An approach on the estimation and temporal interaction of runoff: the band similarity method

The increasing world population, industrialization, agricultural needs, environmental pollution, and global climate change increase water stress and place increasing pressure on water resources (Hssaisoune et al. 2020; Kılıç 2020; Luo et al. 2020). Therefore, it is necessary to protect, develop, and control these resources. However, changes in the climatic and environmental factors and the rainfall–runoff relationship make it difficult to control and use renewable but limited water resources (Kilinc 2022). For this reason, prospective estimation of river flows is of great importance to realize the most rational and sustainable use of water resources (Kilinc & Haznedar 2022). Streamflow is a phenomenon dependent on many geographical (vegetation, topography) and meteorological (temperature, precipitation, evaporation) variables. Therefore, the estimation of streamflow is quite complex due to its interactions with stochastic and nonlinear processes (Yaseen et al. 2019; Akçakoca & Apaydın 2020).

Artificial Intelligence (AI) models and physically based models (PBMs) are used by hydrologists to estimate streamflow (Koycegiz & Buyukyildiz 2019; Yang et al. 2020). However, the complex structure of PBMs, the large amount of data on the basin, and the need for expert knowledge for model use lead PBMs to be less feasible (Peng et al. 2017; Ritto & Rochinha 2021). Therefore, as in different phenomena in hydrology, researchers have focused more on AI methods in streamflow prediction. The most well-known and widely used AI models are support vector machine (SVM), artificial neural networks (ANN), and adaptive-based fuzzy logic inference system (ANFIS) (Mohammadi et al. 2021; Mohapatra et al. 2021; Vadiati et al. 2022).

Hadi & Tombul (2018) employed ANN, SVM, ANFIS, and linear autoregressive (AR) models to simulate the daily flow data of three stations in the Seyhan Basin of Türkiye. In all three stations, AR models showed the lowest performance, while ANN models showed the best performance. Parisouj et al. (2020) used support vector regression (SVR), ANN-back propagation (ANN-BP), and extreme learning machine methods to predict streamflow at monthly and daily scales in four river basins in the United States. SVR (ANN-BP) performed the best (worst) on monthly and daily scales. In the modeling of hydrological processes, nature-inspired models have been widely used by researchers as either single or hybridized with AI models (Riahi-Madvar et al. 2021; Samanataray & Sahoo 2021; Adnan et al. 2022). Samanataray & Sahoo (2021) evaluated the streamflow forecasting capability of ANFIS-particle swarm optimization (PSO) hybridized with PSO to optimize ANFIS parameters for monthly streamflow forecasting of the Barak River in India. The findings showed that the ANFIS-PSO model was more successful than the classical ANFIS models.

Recently, deep learning (DL) (e.g., long short-term memory (LSTM)) and boosting algorithms (e.g., XGBoost, LightGBM, and CatBoost) have gained popularity in modeling hydrological processes (Ni et al. 2020; Ghimire et al. 2021; Szczepanek 2022; Hao & Bai 2023; Vogeti et al. 2024). DL algorithms are neural networks with additional hidden layers, and as the data size increases, the model performance increases (Sarker 2021). Hao & Bai (2023) found that the LSTM technique perform better than SVR and XGBoost in daily flow simulation in the Ao River basin in China.

The band similarity (BS) method was developed by Yilmaz (2022) to improve prediction success using model information generated by optimization algorithms. The BS method produces two main outputs. The first is to increase the modeling success of optimization algorithms. The second is to examine the memory structure of the phenomenon under study. Of course, it would not be correct to state that the BS method produces a new contribution to the literature in the first of these two scopes. However, the BS method produces quite original results in the second case. It is known that modeling, forecasting, and trend studies are carried out in the literature, especially within the flow phenomenon. The BS method provides a different perspective and reveals the memory characteristics of the relevant phenomenon. It is known that many natural phenomena exhibit a cyclical structure. The BS method produces more specific results about this cycle.

In this study, the usability of the BS approach, which is relatively new in the literature, is investigated to improve the monthly flow forecast performance. For this purpose, the performance of the BSPSO model hybridized with the PSO method is evaluated by comparing the results of the single PSO and SVR models. In addition, the memory mechanism in the stream was examined using the BS method. The novelty of this study is that it is the first in which the BS approach was used to estimate the monthly average flow and the memory mechanism in the flow was investigated. In this respect, it is thought that the present study fills a gap in the literature.

The average annual total precipitation of the Seyhan Basin varies between 400 and 1,000 mm (Gumus 2019). Sarız, located in the north of the Seyhan Basin, has an intense continental climate and spends the winter months with rain. Sarız Stream, which forms one of the branches of the Seyhan River, is a headwater. Considering that the Seyhan River makes an important contribution to the development of the region, the importance of the Sarız Stream comes to the fore. Sarız district is at an altitude of 1,560 m above sea level, and its surface area is approximately 1,410 km². Agricultural activities in the district are carried out intensively together with animal husbandry.

In Equation (1), the velocity value obtained for the solution point x_ij is shown. The w value here is the inertia weight. The c₁ and c₂ values in Equation (1) are acceleration coefficients. The r₁ and r₂ values are the parameters that take random values in the (0–1) range and provide the randomness of the PSO. After the velocity values of all particles are calculated, position updates are performed as shown in Equation (2). Subsequently, the Z objective function (Equations (7)–(11)) is run again according to the up-to-date positions of the particles. The quality of the new solution points is determined. If the p_best or g_best values in the previous iteration are improved, the new values are recorded instead of the old ones.

As a result, with the help of the PSO algorithm, values in the range expressing the search space for the β coefficients are produced and the y_M models that will produce the most successful result according to the selected objective function are searched.

The BS method developed by Yilmaz (2022) uses the model information produced by optimization algorithms to increase modeling success and helps reveal the memory characteristics of the phenomenon studied.

Suppose a model is created on any phenomenon. If the temporal change (increase, decrease, acceleration, deceleration, etc.) of the input parameters of the model repeats in the future and if the output parameter of the model can respond to this change similarly, it is accepted that the related phenomenon remembers the changes that occurred in its past. This mechanism is defined as temporal interaction in the BS method. Many phenomena occurring in nature are expected to be cyclical. In the BS method, when all the input parameters of the model exhibit the same change, if the output parameter of the model can also respond to this, it is accepted that the temporal interaction is established.

Since K_i,j values of all x_i,j values are calculated, whether the changes made by testing and training individuals in the data set are similar to each other over time will be determined by K_i,j values. For this purpose, a band structure is created beginning from the first test individual. Among the training individuals, individuals who can place all K_i,j values in this band structure are sought. If this situation has occurred, it is understood that the training individuals related to this test individual behave similarly to each other in a temporal sense.

Here; is the Lagrange multipliers, is the kernel function, and b is the trend value. Error term (ε), regulation factor (C), and kernel function are three factors that affect the performance of SVR models. In this study, the radial basis kernel function is used in SVR applications.

In the models created for monthly average flow estimation (Q_t), it represents the neuron in the output layer as the monthly average flow at time t. Four input combinations (M1, M2, M3, M4) consisting of different delays of Q_t were used in the study. Q_t estimation was performed with Q_t−1 model input for M1, Q_t−1, Q_t−2 for M2, Q_t−1, Q_t−2, Q_t−3 for M3, and Q_t−1, Q_t−2, Q_t−3, Q_t−4 for M4. About 70% of the monthly average flow data for the period 1990–2017 (28 years) were used for training purposes and the rest for testing purposes. In this study, linear models were produced for four different models with the PSO method, as shown in Equations (3)–(6) for the estimation of Q_t values. In the PSO method applications, the limit values expressing the search space are determined as (−5;5). The number of particles is taken as 8 and c₁ = c₂ = 2. The inertia weight (w) was determined to decrease linearly over the iterations in the range of 0.9–0.4. With the PSO method, 10 linear models were produced for each model shown in Equations (3)–(6). The performance criteria of the produced models are shown in Appendix A in detail. The PSO performance criteria of the most successful model produced for each model are shown in Table 1 for the training and testing period.

In the literature, R² and NSE (alone or both) are widely used in evaluating the prediction performance of models. There are also studies where the advantages and disadvantages of both metrics are discussed (Moriasi et al. 2007; Gupta et al. 2009; Ritter & Muñoz-Carpena 2013; Onyutha 2020, 2022). In the methods used in our study, the NSE value was taken into account as the objective function in optimizing the parameters of each method. When the R² and NSE values given in Table 1 are compared, it is seen that the R² values are higher than the NSE values. According to Table 1, in all three methods, it is seen that the best model scenarios determined by the NSE are generally also achieved with the R² metric. In this study, the most successful model in the monthly average flow estimation was decided according to the model input with the highest NSE value in the test period.

In Table 1, although the highest R² value in the PSO models is in M2 with 0.589, the M4 input structure with the highest NSE (0.516) and the lowest MAE (1.280 m³/s) and RMSE (2.372 m³/s) values has been the most successful PSO model in predicting monthly average flow. The M1 scenario with the highest MAE, RMSE, and the lowest R² and NSE values was the PSO model with the lowest success in flow estimation. The PSO-M2 and PSO-M3 models also performed closely to the PSO-M4 model, with NSE values of 0.502 and 0.512, respectively.

The BS method was applied using the model information produced by the PSO algorithm, and the results were recorded as BSPSO. To investigate the temporal interaction between training and testing individuals in BS applications, five different simulation models shown in Equations (13)–(17) were used. The simulation models shown in Equations (13)–(17) are referred to as BSM2, BSM3, BSM4, and BSM5, respectively. In all simulation models, the r value was selected in the range of 1–12. The test NSE and μ_best values of the best results obtained with the BS method for all models are shown in Appendix B.

As a result of the BS method, the most successful result value in the M2 model was obtained for the 4-month difference values of the BSM4 normalized data (μ_best = 0.024, TestNSE = 0.673). The most successful results in both M3 and M4 models were obtained for the 5-month and 9-month difference values ​​of the data in BSM1, respectively. The performance criteria of the most successful results produced for all models as a result of the BS method are also shown in Table 1. Since it carries out its studies on the model results produced based on the PSO method, there are no training results obtained with the BS method. Therefore, there are no performance metrics for the training period in Table 1. However, since the same result was obtained in all yi_M1 models (Appendix A) produced in the M1 model, where only the Q_t−1 input parameter was used, and there was no model difference, the BS method was not applied; therefore, the metrics of the test period are not given in Table 1 for the M1 model.

According to Table 1, the most successful BSPSO model was achieved in the M3 scenario. In the BSPSO-M3 model, performance metrics for the testing period were obtained as R² = 0.734, NSE = 0.691, MAE = 1.205 m³/s, and RMSE = 1.895 during the test period. The lowest success in BSPSO models was the M4 scenario with NSE = 0.646.

In the SVR method, the C, ε, and γ parameters were optimized in the ranges of 1–50, 0.01–0.5, and 0.1–5, respectively. According to the results of the most successful SVR models shown in Table 1, the SVR model with the lowest prediction success in flow prediction was the M1 scenario with NSE = 0.471. In the other three combinations, NSE values are higher than 0.6. As a result, the SVR model with the highest prediction success is the M2 input structure with the highest NSE and R² and lowest MAE and RMSE values (0.659, 0.673, 1.268 m³/s, and 1.991 m³/s, respectively).

When the results of the SVR, PSO, and BSPSO models are compared, it is seen that the BSPSO method is the most successful model in each input structure according to the metrics used. The highest R² obtained in BSPSO is 0.734. However, the highest R² values obtained in PSO and SVR are 0.589 and 0.673, respectively. Therefore, these values indicate that the BSPSO method is more successful than the other two methods. According to the criteria stated in Moriasi et al. (2007), if 0.65 < R² ≤ 0.75, the model success is considered ‘good’ and if 0.75 < R² ≤ 1, it is considered ‘very good’. In our study, the value of 0.734 obtained in BSPSO is in the ‘good’ category. It is even very close to the ‘very good’ category.

Table 2 shows the improvement rates of the BSPSO method on PSO results. According to Table 2, it can be said that the BSPSO method improves the success achieved in PSO by increasing the R² and NSE values and decreasing the values of error metrics. Among the metrics, the highest contribution of BS generally occurred in NSE values. According to NSE values, the performance of the BS method in improving success was 34.10, 34.99, and 25.16% for M2, M3, and M4, respectively.

It has been seen that the BSPSO method is the most successful among all the methods used, and then the memory properties of the flow phenomenon studied are examined. The reason for the success of the BS method is the model updates based on the temporal similarity behavior between the training and test individuals. Especially when the improvement rates of the BS method in PSO results are examined in Table 2, it is seen that this mechanism works clearly.

In the grid system shown in Figures 6–8, the x-axis represents the year information in the training period and the y-axis represents the month information. In this way, it can be observed which test month is temporally similar to which training months. For example, 1992/05 and 1998/05 months within the training period were able to enter the band structure of 2009/04 in the test period. These months are shown as red-colored cells in Figure 6. Based on this, it is understood that 2009/04 exhibits a temporally similar behavior with 1992/05 and 1998/05 months. In this way, the images in Figures 6–8 were obtained by showing which training months all test months remembered with red-colored cells.

In line with the findings obtained by the BS method, two main outputs were obtained about the memory properties of the flow data studied. The first of these can be expressed as the water year effect. As stated before, the most successful result with the BSPSO method was obtained with the BSM1 for the 5-month difference values of the data. Accordingly, when 2009/01–2009/03 and 2011/01–2011/03 periods in Figure 6, 2012/02–2012/04 periods in Figure 7, 2015/01–2015/02 and 2016/01–2016/03 periods are examined, it is seen that the memory is intensified, and the test months remember more training months. However, when Figure 9 is examined, it is seen that the most severe memory structure occurs in January, February, and March. In this context, it should be considered that the most successful result is obtained according to the 5-month difference values and the most severe memory structure occurs approximately in the January–March period. Accordingly, it is seen that the August–October period corresponds to 5 months before the January–March period and the August–October period includes the beginning of the water year. Theoretically, it was concluded that they remembered the changes they had made at a high rate according to the August–October period when the flow data fell to the lowest values in the water year. When the findings obtained are examined in more detail in this context, it is seen that the months of October and March stand out. The fact that the differentiation between the theoretically lowest and highest periods in a water year is remembered shows that this change has a significant impact on the flow dynamics.

The second important output obtained on the determination of memory properties with the BS method was that a more intense remembrance occurred especially in water years when low flow values were observed. Especially when the periods in Figure 7 (2014/01–2014/08) and the period in Figure 8 (2017/01–2017/08) are examined, it is seen that there is a long-lasting and severe remember mechanism. When the number of remembered months shown in Figure 9 is examined, it is seen that there is a severe memory structure in the 2014 and 2017 water years, which continues almost throughout the water year. When the flow data in Figure 9 and the memory properties are examined together, it is seen that the flow data in the 2014 and 2017 water years took quite low values. In this context, it has been observed that a more severe and intense memory structure is formed, especially in water years when low flow values occur. As a result of the studies, it was concluded that memory is stronger in periods when flow values are low. In this context, it is thought that low flow values may contain important information. It is known that in the majority of studies on streamflow phenomena, researchers are interested in high flow values. However, this study suggests that low flow values may also contain important information.

In this study, the performance of the BSPSO model obtained by integrating the BS method, which is applied for the first time in river flow forecasting, with the PSO method is compared with the single PSO and SVR model results. BS also has the potential to estimate the memory properties of the time series. In models created utilizing historical significant lags of streamflow data, the findings show that the BS approach improves model performance.

The first advantage of the BS method is that it is not a preprocessing tool and performs dynamic analysis with the model. In addition, the fact that it is integrated into the model regardless of the characteristics of the data set leads to the fact that it is not affected by outliers and jumps in the data set. Thus, it allows for working with different data sets. The BS method, which is easily hybridized with evolutionary optimization algorithms, takes advantage of the memory feature of the data set. This opens up the opportunity to include physical features in black box models. Based on the different flow forecasting studies in the literature and the findings of this study, the BS method appears to be an effective tool.

As a result of this study, the fact that the BSPSO method has achieved more successful results compared to the SVR, which is a widely used and proven method in the literature, shows that the BS method is powerful in modeling and estimation. In the first study where the BS method was introduced by Yılmaz (2022), Artificial Bee Colony (ABC) with three different input model structures was used in the estimation of urban water consumption. Afterward, with the application of the BS method, an increase of approximately 30% in R² values was obtained according to the model success obtained by ABC. The BS method was also used by Yılmaz & Alpars (2023) to determine the memory properties of the water consumption phenomenon and by Yilmaz (2023) to determine the evaporation phenomenon. In both studies, significantly successful results were obtained.

For the same data set used in the current study, the model successes obtained by Koycegiz & Buyukyildiz (2022b) with four different ANN methods can be compared with the results of this study. It has been determined that BS, which is an innovative method, almost achieves ANN success, and even reaches higher success than some ANN algorithms. There is no other study in the literature in which BS was applied. The results obtained from these studies, which are quite limited in number, show that the BS method can be used as an alternative method in estimation studies.

In this study, in which monthly average flow estimation is performed, although the models are established linearly, different nonlinear models can be produced with the help of optimization algorithms. In this respect, the BS method can be applied to nonlinear models with different characters. In short, being easy to use, applying to different mathematical models, and increasing the success of the optimization algorithms in parallel with the success of the optimization algorithms are the strongest aspects of the BS method. In addition, it is thought that the BS method, which can obtain important findings in the examination of the memory properties of the studied phenomenon, is a method that can produce successful results in different fields.

In this study, the memory properties obtained with the help of the BS method on the flow data were evaluated on two outputs. The first of these is called the water year effect. In line with the results obtained, it was seen that the flow data remembered the change between the periods when the theoretical values were lowest and the periods when they started to increase rapidly. In this respect, the fact that the change compared to the period corresponding to the beginning of the water year is remembered at a high rate shows that especially low flow values have a significant effect on the memory structure. When the memory mechanism is analyzed within the framework of the water year from a holistic point of view, it is the second output reached that there is a continuous and intense memory mechanism, especially in the water years when low flow values are observed. Therefore, these two main outputs can be combined on a single main title as the effect of low flow data on memory. A more intense remembrance mechanism is observed in water years when low flow values are obtained both locally and globally in terms of the water year effect. In this respect, it has been concluded that especially low current values have a significant effect on memory properties. This result was accepted as the main output of this study.

It is known that peak flows attract more attention in the analysis of flow data and researchers concentrate on this area. However, within the scope of this study, the fact that the remembrance is more severe, especially in the periods when low current values are observed, shows that the dynamics in these periods are learned better. Therefore, it is thought that hydrologically more meaningful and original findings can be obtained by examining the periods in which low flow values occur in more detail within the scope of the analysis of flow data.

The present study explores the usability of the BS approach, which is relatively new in the literature, in the estimation of flow data as well as memory properties in the flow phenomenon. For this purpose, the models generated by the PSO method are used together with the BS method to perform the flow estimation. The performance of the hybrid BSPSO in improving the success of the single PSO is evaluated and the results are also compared with the results of the SVR method.

The studies' findings showed that, by updating the models produced by the PSO method according to the similarity of the temporal changes between the data, the BS method both increased the success of PSO and achieved more successful results compared to a valid method such as SVR. As a result of the studies, the highest NSE values for the test period were obtained in different input structures for the singular PSO and SVR models and the hybrid BSPSO model. The highest NSE values for each method are obtained as 0.659 for SVR, 0.516 for PSO, and 0.691 for BSPSO. The SVR method has a higher success than PSO in all input structures. However, the BSPSO method outperformed both SVR and PSO in the input structures used. The improvement rates of BSPSO over PSO vary between 25 and 35% for the NSE metric depending on the input structures.

It can be said that the temporal interaction mechanism used by the BS method worked and produced successful results. It is known that in most of the studies on flow analysis, researchers are interested in peak flow values. However, in line with this study, it is thought that remarkable outputs can be obtained by examining the periods in which low flow values occur in more detail, and it is recommended for future studies.

In future studies, it is considered to compare the performance of the BS method, which can use model information produced by different optimization algorithms, can be easily adapted to different mathematical models, and its operating mechanism is simple and easy, with the prediction performance of newly developed DL models and meta-heuristic algorithms, and it is recommended to researchers.

CK contributed to material preparation and data collection. VY and MB contributed to methodology and analysis. VY and CK contributed to conceptualization and visualization. All authors contributed to the study conception, design, writing, and editing. All authors read and approved the manuscript.

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

The authors declare there is no conflict.