A new criteria for determining the best decomposition level and filter for wavelet-based data-driven forecasting frameworks- validating using three case studies on the CAMELS dataset

Recently, several papers have been published regarding the use of preprocessing models, such as Discrete Wavelet, in Data-Driven Forecasting Frameworks (DDFF). However, these models face unresolved issues, including the use of future data, boundary-affected data, and incorrect selection of decomposition level and wavelet filter, which can lead to inaccurate results. In contrast, the Wavelet-based Data-Driven Forecasting Framework (WDDFF) overcomes these problems. To address the first two issues, we can use Maximal Overlap Discrete Wavelet Transform (MODWT) and a-trous algorithm (AT). Although there is currently no definitive solution for selecting the decomposition level and wavelet filter, we propose a novel approach using Entropy to address these issues. By utilizing the concept of predictability of time series using entropy, we can determine the optimal decomposition level and suitable filter to develop the Maximal Overlap Discrete Wavelet-Entropy Transform (MODWET) and apply it to WDDFF accurately. This study, demonstrates the effectiveness of MODWET through three real-world case studies on the CAMELS data set. In these studies, we will forecast the streamflow of specific stations one month ahead to prove the effectiveness of using preprocessing algorithms for forecasting models. The proposed model combines Input Variable Selection (IVS), preprocessing model, and Data-Driven Model (DDM). We will conclude that MODWET-ANN is the most effective model and highlight how entropy can accurately identify the optimal decomposition level and filter, resolving the concerns associated with using WDDFF in hydrological forecasting problems.


Introduction
Streamflow forecasting plays a critical role in water resources management and hydrological modeling (Chakraborty and Biswas 2023; Yaseen et al. 2015).Several recent studies have aimed to improve forecasting models accuracy considered classic Data-Driven Forecasting Frameworks (DDFFs).Recently, the accuracy of forecasting models has been improved by using Wavelet-based Data-Driven Forecasting Frameworks (WDDFF) (Alizamir et al. 2023;Hazarika et al. 2022;Jamei et al. 2023;Santos et al. 2023;Wu and Wang 2022).However, the application of WDDFF in real-world case studies can be problematic due to misinterpretation of parameters, which can lead to invalid results (Du et al. 2017;Quilty and Adamowski 2018;Zhang et al. 2015).These issues pertain to determining the appropriate decomposition level, wavelet filter, data partitioning, and boundary condition.Quilty and Adamowski (2018) have attempted to address these issues and resolve some of these problems.For instance, they demonstrated that only AT and MODWT are the preprocessing models that do not use future data in the decomposition process.Other models, such as Discrete Wavelet Transform (DWT), cannot be used for real-world forecasting problems due to their use of future data in the decomposition process.When DWT is calculating wavelet and scaling coefficient in time step t, it also use recorded data in time step t + a.The value of a varies and depends on the kind of filter, its length, and decomposition level.
Boundary Condition (BC) is a significant source of error when using WDDFF.When we use MODWT to calculate wavelet and scaling coefficients for a time series at time step t, it uses recorded data at time step t -a.It is important to note that for the initial a time steps there are no previous records.Therefore, the calculated coefficients for initial a time steps are incorrect and we refer them as boundaryaffected data.To achieve accurate results, the boundaryaffected time steps should be removed (Chen et al. 2022;Quilty and Adamowski 2021).
Another problem addressed by Quilty and Adamowski (2018) is the selection of suitable decomposition levels and wavelet filters.As the main objective of this investigation, we propose a novel solution to this problem based on the concept of entropy.Entropy has been used in various problems, but in this study, we use it for the first time to determine the optimal decomposition level and most suitable filter.We will employ the predictability concept to accomplish this.Predictability is an index that shows how well the future time steps of a time series can be predicted (Li et al. 2023).Predictability can be evaluated using measurement indices such as Lyapunov exponents (Palmer and Hagedorn 2006), recurrence measures (Marwan et al. 2002;Pospelov et al. 2019), and information entropic measures (Garland et al. 2014;Guntu et al. 2020).In this paper, we will calculate the entropy of decomposed wavelet and scaling coefficients to determine the decomposition level and filter with the best predictability.
In the following sections we will discuss: (1) A novel approach using entropy to determine a suitable decomposition level; (2) How to determine a suitable wavelet filter using entropy as a novel work; (3) The application of Quilty and Adamowski (2018) achievements in developing an accurate WDDFF using our new Maximal Overlap Discrete Wavelet Entropy Transform (MODWET), including boundary affected data elimination; (4) The implementation of the MODWET in a WDDFF for real-world streamflow one-month ahead forecasting using the CAMELS data set; (5) Finally, comparison of the results obtained from modeling using WDDFF and DDFF to determine which method is more accurate (Addor et al. 2017).

Theoretical basis of MODWET application in WDDFF
Numerous investigations have shown that adding decomposed signals can positively affect the accuracy of the forecasting frameworks (FFs).Different wavelet-based decomposition methods, suchas AT and MODWT can be applied in FFs, while Other methods like DWT cannot be used due to boundary conditions (Quilty and Adamowski 2018;Quilty and Adamowski 2021).
This study will use MODWT to improve the accuracy of forecasting.However, there is a problem in using wavelet-based decomposition methods, namely the selection of decomposition level.Although, there is no absolute solution to this problem, this investigation propose a novel algorithm, MODWET, for the first time.This method is based on the concept of entropy.Entropy is applied in MODWET to determine if the next decomposition level reveals new information or not, which we will discuss in the next subsections.This study will also demonstrate the effectiveness of entropy application in wavelet-based models using MODWET.

An overview on MODWT
In this section, we will provide an overview of MODWT equations (Eqs.(1-3).For more detailed information, please refer to (Quilty and Adamowski 2018;Quilty and Adamowski 2021).One of the addressed concerns, is the avoidance of future data in WDDFF.To address this issue, we select MODWT and AT, two algorithms that do not have this problem.For this study, we have chosen MODWT, which is more commonly used.
These equations (Eqs.(1-3) show the wavelet coefficient, scaling coefficient, and reconstructing coefficient respectively.In these equations t represents the time index, j represents the decomposition level, L represents the length of the wavelet filter, N represents the length of recorded data, and hl ( gl) represents the wavelet (scaling) filter (Addor et al. 2017).

Noticeable issues about using WDDFF
There are significant implementation issues associated with WDDFF that, if overlooked, can result in incorrect forecasting.However, these errors are avoidable.We will address several problems, including the selection of an appropriate filter and decomposition level, using a novel application of the entropy algorithm in MODWET.Additionally, we will account for boundary conditions in our modeling (Quilty and Adamowski 2018).

Solving WDDFF problems of application in hydrology using entropy
The entropy of information is a concept that quantifies the amount of information that could be obtained from a random variable.Equation 4 illustrates the Shannon Entropy (Shannon 1948), where E represents information entropy, X represents a random variable or a time series, N represents the length of time series, and P (x i ) represents the probability of occurrence of x i .
The Shannon entropy has a limitation between [0 ~ 1], and the highest entropy value is presented by a white noise process that is completely random and unpredictable (Doss-Gollin et al. 2019;Li and Zhang 2008;Ravi Kumar Guntu 2020).The entropy value of an occurrence with absolute certainty is zero.Conversely, if an occurrence is completely random and unpredictable, its entropy value will be one.We will utilize this concept by applying three conditions to determine a suitable decomposition level and filter for MODWT.
Condition-1: Following Quilty and Adamowski (2018), the first L(J max ) number of recorded data (Eq.5) should be removed due to boundary condition.Therefore, we must select a filter and decomposition level that its L(J) is much smaller than the length of our time series (N).By using Eq. 5, we can determine the Upper Limitation (UL) for suitable decomposition level.
Condition-2: In each decomposition level (J), we will calculate the entropy value of approximate (a) and detail (d) parts.A suitable J has a lower summation entropy value of approximate and detail part in level J rather than approximate part in level J-1.The minimum suitable decomposition level based on the first J that satisfies this condition will be the minimum decomposition level or Lower Limitation (LL).
Condition-3: Finally, based on limitations presented between LL and UL, we will select the option that has the lowest entropy and provide the most achievable information. (5) To sum up, we can select appropriate J max based on the following three conditions: Con. 1) L J << N , (UL) Con. 2) Entropy a,d (j) < Entropy a (j − 1), (LL) Con. 3) Min(Entropy J max ); LL < Jmax < UL As shown in Eq. 5, Condition-1, depends on J max and the kind of filter that used.Table 1 displays sample wavelet filters with different lengths.For instance, for a time series with a length of 420 using fk8 (J = 4) is not feasible because the first 106 records should be eliminate, which corresponds to approximately %25 of N and is contrary to Condition-1 regarding suitable J and filter selection.In Table 1, highlighted pixels represent the appropriate filter and decomposition level for a time series with N = 420 based on Condition-1.

Input variable selection for a practical WDDFF
This investigation will use original data, one-month lagged data, large-scale climatological data, wavelet, and scaling coefficient to develop a WDDFF.It is essential to employ an IVS to select the appropriate time series and avoid redundancy.IVS automatically eliminates redundant data that do not contribute to the predictability of target variable and makes this process considerably easier.Several investigations have studied various IVS such as Partial Correlation Index (PCI) (Galelli et al. 2014), Mutual Information (MI) (Vergara and Estévez 2014), Edgeworth Approximationsbased (EA) (Hulle 2005;Quilty et al. 2016), Genetic programming (GP) (Garg et al. 2014) and new models such as GCA (Dariane and Behbahani 2022).However, Ren et al. ( 2020) demonstrated the superiority of PCI over seven other IVS in three case studies on CAMELS data, which we will also implement in our models.We have chosen two IVS namely PCI and GP.PCI calculates the partial correlation between different time series and is limited to linear relationships, whereas GP can model various non-linear relationships based on realistic modeling results.We have selected GP as a representative for model-based IVS and PCI as a filter method.After determining PCI for different inputs, we need a Termination Criteria (TC) to determine the best inputs based on their PCI value.We have opted for Hampel Test (HT) as a TC, which Ren et al. ( 2020) determined as the best pair for PCI (May et al. 2008).GP works based on the evolutionary genetic algorithm and tries different operations and inputs to ensemble the best nonlinear model.Additionally, the nonlinear equation represented by GP can be used as a model (will be used in Sect.3.4).As an additional application for GP, we will extract inputs used in the top 10 equations as the selected inputs.To the best of our knowledge, this paper is the first that uses GP for selection between different wavelet and scaling filters.

Data-driven model for a practical WDDFF
All selected data sets will be used to forecast a target value (One-month ahead streamflow).To model the relationship between inputs and target, we can use different linear and nonlinear models.This study has used KNN as a linear regression and ANN and GP as nonlinear models.As described in Sect.2.3, GP represents the appropriate inputs and nonlinear model at the same time.ANN is another wellknown nonlinear selected DDM.In addition, KNN is a linear regression method used for classification and regression, which utilizes K time steps with greater similarity to the objective to find the target value using regression.Since our main objective is determining the suitability of adding entropy to the MODWT and evaluating the effect of adding wavelet and scaling coefficient to the DDM, this investigation have adopted models which are easy to program.For instance, KNN can be programmed easily as a linear model, ANN can be modeled using the MATLAB toolbox and we can use free software Eureqa Pro. for GP (Eureqa 2009).Additionally, KNN and ANN are extensively used in hydrological problems and this study will compare them with GP, which is rarely used in hydrological problems.For more theoretical details refer to (Garg et al. 2014;He et al. 2011;Nigsch et al. 2006).Finally, this paper will pair PCI-Hampel Test (PCIHT), and GP (as IVS) with ANN, KNN, and GP (as DDM) to develop different MODWET-DDFF.The described methods are summarized in the next section.2 displays the monthly statistical characteristics of the used data.The local meteorological data contains daily average precipitation (mm/d), solar radiation (W/m2), minimum temperature (°C), maximum temperature (°C), vapor pressure (Pa), and Monthly streamflow (m3/s); the monthly average was calculated from these daily data.Additionally, among many large-scale signals, the Niño 3.4, Pacific North American (PNA), Arctic Oscillation (AO), North Atlantic Oscillation (NAO), and Pacific Decadal Oscillation (PDO) data sets were selected.Before training, all data were normalized to fall within the range of [0 ~ 1].The large-scale climatological indices can be downloaded from the National Oceanic and Atmospheric Administration (NOAA) website (https://psl.noaa.gov/gcos_wgsp/Timeseries/).

Pre-processing method
Quilty and Adamowski (2018) demonstrated the superiority of MODWT over AT.Therefore, this study will use MODWT to apply entropy to it.Using MODWET, this investigation will calculate wavelet and scaling coefficients for 6 meteorological time series and 5 large-scale climatological indexes.As per the recommendation of Quilty and Adamowski (2018) we need to select a suitable wavelet filter with an appropriate length (Condition-1).We used various wavelet filters with different lengths as shown in Table 1, to determine the best case.Since, the length of our time series is 420, this study will select the filter, length and J max , that removes boundary condition for a maximum of 5% of the records (L J ).In this study study, this value ≈ 22.We selected J max based on Condition-1 (Table 1).In addition, based on Condition-2, the J value interval that demonstrates decrease in entropy value in each filter is shown in Table 3.Using this table this investigation have determined a set of appropriate wavelet filter and length for each time series.
In the next stage, we calculated the entropy of the decomposed signals and compared with the previous level as per Condition-2.The results are shown in Table 3, with highlighted numbers indicating options that satisfy Condition-2.For instance, decomposing NINO using filter d1 will result in decrease in entropy between levels 1 to 3 in Table 3, which satisfies Condition-1 and is smaller than Jmax for filter d1 in Table 1, that its value is 4.
Finally, based on Condition-3 this study selected the option with the minimum entropy value.The entropy value

Summary of the MODWET-DDFF
By using MODWET, we are one step closer to developing an optimal WDDFF suitable for real-world problems.We have utilized some achievements from previous studies and proposed new solutions, including: (1) avoiding using future data (Quilty and Adamowski 2018), and ( 2) selecting appropriate decomposition level and wavelet filter using the entropy concept.This investigation summarizes WDDFF as follows: 1) WDDFF employs MODWET, which does not use future data to forecast future time steps 2) Determine UL using Condition-1 3) Determine LL using Condition-2 4) Condition-3 should be used to select a suitable filter and decomposition level that provides enough data for the calibration part after removing the boundary-affected data.5) The boundary affected time steps should be removed to avoid affecting forecasting accuracy 6) Use GP (as IVS) and PCIHT to select between original data, wavelet, and scaling coefficients 7) Add Cyclic Seasonality Indexes (CSI) to the inputs to simulate monthly anomalies for the model (Nilsson et al. 2006).8) Employ a DDM to forecast the target values

Experimental setup
In this section, the case study and the additive data (Largescale climatological signals and CSI) that this study aim to incorporate to WDDFF will be discussed.Following this, various experiments that were conducted to forecast onemonth ahead streamflow at three selected stations will be outlined.The experiments were run on a Core i7 system with 8GB RAM, and all proposed methods were implemented using MATLAB R2020b.However, instead of coding, ready software Eureqa was used for implementation of GP (Eureqa 2009).

Study area and data
CAMEL data set gathered by Addor et al. (2017) will be used in this study.All the climate indices and hydrological data have been computed for the period spanning from 1 to 1980 to 30 December 2014.Figure 1 shows the selected three sub-basins (Station 01013500, 07083000, and 11264500) that were chosen as study area due to relatively low level of human activities and different hydro-climatic conditions (Ren et al. 2020).The monthly time series used in this study and final selected filter and decomposition level for each time series are displayed in Table 4.For example, in Table 4 filter d6 has the smallest entropy value for decomposing NINO, making it the best option.However, the pixels marked with (-) indicate that one or more conditions have not been met, and decomposition level and filter cannot be selected for them.

Input variable selection
The training portion of the data was utilized for IVS.GP (as IVS) and PCIHT were employed to select between meteorological data, large-scale climatological indexes, their lags, and wavelet and scaling coefficients.Once the selection was completed, CSI was added to represent the cyclic situation of data.

Data-driven models
The DDM models selected were ANN, GP (as DDM), and 5-fold KNN.The data selected using GP (as IVS) and PCIHT were fed into these models.Thus, the following models were  4-8 level shown in last column.However, some time series did not met the specified conditions in Sect.2.2.1, and consequently, we could not determine a suitable filter and decomposition level.Furthermore, since the data's most boundary-affected records across all-time series were 22, the first 22 boundary-affected records from all data in the training part were removed.Another important observation that can be drawn from Table 3 is the minimum appropriate decomposition level.As shown in some time series, initial decomposition levels did not yield any benefits as they increased the entropy value.
Table 5 presents the results of WDDFF and NWDDFF modeling with the best WDDFF for each station highlighted in bold font, and the corresponding NWDDFF in italic font.The best model achieved by Ren et al. ( 2020) at each station indicated with an asterisk (*).Our results demonstrate an improvement compared to the modeling results of Ren et al. ( 2020).However, while the accuracy of all models in the training set was similar, their testing results varied significantly.This indicates that all the models were trained within an acceptable range, but their potential for prediction differed based on their testing results.The best result for onemonth ahead streamflow forecasting were achieved by GP (as IVS)-ANN, MODWET-GP (as IVS)-ANN, and MOD-WET-PCIHT-ANN for stations 01013500, 07083000, and 11264500 respectively.Therefore, for these data sets, these models presented the best WDDFF.It is worth noting that wavelet based models yielded the best results at two stations.Furthermore, the superiority of wavelet-based models over non-wavelet-based models was demonstrated in ANN based and KNN based models.However, in some cases GP based WDDFFs yielded worse results than NWDDFF.This indicates that ANN and KNN can be considered suitable DDM for WDDFF, but GP is not an appropriate choice.As shown in Table 5, GP-based models require at least 1200 s for modeling, while the maximum time required for ANNbased and KNN-based models is 191 and 11 s, respectively.Additionally, it is worth noting that the ANN-based models that used PCI as the IVS technique required fewer neurons than models that used GP for IVS, indicating that modeling using selected data by PCI is simpler than GP.Scatter plots of the best WDDFF and DDFF are shown in Fig. 4, while comparative time series of observed streamflow versus WDDFF and NWDDFF forecasted values for different stations are presented in Fig. 5.
As shown in Table 2, the standard deviation of streamflow is 1504, 43, and 547, respectively, for stations 01013500, 07083000, and 11264500.Additionally, the maximum peaks in Fig. 5 are approximately 8000 (m3/s), 150 (m3/s), and 3000 (m3/s), respectively, for stations 01013500, 07083000, and 11264500.The differences between station 01013500 and the other two stations, 07083000 and utilized: MODWET-GP (as IVS)-KNN, MODWET-PCIHT-KNN, MODWET-GP (as IVS and DDM), GP (as IVS and DDM), MODWET-GP (as IVS)-ANN, MODWET-PCIHT-ANN, GP (as IVS)-ANN, and PCIHT-ANN.In the next setup, we selected the best DDFF and removed wavelet and scaling coefficients from inputs to run a Non-Wavelet Data-Driven Forecasting Framework (NWDDFF).These models have been named in this section, and further details can be found in the results section.The NWDDFFs were Original data-GP (as DDM) and Original data-PCIHT-ANN.In addition, PCIHT-KNN will be used to compare the results of this paper with the results of Ren et al. ( 2020), which was conducted on the same case study.KNN was trained based on Euclidean distance, where ANN and GP were trained based on Nash-Sutcliff Efficiency Index, owing to intrinsic characteristics of the models.For the testing phase, this study employed commonly used criteria namely NSE, Correlation coefficient (R), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE).

Experimental details and objectives
This study conducted two forecasting experiments using DDFF and a novel application of entropy in WDDFF, with the aim of achieving two objectives: 1) Comparing the forecasting result obtained using DDFF and WDDFF to demonstrate the effect of adding wavelet and scaling coefficient on the forecasting accuracy.2) Identifying the best decomposition level and wavelet filter by utilizing a novel application of entropy to achieve the highest forecasting accuracy and predictability for time series.

MODWET preprocessing method
Figure 2 presents the process of mean entropy calculation at different decomposition levels using MODWET for Solar Radiation at station 01013500.In this case the most appropriate decomposition level was determined to be 3 (J max = 3).The wavelet and scaling coefficients of the Solar Radiation decomposition at the third level using MODWET are presented in Fig. 3.In Fig. 2, as the decomposition level increases, the entropy value decreases, but due to Condition-1, level three is the maximum allowable decomposition level.The result of filter and decomposition level selection that satisfies mentioned conditions in Sect.2.2.1 is presented in Table 4, with the selected filter and decomposition

The effect of adding wavelet and scaling coefficients to DDFF
In this section, we will compare the results of waveletbased and non-wavelet-based models, presented in Table 5 to evaluate the effect of adding wavelet and scaling coefficients.For station 01013500, the wavelet-based models, which use PCIHT as IVS, showed better results than nonwavelet-based models, which was anticipated based on our initial hypothesis.However, the models that used GP as IVS showed a different response, with non-wavelet-based models demonstrating a better accuracy than wavelet-based models.As a result, among five comparative pair models, non-wavelet-based models out performed in three cases, while wavelet-based models had better performance in two cases.
Therefore, for the station 01013500, which has a greater streamflow standard deviation, the wavelet-based model did not perform well, and GP (as IVS)-ANN produced the best result with an NSE value of 0.71.
When comparing all five pair models for station 07083000, we observe that wavelet-based models outperform non-wavelet-based models in four cases, while there is only one instance where the non-wavelet-based model performs slightly better than the wavelet-based model.Specifically, the MODWET-GP (as IVS)-ANN model demonstrates the highest accuracy with an NSE value of 0.84 for this station.
In station 11264500, all wavelet-based models demonstrated better performance than non-wavelet-based models.Specifically, the MODWET-PCIHT-ANN model, achieved highest accuracy with an NSE value of 0.83.It is noticeable that in all stations we reached more accurate results than Ren et al. ( 2020) (displayed in Table 4 with *).To summarize, 11264500, illustrate that due to the greater variation in station 01013500, the results for this station were poorer compared to the other two stations.

Determining the best input variable selection for MODWET-DDFF
To determine the effect of different IVSs on different WDDFFs we fed selected data using GP (as IVS) and PCIHT to KNN and ANN.Meanwhile, GP at the same time does the selection and ensembles the models.The results in Table 5 showed that NWDDFF models used GP as IVS, has a better result than WDDFF in most stations.However, in cases that PCIHT is used for IVS, WDDFF has a better result than NWDDFF.As a result, GP is a better IVS for DDFF and PCIHT more suited WDDFF.

Comparing the results to find the best DDM for MODWET-DDFF
Given that wavelet-based forecasting models provide the most accurate results in two out of three stations, this study recommends wavelet-decomposed signals as inputs to DDFF in these stations.Furthermore, the results indicate that non-linear model ANN and linear model KNN outperform non-linear model GP in these case studies.Moreover, ANN exhibited superior performance compared to GP (as DDM) and KNN in all stations, which is consistent with previous studies.Additionally, as hydrological relationships are non-linear, ANN showed a better performance than KNN.Based on these findings, this study concludes that GP is not a suitable IVS or DDM for WDDFF.summation of ranking for samples in each group.Additionally, the degree of freedom can be calculated using Eq. 7.
The results of the Friedman Test for all stations are presented in Table 6.As shown, the Chi-square resulting from the Friedman test is greater than the Chi-square with a significance level of 0.05 in all stations.Therefore, this investigation rejects the null hypothesis and concludes that in all stations, there is at least one model that has a significant difference from the rest.

Summary and conclusion
This study introduced a novel entropy based technique for determining the optimal filter and decomposition level for MODWET.To the best of our knowledge, this is the first study to propose a systematic method for optimal in 11 out of 15 pair cases, wavelet-based models performed better than non-wavelet-based models.

Friedman test
The Friedman Test is a non-parametric statistical test that is commonly used to compare the differences between three or more groups in a repeated measures design.This test involves ranking the data within each group and then calculating the average ranks across all groups.Its purpose is to test the null hypothesis that there are no differences between the groups, while the alternative hypothesis suggests that at least one group differs from the others.If the p-value is significant, it indicates that there are significant differences between the groups (López-Vázquez and Hochsztain 2019).This study employed the Friedman Test to determine whether there are any significant differences between the applied models in the investigation.The test was conducted separately for each station, with K = 10 representing the number of groups (Models) and N = 126 representing the number of samples (Testing sets).The Chi-Square was computed using Eq.6, where N represents the number of samples, K represents the number of groups, and R represents the In summary, the conclusions of this study are as follows: 1) In aspect of time consumption and accuracy, MOD-WET-PCI-ANN presented a better performance than other models decomposing level and filter selection, while also addressing other problems like boundary condition removal.To evaluate the impact of adding decomposed signals to the forecasting models, 10 hybrid models were used, with PCIHT and GP selected for IVS, and KNN, ANN, and GP utilized for one-month ahead streamflow forecasting.The proposed models were compared with, Ren et al. ( 2020) and the performance of the proposed models in this study was found to be better.The Friedman Test was used to determine if there  2) The novel entropy-based technique effectively determined the optimal decomposition level and filter for MODWET 3) Wavelet based models outperformed non-wavelet based models in 11 out of 15 pair models.Additionally, this study concluded that wavelet based models have better performance in targets with lower standard deviation.

Fig. 1
Fig. 1 Study area of CAMELS data set and selected stations for this study

Fig. 2 Fig. 3
Fig. 2 Mean entropy value of solar radiation in different decomposition levels using MODWET with fk4 filter at station 01013500

Fig. 4
Fig. 4 Scatter plot of best models (WDDFF or NWDDFF) in different stations

Fig. 5
Fig. 5The observed streamflow time series versus forecasted streamflow in different stations for test period using WDDFF and NWDDFF

Table 1
Different candidate wavelet filters, their length, related L(J) and UL.covers the period from January 1980 to December 2014.Candidate data contains 12 lags in 5 types of climatological and hydrological time series, 12 lags in 5 types of large-scale climatological indices, and 2 CSI without lag.The sample size is 420 and the number of all time series is 134.The data was partitioned into 70% for training and 30% for testing.Table

Table 2
Regular statistical meteorological characteristics of stations

Table 4
Final selected Wavelet filter and decomposition level based on Con. 3

Table 5
Results of forecasting using different WDDFF and NWDDFF * The result of modeling byRen et al. (2020)