Wavelet-based Artificial Neural Network Research Articles

Comparing wavelet-based artificial neural network, multiple linear regression, and ARIMA models for detecting genuine radon anomalies associated with seismic events

Comparing wavelet-based artificial neural network, multiple linear regression, and ARIMA models for detecting genuine radon anomalies associated with seismic events

Multi-time-step ahead daily global solar radiation forecasting: performance evaluation of wavelet-based artificial neural network model

Multi-time-step ahead daily global solar radiation forecasting: performance evaluation of wavelet-based artificial neural network model

Modeling stage\u2013discharge\u2013sediment using support vector machine and artificial neural network coupled with wavelet transform

Many real water issues involve rivers’ sediment load or the load that rivers can bring without degrading the fluvial ecosystem. Therefore, the assessment of sediments carried by a river is also crucial in the planning and designing of various water resource projects. In the current study, five different data-driven techniques, namely artificial neural network (ANN), wavelet-based artificial neural network (WANN), support vector machine (SVM), wavelet-based support vector machine (WSVM), and multiple-linear regression (MLR) techniques, were employed for time-series modeling of daily suspended sediment concentration (SSC). Hydrological datasets containing the daily stage (h), discharge (Q), and SSC for 10 years (2004–2013) from June to October at Adityapur and Ghatshila station of Subernrekha river basin, Jharkhand, India, were considered for analysis. The Gamma test was used to determine the input variables in the first step. Various combinations were made by lagging the maximum three-day time step for predicting current-day SSC. The outcomes of ANN, SVM, WAAN, WSVM, and MLR models were evaluated with the actual values of SSC based on statistical metrics. Pearson correlation coefficient (PCC), root-mean-square error (RMSE), Nash–Sutcliffe efficiency (NSE), and Wilmot index (WI) as well as visual inspection of time variation, scatter plots, and Taylor diagrams. Our results stated that the WSVM model discovered the best trustworthy models among all existing models. PCC, RMSE, NSE, and WI values were 0.844 and 0.781, 0.096 g/l and 0.057 g/l, 0.711 and 0.591, 0.907 and 0.878, respectively, throughout the training and testing processes at the Adityapur site. Also, at the Ghatshila location, it was the most accurate model. During the training and testing stages, PCC, RMSE, NSE, and WI values were 0.928 and 0.751, 0.117 g/l and 0.095 g/l, 0.861 and 0.541, 0.962 and 0.859, respectively. Our findings showed that the WANN model was the second-best model during the testing phase for both sites. Hence, the WSVM technique can model SSC at this location and other similar (i.e., geomorphology and flow regime type) rivers.

The Superiority of Data-Driven Techniques for Estimation of Daily Pan Evaporation

In the present study, estimating pan evaporation (Epan) was evaluated based on different input parameters: maximum and minimum temperatures, relative humidity, wind speed, and bright sunshine hours. The techniques used for estimating Epan were the artificial neural network (ANN), wavelet-based ANN (WANN), radial function-based support vector machine (SVM-RF), linear function-based SVM (SVM-LF), and multi-linear regression (MLR) models. The proposed models were trained and tested in three different scenarios (Scenario 1, Scenario 2, and Scenario 3) utilizing different percentages of data points. Scenario 1 includes 60%: 40%, Scenario 2 includes 70%: 30%, and Scenario 3 includes 80%: 20% accounting for the training and testing dataset, respectively. The various statistical tools such as Pearson’s correlation coefficient (PCC), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), and Willmott Index (WI) were used to evaluate the performance of the models. The graphical representation, such as a line diagram, scatter plot, and the Taylor diagram, were also used to evaluate the proposed model’s performance. The model results showed that the SVM-RF model’s performance is superior to other proposed models in all three scenarios. The most accurate values of PCC, RMSE, NSE, and WI were found to be 0.607, 1.349, 0.183, and 0.749, respectively, for the SVM-RF model during Scenario 1 (60%: 40% training: testing) among all scenarios. This showed that with an increase in the sample set for training, the testing data would show a less accurate modeled result. Thus, the evolved models produce comparatively better outcomes and foster decision-making for water managers and planners.

Hybrid wavelet-based learning method of predicting effective thermal conductivities of hybrid composite materials

The hybrid composite materials are a new type of composite material. Due to their complex microscopic structures, it is very challenging to predict the equivalent thermal conductivities of hybrid composites. In this paper, an innovative hybrid wavelet-based learning method assisted multiscale analysis is developed to predict the effective thermal conductivities of hybrid composite materials with heterogeneous conductivity by the asymptotic homogenization method, wavelet transform method, and machine learning method. This innovative approach mainly includes two parts: off-line multi-scale modeling and on-line machine learning. Firstly, the material database about thermal transfer performance of hybrid composites is established by the asymptotic homogenization method and off-line multi-scale modeling, and then the off-line material database is preprocessed by the wavelet transform method. Secondly, the artificial neural network and support vector regression method are employed to establish the on-line machine learning model for predicting the equivalent heat conduction properties of hybrid composites. Finally, the effectiveness of the proposed hybrid wavelet-based learning method is verified by numerical experiments on the periodic and random hybrid composites. The numerical results show that the hybrid wavelet-based artificial neural network method owns the optimal capability of parameter prediction and anti-noise. Furthermore, it should be emphasized that the hybrid wavelet-based learning method can not only extract the important features of off-line material database for random hybrid composites with high-dimensional large-scale data features, but also significantly reduce the quantity of input data for ensuring the successful on-line supervised learning and improve the training efficiency and anti-noise performance of the machine learning model. The established hybrid wavelet-based learning method in this paper can not only be used to evaluate the equivalent thermal conductivities of hybrid composite materials, but also further extend to the predicting of the equivalent physical and mechanical properties of composite materials.

Estimation of Daily Stage–Discharge Relationship by Using Data-Driven Techniques of a Perennial River, India

Modeling the stage-discharge relationship in river flow is crucial in controlling floods, planning sustainable development, managing water resources and economic development, and sustaining the ecosystem. In the present study, two data-driven techniques, namely wavelet-based artificial neural networks (WANN) and a support vector machine with linear and radial basis kernel functions (SVM-LF and SVM-RF), were employed for daily discharge (Q) estimation. The hydrological data of daily stage (H) and discharge (Q) from June to October for 10 years (2004–2013) at the Govindpur station, situated in the Burhabalang river basin, Orissa, were considered for analysis. For model construction, an optimum number of inputs (lags) was extracted using the partial autocorrelation function (PACF) at a 5% level of significance. The outcomes of the WANN, SVM-LF, and SVM-RF models were appraised over the observed value of Q based on performance indicators, viz., root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), Pearson’s correlation coefficient (PCC), and Willmott index (WI), and through visual inspection (time variation, scatter plot, and Taylor diagram). Results of the evaluation showed that the SVM-RF model (RMSE = 104.426 m3/s, NSE = 0.925, PCC = 0.964, WI = 0.979) outperformed the WANN and SVM-LF models with the combination of three inputs, i.e., current stage, one-day antecedent stage, and discharge, during the testing period. In addition, the SVM-RF model was found to be more reliable and robust than the other models and having important implications for water resources management at the study site.

Optimal level of wavelet decomposition for daily inflow forecasting

A methodology to select the maximum level of wavelet decomposition to forecast seven days of daily inflows by a hybrid model wavelet-based artificial neural network (WANN) is proposed. The wavelet decomposition was employed to decompose an input time series into approximation and detail components, and the approximations were used as inputs to artificial neural networks (ANN) for WANN hybrid models. In this study, it was used daily inflows from January 1931 to December 2010 to three Brazilian reservoirs with different discharge patterns, and evaluated the accuracy of the WANN models when using seven different mother-wavelets, including Haar, Daubechies, Biorthogonal, Biorthogonal Reverse, Symlet, Coiflet and Discrete Meyer. It was found that the model performance is dependent on the input sets and the selected mother-wavelets. Based on the obtained results, it was observed that the maximum level of decomposition was five, because upper than this level, independently on the inflow magnitude, there is no guarantee that the WANN hybrid models would perform better than the ANN model.

Wavelet based hybrid ANN-ARIMA models for meteorological drought forecasting

Drought prediction is an important subject, particularly in drought-hydrology, and has a key role in risk management, drought readiness and alleviation. Hydrological time series data consists of nonlinear features and various time scales. With this view in mind, this study has combined the strengths of the Wavelet transformation, Autoregressive Integrated Moving Average (ARIMA) and Artificial Neural Network (ANN) to test a new method of a hybrid model for their ability to accurately predict future droughts. A 30-year rainfall data from the year 1986 to 2016 for Malaysia’s Langat River Basin was analyzed. Meteorological drought indices (DI) such as the Standardized Precipitation Index (SPI) and the Standard Index of Annual Precipitation (SIAP) were used to compute historical drought events. At first, each of these computed drought time series went through a process of decomposition to be divided as low frequency and high-frequency sub-series by discrete wavelet transform (DWT). Secondly, the high and low-frequency sub-series were passed through the predictive model of ANN and ARIMA techniques, respectively. Lastly, the predicted sub-series were used to reconstruct and develop a final drought prediction model. It was found that the Wavelet-ARIMA-ANN (which named as W-2A) model outperformed the single ANN and wavelet-ANN predictive models. The ANN model developed by SPI achieved an overall correlation co-efficient R-value of 0.423, but the wavelet-based ANN model decreased in the R-value to 0.415. Finally, two different models, which were established using drought indices SPI and SIAP, and discrete wavelet transformation-based hybrid ANN-ARIMA (W-2A), have achieved improved R values of 0.914 and 0.934 respectively.

Investigating capabilities of machine learning techniques in forecasting stream flow

This paper presents a systematic investigation into modelling capacities of three conventional data-driven modelling techniques, namely, wavelet-based artificial neural network (WANN), support vector regression (SVR) and deep belief network (DBN) for multi-step ahead stream flow forecasting. To evaluate the effectiveness of these modelling techniques, hydro-meteorological hourly datasets from three case-study rivers located in the UK have been used. A heuristic performance analysis of the modelling schemes has been conducted by systematically analysing the key statistics that measure magnitude, scatter and density of model errors. Finally, for each of the modelling techniques, the performance deterioration rate in time was estimated. The results show that the SVR model can forecast quite accurately up to one to two hours ahead but its performance deteriorates gradually from three hours onwards. Further it has been found that the WANN model performs better when the overall non-linearity of the system increases, whereas the DBN model appeared to show consistently poor predictive capabilities when compared to the other models presented herein. The authors conclude by stating that, for any selected model, it is possible to use an identical model structure for up to two steps ahead forecasting. Models need to be re-configured beyond that limit.

A comparative study of wavelet-based ANN and classical techniques for geophysical time-series forecasting

Time-series modeling forms an important area of research in geophysics. Time-series models can be linear, like linear state-space models or non-linear, like artificial neural networks. One way to judge the goodness of different models associated with a given time-series is to assess the prediction capabilities of these models. Some of the important techniques used in time-series forecasting are: (i) Minimum mean squared error (MMSE) forecast obtained using conditional means of ARIMA(p,d,q) models, (ii) Kalman filter approach and (iii) Artificial neural networks (ANN) approach. However, the wavelet-based versions of these techniques, respectively denoted as W-MMSE, W-Kalman and W-ANN, rather than the original techniques themselves, have been found to possess better capabilities in forecasting the highly nonlinear geophysical data. Using the original and predicted data, these observations have been validated by determining the RMSE (root mean squared error) and correlation coefficients between them. The prediction capabilities of both versions of the above techniques are tested on (i) the ionospheric total electron content (TEC) data, (ii) the daily average rainfall data and (iii) the gamma-ray log data from an offshore oil well, off the west coast of India. The TEC and daily average rainfall data sets designate as examples of data with very high correlations pertaining over very large lags. They also have a strong seasonality component associated with them. However, the gamma-ray log data sets show no seasonality component and have no trends associated with them. Therefore, the choice of different nonlinear data sets having diverse sources of their origins are apt to test the forecasting capabilities of these techniques. It has been observed that W-ANN gives the best prediction, when compared with the other algorithms discussed in this paper. This is believed to be due to the use of non-linear activation function by ANNs to produce regression that results in capturing the inherent non-linear dynamics of the process effectively. The results also show the usefulness of discrete wavelet transform (DWT) coefficients as training features for both linear and non-linear forecasting approaches. The better performance of the wavelet-based forecasting algorithms can be attributed to the fact that DWT coefficients are wide-sense stationary sequences. Thus the wavelet-based versions of these models like W-MMSE and W-Kalman provide better fit to these coefficients, compared to the original time-series data itself.

Water Level Prediction Model Based on GRU and CNN

Massive amount of water level data has been collected by using Internet of Things (IoT) techniques in the Yangtze River and other rivers. In this paper, utilizing these data to construct deep neural network models for water level prediction is focused. To achieve higher accuracy, both the factors of time and locations of data collection sensors are considered to perform prediction. And the network structures of gated recurrent unit (GRU) and convolutional neural network (CNN) are combined to build a CNN-GRU model in which the GRU part learns the changing trend of water level, and the CNN part learns the spatial correlation among water level data observed from adjacent water stations. The CNN-GRU model that using data from multiple locations to predict the water level of the middle location has higher accuracy than the model only based on GRU and other state-of-the-art methods including autoregressive integrated moving average model (ARIMA), wavelet-based artificial neural network (WANN) and long-short term memory model (LSTM), because of its ability to decrease the affections of abnormal value and data randomness of a single water station to some extent. The results are verified on an experiment dataset that including 30-year observed data of water level at several collection stations in the Yangtze River. For forecasting the 8-o'clock water levels of future 5 days, accuracy of the CNN-GRU model is better than that of ARIMA, WANN and LSTM models with three evaluation factors including Nash-Sutcliffe efficiency coefficient (NSE), average relative error (MRE) and root mean square error (RMSE).

Drought forecasting through statistical models using standardised precipitation index: a systematic review and meta-regression analysis

Quality and reliable drought prediction is essential for mitigation strategies and planning in disaster-stricken regions globally. Prediction models such as empirical or data-driven models play a fundamental role in forecasting drought. However, selecting a suitable prediction model remains a challenge because of the lack of succinct information available on model performance. Therefore, this review evaluated the best model for drought forecasting and determined which differences if any were present in model performance using standardised precipitation index (SPI). In addition, the most effective combination of the SPI with its respective timescale and lead time was investigated. The effectiveness of data-driven models was analysed using meta-regression analysis by applying a linear mixed model to the coefficient of determination and the root mean square error of the validated model results. Wavelet-transformed neural networks had superior performance with the highest correlation and minimum error. Preprocessing data to eliminate non-stationarity performed substantially better than did the regular artificial neural network (ANN) model. Additionally, the best timescale to calculate the SPI was 24 and 12 months and a lead time of 1–3 months provided the most accurate forecasts. Studies from China and Sicily had the most variation based on geographical location as a random effect; while studies from India rendered consistent results overall. Variation in the result can be attributed to geographical differences, seasonal influence, incorporation of climate indices and author bias. Conclusively, this review recommends use of the wavelet-based ANN (WANN) model to forecast drought indices.

Wavelet-Exponential Smoothing: a New Hybrid Method for Suspended Sediment Load Modeling

In this study, four conventional and a newly proposed method of wavelet-exponential smoothing (WES) - with two presented scenarios (WES1 and WES2) – are employed to estimate daily and monthly suspended sediment load (SSL) in two rivers (Lighvanchai river in Iran and Upper Rio Grande in the USA), which have different hydro-geomorphological characteristics of the related watersheds. In the proposed WES method, first, wavelet transform (WT) is applied to the original observed time series to decompose them into approximation and detailed subseries to separate different components of time series. For the first scenario (WES1), only two time series, i.e., an approximation and a detail time series are utilized as inputs of model, whereas for the second scenario (WES2), all subseries are separately fed into different exponential smoothing (ES) models. The results revealed that for both rivers, the proposed WES2 and wavelet based artificial neural network (WANN) models could lead to superior performance in comparison to the autoregressive integrated moving average (ARIMA), seasonal ARIMA (SARIMA), ES ad-hoc and artificial neural network (ANN). The WES2 method could enhance the overall performance of SSL forecasting both in daily and monthly modeling of the case studies regarding Nash-Sutcliffe (E) efficiency criteria, respectively up to 13%, 42% and 87%, 116% in daily and monthly scales for SSL modeling of the Lighvanchai and Upper Rio Grande Rivers. As a result, combining WT with ES method and ANN led to more accurate modeling.

The utilization of a GR4J model and wavelet-based artificial neural network for rainfall–runoff modelling

Abstract Data-driven models and conceptual models have been utilized in an attempt to perform rainfall–runoff modelling. The aim of this study is comparing the performance of an artificial neural network (ANN) model, wavelet-based artificial neural network (WANN) model and GR4J lumped daily conceptual model for rainfall–runoff modelling of two rivers in the USA. It was obtained that the performance of the data-driven models (ANN, WANN) is better than the GR4J model especially when streamflow data the preceding day (Qt-1) and streamflow data the preceding two days (Qt-2) are used as input data in the ANN and WANN models for the simulation of low and high flows, in particular. On the other hand, when only precipitation and potential evapotranspiration data are used as input variables, the GR4J model performs better than the data-driven models.

Tracking the chaotic behaviour of fractional-order Chua’s system by Mexican hat wavelet-based artificial neural network

In this paper, a process is devised systematically to scrutinize the scrolling chaotic behaviour of fractional-order Chua's system. The process is composed of fractional Laplace transformation, artificial neural network with Mexican hat wavelet as an activation function and simulated annealing. Sequentially, the parametric expansion of fractional Laplace transform is employed to convert the governing fractional system into an ordinary differential system. Next, artificial neural network and simulated annealing approximate and optimize the attained system and produce accurate solutions. The predictability and elaboration of double scrolling chaotic structures of fractional-order Chua's system are also studied using the Lyapunov exponent and fifth–fourth Runge–Kutta method. Moreover, the mean absolute error and root mean square error are measured for the convergence analysis of the proposed scheme. On the whole, the accurate approximate solutions, the phase plots of Lyapunov exponent spectrum and bifurcation maps of the dynamical evolution of fractional Chua's system are a triumph of this endeavour.

Wavelet-ANN versus ANN-Based Model for Hydrometeorological Drought Forecasting

Malaysia is one of the countries that has been experiencing droughts caused by a warming climate. This study considered the Standard Index of Annual Precipitation (SIAP) and Standardized Water Storage Index (SWSI) to represent meteorological and hydrological drought, respectively. The study area is the Langat River Basin, located in the central part of peninsular Malaysia. The analysis was done using rainfall and water level data over 30 years, from 1986 to 2016. Both of the indices were calculated in monthly scale, and two neural network-based models and two wavelet-based artificial neural network (W-ANN) models were developed for monthly droughts. The performance of the SIAP and SWSI models, in terms of the correlation coefficient (R), was 0.899 and 0.968, respectively. The application of a wavelet for preprocessing the raw data in the developed W-ANN models achieved higher correlation coefficients for most of the scenarios. This proves that the created model can predict meteorological and hydrological droughts very close to the observed values. Overall, this study helps us to understand the history of drought conditions over the past 30 years in the Langat River Basin. It further helps us to forecast drought and to assist in water resource management.

Ultrafast Transmission Line Fault Detection Using a DWT-Based ANN

Digital impedance protection of transmission lines suffers from known shortcomings not only as a principle but also as an application. This necessitates developing a new relaying principle that overcomes those shortcomings. Such a principle is offered in this paper and is currently being validated using field data. The principle is a new application of wavelet-based artificial neural networks (ANNs). The application uses high-frequency content of a subset of local currents of one end of a protected line to classify transients on the line protected and its adjacent lines. The scheme can classify transients—including faults—occurring on a protected line, categorize transients on adjacent lines, and pinpoint the line causing the transient event. It is shown that the feature vector of the event can be determined from a subset of local currents without using any voltages altogether. The subset of local currents consists of the two aerial modes of the local current. Modal transformation is used to transform phase currents to modal quantities. Discrete wavelet transform (DWT) is used to extract high-frequency components of the two aerial modal currents. A feature vector is built using the wavelets details coefficients of one level of the aerial modes and is used to train an ANN. Results show that the classes corres-ponding to each transient event type on the protected line and its adjacent lines are almost linearly separable from each other. Results demonstrate that very accurate classification using one-eighth of a cycle of postevent data is possible.

A wavelet-based non-linear autoregressive with exogenous inputs (WNARX) dynamic neural network model for real-time flood forecasting using satellite-based rainfall products

Summary Although flood forecasting and warning system is a very important non-structural measure in flood-prone river basins, poor raingauge network as well as unavailability of rainfall data in real-time could hinder its accuracy at different lead times. Conversely, since the real-time satellite-based rainfall products are now becoming available for the data-scarce regions, their integration with the data-driven models could be effectively used for real-time flood forecasting. To address these issues in operational streamflow forecasting, a new data-driven model, namely, the wavelet-based non-linear autoregressive with exogenous inputs (WNARX) is proposed and evaluated in comparison with four other data-driven models, viz., the linear autoregressive moving average with exogenous inputs (ARMAX), static artificial neural network (ANN), wavelet-based ANN (WANN), and dynamic nonlinear autoregressive with exogenous inputs (NARX) models. First, the quality of input rainfall products of Tropical Rainfall Measuring Mission Multi-satellite Precipitation Analysis (TMPA), viz., TRMM and TRMM-real-time (RT) rainfall products is assessed through statistical evaluation. The results reveal that the satellite rainfall products moderately correlate with the observed rainfall, with the gauge-adjusted TRMM product outperforming the real-time TRMM-RT product. The TRMM rainfall product better captures the ground observations up to 95 percentile range (30.11 mm/day), although the hit rate decreases for high rainfall intensity. The effect of antecedent rainfall ( AR ) and climate forecast system reanalysis (CFSR) temperature product on the catchment response is tested in all the developed models. The results reveal that, during real-time flow simulation, the satellite-based rainfall products generally perform worse than the gauge-based rainfall. Moreover, as compared to the existing models, the flow forecasting by the WNARX model is way better than the other four models studied herein with the TRMM and TRMM-RT rainfalls at 1–3 days lead times. The results confirm the robustness of the WNARX model with only the satellite-based (TRMM-RT) rainfall (without use of gauge data) to provide reasonably good real-time flood forecasts. The utility of the TRMM-RT solves the real-time flood forecasting issues, since this is the only rainfall product disseminated in real-time. Hence, the WNARX model with the TMPA rainfall products can offer an exciting new horizon to provide flood forecasting and early warning in the flood prone catchments.

Groundwater budget forecasting, using hybrid wavelet-ANN-GP modelling: a case study of Azarshahr Plain, East Azerbaijan, Iran

Meticulous prediction of hydrological processes, especially water budget, has an individual importance in environmental management plans. On the other hand, conservation of groundwater, a fundamental resource in arid and semi-arid areas, needs to be considered as a great priority in development plans. Prediction of a groundwater budget utilizing artificial intelligence was the scope of this study. For this aim, the Azarshahr Plain aquifer, East Azerbaijan, Iran, was selected because of its great dependence on groundwater and the necessity of cognizance of its budget in future programs. The long-term fluctuations of the water table in 13 piezometers were simulated by a wavelet-based artificial neural network (WANN) hybrid model, and their statistical gaps were covered. Then, the modelled water table was predicted for the next 12 months using genetic programming. The results of simulation and prediction were assessed by performance evaluation criteria such as R2, root mean squared error, mean absolute error and Nash–Sutcliffe efficiency. Thiessen polygons were then utilized, plotting the predicted unit hydrograph of the study area. The predicted water table from September 2012 to August 2013 revealed about 0.12 m depletion. Regarding the area of the Azarshahr Plain aquifer and its average storage coefficient, the aquifer budget will be reduced by about 0.3557 million cubic metres during this period.

Discussion of “Comparison of Wavelet-Based ANN and Regression Models for Reservoir Inflow Forecasting” by Krishna Budu

The discusser wishes to thank the author for investigating the accuracy of the wavelet neural network (WNN) technique in predicting daily reservoir inflows to the Malaprabha Reservoir in Belgaum, India, using daily values of rainfall, inflow, and streamflow at an upstream gauging station. The WNN results were compared with two standard neural network models along with a multiple linear regression (MLR) model and moving average (MA). It was found that the WNN model performed better than the artificial neural network (ANN) and MLR models in inflow forecasting. The discusser would like to point out some important points that the author and other researchers might consider: • According to the author, the testing data-set extremes (Xmin 1⁄4 0 and Xmax 1⁄4 1,016 m=s) fell within the training data-set limits (Xmin 1⁄4 0 and Xmax 1⁄4 1,036.4 m=s) and therefore the trained neural network models did not face difficulties in extrapolating. It is clear from Table 1, however, that Xmin 1⁄4 0.14 m=s, which was higher than the corresponding value of the testing data set and implies that these models faced difficulties in extrapolating low flow values. This can also be seen from Fig. 10, in which the WNN-BP and WNN-RB models (here BP = back propagation and RB = radial basis) slightly overestimated the low flow values. • In the study, the iteration numbers for the optimal ANN-BP, ANN-RB, WNN-BP, and WNN-RB were not provided. As seen from Table 3, the fitting accuracy of the ANN-BP model in the training period was generally similar to that of the ANN-RB model. However, WNN-BP approximated flow values in the training period much better than WNN-RB did. What might be the reason? • According to the author, it is seen from the scatter plot in Fig. 7 that the WMLR model had more values close to its predicted line and its R value was slightly higher. The discusser agrees with the author on this. However, the fit line coefficients should also have been considered in comparing the two methods. It is clear from the scatter plots (assuming that the fit line equation was y 1⁄4 axþ b) that the WNN-BP a and b coefficients were respectively closer to 1 and 0 (the ideal line is y 1⁄4 x) than were the WMLR a and b coefficients. This was also valid for Fig. 9. • The “Conclusion” section states, “Overall analysis indicates that the performance of ANN and regression models using undecomposed data is relatively lower compared to that of wavelet-decomposed data used NN and regression models; this may be plausible due to the variation in nonlinear dynamics of the rainfall runoff process that mapped effectively by waveletbased models.” In the study, however, the WMLR model was found to be better than the WNN model coupled with the RB and BP methods. The discusser wonders how the linear WMLR model could perform better than the nonlinear WNN models in such a nonlinear rainfall-runoff process, as also stated by the author. This needs clarification.