Introduction

Surface water is essential for ecology, social well-being, and economic growth1,2,3. Water quality (WQ) is influenced by various variables, including natural ones like rainfall and erosion and human ones like urban, agricultural, and industry operations4,5,6. Because surface water is the world’s leading supplier of fresh water, its deterioration may have a considerable impact on the availability of drinking water and, more broadly, on economic growth and long-term plans7,8,9. Water pollution is caused by interactions with their surroundings and the subsequent interchange of toxins from urban, industrial, and agricultural sources along their course10,11,12, which results in polluting the freshwater ecosystems13,14, urban water systems15,16, and agriculture land17, due to present of microplastics and other polluted substances.

In order to assess and classify the quality of ground and surface waters, the WQI has been widely used as a classification indicator and is essential for managing water resources18,19,20. WQI combines several physical and chemical parameters into a single metric to measure the status of WQ21. This indicator’s computation offers a practical method for evaluating the WQ. The WQI application was initially introduced by Horton22 and Brown et al.23, and several practitioners later adopted and modified it24,25. WQI formulations often include extensive computations, which take time and effort. Additionally, the traditional methods for calculating the WQI need significant physical and chemical data, usually at daily intervals. Therefore, alternate methods for accurately and efficiently computing WQI are needed; environmental engineers may find this helpful innovation when monitoring and evaluating water quality.

In the form of machine learning models, soft computing models have been used increasingly in the last several decades to handle various environmental engineering challenges, such as river WQ modeling26,27,28,29,30. According to Yaseen et al.31, soft computing models significantly advance engineering process monitoring and control. Their methods may be used to make precise predictions without requiring intricate programming. Soft computing models are built on data mining and discovering patterns in data. For this, algorithms are built using a portion of the dataset (train), and the performance of predictions is tested using a different subset of the dataset (test)32,33,34. Our literature analysis shows that WQI simulation utilizing soft computing models has received much attention35. Tripathi and Singal36 used the Principal Components Analysis (PCA) model to choose the ideal input variable combination and offer a novel way to compute the WQI in the Ganges River (India). By employing this technique, they could drastically cut the parameters from twenty-eight to just nine. Zali et al.37 investigated the impacts of six primary input factors on the WQI using ANNs. They conducted a sensitivity analysis to determine the relative significance of each parameter in determining WQI, and they concluded that DO, SS, and NO3 are the critical input factors. The ground WQI was calculated using a fuzzy-based model by Nigam and SM38, who also compared its prediction performance to other widely used calculation techniques. They discovered that the fuzzy-based model outperformed them. The Interactive Fuzzy model (IFWQI) was used by Srinivas and Singh39 to construct a unique fuzzy decision-making technique for predicting WQI in rivers. Their findings show that the proposed model performs much better predicting WQI than the conventional fuzzy method. According to Yaseen et al.31, ANFIS-SC (Subtractive Clustering) was the best model for predicting WQI out of three hybrid methods based on the Adaptive Neuro-Fuzzy Inference System (ANFIS). These were ANFIS-FCM (Fuzzy C-Means data clustering), ANFIS-GP (Grid Partition), and ANFIS-SC (Subtractive Clustering).

Environmental scientists have been looking into other strong and reliable data-driven models, even though standard models based on ANN and ANFIS are well known for WQI modeling26,27,35 to show how WQI affects different chemical factors in tropical environments. Another prominent strategy used effectively for different hydrological and environmental issues, such as rainfall forecasting, is tree-based models, such as Decision Trees (DTs)40. For predicting WQ, Granata et al.41 made a Support Vector Regression (SVR) model, a Gaussian Process (GP) model, and a Regression Tree (RT) model. The SVR model worked the best for them. These relate to applying decision-tree and support vector regression models for WQ parameter prediction. Li et al.42 suggested a hybrid SVR model with the FireFly Algorithm (FFA) to predict WQI using monthly data on the WQ parameter. This model was much better at making predictions than the standalone SVR model. Nitrate was discovered to be the most significant parameter for WQI prediction by Kamyab-Talesh et al.43. They explored the optimization of the SVM model to find the parameters that primarily impact the WQI. Wang et al.44 investigated the performance of three machine learning models, SVR, SVR-GA (Genetic Algorithm), and SVR-PSO (Particle Swarm Optimization), to predict WQI using the spectral indicators Difference Index (DI), Normalized DI, and Ratio Index (RI) that were obtained from remote sensing, and found that the SVR-PSO was the best performing model.

Numerous studies have pointed out the uncertainty in soft computing models30,34,45,46,47. Enhancing the reliability and effectiveness of soft computing forecasts is crucial. Techniques such as Artificial Neural Networks (ANN), Fuzzy Logic, and Adaptive Neuro-Fuzzy Inference Systems (ANFIS) often operate within complex, nonlinear problem domains where inaccuracies in data, model parameters, and predictions are inevitable. It facilitates the quantification of model confidence, allowing for the provision of point forecasts and a range of probable outcomes accompanied by associated uncertainty levels48. Furthermore, it improves the reliability of soft computing models, making their predictions more resilient and comprehensible49. The hybrid model effectively addresses uncertainty since the FFA and bagging models enhance the robustness of the ANN and REPT models, respectively, and mitigate uncertainty.

Soft computing methods are advocated due to the infeasibility of performing consistent global monitoring of water quality in all rivers50. This research introduces several methodologies for predicting the WQI of three sub-watersheds in Iran, using soft computing models that have yet to be used in this region. The models included the Gaussian Process (GP), Gene Expression Programming (GEP), REP tree (REPt), Bagging REP tree (BREPt), and a hybrid Artificial Neural Network – FireFly Algorithm (ANN-FFA). The innovation of this research is in the development of hybrid models, namely combining Artificial Neural Networks (ANN) with Firefly Algorithm (FFA) and Bagging Random Ensemble Pruning Technique (REPt). These models have not been used before to predict the WQI in these sub-watersheds. Evaluating the Water Quality Index in a laboratory is costly and labor-intensive due to the processes of sample collection, transportation, and testing. This research introduces a real-time prediction system that employs soft computing models as an alternative method for predicting WQI. The objective is to rectify the deficiency in precise and reliable prediction of WQI by examining the efficacy of advanced soft computing models, namely Artificial Neural Network with Firefly Algorithm (ANN-FFA) and Bagging Random Enhanced Predictive Trees (REPt). These models are undergoing evaluation in comparison to conventional approaches like the Gaussian Process (GP), Gene Expression Programming (GEP), and Randomized Exponential Perturbation Tree (REPt). This work introduces and validates ANN-FFA as a superior model for predicting WQI and enhancing accuracy and reliability in water quality management.

Data-driven models

Recently, many researchers have used soft computing models in civil engineering and water resources51,52,53,54,55,56,57,58,59,60,61,62. The ANN-FFA, BREPt, REPt, GP, and GEP are used in this study, and the details of these models are as follows:

Gaussian process (GP)

GP regression is a state-of-the-art method straight over the function space based on the premise that neighboring studies must exchange information. Gaussian regression is the term for the extension of the Gaussian distribution. In GP regression, the covariance and mean are expressed as the matrix and vector of the Gaussian distribution. The validation for generalization is not necessarily due to prior knowledge of functional dependency and data. The GP Regression models can tell the difference between the forecast distribution and the input test data63. Any finite number of the random variables that make up a GP have a multivariate Gaussian distribution. Assuming p and q represent the input and target domains, x pairs (xi, xj) are selected randomly and equally. The mean function v0 represents a GP on p in regression, assuming that h⊆Re. The radial basis kernel (rbf) and Pearson VII kernel (puk) functions are used in this investigation. Kuss64 is recommended to readers for further details. This investigation uses two kernels, rbf and puk, for model development in GP regression.

$$\:puk=\:\left(1/{\left[1+{\left(2{\sqrt{\Vert\:{x}_{i}-\:{x}_{i}\Vert\:}}^{2}\sqrt{{2}^{\left(\frac{1}{\omega\:}\right)-1}\:/\sigma\:}\right)}^{2}\right]}^{\omega\:}\right)$$
(1)
$$rbf=\:\left({e}^{{-\gamma\:\left|{x}_{i}-\:{x}_{i}\right|}^{2}}\right)$$
(2)

Here, ω, γ, and σ are the parameters of kernels.

GP offers probabilistic predictions, interpolation, versatility, and compatibility with small data sets. It can handle privacy constraints by adding synthetic noise, model complex phenomena, and provide uncertainty estimates by learning noise and smoothness parameters from training data. However, it has several disadvantages, including being not sparse, inefficient in high dimensions, unsuitable for outliers, positive-only variables, computationally expensive, difficult to choose a kernel function, and potentially requiring careful hyperparameter tuning, which can be computationally expensive and require careful optimization techniques65.

GEP (gene expression programming)

GEP is a strategy suggested by Ferreira66 that uses software programs. It is an advanced technique built on the GA foundation frequently used in recent research. The linear chromosomes that comprise the GEP’s software programs are articulated or translated into ETs. The initial step in this program’s problem-solving process is to create the initial population, accomplished through the arbitrary birth of chromosomes. Later, the chromosomes transform into extracellular organisms, which are then evaluated according to performance criteria to represent the solubility of the produced ETs. If the results satisfy the performance standards, population generating ceases. If the outcomes are unsatisfactory, the system regenerates with some improvements to create a new generation with improved value. The benefit of GEP is the ability to contrast chromosomes in a symbolic and linear string of a predetermined length. The step involved in the GEP is depicted in Fig. 1. For this study, the mutation rate is 0.044, inversion rate, incessant and root scale transport rate is 0.1. one -point and two-point crossover rate is 0.3, gene recombination and transportation rate is 0.1, no. of chromosomes is 30 with three head size, no. of gene per chromosomes is 3. Researchers are directed to Ferreira67 and Ebtehaj et al.68 for further information on GEP.

GEP offers flexibility, efficiency, simplicity, and power in evolutionary computation methods. It allows for varying solutions and linear representation of chromosomes and can solve complex problems by evolving intricate models representing nonlinear relationships. Despite its strengths, it faces challenges like complex solutions, computational resources, and parameter tuning, which can be challenging to interpret and optimize.

Fig. 1
figure 1

Flowchart for GEP (created using diagrams.net).

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Reduced error pruning tree (REPt)

The REPt model employs the idea of randomly chosen characteristics determined by computer technology to speed up classification tree logic techniques and reduce variance inaccuracy69. The REPt employs the logistic regression technique and creates many trees through various computation processes; the most straightforward tree was selected from all the created trees. When the conclusion is significant, and the complexity of the tree’s internal structure is minimized, the REPt has provided a flexible and straightforward modeling technique by monitoring training data sets. The pruning algorithm considers the backward over-fitting complexity of this technique. It uses the post-pruning algorithm to push for the most miniature possible representation of the best precision tree logic70. It only chooses values once for numeric characteristics71.

REPt is a simple, intuitive, fast, and easy method for improving model generalization and reducing complexity and tree size. It requires a validation set and accuracy measure, and unnecessary nodes are removed for better prediction. However, a greedy algorithm makes the best decision at each step without considering future consequences, leading to suboptimal solutions. It is sensitive to the choice of validation set, affected by data noise or randomness, and biased towards simpler trees.

Bagging

Bagging is the technique that enhances the results of weak, soft computing models72. The amount of the original database that will be merged depends on the bagging factor73. Bootstrap-resampled observed data are used in each model. Algorithms for bagging include three steps: Bootstrapping produces a fresh training set using replacements. The outputs of the classifier are associated with distributed voting. This strategy enhances classification variance and generalization. The fundamental classifier must be unbalanced for this model to work; otherwise, no classification will result. In this study, bagging is used to investigate the reliability of the Reduced Error Pruning Tree in predicting WQI.

Bagging allows weak learners to outperform strong ones, reducing variance and eliminating overfitting. However, it can introduce bias and be computationally expensive, potentially discouraging its use in certain situations.

Artificial neural network (ANN)

A computational model that replicates how nerve cells in the human brain function is known as an ANN. The multi-layer perceptron (MLP) kind of ANN employed in this study is trained using the backpropagation learning technique. An input layer, a hidden layer, and an output layer comprise the three layers of the MLP. The data are accepted by the input layer, processed by the hidden layer, and then shown by the output layer as the model’s outputs. During the learning process, each layer’s neurons are connected to the layers below it through a weight. The external world provides input to the neurons in the input layer. The calculations in this layer are nonexistent. The input layer sends data to the hidden layer, which performs calculations and sends the results to the output layer. The system’s output is transmitted via neurons in the output layer. The Neural Network is a two-way process. The first process is training the model, which seeks a suitable nonlinear relationship by generating appropriate weights between the various variables. The second process involves processing the sum using a nonlinear transfer function to produce a prediction. Then, an ANN learns by running a backward process to update the weights until the error has been minimized and done in response to the errors between the actual output values and the intended output values.

In the subsequent testing phase, the neural network is fed a different data set. The learned weights-based neural network predictions are tested against the desired output values. It is done to determine if the ANN over- or under-fits a particular quantity of data. The ANNs have performed satisfactorily while dealing with a variety of engineering simulations.

ANNs offer fault tolerance, self-learning, parallel processing, nonlinear modeling, and complete data. They can tolerate long training times, perform quickly, and predict output values for specific input values. They can also detect complex relationships and nonlinear problems. ANNs have drawbacks such as their black-box nature, computational expense, long development time, overfitting, high data requirements, and reliance on numerical input, making them difficult to understand74. For ANN models, a low bias and low variance are crucial. Haykin74 is recommended to readers for further details. One hidden layer of the ANN model is utilized in the current investigation.

FireFly algorithm (FFA)

Yang75 introduced the firefly method at Cambridge University. This method is a swarm intelligence optimization model based on firefly movement and was created using the firefly’s natural behavior and radiance pattern76. Due to its population-based search, it efficiently handles multi-modal functions, allowing candidate solutions to benefit from building blocks from diverse solutions. Fireflies constantly go toward sources of light when they have less of them76. Additionally, fireflies can create a specific pattern. The patterns that fireflies follow are special. The rhythmic light (Flash), light rate (Rate of Flash), and the separation between the light signals are three of the most crucial elements in the absorption of fireflies. The brightness varies in attractiveness with absorption and is proportionate to the distance from its source75.

$$\:P\left(l\right)=\:\frac{{P}_{s}}{{l}^{2}}$$
(3)

For an environment with a fixed light gain coefficient, the intensity of light P fluctuates with l in the above relationship, where Ps is the intensity of the light source75.

$$P=\:{P}_{0}{e}^{-\gamma\:l}$$
(4)

Where P0 is the initial light intensity. Given that a firefly’s attractiveness is inversely correlated with the amount of light it receives from its neighbors, the following definition of β is used:

$$\beta\:={\beta\:}_{0}^{-\gamma\:{l}^{2}}$$
(5)

The hybridization of the artificial neural network-based Firefly algorithm is depicted in Fig. 2. The appeal of l = 0 is that β0. The charm function β(l) in actual implementation can be any uniform descending function, as in the generic form below75.

$$\beta\:\left(l\right)={\beta\:}_{0}^{-\gamma\:{l}^{m}}$$
(6)

However, it also has drawbacks, such as high computational time complexity and slow convergence speed due to its full-attracted model, which focuses on firefly oscillation during movement.

Fig. 2
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Flowchart for FFA (created using diagrams.net).

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Materials and methodology

Study area

The Khorramabad, Biranshahr, and Alashtar sub-watersheds, situated in the Iranian province of Lorestan between 48°030 10′′E and 48°590 07′′E and between 33°110 47′′N and 34°030 27′′N with an area of 3,562.1 km2, were used as the source of flow and water quality data for the study. The catchment area’s elevation ranges from 1,158 to 3,646 m MSL. The data measurements were made between September 2014 and August 2017. The average rainfall for the Khorramabad, Biranshahr, and Alashtar sub-watersheds is 442 millimeters, 484 millimeters, and 556 millimeters, respectively. The study area is depicted in detail in Fig. 3; the red dots in the figure indicate the location from where the samples were collected.

Fig. 3
figure 3

Study area (generated using ArcGIS, v10.8, https://www.esri.com/en-us/arcgis/products/arcgis-desktop/overview).

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WQI and data preparations

The WQI consistently summarizes WQ data for reporting to the public. It provides a straightforward assessment of drinking water quality from a source comparable to the UV or air quality index. The water quality data are compared to “BIS and WHO” to determine the WQI77. The WQI calculates a single score by combining three measurements: the scope, frequency, and amplitude of water quality exceedances. This computation yields a score that ranges from 0 to 100. The better the water quality, the lower the score. The results are then categorized into one of the five groups. If the value of WQI is less than 50, the water quality will be ‘Excellent’. If it comes in 50–100, 100–200, and 200–300, then the quality is ‘Good,’ ‘Poor,’ and ‘Very Poor.’ If the value exceeds 300, the water is “Not Suitable for Drinking’.

The dataset used for the study is collected from Singh et al.54. It comprises water quality measurements from three sub-watersheds in Iran, collected from Sept. 2014 to Aug. 2017. The dataset includes vital water quality parameters such as sulfate (SO4), total dissolved solids (TDS), the potential of Hydrogen (pH), chloride (Cl), electrical conductivity (EC), Potassium (K), bicarbonate (HCO), magnesium (Mg), sodium (Na), and calcium (Ca). The WQI is calculated using the formula given in Singh et al.54. Further, the whole dataset is divided into two subsets: train and test, on the ratio of 70:30. The input variables of the study are SO4, TDS, pH, Cl, EC, K, HCO, Mg, Na, and Ca, while WQI is the output variable. Statistical analysis, including mean, standard deviation, minimum, and maximum values for each parameter, has been performed to understand the dataset characteristics better. This analysis helps understand the data variability and its impact on model predictions. Table 1 gives the values of min., max., ranges, mean, standard error, standard deviation, kurtosis, and skewness of various variables used in this study. Using the kurtosis values, it is observed that EC, pH, HCO, and WQI give the negative values (Platykurtic) and TDS, Cl, SO4, Ca, Na, and k give the positive values (Leptokurtic) for complete data; EC, Mg, and WQI give negative values (Platykurtic) and TDS, Ph, HCO, Cl, SO4, Ca, NA, and K give the positive values (Leptokurtic) for train subset. WQI, pH, and HCO give negative values (Platykurtic), and TDS, EC, Cl, SO4, CA, Mg, Na, and K give negative values (Leptokurtic). The mean, max., min., standard error, and deviation values are approximately the same for full, train, and test data subsets. Finally, the correlation of the various input and output variables is calculated and plotted as a heat map in Fig. 4. The figure suggests that the correlation of WQI with pH is negative, while it is positive for all other variables. The EC and pH have the highest correlation with WQI, while pH gives the lowest correlation.

Table 1 Statistical descriptions of variables.
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Fig. 4
figure 4

Heatmap of input and output variables (generated using Origin Pro, v2024b).

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Statistical parameters

Statistical parameters are a formal and productive procedure to measure and validate the results of data-driven models. Four statistical parameters are used in this investigation, and these parameters are coefficient of correlation (COC), coefficient of determination (COD), root mean square error (RMSE), Nash–Sutcliffe error (NSE), and mean absolute percentage error (MAPE). The range of the COC and COD lies between − 1 and 1; the range of RMSE lies between 0 to ∞, and the output of the MAPE and RRSE is in percentage78,79,80.

Proposed work flow

The workflow of the proposed data-driven models, described in Fig. 5, is as follows:

  • Dataset: The dataset used in the study is collected from Singh et al.54. It contains ten physio-chemical WQ parameters, including SO4, TDS, pH, Cl, EC, K, HCO, Mg, Na, and Ca.

  • Data Splitting: The whole dataset is divided into two subsets, train and test, with a ratio of 70:30. In the training subset, there are 86 observations, while in the test subset, 38 observations are there.

  • Data-Driven Models: Five soft computing models ANN-FFA, BREPt, REPt, GP, and GEP) are used to predict the WQI of three watersheds in Iran.

  • Statistical Parameters: The potential of the soft computing models is assessed using statistical parameters. Five statistics, such as COD, COC, RMSE, NSE, and MAPE, are used.

Fig. 5
figure 5

Proposed workflow of the study (created using diagrams.net).

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Results

River water quality affects groundwater quality due to the direct percolation of water81. Additionally, using river water for irrigation might impact groundwater resources. Hence, areas where aquifers are protected should be used for river water irrigation82. Iran experiences an average annual precipitation of 730 mm due to its humid environment. Its rivers’ daily streamflow varies and occasionally experiences spikes of discharge. Seasonal variations are seen in the measured physio-chemical parameters, with rainy seasons showing the most incredible values. A total of five soft computing models (ANN-FFA, BREPt, REPt, GP, and GEP) are used in this study to predict the WQI of three watersheds (Khorramabad, Biranshahr, and Alashtar) in Lorestan province, Iran. The performance of the soft computing models strictly depends upon the user-defined parameters (UDFs). The UDFs are calculated using trial and error methods. Several sets of UDFs are used; on these sets, the performance of the soft computing models is checked. The set that gives the best results of soft computing models is chosen. These chosen values of UDFs for different soft computing models used to predict WQI in this study are summarized in Table 2.

Table 2 UDFs of soft computing models.
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Statistical parameters are one measure that checks the performance of the data-driven models. This study uses four statistical parameters (Eqs. 7–10). The outcomes of the statistical parameters for various soft computing models are tabulated in Table 3 for the train and test subsets. According to Table 3, ANN-FFA is the model that has the edge over other models in the prediction of the WQI in training and test and test subsets. It gives the most efficient values of statistical parameters i.e. COC = 0.9990 & 0.9989; COD = 0.9612 & 0.9980; RMSE = 0.3036 & 0.3340; NSE = 0.9980 & 0.9979; and MAPE = 0.7259% & 0.7969% for train and test subsets, respectively. Regarding preciseness, the ANN-FFA is followed by GEP, GP_puk, BREPt, REPt, and GP_rbf. In between the kernel function, the puk kernel with GP gives better results than the rbf kernel with GP. The GP-rbf kernel gives the worst result in the prediction of WQI. Thus, the ANN-FFA data-driven model gives the best result of WQI and is supreme among all data-driven models.

Table 3 Results of statistical parameters for various data-driven models.
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Scattered plots, variation plots, box plots, and Taylor diagrams are also used to check the potential of the soft computing models. The scattered plots of WQI with all soft computing models are plotted in Fig. 6 (a to f). It is plotted between the actual values of WQI (WQIActual) vs. predicted values of WQI (WQIPredicted), and the diagonal line is the best fit. The model in which all the points lie on best-fit lines is the best. Figure 6 (a) shows that all the points of WQIActual vs. WQIPredicted lie on the best-fit line compared to other data-driven models. Figure 6 (f) shows the points of GEP, which has the second position; Fig. 6 (d) shows points of GP_puk, which got the third position; Fig. 6 (c) shows points of BREPt, which got the fourth position; Fig. 6 (b) shows points of GP_puk which got the fifth position and Fig. 6 (e) shows points of GP_rbf which got the last position in the prediction of WQI as per the scattered diagram. The scattered diagram (Fig. 6) suggests the same trend, which is suggested in Table 3. Hence, the ANN-FFA model has the highest accuracy in predicting WQI.

Fig. 6
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(a–f) Scattered plots of various soft computing models in the prediction of WQI (generated using MS Office, v2019).

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The variation plots of various soft computing models are depicted in Fig. 7 (a to f). As the name indicates, the variations plot shows the visual interpretation of the variation among the WQIActual and WQIPredicted. Figure 7 (a) shows the variation plot of ANN-FFA; Fig. 7 (b) shows the variation plot of REPt; Fig. 7 (c) shows the variation plot of BREPt; Fig. 7 (d) shows the variation plot of GP_puk; Fig. 7 (e) shows the variation plot of GP_rbf; Fig. 7 (f) shows the variation plot of GEP. Figure 7 suggests a minimum difference between WQIActualand WQIPredicted for ANN-FFA data-driven models, followed by GEP, GP_puk, BREPt, REPt, and GP_rbf. Hence, the variations plots (Fig. 7a to f) also suggest that the ANN-FFA is the model that can predict the WQI accurately and precisely.

Fig. 7
figure 7

(a–f) Variation plots of various soft computing models in the prediction of WQI (generated using MS Office, v2019).

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The distributions of relative errors (%) in the form of an open box plot for all models are plotted in Fig. 8 to illustrate the efficacy of the soft computing models. This figure shows that the ANN-FFA model had the slightest errors compared to the other soft computing models for the training subset. Also, the ANN-FFA models performed well through the test subset. ANN-FFA points are not distributed and present near zero, while the points of other models are distributed from + 10 to -15. Hence, Fig. 8 concludes that the ANN-FFA model is the best data-driven model for predicting WQI.

Fig. 8
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Relative error for various data-driven models (generated using Origin Pro, v2024b).

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Fig. 9
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Taylor diagram for various data-driven models (generated using R, v4.4.1).

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The Taylor diagrams, a graphical method for assessing the performance of a data-driven model, are displayed in Fig. 9. This figure shows that for the testing subset, the red solid circle point from the ANN-FFA model is closer to the actual (black hollow) point than those from the other models based on the distance between the points acquired by the soft computing models and the actual point. The Taylor diagram (Fig. 9) also concluded that ANN-FFA is the best-performing model, followed by the GEP model for predicting WQI. The performance of the GP_rbf (solid orange circle) model is the lowest among all applied models for predicting WQI.

Comparison of obtained results with previous literature

The result of the best model, i.e., ANN-FFA, is compared with the previously published literature. The previously published literature selected for this study are Hu et al.83, Hussein et al.84, Mohseni et al.85, and Kim et al.86. Table 4 shows the comparison of these published models with the best-selected model of the study, ANN-FFA. The comparison is based on four statistical parameters: COD, RMSE, NSE, and MAPE. The results suggest that the obtained model (ANN-FFA) is superior to the models published in the literature based on statistical parameters. Thus, the ANN-FFA model outperforms the comparative soft computing models and has superior results to the model published in the literature. It implies that the ANN-FFA model is a robust and reliable tool for predicting WQI, with potential applications in various fields such as environmental science, water resource management, and public health.

Table 4 Comparison of this study’s results with previously published results based on statistical parameters.
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Discussions

This research aimed to examine the efficacy of several soft computing methods in predicting WQI for three sub-watersheds located in Iran. The investigated models included Gene Expression Programming (GEP), Gaussian Process (GP), Reduced Error Pruning Tree (REPt), Bagging REPt, and Artificial Neural Network optimized with the FireFly Algorithm (ANN-FFA). The ANN-FFA model exhibited exceptional performance due to its strong correlation coefficients and fewer errors. The accuracy of ANN-FFA may be ascribed to its robust and efficient optimization capabilities. The GEP model demonstrated commendable performance. Nevertheless, it attained a different level of accuracy than shown by ANN-FFA. The approach is evolutionary, progressively enhancing solutions over generations, efficiently capturing the data’s intrinsic relationships. Nonetheless, the system’s performance may be influenced by the complexity of the problem and the choice of parameters. The GP models, using radial basis function (RBF) and Pearson VII kernel (PUK) functions, demonstrated robust albeit somewhat worse performance than ANN-FFA and GEP. The efficacy of water quality data may have been constrained by the challenges posed by its noisy and complex nature. The decreased prediction accuracy may also be ascribed to the selection of kernel functions since the RBF and PUK kernels may not have been the most appropriate for this dataset. The REPt and Bagging REPt models had the lowest accuracy relative to the other models analyzed. However, acknowledging that these models may exhibit constrained simplicity and interpretability when used for highly complex and nonlinear data, such as water quality measurements, is essential.

This research’s findings have substantial implications for water resource management, particularly in regions like Lorestan province, where accurate and timely measurement of the WQI is crucial for sustainable water management. The ANN-FFA algorithm’s ability to provide precise and accurate WQI forecasts with few mistakes makes it an excellent option for integration into decision support systems used by water resource managers. The study shows that soft computing models, especially those that are enhanced with optimization algorithms like FireFly, can be used in addition to or instead of traditional laboratory methods for testing WQI. The shift to model-based predictions offers a cost-efficient, time-saving, and scalable method for water quality monitoring, which is particularly beneficial in resource-constrained scenarios.

Conclusion

This research introduces advanced soft computing models for predicting the WQI of three sub-watersheds in Lorestan Province, Iran. This work’s primary contribution is the invention and implementation of hybrid models, namely the Artificial Neural Network optimized by the FireFly Algorithm (ANN-FFA), which has not been used before in this research domain. The paper illustrates the higher prediction accuracy of the hybrid technique by comparing the performance of ANN-FFA with other models, such as the Gaussian Process (GP), Gene Expression Programming (GEP), and REP Tree (REPt).

The main results indicate that the ANN-FFA model surpassed all other models, achieving a correlation coefficient (COC) of 0.9989, a coefficient of determination (COD) of 0.9980, a root mean square error (RMSE) of 0.3340, a Nash–Sutcliffe error of 0.9979, and a mean absolute percentage error (MAPE) of 0.7969%. The research indicates that the Gaussian Process model using the Puk kernel function outperforms the model employing the RBF kernel function for WQI prediction. These results are crucial for the literature on water quality modeling since they provide a new standard for using hybrid models in environmental monitoring. The ANN-FFA model, by providing a cost-efficient, real-time prediction system, can significantly contribute to water resource management and environmental conservation. This method offers a dependable resource for policymakers and environmental managers to make educated choices, particularly in areas with inadequate water quality monitoring equipment.

Notwithstanding the encouraging outcomes, the research had several limitations. The dataset was confined to three sub-watersheds, perhaps failing to capture the heterogeneity in water quality over larger areas or diverse climatic zones. The models depend significantly on historical data, and their accuracy may diminish in regions where such data is less abundant or inconsistent. Subsequent research needs to broaden the investigation to include a more comprehensive array of datasets from other places and environmental situations. The possibility of integrating machine learning with conventional physics-based models to enhance forecast accuracy warrants investigation.