain accuracy levels comparable to more complex approaches, provided they are properly optimized. Predictions optimized for MAE were more stable and moderate, whereas those optimized for MSE, due to its sensitivity, produced more variable and extreme forecasts. This distinction highlights the practical trade-off between stability and sensitivity, which should be carefully considered when selecting the error metric based on the forecasting objective.

Practical implications: Beyond statistical performance, this study offers practical implications for climate risk management. Accurate rainfall forecasting is critical for urban infrastructure planning, flood prevention systems, and early warning strategies, especially in vulnerable cities like Barranquilla. The optimized models presented in this study are computationally efficient and interpretable, making them feasible options for implementation. However, although the optimization approach significantly improved predictive performance, the models themselves are inherently linear and may not fully capture complex nonlinear relationships present in rainfall data.

Future directions: In this sense, future research should explore more advanced and adaptive models, including artificial neural networks, Long Short-Term Memory (LSTM) networks or hybrid CNN–LSTM approaches. These approaches could better accommodate non-stationary behaviors and irregular patterns. Nevertheless, given the technical and infrastructural limitations often present in developing countries, such models should be adopted with caution and only when local conditions permit robust implementation.

In conclusion, this study demonstrates that integrating seasonality-sensitive models with optimized parameter provides a substantial advantage in terms of accuracy and reliability. This methodology, in addition to being statistically robust, is highly applicable to real-world environmental management contexts.