Multi-step-ahead p edictor design for e ff ective long-term forecast of hydrological signals using a novel wavelet-NN hybrid model

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Introduction
Hydrological signal forecasts, especially a long-term forecast, are important for the study and guidance of water resource management.Nevertheless, hydrological signals are highly complex nonlinear systems and have severe variations in time and space, which make accurate forecasts difficult.Generally, the hydrological time series are pre-Figures

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Full dicted with models based on physical considerations or other numerical theories, such as LR (linear regressive) analysis methods (Salas et al., 1980) based on stochastic theory; grey models (Deng, 1992) based on grey information theory; chaos models (Jayawardena and Lai, 1994;Islam and Sivakumar, 2002) based on a local similarity of signals; fuzzy prediction models (Jang, 1993;Jang et al., 1997;Chen, 2005) based on fuzzy theory; TAR (threshold auto-regression), BL (bilinear time series), and SVM (Support Vector Machine) models (Tong, 1990;Jin and Ding, 2002;Liong and Sivapragasm, 2002;Zou et al., 2010) based on nonlinear time series analysis; ANN (artificial neural networks) models (Raman and Sunlikumar, 1995;Yu et al., 2008;Yang et al., 2009) based on black-box theory; and NNB (nearest neighbour bootstrapping) regressive models (Wang et al., 2001) based on nonparametric prediction theory.However, these models are generally not successful enough in producing accurate predictions due to some inaccurate initial conditions, parameterisation schemes of sub-scale phenomena, and limited spatial resolution (Olson et al., 1995).Many hybrid models have been proposed as predictors to improve the accuracy of hydrological time series forecasts, such as the wavelet-ANN model (Anctil and Tape, 2004), the periodic ANN (PANN) model (Wang et al., 2006), the chaotic-ANN model (Karunasinghe and Liong, 2006), the wavelet-based grey model (Chou, 2007), the wavelet-based NF (Neuro-Fuzzy) model (Partal and Kisi, 2007;Engin et al., 2007;El-Shafie et al., 2007), the non-supervised ANN-EA (evolutionary algorithms) model (Cao and Park, 2007;Chang et al., 2007), the fuzzy-SVM model (Hua et al., 2008), the wavelet-based multi-layer perceptron model (Kisi, 2008), and the wavelet-regression (WR) model (Kisi, 2011).These hybrid models have shown different advantages for accurate predictions due to their capabilities of utilising present information effectively.Among these hybrid models, the neural network (NN) models, such as NF (Neuro-Fuzzy) and ANN, are the most popularly utilised sub-models for signal forecast due to their capabilities of effectively learning complex and nonlinear relationships (Maier et al., 2010).The ANN model has been popularly used in hydrological signal forecasts in recent years by a number of researchers (French et al., 1992;Jain et al., 1999; ASCE, Introduction

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Full 2000; Cigizoglu, 2005;Marzano et al., 2006;Zou et al., 2010).The NF model has been successfully used in the hydrological sciences in recent years (Nayak et al., 2004;Kisi, 2005;Chang and Chang, 2006).In addition, the wavelet transform is a strong mathematical tool that provides a good local representation of the signal in both the time and frequency domains, and it has become a useful method for analysing variations, periodicities, and trends in time series (Daubechies, 1994;Torrence and Compo, 1998;Coulibaly and Burn, 2004;Partal and Kucuk, 2006).Among the various types of wavelet transforms, the discrete wavelet transform (DWT) as the data preprocessing method in a hybrid model is popularly used to decompose the original signal input due to its capabilities of effectively classifying a hydro-meteorological time series into distinct time and frequency domains (Smith et al., 1998;Kim and Valdes, 2003;Labat, 2005).
Because of the common Markovian property (Bolch et al., 2006) embedded in hydrometeorological time series, most recent pure and hybrid models use data series at different previous time points as model inputs to forecast the original data series at the current time point.For daily time series, the data series from one day prior to a few days prior are usually used as model inputs, namely using data series S t−1 , S t−2 , . . .as inputs, to forecast S t .The data series at one day prior is always selected as one of the inputs because of the usually high lag-1 autocorrelation (Kisi, 2008(Kisi, , 2011;;Zhou et al., 2008).This selection principle denotes a type of popular used single-step-ahead (SSA) prediction (Parlos et al., 2000), in which each single forecasting step of the Markovian property-based model can only predict the next one-day datum (Fig. 1a).However, SSA prediction may not provide enough information, especially in the situation in which it is desirable to understand the behaviour of multiple steps in the future, such as signal processing and time series prediction.Given this issue, the serial-propagated multistep-ahead (MSA) prediction (Fig. 1b), which attempts to make predictions several time steps into the future without the availability of output measurements, has attracted an increasing number of scientific studies (Su et al., 1992;Schenker and Agarwal, 1995;Coulibaly et al., 2000;Gao et al., 2002;Chang et al., 2007Chang et al., , 2012;;Yong et al., 2010).However, MSA predictors are difficult to develop because the lack of measure-Introduction

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Full ments in the prediction horizon necessitates the recursive use of SSA predictors to reach the end-point on the horizon, especially difficult for a long-range MSA predictor design.Even small SSA prediction errors at the beginning of the horizon accumulate and propagate, often resulting in a poor prediction accuracy.Over the last twenty years, MSA predictor design to increase the MSA prediction accuracy has received much attention, and different types of neural networks have been used successfully for some short-range MSA predictions (Su et al., 1992;Parlos, et al., 2000;Chang et al., 2007).
As mentioned above, the crucial and most difficult point in a long-term forecast of hydrological signal is the development of effective models to reduce the error accumulation and increase the accuracy of the long-range serial-propagated MSA prediction.
In view of this, the present study designed a novel hybrid model CDW-NN, combining continuous and discrete wavelet transforms and neural networks, as MSA predictor for effective long-term forecast of hydrological signals by broadening the prediction range in each forecast step and reducing the total iteration steps in the long-term forecasting process.In the remainder of this paper, the long-term forecast methodologies of the MSA predictor CDW-NN are presented.In the next section, the details of daily river stage data series in different hydro-stations in Yangtze River Estuary, China are presented and CDW-NN hybrid models are applied to the long-term forecasts of different river stage signals.The results are discussed by comparing with the performances of other pure and hybrid models in the subsequent section, and finally, conclusions are drawn.

Continuous wavelet transform (CWT) and discrete wavelet transform (DWT)
Wavelet transform is a mathematical tool that allows the decomposition of the signal f (t) in terms of elementary contributions called wavelets (Sadowskey, 1996;Labat et al., 2005).For the time series f (t) ∈ L2 (R) or finite energy signal, the continuous Introduction

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Full wavelet transform (CWT) of the signal f (t) with the analysing wavelet φ is the convolution of f (t) with a set of dilated and translated wavelets: where φ(t) is the complex conjugate function of φ(t), a is the dilation (scale or frequency) parameter, b is the translation (position or time) parameter, R is the domain of real numbers, and δt is the time interval of the data series.In this paper, the time interval of the data series equals 1.0 day, and the popularly used Morlet wavelet is selected as φ (Mallat, 1989;Daubechies, 1994;Torrence and Compo, 1998).The Morlet wavelet, which is a complex wavelet consisting of a plane wave modulated by a Gaussian function, is defined by: where ω 0 is the non-dimensional frequency (usually taken to be 6 to satisfy the admissibility condition) (Farge, 1992).The global wavelet power spectrum is defined as the power density at different time scale a, which is calculated by: where N is the length of the data.The signal's periodicity can be indicated at the time scale at which the wave crest of wavelet power spectrum is observed.The significance of the global wavelet power spectrum is tested using a white or red noise model by compared with the theoretical global wavelet power spectrum (P ).P is given as (Torrence and Compo, 1998):

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Full where σ 2 is the variance of data series, x 2 v (p) is the inverse of chi-square cumulative distribution with v degrees of freedom at the requested confidence level 1 − p, and p is the distribution fraction.For the lag-1 autocorrelation, r(1) < 0.1P a is the white noise spectrum, and for r(1) > 0.1, P a is the red noise spectrum.For the Morlet wavelet, P a is given as Eq. ( 5), and v is given as Eq. ( 6).In this study, a significance level of 0.005 was selected, e.g., χ 2 2 (99.5 %) = 10.597.
The continuous wavelet (Eq. 1) is often discrete in real applications.When and k and j are integer numbers, the Discrete Wavelet Transform (DWT) of f (t) can be written as: Based on the commonly used Mallat algorithm for calculating discrete wavelet coefficients, the most common and simplest choice for the parameters a 0 and b 0 is 2 and 1 time steps, respectively, and the Daubechies wavelet, which has no explicit mathematical expressions and can be calculated only numerically, is commonly used in the DWT (Mallat, 1989;Daubechies, 1994;Partal and Kucuk, 2006;Kisi, 2011).For a discrete time series f (t) occurring at different times t (e.g., integer time steps are used herein), the DWT can be defined as:

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Full where N is the number of discrete time steps and W f (j , k) is the wavelet coefficient for the discrete wavelet of scale a = 2 j and time b = 2 j k.

Neuro-Fuzzy (NF) and BP-ANN
The popular neural network (NN) model Neuro-Fuzzy (NF), based on the Adaptive Neuro-Fuzzy Inference System (ANFIS) (Jang et al., 1997;Partal and Kisi, 2007), is utilised as the sub-model for the different hydro-meteorological signals forecast in this paper.ANFIS, first introduced by Jang (1993), is a universal approximator and, as such, is capable of approximating any real continuous function on a compact set to any degree of accuracy.The ANFIS is functionally equivalent to the Sugeno first-order fuzzy model (Jang et al., 1997;Drake, 2000), and its typical architecture with five learning layers is shown in Fig. 2. As mentioned by Jang et al. (1993), Partal and Kisi (2007), and Engin et al. (2007), two types of bell-shaped functions are generally used as transfer function in Layer 1 of the ANFIS architecture.The significant advantage of the NF model depends on the hybrid learning algorithm in ANFIS, which combines gradient descent, back-propagation, and the least-squares method and can rapidly train and adapt the ANFIS.Each learning epoch of the ANFIS is composed of a forward pass and a backward pass, and more information for Neuro-Fuzzy and ANFIS can be found in Jang's papers (Jang, 1993;Jang et al., 1997).
Another popular NN model, BP-ANN (back-propagation artificial neural networks), is utilised in our case to compare the forecast performance with the NF model.Based on the back-propagation algorithm, a common three-layer feed-forward type of BP-ANN is considered, the Levenberg-Marquardt methodology, which is more powerful than conventional gradient descent techniques (Hagan and Menhaj, 1994;Kisi, 2011), to adjust the weights of the ANN model, and the Tangent Sigmoid and linear activation functions are used for the hidden and output node(s), respectively.Because there is no theory yet to determine how many hidden layer nodes in the BP-ANN are needed to approximate any given function, the hidden layer node number in BP-ANN is commonly determined by the trial and error approach.Introduction

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Full  3. Firstly, for the original given daily data series x(1) ∼ x(t), the CWT method was utilized to reveal its short-term periodicities, i.e. periods at a1 ∼ ai days in Fig. 3 (a1 < a2 < . . .< ai ).Meanwhile, decomposition of the original signal DWT was conducted to get new data series TD(1) ∼ TD(t), which was constructed by selecting and combining optimal DWT decomposition components.Then, by combining the CWT and DWT results, the new TD series at a1 ∼ ai days ahead (TD(t − a1) ∼ TD(t − ai )) were selected as NN model inputs for model training to forecast the datum x(t).According to the serial-propagated prediction principle, using TD(t −a1+1) ∼ TD(t −ai +1) as model inputs can predict the first future day datum y(t + 1), and using TD(t) ∼ TD(t − ai + a1) as inputs can predict y(t + a1).Here, the first batch of outputs y(t + 1) ∼ y(t + a1) are predicted from the first forecasting step, and then can be used as new observations for the second step DWT decomposing and NN forecasting to predict the second batch of outputs y(t+a1+1) ∼ y(t+2a1).Just as the one day prediction from each SSA forecast involved in conventional MSA forecast, the day number of predictions in the output of each CDW-NN forecasting step is a1.So, after about n/a1 steps of forecasting process, the final long-term prediction series y(t + 1) ∼ y(t + n) can be obtained.Introduction

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Full (QPOs), namely at 12 day and 23 day time scales, and both of their global wavelet power spectrums were prominent at the 99.5 % confidence level.The QPO of estuarine daily river stage at a fine time scale was often nested in a broad time scale.At the 12 day time scale, the average changing periodicity (T ) of river stage time series was 15 days obtained by calculating and averaging the day numbers of each two neighboring high and low river stage periods.At the 23 day scale, the average T was 28 years.At Qinglong Station, the Morlet wavelet transform coefficients of daily river stage series generated obvious two QPOs at 12 day and 22 day time scales, of which corresponding T were 15 days and 28 days same as that at Santiao Station, and both of their global wavelet power spectrums were prominent at the 99.5 % confidence level.Based on the prominent short-term periodic features of estuarine daily river stage time series, estuarine daily river stage at 15 day prior and 28 day prior were determined to simulate and forecast the river stage at the current day, i.e. a1 and a2 in Fig. 3 equaled 15 and 28 respectively.

Decomposition of daily river stage time series and optimal DWT components combination
The The correlation coefficients (R) between each discrete wavelet component (A 10 , D 1 − D 10 ) at 15 day and 28 day prior (t − 15 and t − 28) and original time series at time t(S t ) were computed and presented in Table 1.By single factor and double factors analysis, at both Santiao and Qinglong Stations the D 3 , D 4 and D 8 components at 15 day and 28 day prior showed prominently higher R with S t than the other DWT components, especially significant in double factors analysis.Instead of using each DWT component individually as the model input, employment of the added suitable DWT components is more useful and can highly increase the forecast performance.Based on the revealed dominant DWT components of different hydrological series, the new series (TD) obtained by adding D 3 , D 4 and D 8 at 15 day and 28 day prior were selected as two NN model inputs for the daily river stage forecast at both Santiao and Qinglong Stations.Comparing each DWT component with the original series (S), the new series (TD) showed significantly higher correlations at both 15 day and 28 day delay time nodes with S t .

CDW-NN model training and a long-term forecasting of daily river
where n is the number of data sets, and Y i is the daily river stage.
The forecast performances of CDW-NF and CDW-ANN models at Santiao and Qinglong Stations were shown in Fig. 4. Results showed that the CDW-NF models performed significantly better correlations between observed and predicted river stage data during 2008-2010 with the higher R of 0.533 and 0.283 at Santiao and Qinglong Stations respectively, while CDW-ANN with the lower R of −0.142 and 0.172 at Santiao and Qinglong Stations respectively.And the CDW-ANN hybrid model showed better forecast performances during the first year than that in the last two years.11) and ( 12) respectively.Without the DWT treatment, three other CWT based (CW-) hybrid models were established by using the original daily river stage series at 15 day and 28 day previous (S t−15 and S t−28 ) as input factors to simulate S t .By model training, the optimal CW-ANN structures for Santiao and Qinglong Stations were determined as CW-ANN(2-3-1) and CW-ANN(2-1-1) respectively, and the CW-LR models for Santiao and Qinglong Stations were obtained and showed as Eqs.( 13) and ( 14) respectively.
S t = 0.454TD As mentioned in Fig. 1, many conventional studies on MSA predictor design focused on the methodology of recursive use of SSA predictors, e.g.generally using S t−1 and S t−2 as model inputs to forecast S t .In view of this, 6 types of conventional SSA based long-term forecast hybrid and pure models were established for comparing with the CWT based hybrid models.Among the 6 types of models, DW-R, DW-ANN and DW-NF hybrid models utilized the optimum decomposition components combinations (TD t−1 and TD t−2 ), determined by DWT, as model inputs to forecast S t .With respect to the pure LR, BP-ANN and Neuro-Fuzzy models, the original daily river stage series at 1 day prior and 2 day prior (S t−1 and S t−2 ) were used as model inputs to forecast S t .By model training, the optimal DW-ANN structures for daily river stage forecasts at Santiao and Qinglong Stations were determined as DW-ANN(2-5-1) and DW-ANN(2-7-1) respectively, and the optimal structures of pure BP-ANN models for daily river stage forecasts at Santiao and Qinglong Stations were determined as BP-ANN(2-3-1) and BP-ANN(2-5-1) respectively.The DW-LR models for Santiao and Qinglong Stations were obtained and shown as Eqs.( 15) and ( 16) respectively, and the pure LR models for Santiao and Qinglong Stations were obtained and shown as Eqs.( 17) and ( 18 respectively.
As shown in Table 2, the training and forecasting performances of 12 types of hybrid and pure models were compared with each other in respect to RMSE, MAE and R statistics.Due to the high lag-1 and lag-2 autocorrelations in hydrological time series, in the training periods the 6 types of conventional SSA based long-term forecast models performed better than the 6 types of CWT based hybrid models with higher R and smaller RMSE and MAE.Nevertheless, in the test periods, without the observation data as model inputs in each forecasting step, the CDW-NF hybrid model showed significant performances among all the 12 models, especially prominent at Santiao Station.In addition, due to the prominent ability of decomposition approach based on DWT in filtering weak correlated details from original signal, each DWT based hybrid model performed better than its corresponding model without DWT both in training and test periods.

Driving mechanism of advantages of the CDW-NN models on long-range MSA predictions
Prediction performance details in respect to R for the 12 types of hybrid and pure models at different forecasting steps during the overall 1096 day river stages forecasting were calculated and shown in Fig. 7.According to the serial-propagated MSA prediction theory, the error accumulation increases with the iteration steps increase in a long-term forecast.Therefore, the prediction performances of all 12 types of models had overall decreasing trends with the increasing of predicted data length to 1096 days.Nevertheless, during approximately the first 200 days ∼ 600 days river stage forecasting 9253 Introduction

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Full all the 6 types of CWT based hybrid models performed better than the other 6 types of SSA based models.In particular, at Santiao Station the CDW-NF model showed significantly better performance than the other models covering the overall 3 yr (1096 days) river stage forecasting steps.With respect to the approximately last 500 days river stage forecasting at Qinglong Station, the CDW-NF model shared better performances with the 3 types of SSA based DW-hybrid models than the other models.The main explanations are that the CWT based models reduce the overall forecasting iteration steps to 74 steps by using the 15 day prior data series as the first model input, while the conventional SSA based models needs 1096 steps by using 1 day prior data series as the first model input.The prominent decrease of forecasting steps consequently brings significant reduction of error accumulation in the long-range MSA prediction.In addition, the combination of the advantages of the DWT method and Neuro-Fuzzy system also benefits weakening the noisy dynamics for the model inputs and enhancing the simulation and forecast ability of the complex hydro-system.In view of the above discussion, the CDW-NF hybrid model is proven as an effective 1. Comparing conventional SSA based models, the CWT based hybrid models broadened the prediction range in each forecast step from conventional 1 day to now 15 days and reduced the overall forecasting iteration steps from conventional 1096 steps to now 74 steps by using the 15 and 28 day prior data series as model inputs, which was determined by revealing signal's significant short-time periodicities from CWT.This prominent reduction of forecast steps has created significant decrease of error accumulations and increase of long-term forecast performances in the CWT based hybrid models.Full  Full Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Screen / Esc Printer-friendly Version Interactive Discussion Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 2.3 Architecture of the long-term forecasting based on the MSA predictor CDW-NN In order to reduce the error accumulation and increase the accuracy of the long-range serial-propagated MSA prediction, the present study designed a novel hybrid model CDW-NN, combining CWT, DWT, and NN, as MSA predictor for effective long-term forecast of hydrological signals.The architecture of CDW-NN hybrid model was shown in Fig.
area and dataThe hightide level data at two time points each day during 13 yr (4748 days) covering 1998-2010, supported by the Water Resources Department of Jiangsu Province, were observed and collected from estuarine Santiao Port Hydrologic Station (31.721•N, 121.698 • E) and Qinglong Port Hydrologic Station (31.862 • N, 121.239 • E), which respectively locate about 19.0 km and 70.0 km upstream from Chinese Yangtze River entrance into East China Sea (Fig.4).The daily river stage data series at each station was obtained based on the average value of hightide levels at two time points each day.The first 10 yr of river stage data (3652 days) was used for training and establishing hybrid models, i.e. t = 3652 in Fig.3.The remaining 3 years of river stage data (1096 days) was used for testing the long-term forecasting performance of hybrid models, i.e. n = 1096 in Fig.3.3.2Short-term periodic features of estuarine daily river stage seriesMorlet wavelet transform coefficients of the training data series at relatively fine time scales (from one day to fifty days scales) were calculated by MATLAB language.The real parts of Morlet wavelet transform coefficients at Santiao Station (Fig. 2a) and Qinglong Station (Fig. 2c) clearly indicated the distribution conditions of the river stage signals for different time scales.The solid isograms indicate positive wavelet coefficients and a relatively high river stage period, and the dashed isograms indicate negative coefficients and a relatively low period.Further calculating the global wavelet power spectrums and the corresponding theoretical power spectrums using white noise model, the significances of wavelet power densities of daily river stage series at different time scales at Santiao and Qinglong Stations were calculated and shown in Fig. 2b and d.Results showed that the Morlet wavelet transform coefficients of daily river stage series at Santiao Station generated obvious two kinds of quasi periodic oscillations Discussion Paper | Discussion Paper | Discussion Paper | decomposition process of DWT consists of a number of filtering steps following the Mallat algorithm.The original signal of training data series is first decomposed into an approximation (A 1 ) and details (D 1 ), and the A 1 is then broken down into many lowerresolution components (A i and D i ).The details are the low-scale high-frequency components of the signal, while the approximations are the high-scale low-frequency components.The higher scales consist of the extended version of a wavelet, and the corresponding coefficients refer to the slowly changing coarse features of low-frequency components.The lower scales present the condensed wavelet and follow the rapidly changing details (high frequency components) of the signal(Mallat, 1989).In our case, ten decomposition components (A 10 , D 1 -D 10 ) of the original daily river stage signal(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007) in ten resolution levels were calculated by MATLAB language, and were used to analyze the optimum input factors in our hybrid model.Discussion Paper | Discussion Paper | Discussion Paper | stage signal According to the above CWT and DWT results, two new daily series TD(1) ∼ TD(t −28) and TD(14) ∼ TD(t−15) extracted from the training data series were used as NN model inputs to simulate and forecast the original series x(29) ∼ x(t).Program codes were written in MATLAB language for training the Neuro-Fuzzy and BP-ANN submodels and determine their optimal model structures.At Santiao Station, by many trials the optimal CDW-ANN hybrid model structure was determined as CDW-ANN(2-3-1), which denotes two input layer nodes, three hidden layer nodes and one output layer node in BP-ANN submodel.At Qinglong Station the optimal CDW-ANN(2-4-1) model was obtained.When the training programs of NN submodels were done, two TD series TD(t − 27) ∼ TD(t − 13) and TD(t − 14) ∼ TD(t) were used as model inputs to forecast the future 15 days river stage series y(t + 1) ∼ y(t + 15), i.e. the first forecasting step in Fig. 3.The predicted 15 day river stage series was then treated by DWT and used as 9250 Discussion Paper | Discussion Paper | Discussion Paper | new inputs to forecast the next 15 days river stage in the next forecasting step.Because each single step forecasting using CDW-NN model obtained 15 days predictions, the total 1096 days future river stage values from 2008-2010 were predicted after 74 steps of forecasting.The root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) statistics were used to evaluate the model performance of simulation and prediction.The RMSE and MAE are defined as: performances comparison between CDW-NN models and the other 10 types of hybrid and pure models Similar to the establishing process of CDW-NN models, the hybrid CDW-LR model was established by combining CWT, DWT and a linear regression (LR) model.Using the 9251 Discussion Paper | Discussion Paper | Discussion Paper | training data series, simulating equations of the hybrid models (CDW-LR) at Santiao Station and Qinglong Station were obtained and shown as Eqs.( Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | MAS predictor for long-term forecast of estuarine daily river stage signals.In addition, the CDW-ANN hybrid model can be taken as the second selection for shot-term and mid-term forecasts of estuarine daily river stage because of its high performance during the first year forecast process.It should be noted that the methodology of accurate MSA predictor design by reducing iteration steps and error accumulation is innovated in this study by revealing the short-time periodic features of estuarine river stage dynamics, which is mainly caused by the half-month periodicity involved in astronomical tidal fluctuation in river estuary.With respect to other kinds of hydro-meteorological signals, which have non-significant short-term periodic features, other kinds of algorithms and models for reducing error accumulation in long-term forecasting steps need further study in future research.long-term forecast of hydrological signals have high practical value, but the accurate long-range MSA predictor design is very difficult, especially in conventional SSA based MSA predictions, because of the huge error accumulation in serialpropagated long-term forecast.In this study, we designed a novel hybrid model CDW-NN, combining continuous and discrete wavelet transforms and neural networks, as MSA predictor for effective long-term forecast of hydrological signals.By the application of CDW-NN hybrid models and the other 10 types of hybrid and pure models in estuarine daily river stage series long-term forecasting, the 1096 days estuarine river stage data were forecasted, and the superiorities of CDW-NN models with corresponding driving mechanisms were proven as follows:

2.
Among the CWT based models, one type of CDW-NN model (CDW-NF) has been proven to be the most effective MSA predictor for the accuracy enhancement in the overall 1096 days long-term forecast of estuarine hydro-signal.The other type of CDW-NN model (CDW-ANN) has been proven to be the second selection for shot-term and mid-term forecasts of estuarine hydro-signal.The main explanation is the combination of the advantages of the CWT and DWT methods and Neuro-Fuzzy system in reducing the error accumulation, filtering weak correlated details from original signal and weakening the noisy dynamics for the model inputs, and enhancing the simulation and forecast ability of the complex hydro-system.Discussion Paper | Discussion Paper | Discussion Paper | 3. It should be noted that because the successful application of the novel CDW-NF model in hydro-signal long-term forecast largely depends on the significant short-term periodicities involved in estuarine hydro-signals, some other innovative algorithms and models still need to be further studied in future research for other kinds of hydro-meteorological signals without significant short-Discussion Paper | Discussion Paper | Discussion Paper | Yang, L. N., Peng, L., Zhang, L. M., Zhang, L. L., and Yang, S. S.: A prediction model for population occurrence of paddy stem borer (Scirpophaga incertulas), based on back propagation artificial neural network and principal components analysis, Comput.Electron.Agr. , 68, 200-206, 2009.Yong, S., Li, Y. B., Wang, Q., and Li, C. H.: Multi-steps prediction of chaotic time series based on echo state network, BIC- Discussion Paper | Discussion Paper | Discussion Paper |

Fig. 5 .Fig. 6 .Fig. 7 .
Fig. 5. Real parts of Morlet wavelet transform coefficients at Santiao Station (a) and Qinglong Station (c), and their global wavelet power spectrums and corresponding confidence tests using white noise models (b, d), based on 3652 daily river stage data during 1998-2007.

Table 1 .
Correlation coefficients between each discrete wavelet component series at 15 day and 28 day prior (t − 15 and t − 28) and original river stage data series at the current day (S t ).