Double decomposition with enhanced least-squares support vector machine to predict water level Open Access

In Malaysia, floods have emerged as a recurring and prominent natural disaster over recent decades, inflicting considerable economic devastation, especially in regions like Sri Muda, Sungai Gombak, and Mentakab. These floods result from intense rainfall overwhelming local catchment capacities. Establishing robust early warning systems is crucial to protect urban centers and lives (Sumi et al. 2012). Hydrologists analyze statistical properties in hydrological data, including rainfall–runoff patterns (He et al. 2022) and water level records, to enhance preparedness. Monitoring water level data is vital for anticipating floods, but improving future water level predictions is essential for effective flood warnings (Faruq et al. 2021). Moreover, climate change is causing more frequent and intense extreme weather events like hurricanes, storms, and heavy rainfall. These events lead to sudden and significant changes in water levels in rivers, lakes, and coastal areas. To effectively manage water resources and prepare for disasters, it is crucial to develop advanced forecasting models. Our research focuses on understanding and predicting how climate change impacts water levels. By exploring these connections, we aim to improve forecasting accuracy, support sustainable water management, and help communities cope with the increasing risks associated with changing water levels.

Recent times have seen a surge in the use of machine learning techniques, including support vector machines (SVMs) (Cortes & Vapnik 1995; Sukanya & Vijayakumar 2023) and its variant, the least-squares support vector machine (LSSVM), for effective processing of both linear and nonlinear datasets across diverse domains. In flood forecasting research, SVM has gained attention for robust predictive capabilities in hydrological data prediction. Despite its advantages, SVM has drawbacks like longer computation times. To address this, the improved LSSVM model has been introduced, known for its ability to solve linear matrix equations with fewer constraints (Wang & Hu 2005). The use of the radial basis function (RBF) kernel function with LSSVM has emerged as a favorable alternative, transforming inequality computations into equality computations. Integrated with decomposition methods, LSSVM has proven effective in predicting long-term runoff (Seo et al. 2016). Considering the small sample size challenge, especially pertinent in the case of one-dimensional and limited dataset, LSSVM is a preferred choice (Tang et al. 2018). Consequently, this study employs machine learning models, specifically LSSVM and enhanced LSSVM (ELSSVM), for a comparative analysis. These models are integrated with decomposition methods to forecast daily water levels, facilitating flood prediction within the study area.

In dealing with the inherent inconsistency in hydrological time series data, effective data preprocessing techniques are crucial for noise reduction. Decomposition methods, particularly empirical mode decomposition (EMD), have proven to be powerful tools for enhancing prediction accuracy by mitigating signal or time series noise (Huang et al. 1998). EMD decomposes water level data into components known as intrinsic mode functions (IMFs) and residue, offering a natural mode for each mono-component. The effectiveness of EMD has been acknowledged in various fields, including biomedical data analysis, fault detection, power signal analysis, and medicine. Recent studies highlight the impact of EMD on water level prediction (Loh et al. 2019), demonstrating improved performance when combined with forecasting models like EMD with artificial neural network and EMD with SVM. This article focuses on forecasting river water levels in the context of climate change, which holds significant utility for society. Accurate river water level forecasts are critical for effective water resource management, flood preparedness, and infrastructure planning, particularly in the face of changing climate patterns. The utility of the article lies in its potential to enhance early warning systems, allowing communities to prepare for and mitigate the impacts of floods resulting from climate-induced factors such as altered precipitation patterns and extreme weather events.

The main contribution of the proposed work is as follows:

• i The enhancement of forecasting model accuracy through the incorporation of an additional control term for bias error into the objective function of the LSSVM, referred to as ELSSVM. This modification enables unbiased estimation within the forecasting framework, thereby addressing a crucial aspect of predictive modeling.

• ii Exploring the potential of integrating these decomposition methods into existing forecasting models, such as ELSSVM, with the goal of improving predictive performance while minimizing overfitting issues. Through empirical analysis, this study seeks to evaluate the impact of double decomposition on forecasting river water levels.

• iii Developing hybrid prediction methods, specifically double empirical mode decomposition (DEMD)-LSSVM-GA and DEMD-ELSSVM-GA models, which combine DEMD as a decomposition technique with the forecasting tools such LSSVM and ELSSVM optimized by genetic algorithm (GA). These models are designed to enhance the forecasting accuracy of future water levels, thus contributing to the evolution of early warning systems for floods.

EMD is a powerful procedure that is usually used to analyze the nonlinear (Dehghan et al. 2022). The procedure of the EMD is to decompose the original data into different features to produce numerous sets of inherent IMFs. The produced IMFs include different frequency bands ranging from high to low. Let be an original time series from the dataset. The procedure for EMD methods are as follows (Chowdhury et al. 2008):

Step 1: Find all local extrema that includes local maxima and minima of the time series.

Step 2: Connect all the local maxima and local minima by applying a cubic spline line to obtain the upper envelope and the lower envelope.

Step 3: Determine the mean envelope from the lower and upper envelope.

Step 4: Determine the difference between the original data and the mean envelope determined in Step 3.

Step 5: Verify whether the vector satisfied the characteristics of IMFs. If yes, the vector is the first IMF and the vector is replaced by the residuals. If not, replace the signal with the vector.

Step 6: Repeat Steps 1–5, and then stop the process until the termination condition is satisfied.

In addition, the shifting process of EMD will end its operation as soon as the residual exhibits a monotonous behavior, making it unfeasible to continue extracting IMFs. The final product of decomposition by EMD is a set of IMFs and residuals from the original data (Colominas et al. 2015). For double EMD (DEMD), the first IMF with the highest frequency will undergo EMD again and produce IMFs that will be named SIMF1, SIMF2, etc (Ahmed et al. 2022).

To achieve unbiased estimation for the forecasting model, an additional term for controlling bias error

It is imperative to water level models for the evaluation of flood occurrences and the effective management of water resources within the country. Elevated water levels are indicative of broader impacts stemming from climate change. Changes in precipitation patterns, temperature, and glacial melting contribute to fluctuations in water levels, exerting influence on the flow of rivers, the levels of lakes, and coastal regions.

According to Figure 2, the water level data have been transformed into 10 IMF components and one residual component through the EMD method. Each IMF corresponds to a specific frequency component or oscillatory mode present in the original signal. IMF 1 generally captures the highest frequency component, and the subsequent IMFs represent lower frequency components. The residual component represents the remaining part of the signal that is not captured by the 10 IMFs. It includes fine details, high-frequency noise, and any components that could not be effectively modeled by the decomposition process. Hence, the IMF 1 with the highest frequency has chosen to undergo EMD again by using DEMD to improve predictive performance while minimizing overfitting issues. Figure 4(a) and 4(b) depicts the results of data analysis conducted at multiple decomposition levels. SIMF 1 represents higher frequencies, while SIMF 10 encapsulates the lowest frequencies within the decomposed water level data.

The accuracy of SVM and LSSVM models is significantly influenced by the careful selection of the kernel and its parameters. Previous studies on hydrologic issues have explored LSSVM and ELSSVM with different kernels, and the RBF has consistently emerged as the most accurate and efficient choice. In this study, we adopt the RBF kernel based on this established effectiveness. The kernel parameters of LSSVM and ELSSVM are denoted as and ⁠, respectively, ranged between 0.001 and 10. The penalty factors of LSSVM and ELSSVM are denoted as C and ⁠, respectively, ranged between 0.1 and 10. To optimize the parameters, a GA approach is employed and the optimized parameters are shown in Table 1. Table 2 shows the statistical results pertaining to the hybrid technique employed for the estimation of IMFs utilizing LSSVM and ELSSVM methods.

Table 2

Results of LSSVM and ELSSVM models in estimating IMFs and SIMFs obtained from EMD and DEMD, respectively, in the test phase

.		LSSVM-GA .		ELSSVM-GA .
		RMSE .	.	RMSE .	.
EMD	IMF1	0.4589	0.2489	0.3598	0.2658
	IMF2	0.0569	0.9485	0.04698	0.9789
	IMF3	0.0258	0.9564	0.0245	0.9897
	IMF4	0.0197	0.9548	0.0154	0.9978
	IMF5	0.0199	0.9632	0.0198	0.9975
	IMF6	0.0117	0.9745	0.0104	0.9974
	IMF7	0.0071	0.9787	0.0052	0.9988
	IMF8	0.0067	0.9799	0.0254	0.9824
	IMF9	0.0041	0.9854	0.0035	0.9856
	IMF10	0.0021	0.9956	0.0014	0.9984
	Res	0.0009	0.9995	0.0009	0.9998
DEMD	SIMF1	0.5869	0.3598	0.4578	0.2569
	SIMF2	0.0569	0.9574	0.0485	0.9854
	SIMF3	0.0249	0.9645	0.0152	0.9942
	SIMF4	0.0248	0.9647	0.0121	0.9974
	SIMF5	0.0185	0.9674	0.0075	0.9984
	SIMF6	0.0098	0.9762	0.0045	0.9987
	SIMF7	0.0087	0.9781	0.0054	0.9992
	SIMF8	0.0074	0.9863	0.0005	0.9993
	SIMF9	0.0049	0.9871	0.0035	0.9997
	SIMF10	0.0030	0.9929	0.0015	0.9999
	SRes	0.0025	0.9985	0.0009	0.9999

.		LSSVM-GA .		ELSSVM-GA .
		RMSE .	.	RMSE .	.
EMD	IMF1	0.4589	0.2489	0.3598	0.2658
	IMF2	0.0569	0.9485	0.04698	0.9789
	IMF3	0.0258	0.9564	0.0245	0.9897
	IMF4	0.0197	0.9548	0.0154	0.9978
	IMF5	0.0199	0.9632	0.0198	0.9975
	IMF6	0.0117	0.9745	0.0104	0.9974
	IMF7	0.0071	0.9787	0.0052	0.9988
	IMF8	0.0067	0.9799	0.0254	0.9824
	IMF9	0.0041	0.9854	0.0035	0.9856
	IMF10	0.0021	0.9956	0.0014	0.9984
	Res	0.0009	0.9995	0.0009	0.9998
DEMD	SIMF1	0.5869	0.3598	0.4578	0.2569
	SIMF2	0.0569	0.9574	0.0485	0.9854
	SIMF3	0.0249	0.9645	0.0152	0.9942
	SIMF4	0.0248	0.9647	0.0121	0.9974
	SIMF5	0.0185	0.9674	0.0075	0.9984
	SIMF6	0.0098	0.9762	0.0045	0.9987
	SIMF7	0.0087	0.9781	0.0054	0.9992
	SIMF8	0.0074	0.9863	0.0005	0.9993
	SIMF9	0.0049	0.9871	0.0035	0.9997
	SIMF10	0.0030	0.9929	0.0015	0.9999
	SRes	0.0025	0.9985	0.0009	0.9999

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Based on Table 2, the results underscore the superior predictive capabilities of the LSSVM model when compared to the ELSSVM model, particularly in the context of IMFs generated through the EMD and DEMD techniques. For example, when considering IMF3 of EMD, the RMSE value for the LSSVM model stands at 0.0258, while the ELSSVM model exhibits an RMSE value of 0.0245. Next, consider IMF3 of DEMD, the RMSE value for the LSSVM model stands at 0.0249, while the ELSSVM model exhibits an RMSE value of 0.0152. Therefore, we can conclude that ELSSVM models have shown better results compared to LSSVM models with the help of data decompositions. The IMF 1 and SIMF 1 often contain high-frequency noise or fast-varying components that might not follow a predictable pattern. Attempting to forecast such noise can result in poor performance, as the noise may not have a consistent structure over time. However, combining the forecasting results from multiple IMFs gives less weight to the last IMF and improves overall forecasting performance. On the other hand, the first IMF of EMD tends to capture high-frequency noise or rapid fluctuations in the signal that results in poor forecasting with low values of ⁠. Therefore, DEMD has been implemented only for IMF 1.

In Figure 8, the decomposition method with ELSSVM that added bias error term introduces a bias toward certain predictions, especially on days with extreme measurements. While this bias improves performance on average, it leads to suboptimal predictions on days with extreme values. In contrast, the decomposition method with LSSVM model's ability to adapt its decision boundary without introducing excessive bias results in more accurate predictions for extreme scenarios during November and December 2021. However, despite the existing model showing better performance during this time period, it is essential to consider the overall performance metrics such as RMSE and values. These metrics evaluate the accuracy and goodness of fit of the model across the entire dataset, providing a comprehensive assessment of its predictive capability. The proposed model's superior performance in terms of these metrics indicates its overall effectiveness.

Analyzing the data presented in Table 3 reveals a notable difference in the performance of the LSSVM and ELSSVM models when tested on the dataset. Specifically, ELSSVM demonstrates a higher level of effectiveness compared to LSSVM, indicating that ELSSVM models hold the potential for superior forecasting compared to LSSVM models. This can be attributed to ELSSVM's efficiency in handling large-scale problems, particularly with parameter optimization, and its compatibility with data decomposition methods. The findings indicate that the ELSSVM results are typically satisfactory. Despite a very small difference in performance metrics, ELSSVM generally enhances forecasting accuracy. This study underscores the effectiveness and adaptability of LSSVM-type models in addressing time series challenges. ELSSVM emerges as a promising option for water level forecasting in the Klang River (Malaysia).

In the context of LSSVM and ELSSVM models, the integration of various decomposition methods (EMD and DEMD) that are optimized by GA highlights a significant improvement in the performance of proposed models based on values compared to the single model. The hybrid LSSVM models have outperformed single LSSVM based on the errors presented in Table 3. The EMD-LSSVM-GA and DEMD-LSSVM-GA leverage the complementary strengths of both approaches. While decomposition methods capture nuanced patterns inherent in the signal, LSSVM excels in effectively classifying these patterns. The integration of these methodologies creates a synergistic effect, where the hybrid model benefits from the detailed feature representation obtained through decomposition, leading to improved classification accuracy. As climate change intensifies, the versatility of hybrid models becomes essential for providing reliable forecasts that aid in proactive water resource management, flood preparedness, and sustainable river basin planning. In fact, DEMD-LSSVM-GA outperforms EMD-LSSVM by 36.83%. This substantial enhancement in testing data results suggests that the proposed DEMD-LSSVM-GA models exhibit strong predictive capabilities. DEMD's strength lies in its ability to handle changing patterns in water data due to factors like climate change. First, decomposition separates different frequencies in the data more effectively, helping to capture important features that influence water levels. However, the first IMF has the highest frequency. Hence, only IMF 1 undergoes DEMD. DEMD's approach of making these features independent from each other is particularly helpful when multiple factors, like weather and human activities, affect water levels. This makes DEMD a powerful tool for accurate water level predictions, especially in the face of complex and changing environmental conditions.

On the other hand, the superior performance of the hybrid ELSSVM model, integrating ELSSVM with a decomposition method, can be attributed to several key factors. One primary advantage lies in the enhanced feature representation achieved through the decomposition process. This decomposition method effectively breaks down the input signal into IMFs or components, each representing specific frequency patterns or oscillations. This is particularly beneficial for ELSSVM, as it excels in handling complex, nonlinear relationships between features and labels. The additional information obtained from the decomposition step enables the ELSSVM classifier to better discern intricate patterns in the data, leading to improved generalization and classification accuracy. Moreover, hybrid models excel in capturing the complex interactions influenced by climate change, where shifting precipitation patterns, extreme weather events, and sea-level rise contribute to dynamic water level fluctuations. The combination of decomposition methods (EMD and DEMD) with ELSSVM, are known as EMD-ELSSVM-GA and DEMD-ELSSVM-GA, respectiveky. DEMD-ELSSVM-GA showcased a remarkable 24.67% improvement over EMD-ELSSVM-GA. This outcome underscores the substantial enhancement in daily water level estimation achieved through the hybrid technique involving DEMD data decomposition. In summary, the evaluation of various models reveals that DEMD-ELSSVM-GA exhibits a performance metric close to 1, indicating a highly effective prediction model compared to its counterparts. Therefore, it can be reasonably concluded that the DEMD-ELSSVM-GA model stands as the most suitable technique for forecasting daily water levels based on the justifications.

Table 3 presents insightful comparisons of individual models, revealing notable improvements in performance metrics. Specifically, the EMD-ELSSVM model exhibits superior performance compared to EMD-LSSVM, showcasing reductions in RMSE for testing data by 26.81%. The fusion of the DEMD approach with the ELSSVM model results in heightened prediction efficiency compared to DEMD-LSSVM, with substantial reductions in RMSE value for testing data by 15.1%. Flood dynamics are often influenced by complex, nonlinear interactions between various hydrological variables. ELSSVM, being specifically designed for nonlinear regression tasks, can better capture these intricate relationships compared to LSSVM. It is noteworthy that the adoption of EMD and DEMD methodologies has significantly improved both the accuracy of LSSVM and ELSSVM models in the context of this study. When contrasted with the EMD method, DEMD excels in its ability to efficiently distinguish between tidal signals and noise. This effectiveness can be attributed to DEMD's adaptable decomposition traits, its strong theoretical foundation, and its capacity to effectively mitigate high frequency. When these qualities are coupled with LSSVM's and ELSSVM's capability for forecasting, it results in the attainment of remarkably accurate tidal forecast predictions.

In summary, the utilization of the EMD method for data analysis did not lead to an improvement in the information received by the LSSVM and ELSSVM models. This outcome can be attributed to the inadequate data decomposition performed by EMD. In the context of this research, the DEMD method integrated with LSSVM and ELSSVM to enhance the data preprocessing method that enhanced the performance of LSSVM and ELSSVM models. Furthermore, the findings strongly suggest that the DEMD-LSSVM-GA and DEMD-ELSSVM-GA models proved to perform better in predicting the daily water level.

LSSVM models have been widely applied in the domain of water level prediction. The current study introduces an ELSSVM model, which incorporates a bias error control mechanism. Nonetheless, prior studies often neglected the incorporation of data features when constructing these models, leaving room for potential improvements in prediction accuracy. In this research, we introduced an approach for daily water level prediction, employing data decomposition principles to augment the predictive performance of daily water levels. Decomposition techniques, namely, EMD and DEMD were employed to break down the original daily water level dataset into individual IMF components characterized by reduced complexity and pronounced periodicity. Moreover, the machine learning models that have been utilized to forecast water level in this study are LSSVM and ELSSVM. Some researchers argued that a single LSSVM and ELSSVM model is not the best technique to forecast hydrological data. Therefore, this study has implemented the decomposition method with SVM and LSSVM. By comparing the machine learning methods, ELSSVM-GA, EMD-ELSSVM-GA, and DEMD-ELSSVM have outperformed LSSVM-GA, EMD-LSSVM-GA, and DEMD-LSSVM-GA, respectively. According to the experimental results, the DEMD-LSSVM-GA and DEMD-ELSSVM-GA were chosen as the best models in terms of multiple performance metrics by comparing them to other decomposition models. Overall, by comparing all models, the results demonstrated that the DEMD-ELSSVM-GA model outperformed the other models based on the performance analysis in forecasting the water level for Klang River in Sri Muda, Malaysia, with RMSE = 0.2536 m and = 0.8596 for testing data that indicate the forecasting accuracy. On the other hand, hybrid models relying on EMD displayed poor performance.

Researchers can explore by focusing on three aspects. The ELSSVM demonstrates significant potential to forecast water levels in the Klang River, Malaysia. However, future research should focus on optimizing forecast extrapolation and refining error control mechanisms to enhance its effectiveness further. Furthermore, conducting additional research into water level prediction using further decomposed hydrological data is recommended such as variational mode decomposition-based hybrid models. Finally, a comparison between other machine learning models such as radial basis neural networks and artificial neural networks can be investigated.

All the authors gratefully acknowledged the financial support from the ‘Ministry of Higher Education Malaysia for Fundamental Research Grant Scheme with Project Code: FRGS/1/2022/STG06/USM/03/1’

Vikneswari Someetheram: conceptualization, software, project administration, formal analysis, writing. Muhammad Fadhil Marsani: supervision. Mohd Shareduwan Mohd Kasihmuddin: supervision. Siti Zulaikha Mohd Jamaludin: validation. Mohd. Asyraf Mansor: validation, funding acquisition. All authors have read and agreed to the published version of the manuscript.

The research is fully funded and supported by the Ministry of Higher Education Malaysia for Fundamental Research Grant Scheme, FRGS/1/2022/STG06/USM/03/1.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.