A New Criteria for Determining the Best Decomposition Level and Filter in Wavelet-based Data-driven Forecasting Frameworks- Validating Using Three Case studies on The CAMELS Dataset

doi:10.21203/rs.3.rs-2215714/v1

Recently, numerous papers have been published in the field of using preprocessing models (e.g. Discrete wavelet) in Data-driven Forecasting Frameworks (DDFF). There are some unresolved problems in these models like using future data, boundary affected data, and miss selection of decomposition level and wavelet filter that cause an erroneous result. However, Wavelet-based Data-driven Forecasting Framework (WDDFF) solves these problems. The first two problems could be solved using Maximal Overlap Discrete Wavelet Transform (MODWT) and a trous algorithm (AT). As the best we know, there is no absolute solution for decomposition level and wavelet filter selection. Meanwhile, as a novel investigation, we are going to use Entropy to find a solution for these problems. We are using the concept of predictability of time series using entropy for determining decomposition level and suitable filter, to develop the Maximal Overlap Discrete Wavelet-Entropy Transform (MODWET) to apply in WDDFF correctly. We will reveal the effectiveness of MODWET through three real-world case studies on the CAMELS data set. In these case studies, we will forecast the streamflow of determined stations from one month ahead to prove the effectiveness of using preprocessing models on forecasting accuracy. The proposed model is a combination of Input Variable Selection (IVS), preprocessing model, and Data-Driven Model (DDM). In conclusion, we will show that MODWET- ANN is the best model. In addition, we will realize how good entropy could find decomposition level and filter, which solves the mentioned concerns about using WDDFF in real-world hydrological forecasting problems.

Data-driven Forecasting Framework

Signal Pre-processing

Maximal overlap discrete wavelet entropy transform

Wavelet decomposition level

Wavelet filter

Streamflow forecasting

Streamflow forecasting has an important role in water resources management and hydrological modeling (Yaseen et al. 2015). Recently, numerous studies tried to improve the accuracy of forecasting models (Fahimi et al. 2017; Fang et al. 2019). Meanwhile, because of climate change, urbane developments, human activities, and geographical characteristics, streamflow forecasting has become complicated (Huang et al. 2014). Regarding mentioned complexities, recently, DDMs have been used more than physical models in hydrological studies. This could be because of how much these models are easy to apply in different problems (Liu et al. 2014; Shoaib et al. 2016). There are some trends in hydrological time series that could not be revealed unless you use signal decomposition methods (Quilty and Adamowski 2018).

There are numerous data-driven models, which have been used for time series forecasting like Multiple Linear Regression (MLR) (Abbasi et al. 2021), K-Nearest Neighbor (KNN) (Modaresi et al. 2018), and Artificial Neural Networks (ANN) (Khazaee Poul et al. 2019). The mentioned models are the classic Data-Driven Forecasting Frameworks (DDFF). Recently, using Wavelet-based Data-Driven Forecasting Frameworks (WDDFF) the accuracy of forecasting models has been Improved (Afan et al. 2016; Dixit et al. 2016; Fahimi et al. 2017; Sang 2013; Yaseen et al. 2015). However, there are problems with WDDFF application in real-world case studies that are originated from misinterpretation of parameters, which, result in an invalid response (Du et al. 2017; Quilty and Adamowski 2018; Zhang et al. 2015). These issues are about determining the appropriate decomposition level, wavelet filter, data partitioning, and boundary condition (Quilty and Adamowski 2018). Quilty and Adamowski (2018) tried to address these issues and solve part of these problems. For instance, they showed that only AT and MODWT are the preprocessing models, which do not use future data in the decomposition process. The other models like Discrete Wavelet Transform (DWT), because of using future data in the decomposition process could not be used for real-world forecasting problems (Quilty and Adamowski 2018). When DWT is calculating wavelet and scaling coefficient in time step t, it will use recorded data in time step t + a. Meanwhile, the a value varies and depends on the kind of filter, its length, and decomposition level.

Boundary Condition (BC) causes the main source of error in using WDDFF (Aussem 1998; Bakshi 1999; Maheswaran and Khosa 2012; Quilty and Adamowski 2018). When we use MODWT and calculate wavelet and scaling coefficients for a time series in time step t, it uses recorded data in time step t – a. You should notice that for the a initial number time steps there are no previous records. Therefore, the calculated coefficients for initial a time steps are incorrect and we call them boundary-affected data. To achieve an accurate result the boundary-affected time steps should be removed (Quilty and Adamowski 2018).

Another addressed problem by Quilty and Adamowski (2018) is the selection of suitable decomposition levels and wavelet filters. As the most significant objective of this investigation, we are proposing a novel solution for this problem, which works based on the entropy concept. Entropy has been used in different problems. For the first time, we use it to determine the appropriate decomposition level and the most suitable filter. We will use the predictability concept to solve this problem. Predictability is an index that shows how well the future time steps of a time series could be predicted (Lorenz 1969). Predictability could be measured using measurement indexes like 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 reach the decomposition level and filter with the best predictability.

In the following sections we will discuss about: 1) How entropy helps to determine a suitable decomposition level as a novel work; 2) How to determine a suitable wavelet filter using entropy as a novel work; 3) Use the Quilty and Adamowski (2018) achievements in developing a correct WDDFF using our new Maximal Overlap Discrete Wavelet Entropy Transform (MODWET) like boundary affected data elimination; 4) We will implement the MODWET in a WDDFF to practice WDDFF in a real-world streamflow one-month ahead forecasting case study (CAMELS data set); 5) Finally, we will compare the result of modeling using WDDFF and DDFF to find out which one is more accurate (Addor et al. 2017).

Many investigations show the positive impact of adding decomposed signals on the accuracy of the forecasting framework (FF). There are different wavelet-based decomposition methods like AT and MODWT can be applied in FFs. Other methods like DWT because of boundary conditions could not be used in FF (Quilty and Adamowski 2018; Quilty and Adamowski 2021).

In this study, we will use MODWT to improve the accuracy of forecasting. It is noticeable, that there is a problem in using wavelet-based decomposition methods, which is the selection of decomposition level. While, there is no absolute solution for this problem, for the first time we are proposing novel algorithm MODWET. This method is based on the entropy concept. Entropy is applied in MODWET to determine if the next decomposition level reveals new information or not that we will discuss in the next sub-sections. In addition, we will hire MODWET to show the effectiveness of entropy application in wavelet-based models.

2.1. An overview on MODWT

In this part, we are going to have an overall description of MODWT equations (Eqs. (1–3)). If you need more basic information you can see Quilty and Adamowski (2018). One of the concerns, which is addressed by Quilty and Adamowski (2018) is avoiding future data in WDDFF. Regarding this point, MODWT and AT are two algorithms, which do not have this problem and we select MODWT, which is more popular for this study. These equations (Eqs. (1–3)) respectively demonstrate the wavelet coefficient, scaling coefficient, and reconstructing coefficient. In these equations t is the time index, j is the decomposition level, L is the length of the wavelet filter, N is the length of recorded data, and ${\tilde {h}_l}$( ${\tilde {g}_l}$) is the wavelet (scaling) filter (Addor et al. 2017).

$${\tilde {W}_{j,t}}=\sum\limits_{{l=0}}^{{L - 1}} {{{\tilde {h}}_l}{{\tilde {V}}_{j - 1,(t - {2^{j - 1}}l)\bmod N}}}$$

1

$${\tilde {V}_{j,t}}=\sum\limits_{{l=0}}^{{L - 1}} {{{\tilde {g}}_l}{{\tilde {V}}_{j - 1,(t - {2^{j - 1}}l)\bmod N}}}$$

2

$${\tilde {V}_{j - 1,t}}=\sum\limits_{{l=0}}^{{L - 1}} {{{\tilde {h}}_l}{{\tilde {W}}_{j,(t+{2^{j - 1}}l)\bmod N}}} +\sum\limits_{{l=0}}^{{L - 1}} {{{\tilde {g}}_l}{{\tilde {V}}_{j,(t+{2^{j - 1}}l)\bmod N}}}$$

3

2.2. Noticeable issues about using WDDFF

There are important issues in the implementation of WDDFF that if we do not notice them the result of forecasting will be incorrect, while these errors are avoidable. A couple of problems that we are going to solve is the selection of appropriate filter and decomposition level using a novel application of entropy algorithm in MODWET. In addition, we will consider the boundary conditions in our modeling (Quilty and Adamowski 2018).

2.2.1. Solving WDDFF problems of application in hydrology using Entropy

The entropy of information is a concept that determines the amount of information that could be achieved from a random variable. Eq. 4 demonstrates the Shannon Entropy (Shannon 1948). In this equation E is information entropy, X is a random variable or a time series, N is the length of time series, and $P({x_i})$ is the probability of occurrence of ${x_i}$.

$$E(X)= - \sum\limits_{{i=1}}^{N} {P({x_i})Lo{g_2}[P({x_i})]}$$

4

Shannon entropy limitation is 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 absolute occurrence will be zero. Conversely, if an occurrence is completely random and unpredictable, its entropy value will be one. We will use this concept using three conditions to determine a suitable decomposition level and filterfor MODWT. Condition-1) Based on Quilty and Adamowski (2018), because of the boundary conditions, the first L(${J_{\hbox{max} }}$) number of recorded data (Eq. 5) should be removed. As a result, we have to select a filter and decomposition level that its L(J) is much smaller than the length of our time series (N). Using the 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 the represented limitation between LL and UL, we have to select the option that has the lowest entropy and provide the most achievable information.

$$L(J)=({2^J} - 1)(L - 1)+1$$

5

To sum up, we can select appropriate ${J_{\hbox{max} }}$based on the following three conditions:

Con. 1)${L_J}<<N$, (UL)

Con. 2) $Entrop{y_{a,d}}(j)<Entrop{y_a}(j - 1)$, (LL)

Con. 3)$Min(Entrop{y_{J\hbox{max} }});LL<J\hbox{max} <UL$

As you can see in Eq. 5, Condition-1, depends on ${J_{\hbox{max} }}$and the kind of filter that we are using. Table 1 shows the sample wavelet filters with different lengths. For example, when you are decomposing a time series with a length of 420 you cannot use fk8 (J = 4) because you have to eliminate the first 106 records, that is about %25 of N and this is against the Condition-1 about appropriate J and filter selection. In Table 1, highlighted pixels are the appropriate filter and decomposition level for a time series with N = 420 based on Condition-1.

Table 1

Different candidate wavelet filters, their length, related *L(J)* and UL.
Filter	Scaling Filter Coefficients																J max
																	1		2		3		4		5
																	L(J)
d1	0.707	0.707													2	2		4		8		16		32
d2	0.483	0.837	0.224	-0.129											4	4		10		22		46		94
fk4	0.654	0.753	0.053	-0.046											4	4		10		22		46		94
d3	0.333	0.807	0.46	-0.135	-0.085	0.035									6	6		16		36		76		156
fk6	0.428	0.813	0.356	-0.146	-0.077	0.041									6	6		16		36		76		156
coif1	-0.073	0.338	0.853	0.385	-0.073	-0.016									6	6		16		36		76		156
d4	0.23	0.715	0.631	-0.028	-0.187	0.031	0.033	-0.011							8	8		22		50		106		218
sym4	0.032	-0.013	-0.099	0.298	0.804	0.498	-0.03	-0.076							8	8		22		50		106		218
fk8	0.349	0.783	0.475	-0.1	-0.16	0.043	0.043	-0.019							8	8		22		50		106		218
d5	0.16	0.604	0.724	0.138	-0.242	-0.032	0.078	-0.006	-0.013	0.003					10	10		28		64		136		280
d6	0.112	0.495	0.751	0.315	-0.226	-0.13	0.098	0.028	-0.032	0.001	0.005	-0.001			12	12		34		78		166		342
sym6	-0.008	0.002	0.045	-0.021	-0.073	0.338	0.788	0.491	-0.048	-0.118	0.003	0.015			12	12		34		78		166		342
coif2	0.016	-0.041	-0.067	0.386	0.813	0.417	-0.076	-0.059	0.024	0.006	-0.002	-0.001			12	12		34		78		166		342
d7	0.078	0.397	0.729	0.47	-0.144	-0.224	0.071	0.081	-0.038	-0.017	0.013	0	-0.002	0	14	14		40		92		196		404
fk14	0.26	0.687	0.612	0.051	-0.246	-0.049	0.124	0.022	-0.064	-0.005	0.03	-0.003	-0.009	0.004	14	14		40		92		196		404

2.3. Input Variable Selection for a practical WDDFF

We will use original data, one-month lagged data, large-scale climatological data, wavelet, and scaling coefficient to develop a WDDFF. It is essential to hire an IVS to select appropriate time series and prohibit redundancy. IVS automatically eliminates redundant data that do not help the predictability of target and makes this process considerably easier. Different investigations studied various IVS like Partial Correlation Index (PCI) (Galelli et al. 2014), Mutual Information (MI) (Vergara and Estévez 2014), Edgeworth Approximations-based (EA) (Hulle 2005; Quilty et al. 2016), and Genetic programming (GP) (Garg et al. 2014) and new models like GCA (Dariane and Behbahani 2022). However, Ren et al. (2020) proved the superiority of PCI than seven other IVS which was practiced in three case studies on CAMELS data which we will implement our models on that. We selected two IVS namely PCI and GP. While PCI is limited to the linear relationships and calculates the partial correlation between different time series, GP can model different non-linear relationships based on realistic results of modeling. We select 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 opt Hampel Test (HT) as a TC which is determined by Ren et al. (2020) as the best pair for PCI (May et al. 2008). Meanwhile, GP works based on the evolutionary genetic algorithm and tries different operations and inputs to ensemble the best nonlinear model. Besides, the represented nonlinear equation by GP could be used as a model (will be used in Sec. 3.4). In an additional application for GP, we are going to extract the used inputs in the best 10 equations as the selected inputs. To the best of our knowledge, we are the first that use GP for selection between different wavelet and scaling filters.

2.4. 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 could use different linear and nonlinear models. We used KNN as a linear regression method and ANN and GP as nonlinear models. As we described in Sec. 2.3, using an evolutionary algorithm, GP represents the appropriate inputs and nonlinear model at the same time. ANN is selected as another well-known nonlinear DDM. Besides nonlinear models, KNN is a linear regression, which is used for classification and regression. This model utilizes the K time steps with more 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 we adopted models which are easy to program. For instance, KNN could be programmed easily as a linear model, ANN could be modeled using the Matlab toolbox and for GP we can use free software Eureqa Pro. (Eureqa 2009). In addition, KNN and ANN are extensively used in hydrological problems and we will compare them with GP, which is rarely used in hydrological problems. For more theoretical details, you can see (Garg et al. 2014; He et al. 2011; Nigsch et al. 2006). Finally, we will pair PCI-Hampel Test (PCIHT), and GP (as IVS) with ANN, KNN, and GP (as DDM) to develop different MODWET-DDFF. We will summarize described methods in the next section.

2.5. Summary of the MODWET-DDFF

Using MODWET, we will be another one step closer to an optimal WDDFF that is appropriate for use in real-world problems. We used some achievements of previous studies and proposed new solutions, which includes: 1) avoiding using future data (Quilty and Adamowski 2018), and 2) selecting appropriate decomposition level and wavelet filter using the entropy concept. We summarize WDDFF as follows:

WDDFF uses MODWET that does not use future data to forecast future time steps;

Determine UL using Condition-1.

Determine LL using Condition-2.

Using Condition-3 select a suitable filter and decomposition level that after removing boundary condition affected data it provides enough data for the calibration part.

Remove boundary affected time steps to do not affect forecasting accuracy.

Use GP (as IVS) and PCIHT to select between original data, wavelet, and scaling coefficients.

Add Cyclic Seasonality Indexes (CSI) to the inputs to simulate monthly anomalies for the model (Nilsson et al. 2006).

Hire a DDM to forecast the target values.

In this section, we describe the case study and the additive data (Large-scale climatological signals and CSI) that we are trying to impose to WDDFF. Then we will describe different experiments to forecast one-month ahead streamflow in determined three stations.

3.1. Study area and data

We will use the CAMEL data set gathered by Addor et al. (2017). All the climate indices and hydrological data are computed for the period 1 January 1980 to 30 December 2014. In Fig. 1, selected three sub-basins are shown (Station 01013500, 07083000, and 11264500). These regions that are impacted by a relatively low level of human activities and have different hydro-climatic conditions were selected as the study areas (Ren et al. 2020). The period of monthly time series used in this study is 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. We partitioned data to 70 to 30 percent respectively for training and testing. In Table 2, the monthly statistical characteristics of the used data are shown. The local meteorological data containing daily average precipitation (mm/d), solar radiation (W/m2), minimum temperature (°C), maximum temperature (°C), vapor pressure (Pa), and Monthly streamflow (m3/s); we calculated the monthly average from these daily data. In addition, among many large-scale signals, we selected the Niño 3.4, Pacific North American (PNA), Arctic Oscillation (AO), North Atlantic Oscillation (NAO), and Pacific Decadal Oscillation (PDO) data sets. Before training all data were normalized between the [0 ~ 1] range. You can download large-scale climatological indices from the National Oceanic and Atmospheric Administration (NOAA) website (https://psl.noaa.gov/gcos_wgsp/Timeseries/).

Table 2

Regular statistical meteorological characteristics of stations 01013500, 07083000 and 11264500.
Station		Ave	Min	Max	St. Dev
01013500	prcp(mm/day)	3.1	0.2	8.8	1.4
	srad(W/m2)	301.6	117.9	568.9	110.6
	tmax(C)	9.2	-12.9	25.7	11.2
	tmin(C)	-2.5	-26.2	13.9	10.8
	vp(Pa)	670.4	86.6	1624.0	428.9
	StrFlow(m3/s)	1542.7	65.8	8061.6	1504.5
07083000	prcp(mm/day)	2.1	0.0	6.7	1.2
	srad(W/m2)	467.4	234.6	731.2	128.4
	tmax(C)	5.9	-7.3	20.7	7.6
	tmin(C)	-8.3	-21.8	5.1	7.7
	vp(Pa)	344.0	115.7	798.9	165.3
	StrFlow(m3/s)	30.2	2.2	247.1	43.0
11264500	prcp(mm/day)	2.6	0.0	20.4	3.5
	srad(W/m2)	455.5	211.6	730.8	139.7
	tmax(C)	9.3	-2.7	22.8	6.6
	tmin(C)	-1.6	-11.5	11.4	5.6
	vp(Pa)	422.7	196.2	939.7	123.9
	StrFlow(m3/s)	373.2	3.3	3316.7	547.1

3.2. Pre-processing method

Quilty and Adamowski (2018) showed the superiority of MODWT against AT. Therefore, we will use MODWT to impose entropy to it. Then, using MODWET we will calculate wavelet and scaling coefficients for 6 meteorological time series and 5 large-scale climatological indexes. Based on (Quilty and Adamowski 2018) we have to determine a suitable wavelet filter with an appropriate length (Condition-1). As you can see in Table 1, we used different wavelet filters with different lengths to determine the best case. The length of our time series is 420 and regarding Condition-1 we will select the filter, length and ${J_{\hbox{max} }}$, which use maximum five percent of records as boundary condition removal (${L_J}$). In our case study, this value $\approx$ 22. We selected ${J_{\hbox{max} }}$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. Based on this table we 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. This operation is based on Condition-2. The result is shown in Table 3. The highlighted numbers in this table determine the options, which satisfy Condition-2. For instance, in Table 3, we can see that decomposing NINO using filter d1 will result in decrease in entropy between level 1 to 3. Moreover, this interval satisfies Condition-1 and is smaller than Jmax for filter d1 in Table 1, that its value is 4. Finally, based on Condition-3 we have to select the option with the minimum entropy value. The entropy value and final selected filter and decomposition level for each time series are displayed in Table 4. For example, in Table 4 we can see that filter d6 has the smallest entropy value for decomposing NINO; consequently, it is the best option. Meanwhile, the pixels that are signed by (-) explain that they have not satisfied one or more than one condition, and we cannot select a decomposition level and filter for them.

3.3. Input variable selection

The training part of the data was used for IVS. GP (as IVS) and PCIHT were used to select between meteorological data, large-scale climatological indexes, their lags, and wavelet and scaling coefficients. After selection, CSI was added to represented data.

3.4. Data-driven models

The ANN, GP (as DDM), and 5-fold KNN were selected as DDM, and the selected data using GP (as IVS) and PCIHT were fed to them. Therefore, the following models were used: 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 eliminated wavelet and scaling coefficients from inputs to run a Non-Wavelet Data-Driven Forecasting Framework (NWDDFF). We named these models in this section, but you can see the details 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 done on the same case study. Based on intrinsic characteristics of models, KNN was trained based on Euclidean distance, and ANN and GP were trained based on Nash-Sutcliff Efficiency Index. In addition, for the testing part we used commonly used criteria namely NSE, Correlation coefficient (R), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE).

We described two forecasting experiments using DDFF and novel application of entropy in WDDFF to reach three objectives in this study that contain:

Comparing the result of forecasting using DDFF and WDDFF to show the effect of adding wavelet and scaling coefficient on the forecasting accuracy.

Find the best decomposition level and wavelet filter using a novel application of entropy to achieve the best forecasting accuracy and predictability for time series.

5.1. MODWET preprocessing method

The process of mean entropy calculation in different decomposition levels using MODWET for Solar Radiation in station 01013500 is presented in Fig. 2. As you can see in this case ${J_{\hbox{max} }}$= 3 is selected as the most appropriate decomposition level. The wavelet and scaling coefficients of third level Solar Radiation decomposition using MODWET are presented in Fig. 3. In Fig. 2. By increasing the decomposition level, we will achieve a lower entropy value, however, because of Condition-1, level three is the maximum selectable decomposition level. The result of filter and decomposition level selection that satisfies mentioned conditions in Sec. 2.2.1 is presented in Table 4. You can see the selected filter and the decomposition level in last column of this table. Some time series do not satisfy mentioned conditions in Sec. 2.2.1 and consequently, we could not find an appropriate suitable filter and decomposition level. Besides, since the most boundary-affected records between all-time series were 22, we removed the first 22 boundary-affected records of all data from the training part. Another noticeable message that could be extracted from Table 3 is the minimum appropriate decomposition level. As you can see in some time series, initial decomposition levels do not have any profits for us because it raises the entropy value.

The results of WDDFF, and NWDDFF modelings are shown in Table 5. In this table, the best WDDFF for each station is presented with bold font, and related NWDDFF is presented with an italic font. In addition, the best model in each station that was achieved by Ren et al. (2020) is determined by the * sign. You can see an improvement in our results rather than the results of modeling by Ren et al. (2020). As you can see, GP (as IVS)-ANN, MODWET-GP (as IVS)-ANN, and MODWET-PCIHT-ANN respectively for stations 01013500, 07083000, and 11264500 presented the best result of one-month ahead streamflow forecasting. Therefore, for these data sets, mentioned models are the best WDDFF. It is noticeable that in two stations wavelet based models had the best results. Also, in ANN based and KNN based models the superiority of wavelet-based models against non-wavelet-based models was proved. It is noticeable that in some cases GP based WDDFFs had the result worse than NWDDFF. This shows that we can thrust ANN and KNN as a suitable DDM for WDDFF, but GP is not an appropriate choice. Scatter plots of the best WDDFF and DDFF are shown in Fig. 4. In addition, comparative time series of observed streamflow versus WDDFF and NWDDFF forecasted values for different stations are presented in Fig. 5.

5.2. 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.

Table 5

Results of forecasting using different WDDFF and NWDDFF
Station	Wavelet-based	Model	Performance of testing set
Station	Wavelet-based	Model	RMSE(m3/s)	NSE	R	MAE(m3/s)
1013500	No	PCIHT-KNN*	24.7059	0.683	0.834	18.4153
	No	PCIHT-KNN	0.1319	0.5299	0.766	0.0889
	Yes	MODWET-PCIHT-KNN	0.1266	0.5663	0.7857	0.0856
	No	GP(as IVS)-KNN	0.1134	0.6522	0.836	0.0762
	Yes	MODWET-GP(as IVS)-KNN	0.1188	0.6185	0.8112	0.0791
	No	GP(as IVS and DDM)	0.1155	0.6395	0.8207	0.0778
	Yes	MODWET-GP(as IVS and DDM)	0.1303	0.541	0.607	0.0837
	No	GP(as IVS)-ANN	0.1023	0.7119	0.8569	0.0745
	Yes	MODWET-GP(as IVS)-ANN	0.1158	0.6373	0.8237	0.0776
	No	PCIHT-ANN	0.1256	0.5731	0.7894	0.0869
	Yes	MODWET-PCIHT-ANN	0.116	0.636	0.8135	0.0847
7083000	No	PCIHT-KNN*	0.4491	0.834	0.915	0.2158
	No	PCIHT-KNN	0.0735	0.7836	0.8862	0.0382
	Yes	MODWET-PCIHT-KNN	0.0743	0.7796	0.8877	0.0362
	No	GP(as IVS)-KNN	0.0655	0.8281	0.9132	0.0353
	Yes	MODWET-GP(as IVS)-KNN	0.0767	0.8171	0.905	0.0341
	No	GP(as IVS and DDM)	0.0886	0.685	0.8755	0.0416
	Yes	MODWET-GP(as IVS and DDM)	0.0487	0.720	0.8645	0.0418
	No	GP(as IVS)-ANN	0.0734	0.7842	0.8875	0.0391
	Yes	MODWET-GP(as IVS)-ANN	0.0635	0.8387	0.9161	0.0318
	No	PCIHT-ANN	0.0834	0.7216	0.8677	0.045
	Yes	MODWET-PCIHT-ANN	0.07	0.7988	0.8992	0.04
11264500	No	PCIHT-KNN*	8.8552	0.687	0.846	4.3517
	No	PCIHT-KNN	0.0847	0.7435	0.8709	0.0405
	Yes	MODWET-PCIHT-KNN	0.0726	0.8114	0.9172	0.0383
	No	GP(as IVS)-KNN	0.0859	0.7366	0.881	0.0393
	Yes	MODWET-GP(as IVS)-KNN	0.0947	0.6793	0.829	0.0466
	No	GP(as IVS and DDM)	0.0781	0.782	0.8853	0.0384
	Yes	MODWET-GP(as IVS and DDM)	0.0929	0.6913	0.8582	0.046
	No	GP(as IVS)-ANN	0.0704	0.8228	0.9107	0.0433
	Yes	MODWET-GP(as IVS)-ANN	0.0762	0.7923	0.8909	0.042
	No	PCIHT-ANN	0.0745	0.8015	0.8956	0.0394
	Yes	MODWET-PCIHT-ANN	0.0683	0.8334	0.9129	0.0387
* The result of modeling by Ren et al. (2020)

5.3. Comparing the results to find the best DDM for MODWET-DDFF

Since the wavelet-based forecasting models provide the most accurate results in two out of three stations, we suggest feeding wavelet-decomposed signals to DDFF in these stations. In addition, the results show that non-linear model ANN and linear model KNN have a better performance than non-linear model GP in these case studies. In addition, ANN was superior to GP (as DDM) and KNN in all stations. This result was predictable and many previous studies have proved that. In addition, as hydrological relationships are non-linear, ANN showed a better performance than KNN. Besides, we can conclude that GP is not a suitable IVS nor a DDM for WDDFF.

5.4. The effect of adding wavelet and scaling coefficients to DDFF

In this section, we will compare the results of wavelet-based and non-wavelet-based models in Table 4 to evaluate the effect of adding wavelet and scaling coefficients. In station 01013500, the wavelet-based models, which use PCIHT as IVS, have better results than non-wavelet-based models. We have anticipated this result for all models; meanwhile, we saw a reverse response from models that used GP as IVS. In these models, non-wavelet-based models had a better accuracy than wavelet-based models. As a result, between five comparative pair models in three cases non-wavelet-based models and in two cases wavelet-based models had better results. Therefore, we can see in station 01013500, the wavelet-based model does not work well and GP (as IVS)-ANN poses the best result with an NSE value of 0.71.

In station 07083000, if we compare all five pair models we will realize that in four cases, wavelet-based models are superior to non-wavelet-based models, and in one pair, the non-wavelet-based model is slightly better than wavelet-based model. In this station MODWET-GP (as IVS)-ANN has the best accuracy with an NSE value of 0.84.

Finally, in station 11264500 all wavelet-based models displayed better performance than non-wavelet-based models. In this station MODWET-PCIHT-ANN, with an NSE value of 0.83 has the best accuracy. In addition, in all stations, we could achieve more accurate results than Ren et al. (2020) (displayed in Table 4 with *). To sum up, in 11 pair cases out of 15 pair cases wavelet-based models had a better result than non-wavelet-based models.

Numerous WDDFFs have been used in different studies to show the effect of adding wavelet and scaling coefficients to the DDFF. Nevertheless, there was some misused that caused the error in the modeling. Some of these errors like using future data have been solved by using MODWT and AT. However, still, there are unresolved problems in WDDFF and we tried to solve some of them by proposing MODWET-DDFF. Using MODWET-DDFF we suggested a solution and proved its effectiveness for finding decomposition level and a suitable filter. Besides, we showed the superiority of the wavelet-based forecasting framework over the non-wavelet-based forecasting framework by applying the proposed methods to three streamflow forecasting case studies in the CAMELS data set.

In addition, by adding Entropy to the MODWT we developed the innovative MODWET. Using the entropy concept, we could talk about the predictability concept of wavelet and scaling coefficient for the first time. We realized that predictability on average in the initial decomposition levels is low and after that, wavelet and scaling coefficients become more predictable. This is a profitable achievement and it can help the improvement of DDFF accuracy.

In another achievement, we determined the appropriate IVS between linear PCIHT and a nonlinear (GP) model for KNN and ANN. However, we realized the linear method PCIHT is a better fit for KNN, which is a linear model, and ANN, which is a nonlinear model.

Finally, we realized that there is not an absolute WDDFF for different case studies. In station 01013500, GP (as IVS)-ANN with an NSE value of 0.71, in station 07083000, MODWET-GP (as IVS)-ANN with an NSE value of 0.84, and in station 11264500, MODWET-PCIHT-ANN, with an NSE value of 0.83 had the best performance. It is noticeable that in all cases the best DDM was nonlinear model ANN. Since the hydrological relationships mainly are nonlinear, we have anticipated this result. In addition, comparing the result of wavelet-based and non-wavelet-based forecasting frameworks in all stations showed that among 15 comparative pairs in 11 cases WDDFFs had a better performance than NWDDFFs. The results could be seen in the Table 5.

Further research will focus on using entropy as statistical criteria that use the predictability concept to select the best inputs to be fed into the DDFF. This objective could be reached by linking entropy with a correlation index like PCI.

Author Contributions A.M contributed to the study conception and material preparation. Analyses were performed by M.M.B. The first draft of the manuscript was written by M.M.B.

Data Availability The case study presented in this paper is available online in accordance with funder data retention policies (https://ncar.github.io/hydrology/datasets/CAMELS_attributes).

Code Availability The codes that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgements

We acknowledge Dr. John Quilty from Waterloo University for providing helpful idea and technical feedback on this paper.

Ethical Approval Not applicable.

Consent to Participate Not applicable.

Consent to Publish Not applicable.

Competing Interests The authors have no relevant financial or non-financial interests to disclose.

Declaration of interest The authors declare that they have no known personal relationships that could have appeared to influence the work reported in this paper.

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Table 3 and 4 are available in the Supplementary Files section.

Table3and4.docx

A New Criteria for Determining the Best Decomposition Level and Filter in Wavelet-based Data-driven Forecasting Frameworks- Validating Using Three Case studies on The CAMELS Dataset

Status:

Version 1

Abstract

Figures

1. Introduction

2. Theoretical Basis Of Modwet Application In Wddff

2.1. An overview on MODWT

2.2. Noticeable issues about using WDDFF

2.2.1. Solving WDDFF problems of application in hydrology using Entropy

2.3. Input Variable Selection for a practical WDDFF

2.4. Data-driven model for a practical WDDFF

2.5. Summary of the MODWET-DDFF

3. Experimental Setup

3.1. Study area and data

3.2. Pre-processing method

3.3. Input variable selection

3.4. Data-driven models

4. Experimental Details And Objectives

5. Results And Discussion

5.1. MODWET preprocessing method

5.2. Determining the best input variable selection for MODWET-DDFF

5.3. Comparing the results to find the best DDM for MODWET-DDFF

5.4. The effect of adding wavelet and scaling coefficients to DDFF

6. Summary And Conclusion

Declarations

References

Tables

Supplementary Files

Status:

Version 1