Prediction of daily streamflow using artificial neural networks (ANNs), wavelet neural networks (WNNs), and adaptive neuro-fuzzy inference system (ANFIS) models

In recent years, the prediction of hydrological processes for the sustainable use of water resources has been a focus of research by scientists in the field of hydrology and water resources. Therefore, in this study, the prediction of daily streamflow using the artificial neural network (ANN), wavelet neural network (WNN) and adaptive neuro-fuzzy inference system (ANFIS) models were taken into account to develop the efficiency and accuracy of the models’ performances, compare their results and explain their outcomes for future study or use in hydrological processes. To validate the performance of the models, 70% (1996–2007) of the data were used to train them and 30% (2008–2011) of the data were used to test them. The estimated results of the models were evaluated by the root mean square error (RMSE), determination coefficient (R), Nash–Sutcliffe (NS), and RMSE-observation standard deviation ratio (RSR) evaluation indexes. Although the outcomes of the models were comparable, the WNN model with RMSE1⁄4 0.700, R1⁄4 0.971, NS1⁄4 0.927, and RSR1⁄4 0.270 demonstrated the best performance compared to the ANN and ANFIS models. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/). doi: 10.2166/ws.2020.062 om http://iwaponline.com/ws/article-pdf/20/4/1396/705088/ws020041396.pdf 2020 Hüseyin Yıldırım Dalkiliç (corresponding author) Dept. Civil Engineering, Faculty of Engineering, Erzincan Binali Yıldırım University, Erzincan 24000, Turkey E-mail: hydalkilic@erzincan.edu.tr Said Ali Hashimi Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan 24000, Turkey


INTRODUCTION
In recent years, streamflow prediction has been considered to be one of the most important issues in the fields of hydrology, water resources, and water resources management. An accurate streamflow estimation can play a positive role in enhancing the capacity of reservoirs, flood prevention, water supply, design of hydroelectric projects, and water resources management. It can therefore have a significant impact on reducing the effects of climatic events on the environment and improve the efficiency of the outcomes.
In the past few years, data-driven models, including artificial neural networks (ANNs), wavelet neural networks (WNNs), and adaptive neuro-fuzzy inference systems (ANFIS) models, have been applied as effective tools for modeling nonlinear and complex hydrological systems (Wu et al. ; Seo et al. a, b; Seo & Kim ). Despite the wide usage of the ANN method to predict hydrological variables, it may not give accurate and reliable results in the estimation of unstable data. Therefore, to estimate a hydrologic time series comprising nonlinear relationships, the use of data pre-processing techniques is needed to improve the performance of ANNs (Okkan ). One of these methods is the wavelet transform, which is a signal processing technique. In These significant models used together in this study are better than a single model, as the models' forecasted results can be used to compare their forecasting ability and therefore recognize the powers and precision of each model for a given dataset. The multi artificial intelligence models for hydrological modeling that show the differences and compare the models' performances were applied in this study. To have a better conception of the estimated data through the models mentioned, some highly recommended and effective model evaluation indexes were applied in this study. Using these evaluation indexes, the root mean square error (RMSE), determination coefficient (R 2 ), Nash-Sutcliffe (NS), and RMSEobservation standard deviation ratio (RSR), the results can be readily interpreted and used as a reliable outcome for future studies or in hydrological processes.  Figure 1) and were chosen as they had not been much used in previous studies. This significant area has not been researched sufficiently to date, so the aim of this paper is to improve the productivity of the Büyük Menderes River as much as possible. The dataset for this study covers the period 1996 to 2011, as it is the most recent dataset available.

ANNs
ANNs are a new information processing approach and they are one of the most popular subjects of contemporary science with their learning ability, adaptation, rapid operation, and ease of identification. ANNs are completely parallel distributed data processing systems that consist of a wide number of highly interconnected elements, called artificial neurons or nodes, which resemble the biological neural networks of the human brain (Tsoukala & Uhrig ). Neurons work on the principle of generating results by learning knowledge and increasing experience and information. Every neuron receives an input signal from another neuron that sends it through an activation or transfer function and utilizes an altered output to the other outputs ( Figure 2). Even though each neuron implements its function rather slowly and imperfectly, a network can perform a wide number of tasks quite efficiently if they combine to work collectively. ANNs, which are nonlinear black-box models, are accepted as being amongst the most popular methods, especially in modeling, because of their precise and reliable results for the prediction of hydrological variables. Recently, ANNs have not been used regularly, as they do not present hydrological modeling results that are as successful and accurate as WNNs and ANFIS models.
There are a few ANN models that have been proposed since the 1980s, and probably the most influential of them are the multi-layer perceptron (MLP), the Hopfield network, and Kohonen's self-organizing networks.

Feed forward neural network (FFNN)
The FFNN is the most commonly used neural network in water resources. It is generally used for forecasting hydrolo- There are various backpropagation algorithms that have developed relevant techniques to train FFNN models.

Backpropagation was developed by Paul Wrbos in 1974
and is a supervised learning technique, based on the gradient descent (GD) method, which tries to minimize the errors produced by the neural network when it guesses data. The target of the backpropagation training process is to regulate the weights of the system to reduce the errors of the network.
Here, T is the sum of training samples, the number of output layers is m, weights in the network are represented by W, and y P and d P are, respectively, the actual and desired outputs of the network. The alteration of weights can be calculated as below when the model is trained with the Levenberg-Marquardt algorithm, which is the most widely used optimization algorithm and is specifically designed to minimize the sum of square error functions. It has become a standard technique for nonlinear least-squares problems, widely adopted in various disciplines for dealing with data fitting applications. Furthermore, it gives the best performance in the prediction of daily streamflow compared to any other backpropagation algorithm.
Next, the upgrade of the weights is modified as below: Here, J, I, e, μ are, respectively, the Jacobian matrix, identity matrix, network error, and Marquardt parameter that are to be updated using the decay rate β relay on the result. Further, μ is multiplied by the decay rate β With more rounds of optimization, the error in the training data will reduce further and further until there is a constant  Table 1 give a close correlation value. The third statistical parameter has the biggest error value among the other parameters used in Station E07A0037.

WNNs
The WNN model is formed by the application of discrete wavelet transform (DWT), which is an implementation of the wavelet transform using a discrete set of the wavelet scales and translations acting by some predetermined rules. In other words, the significant features of many natural signals are captured by a subset of DWT coefficients that is normally much smaller than the original signal, and the ANN method, and it is a neural network prediction algorithm that stands based on wavelet theory. Among the It has also the ability of self-learning, adaptivity, and calculating the nonlinear parameters of a system (Hou et al. The main advantage of using wavelets is that they are localized in space. Generally, their applications are data processing, data compression, and the solution of differential equations. In this study, the Haar wavelet is used for the WNN model, and it is the simplest, most easily imaginable and the earliest of the wavelet family. The wavelet function of the Haar wavelet is as follows: The scaling function of Haar wavelet is: The working rules of WNNs where non-fuzzy x and y entries to nodes A and B are linguistic labels that are specified by μAi and μBi membership functions respectively. These functions are symbolized by O.
The second layer, base nodes: Each neuron is fixed in this layer and the AND or the OR operator is applied to obtain one output that represents the result of the antecedent for that rule, i.e., firing strength. This layer is the product of the degrees of the first layer of O 2 k outputs, whose equation is as follows: The third layer, medium nodes: The main purpose of the third layer is to determine the ratio of each ignition factor I to the sum of all ignition laws. As a result, w is obtained as a standardized ignition source.
The fourth layer, results nodes: the output of each layer is equal to: In this relation wi is the output of the antecedent layer and pi, ri, qi are the linear coefficients of combination. Also, the total parameters of the tail section are Takagi-Sugeno fuzzy models. The fifth layer, output nodes: This layer calculates the signal node of the total output by collecting all the input signals. Therefore, in this layer of inactivity, the results of each fuzzy rule are transformed into non-fuzzy output.
The network is trained based on learning with supervision. So, the target is to teach ANFIS to estimate uncertain functions derived from instructional information and accurately estimate the unknown parameters ( Figure 3).
To design an ANFIS model, it was not easy to select a better fuzzy inference system for a specific goal. Different types of FIS have been stated in the literature (Takagi & Sugeno ). The adaptive neuro-fuzzy system is usually used with the Sugeno fuzzy system as a progressive network structure, as presented in Figure 4. Here, for generating an FIS, various membership function types were evaluated, and the most effective membership function type (trimf) for input data and output data (linear) shows the best results for all four different stations.
To train the FIS, the backpropagation method was used to cope with the parameter recognition problem in an FIS, which is the basic rule of an adaptive network, and based on a descent rule that can successfully estimate the parameters. The outcomes for the ANFIS model gained are shown in Table 4. On the other hand, the characteristics of the model's performance criteria, that the outputs of the models examined by the evaluators, fortunately remained between those ranges that are shown in Table 5.

RESULT AND DISCUSSION
In this study, to estimate daily streamflow values, three models, ANN, WNN, and ANFIS, were used in four different stations located near the Büyük Menderes River.
RMSE, R 2 , NS, and RSR were used as the evaluation indicators of the models' outcomes to represent the satisfaction or dissatisfaction ratio of the models. Therefore, the best consequence from different parameters of the models' tests for the ANN are listed in Table 6, for the WNN in Table 7, and for the ANFIS in Table 8. The input data were classified into five parameters and each parameter had one or more than one combination of the input data, which were considered to be the same for the four stations used in this study. Despite presenting the most important details about each model's structures and development processes for estimating daily streamflow, it is better to sum over each model's performance and quality individually.
The most important step in modeling is choosing the right combination of input variables. Thus, the correlation between input variables and output was calculated, and the input variables were selected to model the system to estimate the daily streamflow of the Büyük Menderes River, as shown in Table 6.
In this table, the parameters of the flow of the previous 1, 2, and 3 days, precipitation, average temperature, and  relative humidity were the inputs in a period of (t) and flow as an output in a period of (t) were described. According to the meaningful correlation between the inputs and output, which are shown above, a few different combinations of input parameters were used to estimate the daily streamflow values efficiently that are shown in Table 7.
As is shown in Table 8, the best result was for the ANN, as demonstrated by the minimum error rate, RMSE ¼ 4.466, and by the best correlation rate, R 2 ¼ 0.932, which includes three input layers, 10 hidden layers, and one output layer.
Additionally, as can be seen from  Figure 5 shows the time series and plot graphs of the models. The correlation and proximity of all the models' outcomes (output) with the observed (original) output of the data once again can be observed in Figure 6.
The study aimed to find the minimum error rate, and conversely the highest correlation rate, for the observed output through modeling the data.

CONCLUSIONS
The main purpose of the study was to achieve the minimum error rate, and conversely the highest correlation coefficient between the calculated data and the observed data using ANN, WNN and ANFIS models. As shown, the outcomes of the models were meaningful and successful by the end of the calculations. Simultaneously, assembling the most recommended models that are commonly cited in the literature was the other reason to understand and show the performances of the models through sequencing their outcomes by their precision rates.
The data were divided into 70% for training, 30% for testing the network, and the number of hidden neurons were selected by the trail-test method, which is considered to be 10 neurons in this study. The training algorithm selected for this model was the Levenberg-Marquardt because it is quicker than other algorithms, and it is the preferred algorithm for estimating a large number of daily intervals of data. To increase the performance and accuracy of the ANN model, the original data were decomposed into sub-series, their noises removed and the most efficient data were selected by calculating the correlation coefficients for each of the given parameters. ANFIS is a combination model of FIS and ANN. Each one has its positive aspects, and when they are joined together, the ANFIS model performs better. As a result, in this study, the WNN showed the most accurate result, with R 2 ¼ 0.971, compared to the two other models. The ANFIS model, with R 2 ¼ 0.947, and the ANN model, with R 2 ¼ 0.932, were second and third in this study. Briefly, the main reason that WNN gave the most accurate result was that decomposing the original data into sub-series and removing their noises, and again using them as clean data, significantly affected the outcomes.