Neural Comput & Applic DOI 10.1007/s00521-013-1443-6
ORIGINAL ARTICLE
Flood flow forecasting using ANN, ANFIS and regression models M. Rezaeianzadeh • H. Tabari • A. Arabi Yazdi S. Isik • L. Kalin
•
Received: 18 March 2013 / Accepted: 6 June 2013 Springer-Verlag London 2013
Abstract Flood prediction is an important for the design, planning and management of water resources systems. This study presents the use of artificial neural networks (ANN), adaptive neuro-fuzzy inference systems (ANFIS), multiple linear regression (MLR) and multiple nonlinear regression (MNLR) for forecasting maximum daily flow at the outlet of the Khosrow Shirin watershed, located in the Fars Province of Iran. Precipitation data from four meteorological stations were used to develop a multilayer perceptron topology model. Input vectors for simulations included the original precipitation data, an area-weighted average precipitation and antecedent flows with one- and two-day time lags. Performances of the models were evaluated with the RMSE and the R2. The results showed that the areaweighted precipitation as an input to ANNs and MNLR and the spatially distributed precipitation input to ANFIS and MLR lead to more accurate predictions (e.g., in ANNs up to 2.0 m3 s-1 reduction in RMSE). Overall, the MNLR was shown to be superior (R2 = 0.81 and RMSE = 0.145 m3 s-1) to ANNs, ANFIS and MLR for prediction of maximum daily flow. Furthermore, models including
M. Rezaeianzadeh (&) L. Kalin School of Forestry and Wildlife Sciences, Auburn University, 602 Duncan Drive, Auburn, AL 36849, USA e-mail:
[email protected] H. Tabari Department of Water Engineering, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran A. Arabi Yazdi Department of Water Engineering, University of Ferdowsi, Mashhad, Iran S. Isik Turgut Ozal University, Ankara 06010, Turkey
antecedent flow with one- and two-day time lags significantly improve flow prediction. We conclude that nonlinear regression can be applied as a simple method for predicting the maximum daily flow. Keywords MLP Neuro-fuzzy Regression analysis Area-weighted precipitation Antecedent flow Iran
1 Introduction Floods are widespread hydrologic extremes that result in significant property and environmental damages as well as loss of life across the globe. Between 1985 and 2004, floods resulted in more than $15 billion in property damage and nearly 7,500 fatalities [35]. Heavy rainfalls can generate flooding conditions in vulnerable areas such as lands with steep slopes and without enough vegetal covers. This problem can also take a priority where there is no adequate data such as soil moisture, soil storage and percolation rates, snow coverage, etc. for a comprehensive hydrologic modeling in developing countries such as Iran. Official reports indicate that many people lost their lives, animals and agricultural crops as a result of destructive floods, especially in the southwest of Iran [32]. The significance of floods has raised the importance of state-of-the-art hydrologic models and mathematical techniques to simulate streamflows. The most common method for river flow forecasting is by means of application of physical, conceptual and/or statistical rainfall-runoff models [42]. In recent years, artificial neural networks (ANNs) and other artificial intelligence-based computational methods have received a great deal of attention in streamflow forecasting [7, 10, 14–16, 18, 21, 27, 29, 36, 40]. Given their intrinsic nature, ANNs do not require
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Neural Comput & Applic
detailed knowledge of internal processes of a system in order to relate inputs and outputs [27]. Shamseldin [33] compared feedforward neural network (FFNN) models named multilayer perceptron (MLP) with a simple linear model and showed that FFNN had a better capability in rainfall-runoff modeling. Tokar and Johnson [40] employed a FFNN model to forecast daily runoff as a function of daily precipitation, temperature and snowmelt. In addition, nonlinear methods such as ANFIS are widely applied in hydrology and water resources applications (e.g., [1, 6]. Moradkhani et al. [24] explored the applicability of a self-organizing radial basis (SORB) function to one-step ahead forecasting of daily streamflow. SORB outperformed the two other ANN algorithms, the well-known multilayer feedforward network (MFN) and self-organizing linear output map (SOLO) neural network in simulating daily streamflow in the semi-arid Salt River basin. Chau et al. [5] applied two hybrid models based on recent artificial intelligence technology, namely the genetic algorithm-based artificial neural network (ANN-GA) and the ANFIS models, for flood forecasting in a channel reach of the Yangtze River in China. Shamseldin [34] dealt with exploring the use of ANN for forecasting the Blue Nile river flows in Sudan. The related findings indicated that the ANN has considerable potential to be used for river flow forecasting in developing countries. In the other study, flood forecasting at Jamtara gauging site of the Ajay River Basin in Jharkhand, India, was carried out using an ANN, an ANFIS and an adaptive neuro-GA integrated system (ANGIS) models [26]. Recently, Kisi et al. [17] compared the accuracy of several data-driven techniques, that is, ANFIS, ANN and support vector machine (SVM) for forecasting daily intermittent streamflows. Due to difficulties in working with artificial intelligence methods, investigators have turned to fast, simple and straightforward statistical methods such as regression models. These have been successfully employed in modeling a wide range of hydrologic processes like soil temperature [4, 39], snow water equivalent [22, 37], low and flood flows [3, 8, 9, 41] and evaporation and evapotranspiration [31, 38]. Despite a plethora of studies on rainfall-runoff modeling using artificial intelligence-based models such as ANNs, there are still several issues that need further detailed consideration. When designing a model, the type of input data set, used in the networks, plays an important role. For example, when multiple rain gauges are available, streamflow forecasting using all gauges as opposed to an area-weighted average precipitation vector may lead to different results. This study attempts to capture the behavior of ANNs and ANFIS techniques with different types of input precipitation data sets. In fact, the first objective of this study is to
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evaluate the application of MLP network and ANFIS for peak discharge prediction using spatially distributed precipitation and area-weighted average precipitation data as inputs to the networks. Furthermore, antecedent streamflows are included as additional inputs to investigate the potential improvements in streamflow simulations. The Levenberg–Marquardt (LM) training algorithm for MLPs and the subtractive clustering (genfis2; the related function in MATLAB) and fuzzy inference system (FIS) using fuzzy c-means (FCM) clustering (genfis3; the related function in MATLAB) are applied for the training of ANFIS. While many studies have intended to derive the best combinations of input variables to the networks, the main focus of this study is to investigate whether spatially distributed rain gauge inputs produce superior results in predicting peak flows, compared to area-averaged precipitation input. This is analogous to a comparison between lumped (here, areaaveraged) and distributed (here, spatially distributed precipitation) hydrologic models. The second objective of this study aimed at comparing artificial intelligence approaches with multiple linear regression (MLR) and multiple nonlinear regression (MNLR) for maximum daily flow forecasting. The input variables to the MLR and MNLR models were the same as the ones used in the ANN and ANFIS models. The predictive capabilities of the models were evaluated using data from the flood vulnerable watershed of Khosrow Shirin (K–S) located in the southwest of Iran.
2 Methodology 2.1 Multilayer perceptron MLP is perhaps the most popular ANN architecture [7]. It is a network formed by simple neurons called perceptron. The perceptron computes a single output from multiple real-valued inputs by forming combinations of linear relationships according to input weights and even nonlinear transfer functions as schematically shown in Fig. 1. Mathematically, the MLP can be represented as: ! n X y¼f wi pi þ b ð1Þ i¼1
where wi represents the weight vector; pi is the input vector (i = 1,…, n); b is the bias, f is the transfer function; and y is the output. In a recent study, Rezaeian Zadeh et al. [29] showed that the tangent sigmoid transfer function performed better than the logistic sigmoid transfer function. Yonaba et al. [43] found that the application of tangent sigmoid was more efficient for streamflow forecasting. In addition to these studies, Maier and
Neural Comput & Applic Fig. 1 Schematic view of a typical MLP
Input Layer
w 11 ,1
P 1 P
n11
∑ b11
2
∑
P ⋅3 ⋅ ⋅ ⋅ ⋅ ⋅ P R
Hidden Layer
f
∑
2 1 ð1 þ e2s Þ
n12
n1 s1
⋅ ⋅ ⋅
f1
b1 s1
ð2Þ
2.1.1 Neural networks training algorithm Previous studies indicated that the Levenberg–Marquardt algorithm produces reasonable results for most ANN applications [15], and thus, it is selected for this study. The objective of the training algorithm is to minimize the global error E defined as: E¼
M
Em
ð3Þ
m¼1
where M is the total number of training patterns, and Em represents the error for training pattern M. Em is derived as: Em ¼
1 2
n X
ðok tk Þ2
1
f
MLPs are typically trained using the back error propagation algorithm in which the network’s interconnecting weights are iteratively changed to minimize the predefined error, which is root-mean-square error (RMSE) [36].
M X 1
∑
n 21
f2
w 3 1 ,1
a 21
ð4Þ
k¼1
where n is the total number of output nodes, ok is the network output at the kth output node and tk is the target output at the kth output node. In the training algorithm, described in the next section, the weights and biases are adjusted in order to reduce the global error [15].
2
∑
⋅ ⋅ ⋅ a1 s1
n2
3
f
a 31
b 31 2
b2 2 n2s2
1 w2s2,s
n3 1
∑
b 21
Dandy [20] also indicated that not only is the training with the hyperbolic tangent function faster than the training with the logistic sigmoid transfer function, but also the predictions obtained using networks with the hyperbolic tangent are slightly better than those with the logistic sigmoid transfer functions. Thus, in light of these studies, we used the tangent sigmoid transfer function in this study, which is defined for any variable s as: f ðsÞ ¼
w 2 1 ,1
a11
a1
b1 2
w1 s 1 , R
1
Output Layer
∑
b2s2
f2 ⋅ ⋅ ⋅
f
a2
n32
∑
2
⋅ ⋅ ⋅ 2 2 a s 2
w3s3, s
2
∑
b3 2 n3s3
f3 ⋅ ⋅ ⋅
f
a 32
⋅ ⋅ ⋅ a3s 3 3
b3s3
2.1.2 Levenberg–Marquardt algorithm The Levenberg–Marquardt algorithm is designed to approach the second-order training speed without having to compute the Hessian matrix [25]. Please refer to Rezaeianzadeh et al. [29, 30] for more details. 2.2 Adaptive neuro-fuzzy inference system Although ANN is a powerful technique for modeling various real-world problems, it has its own shortcomings. If the input data are ambiguous or subject to a relatively high uncertainty, a fuzzy system such as ANFIS may be a better option [23]. Jang [11] first proposed the ANFIS method and applied its principles successfully to many problems [19]. This system is a fuzzy Sugeno by a forwarding network structure. This sort of models are typically developed and placed into the framework of a neural network model to enable adaptation [12]. Figure 2 shows an ANFIS system with two inputs, one output and two rules. This system has two inputs x and y and one output, where its rule is: If x is A1 and y is B1
then f ¼ p1 x þ q1 y þ r1
ð5Þ
ð6Þ If x is A2 and y is B2 then f ¼ p2 x þ q2 y þ r2 Ai and Bi are fuzzy sets, fi is the output within the fuzzy region specified by the fuzzy rule, pi , qi and ri are the design parameters that are determined during the training process. Figure 2 shows that every node in this layer is a fuzzy set and any output of any node in this layer corresponds to the membership degree of input variable in this fuzzy set. In this layer, shape parameters determine the shape of the membership function of the fuzzy set [44]. In this study, the bell-shaped membership function is employed. For a bell-shaped membership function, lAi is given as:
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Neural Comput & Applic Fig. 2 ANFIS Structure with two inputs, one output and two rules [2]
Layer1
Layer2
Layer4
Layer3
x
Layer5
y
M1
W1
x
N
W1 f1
W1
M2 ∑
f
W2
N1 N
y
W2 f 2
W2
N2
Legend:
x
y
Adaptive Node Fixed Node
lAi ¼
h
1
1 þ ðx ci Þ=ai
i2bi
ð7Þ
where x is value of input to i node, and ai, bi and ci are the parameters of membership function of this set. These parameters are usually called conditional parameters. In the second layer, every node computes the degree of activation of any rules. The membership functions are then multiplied in this layer: wi ¼ lAi ðxÞlBi ðyÞ
ði ¼ 1; 2Þ
ð8Þ
where lAi ðxÞ is membership degree of x in Ai set, lBi ðyÞ is the membership degree of y in Ai set. The ith node computes (third layer) the ratio of activity degree of i rule to the sum of activation degrees of all rules. i is normalized membership degree of i rule. w i ¼ wi=ðw1 þ w2 Þ ði ¼ 1; 2Þ ð9Þ w The output of any node is calculated in the fourth layer: i ðpi x þ qi y þ ri Þ i fi ¼ w w
ð10Þ
where p, q and r are changeable consequent parameters. The final network output f is produced by the node of the fifth layer as a summation of all incoming signals. The final outputs (number of nodes equals to output parameters) of all nodes are derived in the fifth layer (Fig. 2): P X wi fi i fi ¼ P w ð11Þ wi A two-step process is used for a fast training and to adjust the network parameters to the above network. In the
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first step, the premise parameters are kept fixed, and the information is propagated forward in the network to layer 4. In layer 4, a least squares estimator identifies the important parameters. In the second step, the backward pass, the chosen parameters are held fixed, while the error is propagated. The premise parameters are then modified using the gradient descent method. Apart from training patterns, the only user-specified information required is the number of membership functions. A comprehensive description of the learning algorithm is given in [13]. 2.2.1 Training of the ANFIS model using genfis2 and genfis3 In order to begin the training of the fuzzy inference system (FIS), a structure is needed first. The FIS structure specifies the parameters of FIS system for learning. The genfis2 function meets these requirements since it generates a Sugeno-type FIS structure using subtractive clustering and requires separate sets of input and output data as input arguments [12]. When there is only one output, the genfis2 function may be used to generate an initial FIS for ANFIS training. The genfis2 function accomplishes this by extracting a set of rules that models the data behavior. The rule extraction method first uses the subclust function (MATLAB functions) to determine the number of rules and antecedent membership functions and then uses linear least squares estimation to determine each rule’s consequent equations. The function returns a FIS structure that contains a set of fuzzy rules to cover the feature space. Another fuzzy inference system that can be used for training of the ANFIS model is genfis3. It generates a FIS
Neural Comput & Applic
using fuzzy c-means (FCM) clustering by extracting a set of rules that models the data behavior. The function requires separate sets of input and output data as input arguments. When there is only one output, genfis3 can be utilized to generate an initial FIS for ANFIS training. The rule extraction method first uses the FCM function to determine the number of rules and membership functions for the antecedents and consequents.
four meteorological stations and one streamflow gauging station at its outlet (see Fig. 3). Precipitation events data from meteorological stations and the maximum daily discharge values were obtained for the period 2002–2007. Using Thiessen polygons, the weights of the rainfall stations Khosrow Shirin, Dozd Kord, Dehkade Sefid and Sedeh are derived as 0.68, 0.16, 0.08 and 0.08, respectively.
2.3 Regression analysis One of the classical problems in statistical analysis is to find a suitable relationship between a response variable and a set of regressor variables [39]. Regression analysis is commonly used to describe quantitative relationships between a response variable and one or more explanatory variables [38]. In MLR, the function is a linear equation, i.e., straight-line, in the form: Y ¼ b0 þ b1 X1 þ b2 X2 þ þ bn Xn
ð12Þ
where Y is the dependent variable, b0 ; . . .; bn are the equation parameters for the linear relation, and X1 ; . . .; Xn are the independent variables for this system [28]. Nonlinear regression is a form of regression analysis in which observational data are modeled by a function, which is a nonlinear combination of the model parameters and depends on one or more independent variables [4]. Unlike traditional MLR, which is restricted to estimating linear models, MNLR can estimate models with arbitrary relationships between independent and dependent variables [22]. The general appearance of the nonlinear relation is assumed to be: Y ¼ a0 X1a1 X2a3 . . . Xnan ð13Þ where the a0 ; . . .; an are the parameters for the nonlinear relation. Some MNLR problems can be moved to a linear domain by a suitable transformation of the model formulation. Taking the log of Eq. 13, the relationship becomes linear: logðYÞ ¼ logða0 Þ þ a1 logðX1 Þ þ a2 logðX2 Þ þ þ an logðXn Þ
ð14Þ
and so a regression of logðYÞ on logðX1 Þ; logðX2 Þ; . . .; logðXn Þ is used to estimate the parameters a0 ; a1 ; . . .; an [4].
3 Study area and data sets The study area is the Khosrow Shirin watershed, located in the Fars Province of Iran (between 51490 2300 and 52120 1600 E and 30370 0200 and 30590 3400 N). The watershed has a drainage area of 610 km2 and is equipped with
4 Model development Precipitation data are used to develop the MLP, ANFIS, MLR and MNLR models. The first input data set includes the original precipitation data for training. The second input vector utilizes area-weighted precipitation and related hydrographs in the MLP, ANFIS, MLR and MNLR developments. In addition to precipitation, antecedent flows with one- and two-day lag times are also tested as input data set. The performance of each of these input vectors is evaluated using the root-mean-square error (RMSE) and coefficient of determination (R2). Table 1 summarizes the combination of input data of precipitation and discharge used in simulations. In the following, P(t) and Q(t) represent the daily precipitation and discharge, respectively. As shown, six input vectors are employed for MLP, ANFIS, MLR and MNLR, leading to six models labeled as Case 1 through 6, where the first three models are based on the spatially distributed precipitation data set and the rest are based on the area-weighted precipitation data. We first wanted to explore the effect of differences in input data sets including spatially varied precipitation and area-weighted precipitation on model outputs. Secondly, we wanted to see if the consideration of 1- and 2-day antecedent flows would improve the model results. For MLPs, the network structure is optimized using the Levenberg–Marquardt algorithm and tangent sigmoid activation function. Before applying the ANN and ANFIS models, the data were normalized to [0.05, 0.95] using a linear transformation [29, 30]: Xn ¼ 0:05 þ 0:9
Xr Xmin Xmax Xmin
ð15Þ
where Xn and Xr are the normalized input and the original input and Xmin and Xmax are the minimum and maximum of input data, respectively. In order to determine the optimum input combination to the network, various epochs and neuron numbers were examined. Through this process, extracted training and testing records with various proportions are used. The architecture that produced the smallest error was used for the development of networks employed to perform daily
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Neural Comput & Applic
Fig. 3 Map of study site location
Table 1 Combinations of MLPs to forecast Q(t ? 1)
Station precipitation
Area-weighted precipitation
Type of precipitation Model no.
Input combinations
Case 1
MLP1
P(t)Khosro
Shirin,
P(t)DejKord, P(t)Dehkade
Sefid,
P(t)Sedeh
Case 2
MLP2
P(t)Khosro
Shirin,
P(t)DejKord, P(t)Dehkade
Sefid,
P(t)Sedeh, Q(t)
Case 3
MLP3
P(t)Khosro
Shirin,
P(t)DejKord, P(t)Dehkade
Sefid,
P(t)Sedeh, Q(t), Q(t - 1)
Case 4
MLP1
W–P(t)
Case 5
MLP2
W–P(t), Q(t)
Case 6
MLP3
W–P(t), Q(t), Q(t - 1)
P = precipitation, Q = river flow, t = time (daily scale), W = weighted
flow forecasting. For Cases 1 through 3 (spatially distributed precipitation as input), the neuron numbers in input layer were 4, 5 and 6, respectively (for MLP1, MLP2 and MLP3). The numbers of neurons for area-weighted precipitation models were 1, 2 and 3, respectively. Furthermore, the optimum numbers of neurons in hidden layer for all models were 13. Finally, one neuron in the output layer was selected to train and test the models. The target error for the training of networks was set to 10-4, with 50 iterations for all of 6 models. The training of the networks was stopped when their performances reached the target error. The best proportion of data was 70 % (165 daily training patterns) for training and 30 % (70 daily values) for testing phases. Here, the tansig and purelin transfer functions were
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Table 2 The correlation of stations between themselves Variables
PKhosro
PDejKord
Shirin
PDehkade
PSedeh
Means (mm)
SD (mm)
Sefid
1.00
0.83
0.82
0.91
10.24
12.68
PDejKord
0.83
1.00
0.77
0.84
14.22
17.74
PDehkade
0.82
0.77
1.00
0.89
11.85
16.64
0.91
0.84
0.89
1.00
12.03
16.34
PKhosro Shirin
Sefid
PSedeh
used in the hidden and output layers, respectively. The normalized data were employed to train each of the 6 MLP models, all of which being three-layered networks.
Neural Comput & Applic
5 Results and discussion First, the hypothesis of gauges similarity was examined through testing the correlations (p \ 0.05) of daily precipitations from gauges with each other as well as the area-averaged rainfall data. The correlation matrix of the stations is presented in Table 2. Figures 4, 5, 6 and 7
display the correlations between each station and the areaweighted average precipitation. The correlation coefficients (r values) were rather high. The highest value was 0.91 between Khosrow Shirin and Sedeh stations, and the lowest was 0.77 between Dozd Kord and Dehkade Sefid stations. That indicates the rainfall stations are roughly similar.
Fig. 4 The correlation between Khosrow Shirin and areaweighted precipitation values
Fig. 5 The correlation between Dozd Kord and area-weighted precipitation values
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Neural Comput & Applic Fig. 6 The correlation between Dehkade Sefid and areaweighted precipitation values
Fig. 7 The correlation between Sedeh and area-weighted precipitation values
5.1 Results of the artificial intelligence approaches The performances of these MLPs in terms of the RMSE and R2 for training and testing phases are shown in Table 3. The RMSE values for MLP3 (Case 3 and Case 6) are lower than those of the MLP1 and MLP2. The model outputs were transformed back to the original range, and the RMSE and R2 were computed for the training and test data sets.
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While the MLP1 values are almost similar for spatially varied precipitation and area-weighted methods, the RMSE values of the area-weighted methods are significantly lower than the RMSE value of spatially varied precipitation for MLP2 and MLP3. This is mainly due to additional input data used for simulations. Furthermore, in Cases 4–6, in which the area-weighted precipitation was used as input, higher prediction efficiency has been observed as opposed
Neural Comput & Applic
to Cases 1–3 where individual stations were used. This indicates that including all gauges does not necessarily improve the flow estimates. It is acknowledged that this finding cannot be generalized and should be tested for each case study. The results of Table 3 indicate that often a model with simpler structure in terms of input may lead to more representative predictions. In general, higher variability in input data sets may reduce the performance of the model. On the other hand, if the area-averaged precipitation is not representative due to significant spatial variability, the model may not be well trained with an average or input data. The results indicate that for each case study, various combinations of input data should be tested in order to achieve the best model performance. Table 3 indicates that including antecedent discharge along with the precipitation data leads to improved streamflow forecasting. Case 6, which includes prior knowledge of discharge and area-averaged precipitation, outperforms the other combinations. The results indicate that while including the two antecedent values of flow is effective in improving the flow forecasting, even 1-day lag of antecedent flow can be significant. This result confirms the finding of the previous studies (e.g., [16, 18]). Figure 8 displays a comparison between observed and simulated values of maximum daily flow using MLP2 and Table 3 Performance of MLPs for training and testing phases Coefficient of determination (R2)
RMSE (m3 s-1)
Training
Training
Validation
MLP3 with area-weighted precipitation as input (Cases 5 and 6). These two ANN models produced the highest R2 and RMSE values among the tested combinations. As shown in the figure, the addition of two antecedent flows slightly reduced the scatter between measured and predicted streamflows. As mentioned earlier, this study presents simulation using MLP and ANFIS. For two combinations of subtractive clustering and FCM clustering as well as two types of input variables (all stations and area-averaged data), the performances of ANFIS models are presented in Table 4. As shown, the values of RMSEs of ANFIS are typically lower than those of MLPs, indicating that ANFIS can predict MDF more accurately than MLPs. This confirms the results of Aqil et al. [1] where they found the neurofuzzy model to be superior to ANNs in predicting the daily and hourly behavior of runoff. The presented case study indicates that ANFIS model is superior to ANNs in runoff forecasting. It is worth pointing out that when applying ANFIS, having station data sets as input lead to the best results, while in ANNs, area-weighted precipitation exhibits the best performance. Investigation of the results derived based on the subtractive and FCM clustering shows that ANFIS models with subtractive clustering (genfis2) are superior to FCM clustering (genfis3). Similar to the case of MLPs, the results indicate that including the two antecedent values of flow can improve the predictions. 5.2 Results of regression models
Validation
Station precipitation Case 1
MLP1
0.89
0.55
4.5
13.8
Case 2
MLP2
0.99
0.56
0.9
11.3
Case 3
MLP3
0.99
0.91
1.6
5.1
Area-weighted precipitation Case 4
MLP1
0.70
0.55
7.4
11.5
Case 5
MLP2
0.99
0.91
1.6
5.1
Case 6
MLP3
0.99
0.96
1.7
3.3
In order to compare ANN and ANFIS results with an empirical model, we also developed MLR and MNLR models for MDF forecasting. The MDF value was selected as the dependent variable, and the input variables of ANN and ANFIS were selected as independent variables. Again, six cases (three with station and three with area-weighted precipitation) were constructed. Similar to the ANN and ANFIS models, R2 and RMSE values were used to evaluate the MLR and MNLR models.
Fig. 8 Scatter plots of MLP2 and MPL3 for the testing phase
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Neural Comput & Applic
The results of the MLR and MNLR models are presented in Tables 5 and 6, respectively. As shown, the performances of the MLR models are almost similar for spatially distributed precipitation and area-weighted precipitation vectors (Table 5). In both classes, MLR3 model is the best model for flow forecasting because of maximum R2 (0.53 and 0.52 for spatially distributed precipitation and area-weighted precipitation vectors, respectively) and minimum RMSE values (7.88 and 7.97 m3 s-1 for spatially distributed precipitation and area-weighted precipitation vectors, respectively). Similar patterns were observed with the MNLR models (Table 6). Among the MNLR models, the MNLR3 model had the best performance for both spatially distributed precipitation (R2 = 0.78 and Table 4 Performance of fuzzy model for training and testing phases Models
Coefficient of determination (R2)
RMSE (m3 s-1)
Training
Training
Validation
Validation
Station precipitation Genfis2 (subtractive clustering) M1
0.69
0.68
0.77
1.49
M2
0.89
0.78
0.65
0.91
M3 0.96 0.91 Genfis3 (FCM clustering)
0.43
0.57
M1
0.54
0.46
0.93
1.72
M2
0.90
0.72
0.73
0.91
M3
0.89
0.72
0.72
0.93
Area-weighted precipitation Genfis2 (subtractive clustering) M1
0.35
0.29
1.00
1.91
M2
0.90
0.73
0.72
0.91
M3
0.93
0.80
0.63
0.76
Genfis3 (FCM clustering) M1
0.30
0.29
1.00
1.94
M2
0.90
0.67
0.78
0.94
M3
0.89
0.68
0.77
0.96
RMSE = 0.166 m3 s-1) and area-weighted precipitation (R2 = 0.81 and RMSE = 0.145 m3 s-1) vectors. The performances of the MNLR models for the area-weighted precipitation cases were slightly better than those for the spatially distributed precipitation cases. Comparison of the MDF predicted by the MLR3 model and measured values is showed in Fig. 9, and the same comparison for MNLR3 model is depicted in Fig. 10. In all cases, this study showed that the nonlinear regression analysis resulted in more accurate results than the linear analysis, when R2 and RMSE are considered. As can be seen from Tables 5 and 6, the coefficient of determination (R2) increased from about 0.52 for the MLR3 model to 0.82 for the MNLR3 model. Furthermore, the RMSE values of the MLR models are much higher than those of the MNLR models. In fact, the MNLR models produced RMSE values less than 1.0 m3 s-1, indicating the superiority of these models even over ANN and ANFIS models. In both linear and nonlinear regression analyses, the inclusion of antecedent flows resulted in much better flow estimates. These results are in good agreement with the results of the ANN and ANFIS models. Overall, we suggest the use of the MNLR equations (Table 6) for flow forecasting due to their simplicity and good performances. Results of this study cannot be generalized without testing the findings over different watershed sizes. Intuitively, one area-averaged vector of precipitation may not be representative for a very large watershed with high spatial variability in precipitation. In such case, the area-averaged data as input may not necessarily lead to a more accurate MLP and MNLR models in comparison with spatially distributed precipitation as inputs. This study was limited to one watershed, and further in-depth studies are required to investigate these findings. In future studies, the authors will evaluate the uncertainty of input data in other artificial intelligence (AI) models such as fuzzy and SVM over different spatial scales. It is hoped that the current and future studies in this direction will provide a
Table 5 Performance of MLR models for MDF forecasting Model
R2
RMSE (m3 s-1)
Equation
Station precipitation Case 1
MLR1
0.098
10.86
Case 2
MLR2
0.504
8.05
Q = 1.13 ? 0.227P1 ? 0.105P2 ? 0.0783P3 - 0.185P4 ? 0.789Q (t - 1)
Q = 8.87 ? 0.525P1 ? 0.0705P2 ? 0.117P3 - 0.441P4
Case 3
MLR3
0.525
7.88
Q = -0.37 ? 0.249P1 ? 0.0976P2 ? 0.0793P3 - 0.198P4 ? 0.623Q(t - 1) ? 0.382Q (t - 2)
Area-weighted precipitation Case 1
MLR1
0.050
11.15
Case 2
MLR2
0.495
8.13
Q = 1.19 ? 0.207Pw ? 0.805Q(t - 1)
Case 3
MLR3
0.515
7.97
Q = -0.32 ? 0.206Pw ? 0.644Q(t - 1) ? 0.375 Q(t - 2)
P1 = PKhosro
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Shirin,
P2 = PDejKord, P3 = PDehkade
Q = 9.20 ? 0.190Pw
Sefid,
P4 = PSedeh
Neural Comput & Applic Table 6 Performance of MNLR models for MDF forecasting Model
R2
RMSE (m3 s-1)
Equation
Station precipitation Case 1
MNLR1
0.101
0.334
Q = 0.810P10.412P2-0.124P30.179P4-0.295
Case 2
MNLR2
0.779
0.166
Q = -0.118P10.0808P20.122P30.0700P4-0.0567(Q(t - 1))0.962
Case 3
MNLR3
0.779
0.166
Q = -0.118P10.0807P20.122P30.0699P4-0.0566(Q(t - 1))0.964(Q(t - 2))-0.002
Area-weighted precipitation Case 1
MNLR1
0.053
0.323
Q = 0.151P0.795 w
Case 2
MNLR2
0.807
0.146
Q = 0.0212P0.134 (Q(t - 1))0.915 w
Case 3
MNLR3
0.811
0.145
Q = 0.0003P0.137 (Q(t - 1))0.786(Q(t - 2))0.159 w
Shirin,
P2 = PDejKord, P3 = PDehkade
Sefid,
P4 = PSedeh
MLR3-Area-weighted Precipitation
MLR3-Station Precipitation 100 80 60 40 20 0 0
20
40
60
80
100
Predicted Flow, m 3/s
Predicted Flow, m3/s
P1 = PKhosro
100 80 60 40 20 0 0
20
40
60
80
100
Measured Flow, m 3/s
3
Measured Flow, m /s
MNLR3-Station Precipitation 100 80 60 40 20 0
0
10
20
30
40
50
60
Predicted Flow, m3/s
Predicted Flow, m3/s
Fig. 9 Comparison of the MDF predicted by the MLR3 model and measured values
MLNR3-Area-weighted precipitation 100 80 60 40 20 0
0
3
10
20
30
40
50
60
3
Measured Flow, m /s
Measured Flow, m /s
Fig. 10 Comparison of the MDF predicted by the MNLR3 model and measured values
framework for the choice of precipitation input into different AI and regression models.
6 Conclusions In this study, the capabilities of MLPs, ANFIS, MLR and MNLR models for predicting the maximum daily flow were evaluated. Different types of input vectors were considered as model inputs. The results suggest that the choice of input vector to the models has a significant impact on forecasting accuracy. The results showed that the area-weighted precipitation is superior when applied as input to ANNs and MNLR, whereas spatially varied
precipitation input to the ANFIS and MLR models exhibits more accurate forecasts. Furthermore, the results indicate that including antecedent flow with one- and two-day time lags as input vectors can improve the ANNs, ANFIS and regression models outputs. In the presented case study, ANFIS seems to predict maximum daily flow more accurately compared to MLPs. The results further highlight that subtractive clustering (genfis2) is superior to FCM clustering (genfis3). Overall, the MNLR models with the RMSE values less than 1.0 m3 s-1 have a superior performance over the ANN, ANFIS and MLR models. With reference to our findings, we propose the nonlinear regression as a simple way for predicting the maximum daily flow. If there are missing data for some days due to
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Neural Comput & Applic
operational problems of the measuring instrument, the models developed here can be used to fill the maximum daily flow gaps depending on the independent variables. Acknowledgments The data used to carry out this research were provided by the Islamic Republic of Iran Meteorological Office (IRIMO) and Surface Water Office of Fars Regional Water Affair. The first author would like to especially thank Mr. Behzad Shifteh Somee, Prof. Alfred Stein, Dr. Amir AghaKouchak and Prof. Dawei Han for their gracious helps. The first author was partially funded by the Center for Forest Sustainability through the Peaks of Excellence Program at Auburn University.
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