ABSTRACT

This study uses four modeling techniques, Gene Expression Programming (GEP), Support Vector Machine (SVM), and Multiple Linear Regression (MLR), to estimate streamflow in the Brahmani River in India. The objective is to develop accurate models that can predict streamflow based on two different hydroclimatic parameters and one river physical parameter. The study utilizes historical data of streamflow and corresponding hydroclimatic variables, including rainfall, temperature, and physical parameter river stage. The dataset is split into training and testing sets to facilitate the creation and validation of models. Statistical measures like Mean Square Error (MSE), Root Mean Square Error (RMSE), and coefficient of determination (R²) are utilized to assess the efficacy of the GEP, SVM, and MLR models in estimating streamflow. The results indicate that all the models can effectively estimate streamflow in the Brahmani River. On the other hand, the GEP model performs better than the MLR and SVM models. Its capacity to capture the intricate interactions between hydroclimatic parameters and streamflow is demonstrated by its lower error values and higher R² values. The analysis of the models reveals that rainfall, temperature, and river stages can be significant predictors for streamflow estimation of the Brahmani River. These findings emphasize the importance of incorporating multiple hydroclimatic parameters to enhance the accuracy of streamflow predictions. The study also emphasizes the benefits of employing GEP as a modeling tool because of its capacity to handle complicated patterns and non-linear connections. Overall, this research provides valuable insights into streamflow estimation in the Brahmani River using GEP, SVM, and MLR models with different  hydroclimatic parameters. The findings contribute to developing reliable tools for water resource management and hydrological forecasting in the region, facilitating informed decision-making based on accurate streamflow predictions.

1 INTRODUCTION

Streamflow forecasting is essential for a range of applications, particularly in hydrology and water resource management. Accurate forecasts are crucial for water supply and reservoir operations, hydropower generation, environmental sustainability, irrigation, and disaster preparedness. Reliable streamflow estimates support long-term water resource planning and ensure resilience across industries reliant on water availability.

Discharge, the volume of water flowing through a river at a specific location and time, is a fundamental hydrological parameter. It is closely linked to hydroclimatic variables such as precipitation, runoff, infiltration, evapotranspiration, and groundwater flow, which influence water movement within a watershed. Precipitation supplies water to the system, while runoff directs excess water to streams. Infiltration and groundwater recharge regulate base flow, and evapotranspiration removes water from circulation. Hydrologists use stream gauging stations, flow meters, and predictive models to estimate discharge. Understanding these interactions is vital for water management, flood forecasting, and ecological sustainability.

This work aims to create mathematical models for streamflow estimation based on GEP, ANN, and MLR. For this objective, the Brahmani River in Odisha is used as a case study. The GEP models' performance was compared to that of many linear regression techniques. Historical streamflow data, as well as pertinent hydroclimatic parameters such as rainfall, temperature, and river stage, are used. Because of the wide and dynamic riverbed, inaccessible measurement sites, and high-water levels during floods, measuring streamflow in the Brahmani River is extremely challenging. The novel aspect of this study is to compare simple and inexpensive discharge predictions using soft-computing methods such as Gene Expressing Programming (GEP) along with Machine Learning (ML) techniques like Support Vector Machine (SVM) and show how it is more beneficial than other traditional methodologies.

1.1 Literature review

When it comes to water resource planning, streamflow forecasting is essential for comprehending the substantial shift in climatic and hydrologic elements over time, particularly during extreme weather events like floods and droughts. Hydrologists, planners, watershed managers, and stakeholders must accurately estimate streamflow to plan for flood management and build engineered projects (Hosseinzadeh et al. 2023). Because the atmosphere is turbulent and nonlinear, which impacts groundwater and streamflow, forecasting streamflow is a challenging undertaking (Lorenz 1969). Several models have been developed using hydroclimatic factors to determine stream flow. GEP is one of the appropriate tools for the job, and the MLR approach can also be utilized to investigate the relationship. Discharge in open channel flow may be estimated using several AI approaches. These solutions employ data-driven algorithms to anticipate flow rates and are especially beneficial when direct measurements or traditional approaches are unavailable or unfeasible. Among the artificial intelligence approaches used in open channel flow discharge computation are: Artificial Neural Networks (ANNs), Decision Trees, Gaussian Processes (GPs), Genetic Programming, and the Multilinear Regression Technique (MLR). It is possible to simulate runoff using data-driven models like genetic programming (GP) based on available hydrometeorological data. Generating rainfall-runoff models using genetic programming, both in conjunction with conceptual models and alone based on data (Keijzer 2001). According to Adamowski and Sun (2010), in river flow forecasting applications, data-based hydrological methods are becoming increasingly popular due to their rapid development times and minimum information requirements. For instance, Guven and Aytek (2009) anticipates the time series of daily flow rates using two types of neural networks (NNs) and a genetic programming technique called Linear Genetic Programming (LGP). As a transparent model, GEP may give mathematical expressions describing the link between input (climate indices) and output (streamflow) variables (Esha et al. 2019). Following that, comparison studies of various AI approaches have surfaced in the relevant literature, with the goal of determining the best suited. Ghorbani et al. (2010) describes the fundamental hydrological mechanism between climatic mode and streamflow without extensive understanding of the programme employed. Guven (2009) demonstrates the use of gene-expression programming, a genetic programming extension, as a substitute method for simulating the stage-discharge relationship. Stage rating curves and multiple linear regression procedures are two conventional methods that are compared with the results. For forecasting short-term and long-term streamflow, a new combination approach (wavelet-genetic programming) was presented. To anticipate streamflow, the new combination approach combines the discrete wavelet transform with genetic programming methodologies. The resulting findings demonstrated very good agreement between the observed and modeled values (Karimi et al. 2015). Meshgi et al. (2014) suggests a straightforward empirical formula that retains physical information and calls for the least amount of data for estimating baseflow time series using GP. The first step was simulating baseflow for a small semi-urban watershed (0.043 km²) in Singapore using a groundwater numerical model. After that, groundwater level variations—which are relatively straightforward to monitor and physically associated to baseflow production—were linked to baseflow time series using GP to develop an empirical equation (Kisi et al. 2011). In order to estimate daily intermittent stream flows, his research assesses the accuracy of several data-driven methodologies, such as support vector machines (SVM), artificial neural networks (ANNs), and adaptive neuro fuzzy inference system (ANFIS). Additionally, the results are contrasted with those derived using dynamic local linear regression (DLLR) and local linear regression (LLR). Comparative results showed that ANFIS, ANN, and SVM performed better in daily intermittent streamflow forecasting than LLR and DLLR models. The ANN and ANFIS produced the most accurate forecasts. Lafdani et al. (2013) showed complex nonlinear interactions between these input factors, and how runoff may be captured by ANNs with good performance. The fact that stage and temperature are included with precipitation highlights how important these factors are to increase the accuracy of discharge predictions. Ostad-Ali-Askari (2022) showed that runoff depends on changes in such meteorological parameters as rainfall, cosmological energy, airstream, moisture, and temperature.

2 STUDY AREA

The Brahmani is an important cross-state east-flowing river among India's peninsular rivers. This basin may be found roughly between north latitude 20°28', north longitude 83°52', and east longitude 87°03'. The basin is bordered to the east by the Bay of Bengal, to the west and south by the Mahanadi basin, and to the north by the Chhota Nagpur plateau. The basin drains 39,033 square kilometres of land before emptying into the Bay of Bengal, as seen in Figure 1, passing through the states of Jharkhand, Chhattisgarh, and Orissa. The climate of the basin is tropical, with hot summers and moderate winters. From June to October, this basin is influenced by the south-west monsoon, with some infrequent downpours in the lower levels owing to cyclonic depressions in the Bay of Bengal. The average annual rainfall for this basin is 1,460 mm. The average maximum temperature ranges from 38 to 43°C, while the lowest temperature ranges from 10 to 15°C.

Figure 1 Brahmani River basin, Panposh Gauging Station.

3 DATA COLLECTION

The discharge and stage data used for the study were extracted from the Brahmani River Water Yearbook 2018–2019, and 2019–2020, published by CWC, available in the public domain. The mean temperature and precipitation data was collected from the NOAA (National Oceanic and Atmospheric Administration) website as enumerated in Table 1. The amount and timing of streamflow and discharge are influenced by interrelated factors such as temperature, stage, and precipitation. Daily discharge data was collected for the year 2019–2020, shown in Figure 2. This shows the significance of several input variables, including temperature, stage, and precipitation, and is emphasized in relation to discharge prediction. Since the stage accurately represents the amount of water flowing across a cross-section of the river channel, it is an essential factor in determining discharge. From June 2018 to May 2020, a total of 730 daily data points for precipitation, mean temperature, and stage have been considered.

Figure 2 Observed daily streamflow at Panposh gauging station for 2019–2020.

Table 1 Details of the dataset.

Sl No	Data Type	Data Source	Frequency
1	Stage (H)	Brahmani River Water Yearbook	Daily
2	Discharge (Q)	CWC Water Yearbook (CWC 2017) https://cdrc.cwc.gov.in/
3	Precipitation (P)	National Oceanic and Atmospheric Administration https://www.noaa.gov/
4	Temperature (T)

4 METHODOLOGY

4.1 Gene Expression Programming

Genetic programming and evolutionary computation both employ gene expression programming (GEP), a type of evolutionary algorithm. In the late 1990s, Candida Ferreira introduced it as a development of traditional genetic programming (GP).

The procedure begins with the creation of an initial population of potential solutions or people. Each person is represented as a linear string of symbols (chromosome), which comprises genes, operators, constants, and variables. To make an expression, these symbols are joined using a predetermined language that is called Initialization. The fitness of everyone is assessed by applying the expression (chromosome) to the issue domain. The fitness function assesses the expression's ability to do the intended goal. The better the individual's solution, the greater the fitness value, called Evaluation. Individuals are chosen from the population for the following generation depending on their fitness levels. Individuals with greater fitness levels are more likely to be chosen, while other selection procedures such as tournament selection or fitness proportional selection might be utilized under the process of Selection. During reproduction, genetic operators change the chromosomes. The GEP system contains many genetic operators on specific chromosomes.

1. Mutation: Some individuals in the chosen group have random alterations in their chromosomes. Mutation is the process of randomly altering one or more symbols on a chromosome, resulting in genetic variation in the population.
2. Recombination (Crossover): Two parent individuals are picked from the selected population, and a crossover site is determined along their chromosomes. The pieces beyond the crossing point are switched between the parents to produce new offspring.
3. Transposition: In GEP, the chromosome is separated into segments (e.g., head and tail). Transposition is the process of transferring segments inside an individual to produce genetic variety.
4. Inversion: A single individual is selected, and a section of its chromosome is identified. This section is then reversed in order while keeping the rest of the chromosome unchanged. Inversion helps maintain useful gene structures while promoting genetic diversity, improving exploration in the search space.

The new offspring created by these genetic operators (mutation, crossover, and transposition) are added to the population, and the operation is continued indefinitely or until a stopping criterion is met. The evolution process comes to an end when a termination condition is met, which might be a maximum number of generations achieved, a satisfying solution found, or a certain time restriction fulfilled known as Termination. Following the completion of the evolutionary process, the best chromosome (or one of the best individuals) with the highest fitness value is extracted as the solution to the issue. The acquired solution is then applied to a real-world situation, and its generalizability and performance are assessed using fresh, previously unknown data. A brief idea about the GEP is provided in Figure 3. Karva language or Karva expression expressed as an expression tree (ET) is the standard notation used in GEP models (Ferreira 2006).

Figure 3 Brief algorithm of gene-expression programming.

The fitness of each of these chromosomes is evaluated based on a fitness function, as shown in Equation 1:

(1)

Where:

Xj	=	value returned by the chromosome for the fitness case j, and
Yj	=	expected value for the fitness case j.

A fitness function quantitatively assesses how well the model predicts the predicted value. In Equation 1, the function f yields the total number of errors in the target value for which the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Coefficient of Determination (R²) are calculated. Other fitness functions are available; however, in this work, RMSE, MSE, and R² are used to create the GEP model. To simulate the gene expression programming, GeneXproTools 5.0 is used. Models that GeneXproTools uses in the population are selected based on their fitness. The selected models are reconstructed by introducing genetic variations using one or more genetic operators, such as recombination or mutation. A more improved model is produced once this process is repeated for a predetermined number of generations.

4.2 Methodology for Support Vector Machine (SVM)

The Support Vector Machine (SVM) model is implemented following a structured approach to ensure accurate classification or regression. First, data is collected and pre-processed by handling missing values, normalizing numerical features, and encoding categorical variables. The dataset is then split into training and testing sets to evaluate model performance. For our case, 80% of the available data points are used for training and remaining 20% for testing Feature selection techniques may be applied to reduce dimensionality and improve efficiency.

The SVM model here is trained using radial basis function (RBF) kernel function for our data set. The hyperparameters, including the regularization parameter (C) and kernel-specific parameters like gamma (γ), are tuned using random search with cross-validation to optimize model performance. During training, SVM identifies an optimal hyperplane that maximizes the margin between different classes, ensuring better generalization. For non-linearly separable data, a kernel trick is used to transform data into a higher-dimensional space where it becomes linearly separable.

The model is evaluated using performance metrics such as Mean Squared Error (MSE) and the R² score for regression tasks.

4.3 Multiple Linear Regression (MLR)

Multiple linear regression (MLR) is a statistical technique that models the relationship between a large number of independent variables (also called features or predictors) and a dependent variable (sometimes called the objective or result). It's a straightforward extension of linear regression where the dependent variable is predicted using just one independent variable.

Equation 2 shows the general form of MLR:

(2)

Where:

y	=	dependent variable (value to be predicted),
x1, x2, xn	=	independent variables (predictors),
b0	=	y-intercept or constant term,
b1, b2..., n	=	coefficients of the independent variables, which describe the degree and direction of their influence on the dependent variable, and
ε	=	error term, representing the difference between the predicted value and the actual value of the dependent variable.

The purpose of MLR is to estimate the coefficient values (b₀, b₁, b₂..., n) so that the projected values (ý) based on the model are as close to the actual values of the dependent variable (y) as feasible.

The model uses Ordinary Least Squares (OLS) regression to estimate the coefficients. The goal of OLS is to minimize the sum of the squared discrepancies between the observed and predicted values.

The MLR modeled equation developed here is shown in Equation 3:

(3)

The present study adopts a structured approach for predictive modeling using multiple machine learning techniques. Initially, relevant data were collected from reliable sources, capturing various parameters that influence the target variable. Based on literature review and preliminary analysis, the key influencing factors—Stage (H), Temperature (T), and Precipitation (P)—were identified. The collected data were then segregated into two subsets: training and testing datasets. Three different modeling techniques were employed: Multiple Linear Regression (MLR), Gene Expression Programming (GEP), and Support Vector Machine (SVM).

Additionally, a validation step was performed to ensure model robustness and to avoid overfitting. The performance of each model was assessed using several statistical indices such as Coefficient of Determination (R²), Root Mean Square Error (RMSE), Nash-Sutcliffe Efficiency (NSE), and Root Mean Square Error (RMSE). Finally, a comparative analysis of these indices facilitated the selection of the most accurate and reliable model, as shown in Figure 4.

Figure 4 Visual representation of the methodology.

5 PERFORMANCE ANALYSIS OF THE MODEL

The performance of the GEP model in training and testing sets was measured using the Coefficient of Determination (R²), Mean Square Error (MSE), and Root Mean Square Error (RMSE). Equation 4 measures the average difference between values predicted by a model and the actual values. It provides an estimation of how well the model can predict the target value (accuracy). The lower the RMSE value, the better the model. Equation 5 is a measure of errors between paired observations expressing the same phenomenon. A lower RMSE value indicates a more accurate prediction, while a higher RMSE value indicates a less accurate prediction. Equation 6 quantifies the average of the squared differences between predicted and actual values. It's commonly used in regression analysis to assess the accuracy of a model by indicating how well it fits the observed data. In a regression model, the Coefficient of Determination, or R², is a statistical metric that determines how much of the variation in the dependent variable can be accounted for by the independent variable. A strong effect on the dependent variable is indicated by a value > 0.7.

(4)

(5)

(6)

Where:

n	=	number of data points,
	=	actual (true) value,
	=	predicted value, and
	=	mean of actual values.

6 RESULTS AND ANALYSIS

6.1 Development of GEP Model for stream flow prediction

The GEP technique is used here to model the discharge Q in terms of stage, precipitation, and temperature. The "training set" is chosen from the entire data set initially, and the remainder is utilized as the ‘testing set’. The training and testing datasets are divided as 80% and 20%, respectively. The following connection describes streamflow Q or discharge as a function of previously described hydroclimatic variables. Equation 7 shows the dependency of discharge on precipitation, temperature, and stage.

(7)

Where:

P	=	precipitation,
T	=	temperature, and
H	=	stage.

In this work, streamflow is designated as the output and the three independent parameters in Equation 2 as the input. Five fundamental arithmetic operators (+, -, *, /, and sqr) were employed in the model construction as a function set. Multigenic programming employs three genes with addition as the connecting function. Several generations were put to the test. The functional set and operational parameters employed in this work's GEP modeling are summarized in Table 1. A total of 380 datasets were uploaded, 304 of which were utilized for training, and the remaining for testing. Because the fitness function creates an equation to determine n from the dependent parameters, the datasets do not need to be normalized in the analysis. These parameters, whether dimensional or non-dimensional, can be utilized directly to compute streamflow in their typical form, that is, the one used during model creation. The output from the GEP model 3 expression trees presented in Figure 5 are obtained to form the model equation.

The suggested GEP model is represented in its simplified analytical form in Equation 8:

(8)

Where:

K

=

Figure 5a, b, c Expression Tree for the study area.

Table 2 shows the parameters used in GEP, and Table 3 represents all the strategical data used during the model run.

Table 2 Parameters used in GEP.

Models	RMSE	MSE	R2
SVM	0.067	0.004	0.67
GEP	0.036	0.001	0.78
MLR	0.391	0.153	0.74

Table 3 Strategic values used in GEP.

Description of parameter	Parameter setting
Functions used	+, -,*, /, sqr
Number of chromosomes	256
Head size	7
Number of genes	3
Function linked	Addition
Fitness function	RMSE
Program size	41
Number of generations	20,000

The influence of parameters on streamflow was tested by creating a model in GeneXproTools 5.0. The error analysis of the created GEP model for the complete dataset is shown in Table 4. Each of the 3 independent characteristics has a considerable impact on the streamflow. As a result, the functional connection described in Equation 7 is employed in this investigation.

The MSE, R², and RMSE of daily observation flows between the GEP, SVM, and MLR approaches were computed to assess the effectiveness of the studied methodologies. To quantify the divergence from and approximation to observed flows acquired from the gauging station, the RMSE, MSE, and R² values were computed. The overall statistical properties of the three models are shown in Table 4. The effectiveness of the used technique is higher when the RMSE is smaller.

Table 4 Error analysis for models.

Parameter	Value
Mutation rate	0.00138
Inversion rate	0.00546
Insertion sequence transposition rate	0.00546
Root insertion sequence transposition rate	0.00546
Gene transposition rate	0.00277
One-point recombination rate	0.00277
Two-point recombination rate	0.00277
Gene recombination rate	0.00277

To provide further discussion on the models’ performance against peak flow estimation, the forecasting results obtained by the GEP, SVM, and MLR were plotted and compared to those of the observed in Figure 6. In general, Figure 6 displays that GEP models can capture the strong periodicity of the normalized streamflow.

Figure 6a, b, and c Scatter plots for GEP, SVM, and MLR model

7 DISCUSSION

This study shows that Gene Expression Programming (GEP) outperforms Support Vector Machine (SVM) and Multiple Linear Regression (MLR) as the best model for streamflow prediction in the Brahmani River. Rainfall, temperature, river stage, and streamflow all have intricate, non-linear correlations that GEP captures with maximum accuracy (R² = 0.78), and the lowest errors (RMSE = 0.0364, MSE = 0.0013). However, MLR had the biggest inaccuracy (RMSE = 0.3916, MSE = 0.1534), whereas SVM did relatively well (R² = 0.67, RMSE = 0.0675), suggesting that linear models are less accurate for hydrological forecasting. Better prediction accuracy is indicated by lower RMSE and MSE values in GEP, which calculate the average variance between observed and anticipated values. The model's predictions are more likely to match real streamflow values when the RMSE is smaller, which lowers forecasting uncertainty. Likewise, GEP is a reliable option for streamflow modeling since a lower MSE indicates less significant prediction mistakes.

8 CONCLUSION

This work effectively used Multiple Linear Regression (MLR), Support Vector Machine (SVM), and Gene Expression Programming (GEP) models to forecast streamflow in the Brahmani River based on hydroclimatic factors such as temperature, rainfall, and river stage. Although all three models are capable of accurately estimating streamflow, the findings show that GEP performs better than SVM and MLR in terms of accuracy, as evidenced by lower RMSE and MSE values, as well as a higher R² value. GEP's exceptional performance demonstrates its capacity to capture intricate, non-linear interactions between streamflow and hydroclimatic variables, which makes it an invaluable tool for hydrological forecasting.

The significance of combining several hydroclimatic factors for more accurate streamflow forecasts is emphasized by this study. The results offer a reliable model for forecasting river discharge, which advances water resource management techniques. Future research can improve model accuracy even more by investigating hybrid models that integrate many AI techniques and by adding more influencing elements, like changes in land use and soil moisture. All things considered, this study supports the potential of soft computing techniques in hydrology and their usefulness in managing river basins.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the guidance and support of Kishnajit Kumar Khatua, whose expertise was invaluable in shaping this research. They also thank Department of Science and Technology (DST), Science and Engineering Research Board (SERB), Government of India (File No.: CRG/2021/003150, and SERB Qualified Unique Identification Document: SQUID-1969-KK-5771) for their invaluable assistance. We would also like to appreciate NIT Rourkela for their resources.

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