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2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

Daily Runoff Time-series Prediction Based on the Adaptive Neural Fuzzy Inference System

Qiaofeng Tan

Institute of Water Resources and Hydropower Engineering School of Sichuan University Chengdu 610065, P.R.China

Abstract-Artificial neural network and fuzzy inference technology have been successfully used in various fields in the last decades. In order to combine the advantages of these approaches, the previous researchers came up with a new model named adaptive neural fuzzy inference system (ANFIS), which has been applied to signal processing and the related fields. Hydrological prediction is an important aspect of hydrological services for economy and society. The prediction result not only provides decision support for generation optimization, but also is of great significance to the economical operation of hydropower systems, navigation, flood control and so on. This paper presents the application of adaptive neural fuzzy inference system (ANFIS) on daily runoff time-series prediction at Tongzilin station. To evaluate the performances of the selected ANFIS, comparison was made with the ANN and autoregressive (AR) model. Previous inflows were chosen as input vectors of the three different models. Nash-Sutcliffe efficiency coefficient (NS coefficient), root mean square error (RMSE) and mean absolute relative error (MARE) were chosen to evaluate the performances of our models. The results show that ANFIS not only keep the potential of the ANN whose advantage is to deal with nonlinear problem, but it also eases the model building process and makes the result more stable. As a result, ANFIS can be a recommended daily runoff time-series prediction model. Keywords-adaptive neural fuzzy inference system; artificial neural network; autoregressive model; daily runoff prediction

Xu Wang , Siyu Cai, Xiaohui Lei

State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin Institute of Water Resource and Hydraulic Research Beijing 100038, P.R.China computing tools and our ability to obtain hydrological data, data-driven model is gaining more and more attention and application in hydrological prediction. Autoregressive model (AR model) is the most widely used data-driven model in hydrology because such model is simple and easy to obtain expressions. However, the relationship between the previous inflows and the forecasted inflows is complex. AR model cannot describe the non-linear relationship in the hydrological process, and may not always perform well [1]. Artificial neural networks (ANN) [2], as a growing datadriven model is quite suitable to solve non-linear problems due to its powerful parallel processing and fault tolerance [3]. In the last decades, ANN was widely applied in rainfall-runoff modeling [4,5,6], underground water level prediction [7,8,9], reservoir inflow prediction [10,11] and other aspects in hydrology. The more application of ANN in hydrology and water resources field can be found in [12,13]. Adaptive neural fuzzy inference system is a powerful tool to carry out the hydrological forecast which combines the advantages of artificial neural networks and fuzzy systems. The model uses the back-propagation gradient descent and the least squares method to identify a set of parameters. The attractive features of ANFIS include: easy-implementation, fast-learning, high computing precision and strong generalization abilities [14]. During the past decades, ANFIS has been successfully applied to various hydrological problems, such as precipitation forecasting [15,16], streamflow predictions [17,18,19,20]. For a given hydrological time-series, we can choose AR model or ANN or ANFIS model to simulate and predict. It is necessary to consider which model is the best in actual application. Should we choose the simple and convenient AR model or ANN and ANFIS which have strong nonlinear mapping ability? The main purpose of this paper is to study the potential of neuro-fuzzy systems in daily runoff time-series prediction, and compare the performance of ANFIS with ANN and AR model, thus providing a recommended model for daily runoff time-series prediction. The applicability of this method is demonstrated by modeling river flow at Tongzilin station, a part of the Ya-lung River basin.

I.

INTRODUCTION

Hydrological prediction is the base of flood control, drought resistance, water resources utilization and other important decision-makings. It is the fastest growing branch of operational hydrology, which has gain attention from various aspects. Currently the widely used hydrological prediction model can be divided into two sub-models: the data-driven model and the process-driven model. The process-driven model, a kind of mathematical model, based on the concept of hydrology, predicts runoff by simulating runoff processes and river flood routing processes. The data-driven model is a black box method, which is used to establish the optimal mathematical relationship between input and output data without considering the physical mechanism of hydrological processes. Regression model is the most widely used datadriven model. New prediction methods have been developed quickly in recent years, such as neural network model and fuzzy mathematics method, etc. As the rapid development of

Fund project: the National Science & Technology Pillar Program during the 12th Five-year Plan Period (2013 BAB05B00)

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978-1-4673-7682-2/15/$31.00 ?2015 IEEE

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II.

A.

METHODOLOGY

Adaptive Neural Fuzzy Inference System

order polynomial . Thus, the output of the ith node is given by the following formula: (5) There is only one node in the fifth layer. This node calculates the sum of all incoming signals. Hence, the final output of the ANFIS model is given by the following formula:

The adaptive neural fuzzy inference system has been proposed by Jang in the nineties [21]. As a simple example we assume a fuzzy inference system with two inputs x1 and x2 and one output y. The first-order Sugeno fuzzy model, a typical rule set with two fuzzy If-Then rules can be expressed as:

Rule 1: If x1 is A1 and x2 is B1 ， then f1 =

p1 x1 + q1 x2 + r1

o4i = w 'i * f i = w 'i ( pi x1 + qi x2 + ri ); i = 1, 2

y = ∑ i w 'i * f i = ∑ i wi *

Rule 2: If x1 is A2 and x2 is B2 ， then f 2 = p2 x1 + q2 x2 + r2 where p1 , q1 , r1 and p2 , q2 , r2 are the parameters in the consequent part of the first-order Sugeno fuzzy model. The framework of adaptive neural fuzzy inference system with two inputs and one output is presented in figure 1. A brief introduction of this model is as follows: The outputs of layer 1 are the fuzzy membership grade of the inputs, which can be expressed as:

i (6) In this ANFIS structure, the first layer and the fourth layer both are adaptive layers. The three parameters {ai , bi , ci } in

∑w

fi

; i = 1, 2

i

the first layer are called premise parameters. And the three

(1) where x1 and x2 are the inputs of the node, o1i and o2i are the outputs of the node. Ai and Bi are the linguistic labels characterized by appropriate membership function u Ai , uBi , respectively. For instance, if we employ the bell shaped membership function, u Ai ( xi ) is calculated by the following formula:

?o1i = u Ai ( xi ) ; i = 1, 2 ? ?o2i = u Bi ( xi )

modifiable parameters { pi , qi , ri } in the fourth layer are called consequent parameters. The shortcomings of ANFIS is that the number of control rules increases rapidly as the number of fuzzy variables increases. In order to overcome this problem, the ANFIS based on a learning algorithm of subtractive clustering were used in our study. The cluster centers generated by subtractive clustering would be the centers for the fuzzy rules’ premise in the ANFIS, so parameters of the adaptive neural fuzzy inference system can be adjusted intelligently.

u Ai ( xi ) =

(2) where ai , bi and ci are the parameters of the membership function. Each node in the second layer calculates the weight value wi of a rule. The output of ith node is given by the following formula： (3) The nodes in the third layer are fixed nodes. Each ith node calculates the ratio of the ith rule’s weight value to the sum of weight value of all the rules. The output of the ith node is given by the following formula:

1 x ? ci 2 bi 1 + [( i ) ] ai

wi = u Ai ( xi ) * uBi ( xi ); i = 1, 2

AR model and ANN AR model as the most widely used model in hydrological time-series prediction, has become a criterion of comparison for other models. Generally, in order to prove that the proposed models have good performances, they should have better performances than the AR model which is more simple and mature. The AR model is built to predict the future change of a variable based on the analysis of the changing rule of the variable itself in the past. Firstly, we need to determine the order of an AR model. Then generally we use least square method to determine the parameters of our model. Finally we use the model to forecast the change in the future.

B.

w 'i =

(4) The output of each node in the fourth layer is simply the product of the normalized weight value and a first

wi ; i = 1, 2 w1 + w2

ANN simulates several basic characteristics of human brains by using mathematical methods and these techniques have been widely applied in the hydrological field. Among many different types of ANN, the back propagation neural network (BP neural network) has been proven to be a very powerful tool for many engineering problems, because of its simple and easily implemented architecture. Therefore we

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choose the BP neural network to model our hydrological timeseries. III. THE STUDY AREA AND DATA

Tongzilin hydropower station, located in Yanbian county, Panzhihua, Sichuan province, is the last cascade hydropower station in the downstream of the Ya-lung River. It is 18 km away from the Ertan hydropower station in the upstream and 15 km away from the confluence of the Ya-lung River and the Jinsha River respectively. The reservoir has daily regulation performance, whose installed capacity is 600,000kW and average output in the dry season of design low flow year is 227,000kW. Normal water level of Tongzilin reservoir is 1,015.00m. Total storage capacity is 0.912 million m3. The reservoir gives priority to power generation, and also considers comprehensive water requirement in the downstream. This

12000 10000

study chose daily runoff data from the year 1999 to 2012. The daily streamflow hydrograph is shown in figure 2. The runoff distribution variety in a year because of the uneven distribution of precipitation. There is a great difference on runoff between flood and dry seasons, and mainly concentrates on the flood season. In order to establish daily runoff time-series prediction model, we analyzed the autocorrelation and partial autocorrelation of the historical inflow data. Figure 3 and figure 4 showed the changes of autocorrelation and partial autocorrelation with the lag time respectively. It is obvious to see from figure 3 that there is a strong correlation between the inflows in consecutive months, which suggests that prediction of inflows at certain months should be based on the monitored inflows at previous months. Figure 4 shows that the partial correlation coefficient is relatively large from lag 1 to lag 7, while very close to zero from lag 8 until the end.

Runoff value(m3/s)

8000 6000 4000 2000 0

0

500

1000

1500

2000

2500 Time(Day)

3000

3500

4000

4500

5000

Figure 2. Daily runoff hydrograph from year 1999 to 2012 at Tongzilin. 1

0.8 0.7 0.6 0.5 1 2 3 4 5 6 7 8 9 101112 131415 Lag(Day)

Partial autocorrelation

Autocorrelation

0.9

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

1 2 3 4 5 6 7 8 9 101112131415 Lag(Day)

Figure 3. the variety of autocorrelation coefficient with time lag

Figure 4. the variety of partial correlation coefficient with time lag

IV.

A.

MODEL DEVELOPMENT

Input variables We divided the data into two sets, a training data set from the year 1999 to 2006 and a validation data set from the year 2007 to 2012. To research the representation of daily runoff data during the training period, we analyzed the statistical parameters of the two data sets as shown in table I. The maximum daily runoff during the validation period is smaller than that during the training period, which means a variety of hydrological conditions in training period enabled inclusion of various hydrological conditions that are observed during validation period. Several input combinations were tried using three different models to estimate daily flow. The number of

lags was selected according to the autocorrelation function and partial autocorrelation function of daily flow data which were shown in figure 3 and figure 4. It is clear from figure 3 and figure 4 that first seven lags have significant effect on Qt +1 . Thus, seven previous lags were considered as inputs of the models in this study. Qt+1 represents the flow at time t + 1 , and Qt , Qt ?1 , Qt ? 2 , Qt ? 3 , Qt ? 4 , Qt ? 5 and Qt ? 6 represent the flow at time t , t ? 1, t ? 2, t ? 3, t ? 4, t ? 5, t ? 6, respectively. As a result, we developed the following seven models:

M (1) : Qt +1 = f M

( Qt ) ( Qt , Qt ?1 )

( 2) :

Qt +1 = f

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508

M M

M M M

( 3) : ( 4) :

(5) : ( 6) :

Qt +1 = f Qt +1 = f

Qt +1 = f Qt +1 = f Qt +1 = f

( Qt , Qt ?1 , Qt ? 2 ) ( Qt , Qt ?1 , Qt ? 2 , Qt ?3 )

( Qt , Qt ?1 , Qt ? 2 , Qt ?3 , Qt ? 4 ) ( Qt , Qt ?1 , Qt ? 2 , Qt ?3 , Qt ? 4 , Qt ?5 )

(7) where a is an arbitrary constant, Q’ is the transformed value of river flow Q , and b was set to 1 to avoid the entry of zero river flow in the log function. The final forecasting results were then back transformed using the following equation:

Q ' = a log10 (Q + b)

(7) :

( Qt , Qt ?1 , Qt ? 2 , Qt ?3 , Qt ? 4 , Qt ?5 , Qt ? 6 )

Q = 10

(Q '

a

)

?b

(8)

TABLE I. Qmax training test 11700 9260

THE DAILY-RUNOFF STATISTICAL PARAMETERS Qmin 114 272 Qmean 2002 1765 Qstdev 1845 1530 Qske 1.77 1.93

D. Models performance criteria We choose three evaluation indexes to assess the model performance: Nash–Sutcliffe efficiency coefficient (NS coefficient), root mean squared error (RMSE), and mean absolute relative error (MARE). They can be calculated as follows:

Qmax, Qmin, Qmean, Qstdev, Qske denote the maximum, minimum , mean, standard deviation and skewness coefficient of the flow data in the training and test set.

B.

Model structure

NS = 1 ?

∑ (Q

i =1 n i =1

n

o

i o

? Qi p ) ? Qi o )

2

The rule extraction method first uses the subtractive clustering [22] to identify the number of rules and antecedent membership functions and then uses linear least squares estimation to determine each rule's consequent equations. This function returns a fuzzy inference system structure that contains a set of fuzzy rules to cover the feature space. The BP neural network we used includes an input layer, a hidden layer and an output layer. The number of the neurons in the input layer is the number of variables. The number of the neurons in the output layer is 1. And we used trial and error method to determine the number of hidden layer neurons. We used m, s, n to represent the neurons number of input layer, hidden layer and output layer respectively. And (m, s, n) represents the structure of ANN. The ANN structure and the number of fuzzy rules of ANFIS are shown in table II.

TABLE II.

M (1)

M ( 2)

∑ (Q

2

i

(9)

RMSE =

2 1 Qi o ? Qi p ) ( ∑ n i =1

n

MARE =

THE STRUCTURE OF ANN AND ANFIS

M ( 3)

M ( 4)

M (5)

M (6)

M (7)

(m, s, n) fuzzy rules

(1,3,1) 5

(2,5,1 5

(3,8,1) 6

(4,9,1) 6

(5,7,1) 7

(6,7,1) 6

(7,13,1) 5

(11) where n is the number of input samples; Qi0 and Qip are the observed and predicted flow at time i, Qi o is the mean of the observed river flow. The best fitting effect will be obtained between observed and calculated values when NS = 1 , RMSE = 0 and MARE = 0 . NS coefficient ranges from negative infinity to 1, and observed values and simulated values are exactly the same when it is equal to 1. So the closer the NS coefficient is to 1, the better the model’s performance is. RMSE and MARE range from 0 to positive infinity and observed values and simulated values are exactly the same when they are equal to 0. So the closer RMSE and MARE are to 0, the better the model’s performance is. V. RESULTS AND DISCUSSIONS

1 n ( Qi ? Qi ∑ Qo n i =1 i

o

p

)

(10)

C. Data preprocessing

Shank et al. [23] and Luk et al. [24] have drawn a conclusion that networks trained using transformed data could obtain a better performance in general . An appropriate transformation method can be used to transform abnormal distributed data to normal distributed ones. As we can see from table I, the flow data shows a significantly high skewed distribution (1.77 and 1.93 for training and test data set, respectively). In our study, transformation is performed on all time-series data independently using the equation (7).

In order to compare the ANFIS with ANN and the traditional autoregressive model (AR model), we used AR model to model and predict the daily runoff in Tongzilin station with same input vectors. The results of ANFIS, ANN and AR model are listed in table III. To compare the performances of three different models more clearly, we draw the process curve of RMSE, NS and MARE over model order during the training period and test period. The results were summarized in figure 5(a)-5(f). We can get the following conclusions from table III and figure 5(a)-5(f):

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TABLE III. Model

M(1)

THE PERFORMANCE INDICES OF ANFIS, ANN AND AR MODEL IN TRAINING PERIOD AND TEST PERIOD Training Period

ANFIS ANN 424.9061 0.9470 10.5596 430.8051 0.9455 10.6417 420.3236 0.9481 10.3637 397.2593 0.9537 10.4283 379.8013 0.9577 10.4236 408.9107 0.9509 10.9036 397.9204 0.9535 10.9799 AR 421.5896 0.9478 11.0362 406.2949 0.9515 11.3958 405.3356 0.9518 11.3270 405.5693 0.9517 11.2986 405.2827 0.9518 11.2627 404.5429 0.9520 11.2449 403.4108 0.9523 11.1891

340 ANFIS ANFIS ANN 330 AR ANN AR

Performance Indices

RMSE NS MARE 423.5771 0.9473 10.5380 405.8743 0.9516 10.3906 408.9349 0.9509 10.4422 399.0795 0.9533 10.4034 396.8160 0.9538 10.4221 396.5406 0.9539 10.4585 396.9328 0.9538 10.4616

440 430

Test Period

ANFIS 334.3926 0.9501 11.5749 314.5886 0.9562 11.5218 314.9108 0.9552 11.5293 318.3848 0.9543 11.6156 314.9825 0.9569 11.5745 315.7593 0.9556 11.5675 316.9428 0.9553 11.6181 ANN 335.1630 0.9499 11.6097 336.5152 0.9495 11.7703 324.2500 0.9532 11.5649 320.0131 0.9545 11.7412 321.7062 0.9540 12.0471 322.1814 0.9538 11.8853 337.9202 0.9492 11.7230 AR 337.6054 0.9493 11.9637 321.6603 0.9540 12.4570 318.6656 0.9548 12.4269 318.8706 0.9548 12.4178 317.5912 0.9552 12.3692 317.1315 0.9553 12.3497 319.1931 0.9547 12.3730

M(2)

RMSE NS MARE

M(3)

RMSE NS MARE

M(4)

RMSE NS MARE

M(5)

RMSE NS MARE

M(6)

RMSE NS MARE

M(7)

RMSE NS MARE

RMSE

410 400 390

RMSE

320 310 1 2 3 4 model 5 6 7 1

420

2

3

4 model

5

6

7

(a) the variety of RMSE in the training period

(b) the variety of RMSE in the test period

0.956 0.954

0.96 0.958

NS coefficient

NS coefficient

0.952 0.95 ANFIS 0.948 0.946 0.944 ANN AR

0.956 0.954 0.952 0.95 0.948 1 2 3 4 model 5 ANFIS ANN AR 6 7

1

2

3

4 model

5

6

7

(c) the variety of NS coefficient in the training period

(d) the variety of NS coefficient in the test period

5

510

12 ANFIS 11.5 ANN AR

13 ANFIS ANN 12.5 AR

MARE

11

MARE

12 11.5 1 2 3 4 model 5 6 7 1

10.5

10

2

3

4 model

5

6

7

(e) the variety of MARE in the training period

(f) the variety of MARE in the test period

Figure5. The variety of performance indices of ANN, ANFS and AR model in training period and test period

(1) In figure 5(a) and figure 5(b), RMSE of AR model and ANFIS during training period in model M(1) - M(7) has a tendency to decrease, which means AR model and ANFIS can obtain a continuous improvement in performance with the increasing of model order. However, the RMSE of AR model decreases more slowly than that of ANFIS, especially in model M(4)-M(7). RMSE of ANN in model M(1) - M(4) has a tendency to decrease, but there are some fluctuations from M(5) to M(7) . The fluctuation characteristics of ANN show that ANN is unstable. And the RMSE of ANN is always higher than that of ANFIS, which indicates that ANFIS has better application effect compared with ANN. Meanwhile, the RMSE process curve of AR model is consistent in training period and test period, and AR model has more stable performance compared with the ANN and ANFIS. (2) In figure 5(c) and figure 5(d), NS coefficient of AR model and ANFIS in model M(1)-M(7) keep an increasing trend in training period. RMSE of ANN in M(1) - M(4) has a tendency to increase. But there are some fluctuations in the next models. The NS coefficient of ANFIS is higher than that of AR model in training period except for M(3). Moreover, the NS coefficient of ANN is lower than that of ANFIS in most training period except for M(4). In figure 5(d), NS coefficient of ANFIS can even come up to 0.956, whereas the maximum of NS coefficients of AR model and ANN is always below 0.956. As a result, a better result can be obtained from ANFIS compared to ANN and AR model by selecting proper input vectors and net structures. (3) In figure 5(e) and figure 5(f), MARE of AR model is much higher than that of ANN and ANFIS. And there is little difference between ANN and ANFIS. However, MARE of ANFIS is lower than that of ANN in both the training period and test period. In conclusion, AR has larger prediction error than ANN and ANFIS, and the accuracy of ANFIS is higher than that of ANN. (4)From figure 5(a) to figure 5(f), we can found: considering the model works well and the model order is as small as possible, M(3) is an optimal AR model. And ANN and

ANFIS can obtained a good performance in training period and test period in model M(4) and model M(2),respectively. Compare the optimal model of three different models, we can find that AR model can get a good performance by using a small model order and can keep a stable performances in test period. But when it comes to MARE, the performance of AR model cannot compare to ANN and ANFIS. At the same time, after comparing the optimal model of ANN and ANFIS, we can find that the ANFIS shows an improvement in accuracy and stability over the ANN. Due to the uncertainty of hidden layer nodes, ANN is unstable and will spend more time training the network. VI. SUMMARY AND CONCLUSION

In this study, the application effect of adaptive neural fuzzy inference system for modeling hydrological time-series is studied in the flow forecasting in Yalong basin. In order to compare the performance of the ANFIS with ANN and AR model, ANN and AR model are also developed for the same basin. Due to the strong nonlinear processing ability of ANN and ANFIS, ANN and ANFIS can obtain better results compared to AR model when they are used properly. However, regardless of its accuracy, AR model has more stable performance with a lower order model. The comprehensive analysis above suggests that the ANFIS not only keep the potential of the ANN whose advantage is to deal with nonlinear problem, but it also outperforms ANN and AR model in various performance indices. Though ANN can obtain a similar performance as ANFIS sometimes, ANN is not as stable as ANFIS. Unless carefully trained, ANN might show poor performance due to its instability. And the trial and error procedure for choosing a proper ANN structure usually cost considerable computational time. But the model building process of ANFIS is much simpler and the model can get a better precision. In short, ANFIS can be a recommended model for daily runoff time-series prediction.

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(ANN) and *adaptive* *neural* *fuzzy* inference system (ANFIS) have been widely applied in *the* *daily* *runoff* *time* *series* *prediction*.In this paper,these three ...

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