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��������������$�������P?d�\��4��LKKKK��d<� �3�3�3�3�3�3�3$96��8X 49������ 4��KK�Y4�)�)�)��K�K�3�)��3�)�)�)K�����������%��)�3o40�4�)39l)(39�)�)39��*8	���)����� 4 4�)����4������������������������������������������������������������������������39����������	�:	Need to criteria for the evolution of performance and categorize models in hydrology

Hadi Delafrouz
School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
E-mail: delafrooz@gmail.com

Abbas Ghaheri
School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
E-mail: ghaheri@iust.ac.ir


Mohammad Ali Ghorbani
Department of Water Engineering, University of Tabriz, Tabriz, Iran
E-mail: HYPERLINK "mailto:m_ali_ghorbani@ymail.com"m_ali_ghorbani@ymail.com, ghorbani@tabrizu.ac.ir

Abstract 
a lot of models have been developed from the past until now to analysis rainfall-runoff process and the complexity of the models are increasing. Artificial Intelligent technique is one of the most developed methods as a branch of computer science. In these methods, ANN is widely used among those which are employed alone or in combination with others. In this study, daily runoff time series of 60 stations from different parts of Iran are simulated using both auto regressive moving-average (ARMA) as a simple and classic model, and ANN as an advanced model. The results indicate that in most of the time series, there is no significant difference between the results of two models. In the absence of clear criteria for evaluating the model performance to prediction, only complexity of the models were added daily. There has been no significant change in the results of the models and both of the models encountered problems with the prediction of some datasets.
Keywords:  ARMA, Artificial Neural Network, categorize , complexity, daily runoff 

1. Introduction
Rainfall-Runoff forecasting at multiple time scale is an important issue in water resource applications. In last few decades a lot of linear and nonlinear rainfall�runoff models have been proposed (Nourani et al., 2009) and improved along time with developing human knowledge. 
 In recent years, the number of models based on time series analysis has grown rapidly. These models can be classified into three groups: regression based methods, time series models and AI-based methods (Wang et al., 2009).  
Autoregressive moving-average models (ARMA) presented by Box and Jenkins (1970) and developed by Carlson et al. (1970). From 1970 until now, extensive applications of such models in water resource time series has been reported and compared with other methods (Chen and Rao, 2002; Salas, 1993; Srikanthan and McMahon, 2001; Toth et al., 2000; Arena et al., 2006; Komornik et al., 2006; Paolo Burlando et al., 1993; Jayawardena and Lai, 1994; kisi et al., 2012; Nayak et al., 2004; Hsu et al., 2003; Dawson et al., 2006).
Among methods based on AI, artificial neural networks (ANNs) has been widely applied and accepted tools for complex nonlinear phenomena in various sciences like pattern recognition, non-linear modeling, classification, association, and control ( Paliwar and Kumar, 2009; Wen et al., 2009; Piotrowski and Napiorkowski ,2012).
Recently, ANN are used for rainfall�runoff modeling and other fields in hydrology. Several applications of this method for modeling in hydrologyinclude [ASCE,( 2000a,b); Chua & Holz (2005); Sudheer et al.,2002, Campolo et al., 2003, Cherkassy et al. (2006),  Zhang (2003), Solomatine and Ostfeld (2008), Nayak et al. (2006) and Wu and Chau (2011) ,  Wang et al. (2009) and Elshorbagy et al. (2010)]. 
 In fact, presented models in most of the studies have not been tested in different conditions and have been only used for a few time series and concluded on that basis. Thus, the other researchers cannot decide that presented model is better than the other one.
In this study, we try to test the efficiency of the new method versus the classic method and find answers to the above questions. We simulate some daily runoff time series with different conditions with ANN and ARMA models. ARMA, as a classic model, and ANN, as a new model, that has been used alone or combined with other methods in recent years, have been selected.

2. Proposed model
2.1. Artificial Neural Network (ANN) 
Artificial Neural Network ( McCulloch and Pitts (1943)) is a nonlinear data driven model that mimic the brain's interconnected system of neurons (ASCE Task Committee (2000a)). ANN can be viewed as universal approximates black box model that is made of many nonlinear and densely interconnected processing elements or neurons (Raghuwanshi et al. 2006), that estimate the values of output variables (y), as dependent parameters, versus independent input variables (x) by finding an optimal set of weights for the connections and threshold values for the nodes. Among various type of ANN, multi-layer perceptron neural networks (MLPs) are very commonly due to their simplicity, relatively low number of parameters, clear biological inspirations [Adam P. Piotrowski et al. (2013)] (Fig. 1).  Neurons and element in multi-layer perceptron neural networks (MLPs) is arranged in groups called layers. Usually three layers are adequate for forecasting process, an input layer where the data are introduced to the network; a hidden layer where data are processed, and an output layer , where the results of given input are produced , (Haykin, (1999), Siou et al., (2011)). The number of nodes in input and output layers is depended on the number of input and output variables and the number of nodes in hidden layer determined by trial and error process. Each node filters input data through the so-called activation function. As usual, Input and output data are standardized to the limit and small interval such as [0,1]. 

Fig. 1: A sample of MLP model architecture 
2.2. ARMA 
The ARMA (Box and Jenkins (1970)) model is used when the time series are stationary and normal distribution but it can cope with some dependence among successive data points. ARMA(p,�q) combines both AR(p) and MA(q) models, as follows:
 EMBED Equation.DSMT4 
Where  �1, & �P, �1,& ,�P  are model parameters,  [t ,[t-1,&  are white noise and Qt  is the dependent variable and, t is the time and p is the lag. 

3. Model performance evaluation
the performance of the proposed models between observed and calculated data is evaluated by the Nash and Sutcliffe index (E) and correlation coefficient (R) . These measures are defined as:

 EMBED Equation.DSMT4 (1) EMBED Equation.DSMT4 (2)where  Qi,obs is the observed discharge at t=i;  Qi,sim  is the predicted discharge at t=i ; No is the number of observed data. Nash�Sutcliffe criterion ranges between (((, 1): E = 1 indicates the best match of simulated value to the observed data; E = 0 shows that the predicted value are the same as the mean value of the observed data; and (( < E < 0 occurs when the mean observed value is a better predictor than the model simulated value, which indicates unacceptable performance [Wang et al. (2009)]. R varies from -1 to 1and this is index of the degree of linear relationship between observed and predicted data.

4. Study area and data
Iran is divided into 6 major catchments, including Sarakhs, Caspian Sea, Urmia Lake, Persian Gulf, Oman Sea and Central Basin. The climate in each of these catchments is completely different from the others. The variation of the mean rainfall (per year) of these catchments is 337 mm in Caspian Sea and 102 mm in Oman Sea which is widely distributed over Iran. In this study, data of daily runoff from 60 synoptic station of Iran has been collected.these stations are distributed along Iran, and The drainage areas of the stations vary from as small as 8 km2 ( station 19-148 in Ardebil state) to as large as 36500 (station 24-029 in Boushehr state). As average, the 32 years of daily runoff record in each station are used in this study. The value of runoff ranges greatly betweenstations. Some notable attribute of the flow variations are as follows:
The mean flows varies from as low as 0.07 m3s-1 at station 19-148 in Ardebil state to as high as 48.33 m3s-1 at station 21-183 in Lorestan state.
The maximum flow observed was 2866 m3s-1 at station 24-029 in Boushehr state).
The standard deviation values vary from as low as 0.06 m3s-1  at station 19-148 to as high as 116 m3s-1 24-029 at station 24-029 in Boushehr state.
In Fifteen stations were observed flow over 350 m3s-1 an also less than0.2 m3s-1. All these evidences show clearly the extreme variability in stream flow among selected stations. The spatial distribution and some characteristics of selected stations are presented in Fig. 2.

Fig. 2: Scattering Map of selected Stations
Table 1: Details and results of simulation in study Stations



5. Discussion
The ANN and ARMA models are applied to analyze 60 stream flow time series from different part of Iran. The objective of modeling runoff is finding a relation between next runoff value (Qt+1) and past runoff records (Qt, Qt-1, Qt-2,�) where the subscript t represents the time step. In order to perform the simulation, the data from each of the stations are divided into two categories: training and verification dataset. 75 percent of total amount of the data from each stations is assigned as training set and the remaining data is utilized for validation purpose. 
data is normalized between 0.2 and 0.8 for ANN modeling. This interval scaling increase the efficiency of the ANN models (Cigizoglu (2003)). The Levenberg-Marquardt LM algorithm, a standard second-order nonlinear least-squares technique, based on the back propagation process [Hagan and Menhaj (1994)] was used for training the ANN models. The three-layer networks with sigmoid transfer function in the hidden layer and linear transfer function in the output layer were employed. The number of nodes in input and hidden layer has been determined iteratively to obtain best results (Table 1). 
 In order to choose the appropriate ARMA (p , q) model, the  Akaike information criteria (AIC) (Yurekli and Kurunc (2005)) are used to select the value of p and q, which represent, respectively the number of autoregressive orders and the number of moving-average orders of the ARMA model (Table 1).
In Fig. 3 the Nash-Sutcliffe (NE) and correlation Coefficient (R) criteria of simulated training and verification dataset with bisector line (for two models) are presented. Here, the horizontal axis represents ARMA criterion value and the vertical axis represents the ANN. In this figure, an inclination of the points to the vertical axis indicates the fact that the quality of ANN model is better than ARMA model and converse inclination shows the better quality of the ARMA model. The concentration of the points in the neighbor of the bisector line shows that both of the models have similar accuracy and precision in simulation results.
Fig. 3: Correlation Coefficient and Nash-Sutcliffe criteria ARMA V.S. ANN
Fig. 3 shows the variation of accuracy of the results among the 60 stream flow time series. This diagram display ranging from very good accuracy such as station 21-259 to very bad accuracy such as station 13-038 and 19-119 (for both model) and the others are between two extremes.
Fig. 3 could be shown that most of the stations with good accuracy prediction (over 0.85) are placed in the neighbor of the bisector line. This means that in this area, both models have had approximately the same accuracy and not many differences are observed in the results from the two models. Outside this area, especially in the ANN model, training dataset is better qualified. But in the verification dataset, which is of much more importance in the prediction model, the stations are close to the bisector line. Based on the minimum value of Nash-Sutcliffe criteria, ANN models are divided into three categories, shown in table 2. In table 2 the characteristics of each group are listed. The difference in accuracy (in training dataset) between the two models in the above three groups is shown in Fig.4. As presented in this table and referred previously, in the station belonging to group 1 (where the correlation coefficient is more than 0.85), there isn�t much difference between the two models. It could be seen that the average difference between the two models is limited just to 0.01 while the difference in the accuracy of prediction between them in the worst case is only 0.03. It seems that the stations belonging to this group have little need to employ developed models. In other word, it means that the classic models can predict and simulate the time series from this group with sufficient accuracy. In the case of group 2, in which the criterion is between 0.85 and 0.75, the average difference value of the criterion between the two models is about 0.02. In this group, although the two models do not have good prediction accuracy, but the ANN is better than the ARMA. In group 3, there is an increase in the difference value of accuracy. However, according to the Nash-Sutcliffe criterion, neither of the two models have satisfactory results and none of them can be used for modeling the data in this group.
Fig. 1 presents the grouping of the 60 stream-flow time series in the map, according to above categorize. This figure show �homogeneity� in some region. In southwestern, stations belong to group 3 and none of the stations in south does not belong to group 1. In western with wet climate, stations belong to group 1. However, the number of station are not enough and this �homogeneity� isn�t true for every region but we can get overview from condition of time series modeling with these model.  


Fig 4. The difference R criteria between ANN and ARMA Model in three Categories

Table 2: Statistical characteristics of the difference Nash-Sutcliffe criteria between ANN and ARMA Model


6. Conclusion
Modeling hydrological phenomena play a crucial role in the watershed management. Hence, many researchers have noted to select appropriate model. Efforts to develop more complex models are continuing without finding a suitable answer for many questions such as �which type of the data are the presented models useful for� and �if the new methods present good results for time series where the classic models don't� [sivakumar B. (2008)]. The present study showed that aren�t big difference between the value of model accuracy in accepted area and both of them fail in the face of some kind of data. In other words, in dealing with some time series, we don�t need more complex model and simple model is enough. The present study clearly are showed that the absence of a generic framework and criteria cannot be presented for the performance models. This makes additional cost and time which is not necessary. Hence, we must recognize what we need before developing suitable models, and this statement was aimed by other authors (see also, sivakumar B. 2008; Beven K. 2008; Sivakumar B. 2004; McDonnell JJ, Woods RA. (2004)). For instance, some hydraulic criteria, like Froude etc., can help users to select the suitable model and verify the data type. The criteria can show the user which level of complexity (one-dimensional, two dimensional, etc.) is needed for modeling. Some criteria with other accuracy levels like correlation coefficient, Nash-Sutcliffe, etc. have also been presented in papers to let the other users be aware of the validity of the model in addition to the accuracy (This helps generalizing the model). To my knowledge, there is no such criterion in hydrology and researcher must be notice than develop complex model. 
We hope that future studies can identify such as criteria that help researcher to study catchments more effectively and efficiently and develop more appropriate strategies, in terms of simplification in models/model development, generalization in our modeling approach as much as possible.     
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Zhang G.P., Time series forecasting using a hybrid ARIMA and neural network
Model. Neurocomputing 50 (2003) 159�175










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