Real-time recurrent learning neural network for
stream-flow forecasting
Fi-John Chang,* Li-Chiu Chang and Hau-Lung Huang
Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, ROC
Abstract:
Various types of neural networks have been proposed in previous papers for applications in hydrological events. However, most of these applied neural networks are classified as static neural networks, which are based on batch processes that update action only after the whole training data set has been presented. The time variate characteristics in hydrological processes have not been modelled well. In this paper, we present an alternative approach using an artificial neural network, termed real-time recurrent learning (RTRL) for stream-flow forecasting. To define the properties of the RTRL algorithm, we first compare the predictive ability of RTRL with least-square estimated autoregressive integrated moving average models on several synthetic time-series. Our results demonstrate that the RTRL network has a learning capacity with high efficiency and is an adequate model for time-series prediction. We also investigated the RTRL network by using the rainfall–runoff data of the Da-Chia River in Taiwan. The results show that RTRL can be applied with high accuracy to the study of real-time stream-flow forecasting networks. Copyright 2002 John Wiley & Sons, Ltd.
KEY WORDS recurrent neural networks; stream-flow forecasting; rainfall–runoff modelling
INTRODUCTION
Building a real-time stream-flow forecasting model has always been one of the most challenging and important tasks for hydrologists in Taiwan, which has the attributes of subtropical climate and high mountains with steep slopes all over the island. In addition, typhoons hit Taiwan around four times a year, bringing heavy rainfalls, which can flood downstream cities within a few hours. Owing to these particular watershed–rainfall characteristics, accurate site-specific predictions of the real-time stream-flow remain a difficult task. Various types of deterministic and stochastic models, as described by Sherman (1932), the Hydrological Engineering Center (1990) and Salas et al. (1985) have been used to construct the rainfall–runoff processes in Taiwan. However, some unrealistic assumptions and the lack of verified data/parameters in these models have limited their application and practicality.
Capable of modelling non-linear and complex systems, artificial neural networks (ANNs) provide an alternative approach for accurate stream-flow forecasting. Generally speaking, neural networks are information processing systems devised via imitating brain activity. After McCulloch and Pitts (1943) established the first neural network, many ANNs, such as the back-propagation neural network (Rumelhart et al., 1986), the Hopfield neural network (Hopfield, 1984) and the fuzzy neural network (Nie and Linkens, 1994). were developed to solve different problems. More recently, they have been used to deal with stream-flow prediction (Hsu et al., 1995; Shamseldin, 1997; Chang and Hwang, 1999; Sajikumar and Thandareswara 1999; Chang et al., 2001; Chang and Chen, 2001). Although satisfactory results have been reported, the applied neural networks for hydrological processes, to the best of our knowledge, are all based on batch processes, by * Correspondence to: Professor F.-J. Chang, Department of Bioenvironmental Systems Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan, ROC. E-mail: [email protected]
which the update action takes place only after the whole training data set has been presented. These neural networks are classified as static neural networks, which can only simulate the short-term memory structures within processes. The extraordinary time variate characteristics of hydrological time series, especially in our case, could not be modelled well. Therefore, we present an alternative approach of the ANN for streamflow forecasting—the real-time recurrent learning (RTRL) algorithm (Williams and Zipser, 1989), which provides a representation of dynamic internal feedback loops to store information for later use and to enhance the efficiency of learning. The application of feedback enables recurrent networks to acquire state representation, which make them suitable devices for diverse applications, such as non-linear prediction, speech processing and plant control (Haykin, 1999).
We began with a series of simulation experiments to investigate the properties of the RTRL algorithm, and then implemented it to model a watershed rainfall–runoff process for real-time stream-flow forecasting.
REAL-TIME RECURRENT LEARNING (RTRL) ALGORITHM
Figure 1 shows the architecture of a multiplayer perceptron neural network, which includes a concatenated input– output layer, a processing layer and an output layer. There are M external inputs and K outputs. Let
x t denote the M ð 1 input vector to the network at discrete time t, z t C 1 denote the corresponding K ð 1
output vector and yt C 1 denote the corresponding N ð 1 vector one step later at time t C 1 in the processing layer. The input x t and one-step delayed output vector in the processing layer yt are concatenated to form the (M C N ð 1 vectormt, in which the ith element is denoted by mit. Let A denote the set of indices i for which xitis an external input, and B denote the set of indices i for which yitis the output of a unit in the network. We thus have
mit D
xit if i 2 A
yit if i 2 B
The network is fully interconnected. There are M ð N forward connections and N ð N feedback connections. Let W denote N ð M C N, the recurrent weight matrix of the network. To allow each unit a bias weight, we simply include, among the M input lines, one input with a value that is always 1. W $ wji and V $vkj are the matrix form.
The processing and output layers are also fully connected. Let V denote the N ð K weight matrix. The net activity of neuron j at time t, for j 2 B, is computed by
netjt D i2A[B wjit 1mit 1 1 K 1 N 1 N 1 M Input x(t) .... .... .... .... y(t) time delay Output z(t+1) Output Layer Processing Layer Concatenated input-output Layer WNx(M+N) VNxK
The output of neuron j is given by passing netjtthrough the non-linearity f(.), yielding
yjt D fnetjt
The net output of neuron k at time t is computed by netkt D
vkjtyjt
zkt D fnetkt
Notice that the external input at time t does not influence the output of any neuron until time t C 1. The above system of equations constitutes the entire dynamics of the network.
Let dkt denote the target value of neuron k at time t. Then we define a time-varying K ð 1 error vector
et, whose kth element is
ekt D dkt zkt
Define the instantaneous overall network error at time t as
Et D 1 2 K kD1 ek2t
The cost function is obtained by summing Et over all time T
EtotalD T
tD1 Et
To minimize the cost function, the gradient descent method is applied to adjust the weights (V and W) along the negative of rEtotal. Because the total error is the sum of the errors at the individual time steps, one way to compute this gradient is by accumulating the value of rE for each time step along the trajectory. The weight change for any particular weightvkj can thus be written as
VkjD t1 tDt0C1 vkjt vkjt D 1 ∂Et ∂vkjt where 1 is the learning-rate parameter. Now
∂Et ∂vkjt
D ektf0netktyjt
The same method is also implemented for weight wmn, where wmnt 1 D 2
∂Et ∂wmnt 1
and 2 is the learning-rate parameter. The partial derivative ∂Et
∂wmnt 1 can be obtained by the chain rule for differentiation as follows
∂Et ∂wmnt 1 D K kD1 ektf0netktvkjt ∂yjt ∂wmnt 1 ) ∂yjt ∂wmnt 1 Df 0 netjt ∂netjt ∂wmnt 1 ) ∂netjt ∂wmnt 1 D i2A[B ∂wjit 1mit 1 ∂wmnt 1 ) ∂netjt ∂wmnt 1 D i2A[B wjit 1 ∂mit 1 ∂wmnt 1C ∂wjit 1 ∂wmnt 1mit 1
The derivative ∂wjit 1 ∂wmnt 1
is equal to 1 only when j D m and i D n; otherwise, it is zero. We may therefore rewrite the above equations as
∂netjt ∂wmnt 1 D i2A[B wjit 1 ∂mit 1 ∂wmnt 1 Cυmjmnt 1
where υmj is the Kronecker delta with value 1 if and only if j D m; otherwise zero. From the definition ofmit, we also note that
∂mit 1 ∂wmnt 1 D 0 if i 2 A ∂yit 1 ∂wmnt 1 if i 2 B
We assume that the initial state of the network at time t D 0 has no functional dependence on the synaptic weights, so that ∂yj 0 ∂wmn0 D0 ∂yjt ∂wmnt 1 Df0netjt i2B wjit 1 ∂yit 1 ∂wmnt 1 Cυmjmnt 1 Let ∂yjt ∂wmnt ³ ∂yjt ∂wmnt 1
and define a dynamic system described by a triple indexed set of variables fj mng, where j
mnt D ∂yjt ∂wmnt
for all j 2 B, m 2 B and n 2 A [ B.
For each time step t and all appropriate m, n and j, the dynamics of the system are governed by jmnt D f0netj i2B wjit 1imnt 1 C υmjunt 1
with initial condition j
mn 0 D 0. Then the weight changes can be computed as wmnt 1 D 2 ektf0netktvkjt jmnt vkjt D 1ektf0netktyjt
SIMULATING SYNTHETIC TIME SERIES
Statistical approach to the hydrological time-series forecasting involves constructing stochastic models to predict the value based on previous observations. This often is accomplished by using linear stochastic differential equation models with random inputs. By far the most profound models are the autoregressive integrated moving average (ARIMA) models. Detailed discussions and applications of these models may be found in Box and Jenkins (1976) and Salas et al. (1985). To show the robustness of RTRL, we compare the predictive ability of the least-squares estimated ARIMA predictors and RTRL on five different time-series synthetic data. The five time-series models are shown as:
1. AR(1) xt 0Ð4 D 0Ð75xt 1 0Ð4 C et, with eti.i.d³N0, 1Ð0;
2. AR(2) xt 0Ð4 D 0Ð6xt 1 0Ð4 C 0Ð2xt 2 0Ð4 C et, with eti.i.d³N0, 1Ð0; 3. IMA(1,0) xt D xt 1 C et, with eti.i.d ³N0, 1Ð0;
4. IMA(1,1) 1 Bxt D 1 0Ð6Bet, with eti.i.dN ³ 0, 1Ð0;
5. ARIMA(1,1,1) 1 Bxt D 0Ð41 Bxt 1 C et0Ð3et1, with eti.i.d ³N0, 1Ð0
The et values are chosen from an independent and identically normal distribution (i.i.d) with mean zero and variance one. Each series is generated by MATLAB the Maths Works, Inc. After the first 50 generated data are skipped, the initial 100 data are used to construct the RTRL network and to estimate the optimal parameters of the ARIMA models. For simplicity, without model identification, we use the given types of the ARIMA models, and only their parameters are estimated. The least-squares estimator in MATLAB is used to estimate the ARIMA models’ parameters. The architecture of the RTRL network for one-step ahead prediction is illustrated in Figure 1. There is only one single input, the current value x t, and the processing layer has been investigated through the training data set and identified to have five to seven nodes for different times series. Both learning rates (1 and 2) are 100Ð0. Because we only perform one-step ahead prediction,
O
x t C 1, the output node is, then, set as 1.
The one-step ahead prediction is performed based on (i) the constructed RTRL networks and (ii) the fitted ARIMA models for each case, respectively. As the prediction focuses on optimum prediction by the minimum mean squared error, the criteria of mean square error (MSE) and mean absolute error (MAE) are used for the purpose of comparison.
To learn the models’ performance at different stages, the models’ predictions in (i) training sets (100 data) and (ii) testing sets (200 data) at three different periods are presented, respectively. The results are summarized in Tables I– V. To illustrate the models’ performances in general, the series of ARIMA(1,1,1) and its corresponding prediction errors by (i) the fitted ARIMA model and (ii) the constructed RTRL networks are shown in Figure 2. Apparently, the results show that the RTRL networks have very similar performances as those of the fitted ARIMA models and the networks could persist with the main statistical properties, in terms of reasonable MAE and MSE values, of the generated ARIMA series. The results of RTRL in the
Table I. The performance of fitted AR(1) and RTRL at different stages
Set Time-step AR(1) RTRL
MAE MSE MAE MSE
Training 1– 30 1Ð016 1Ð569 1Ð640 4Ð906 1– 100 0Ð836 1Ð107 1Ð053 2Ð059 Testing 101–130 0Ð676 0Ð829 0Ð821 1Ð152 151–180 0Ð741 0Ð848 0Ð852 1Ð197 271–300 0Ð614 0Ð657 0Ð667 0Ð789 101–300 0Ð722 0Ð876 0Ð809 1Ð114
Table II. The performance of fitted AR(2) and RTRL at different stages
Set Time-step AR(2) RTRL
MAE MSE MAE MSE
Training 1– 30 0Ð879 1Ð155 1Ð115 2Ð994 1– 100 0Ð853 1Ð087 1Ð015 1Ð626 Testing 101–130 0Ð698 0Ð733 0Ð736 0Ð777 151–180 0Ð709 0Ð807 0Ð787 0Ð955 271–300 0Ð620 0Ð700 0Ð782 0Ð809 101–300 0Ð745 0Ð880 0Ð846 1Ð114
Table III. The performance of fitted IMA(1,0) and RTRL at different stages
Set Time-step IMA(1,0) RTRL
MAE MSE MAE MSE
Training 1–30 0Ð721 0Ð737 1Ð112 3Ð379 1–100 0Ð654 0Ð672 0Ð998 2Ð770 Testing 101–130 0Ð891 1Ð1819 0Ð976 1Ð594 151–180 1Ð023 1Ð6276 1Ð033 1Ð588 271–300 0Ð825 1Ð0487 0Ð809 1Ð011 101–300 0Ð891 1Ð2053 0Ð923 1Ð271
Table IV. The performance of fitted IMA(1,1) and RTRL at different stages
Set Time-step IMA(1,1) RTRL
MAE MSE MAE MSE
Training 1–30 0Ð878 1Ð097 1Ð627 11Ð735 1–100 0Ð828 1Ð060 0Ð988 1Ð596 Testing 101–130 0Ð964 1Ð452 0Ð929 1Ð455 151–180 0Ð951 1Ð278 0Ð931 1Ð280 271–300 0Ð777 0Ð837 0Ð788 0Ð830 101–300 0Ð821 1Ð096 0Ð853 1Ð170
Table V. The performance of fitted ARIMA(1,1,1) and RTRL at different stages
Set Time-step ARIMA(1,1,1) RTRL
MAE MSE MAE MSE
Training 1–30 0Ð604 0Ð596 1Ð686 12Ð828 1–100 0Ð790 0Ð966 1Ð243 3Ð257 Testing 101–130 0Ð772 0Ð880 0Ð952 1Ð545 151–180 0Ð764 0Ð898 0Ð742 0Ð871 271–300 0Ð797 0Ð909 0Ð774 0Ð916 101–300 0Ð781 0Ð954 0Ð808 1Ð059
0 50 100 150 200 250 300 350 400 450 500 −30 −20 −10 0 10 20 0 50 100 150 200 250 300 350 400 450 500 −10 −5 0 5 10 0 50 100 150 200 250 300 350 400 450 500 −10 −5 0 5 10 ARIMA RTRL Series
Figure 2. The series of ARIMA(1,1,1) and its corresponding errors using a fitted ARIMA model and a constructed RTRL network training set indicate that after about 30 steps on-line input training, RTRL can grasp the major trends in all cases. All of them indicate that RTRL has an efficient ability to learn and a high accuracy.
APPLICATION
The above method is applied to the upstream of the Da-Chia River for predicting real-time stream flow. The Da-Chia River is located in central Taiwan and has a total catchment size of 1236 km2. Being the steepest channel in Taiwan, the Da-Chia River has a length of 140 km and an average channel slope of 1/39. A series of hydraulic structures were constructed for power generation. Locations of the basin studied (area D 514 km2) and gauge stations used are shown in Figure 3. A station for the stream-flow data is denoted by a triangle and the precipitation stations are shown as circles. Son-Mou gauge station was established to measure the inflow to the De-Chi Reservoir, the upmost and pivotal reservoir in the Da-Chia River. Accurate stream-flow forecasting is extremely important for the operation of the De-Chi Reservoir. The stream flow (m3/s) and precipitation (mm/h) data used here are gathered by the Taiwan Power Company.
There are four rainfall gauges above the Son-Mou flow gauge. To construct a one-step ahead stream-flow forecasting, we investigate the RTRL and ARMAX (Autoregressive moving average with exogenous inputs) models’ performance based on current rainfall at four gauges and stream-flow data. The performances of these two methods are presented based on the criteria of annual peak-flow estimation, mean absolute error (MAE) and relative mean absolute error (RMAE) as shown below
MAE D n iD1 OQiQi n
R1 S Son-Mou R2 R3 R4 De-Chi Reservoir
Figure 3. Location of gauge stations in the upstream part of the Da-Chia River
Table VI. Table of model results for different years: Qo is annual mean of observed stream-flow (cms), Qp is observed annual peak flow (cms) and OQp is estimated annual peak flow (cms)
Year Qo Qp ARMAX(1,1,3) RTRL
MAE RMAE QOp MAE RMAE QOp
1992 45Ð87 699 0Ð792 0Ð0173 695 0Ð648 0Ð0141 703 1993 15Ð85 214 0Ð215 0Ð0136 194 0Ð190 0Ð0120 199 1994 34Ð66 1120 0Ð930 0Ð0268 1075 0Ð741 0Ð0214 1190 1995 19Ð01 105 0Ð239 0Ð0126 102 0Ð223 0Ð0118 102 1996 24Ð04 1090 0Ð476 0Ð0198 1020 0Ð415 0Ð0172 1062 RMAE D MAE Q
The hourly data of 1992 are used to construct the ARMAX and the RTRL network. The ARMAX(1,1,3) model is shown by the following
St D a1St 1 C 3 iD1 b1iR1t i C 3 iD1 b2iR2t i C 3 iD1 b3iR3t i C 3 iD1
b4iR4t i C et C c1et 1
where St is stream flow of the Son-Mou stream-flow gauge at time t; R1t, R2t, R3t and R4t are precipitation of four rainfall gauges at time t; et is the model error at time t; and a1, bij, c1 are the model’s parameters to be estimated. The model and its parameters can be obtained by using MATLAB.
800 700 ESTIMATION (CMS) 600 500 400 300 100 200 0 0 100 200 300 OBSERVATION (CMS) 400 500 600 700 800
Figure 4. Comparison of observed and forecast stream flow at the Son-Mou gauge in 1992 250 200 ESTIMATION (CMS) 150 100 50 0 0 50 100 150 OBSERVATION (CMS) 200 250
Figure 5. Comparison of observed and forecast stream flow at Son-Mou gauge in 1993
The processing layer of RTRL is constructed to have five nodes. Both learning rates (1 and 2) are 30Ð0. We found that after 30 steps on-line input training, RTRL can appropriately predict the one-hour ahead stream flow; that is, the network tends to be stable in its forecasting ability. The structures of ARMAX and RTRL are then applied to four different years without any further modifications. Summarized results of both methods are presented in Table VI. Apparently RTRL has a better performance than ARMAX. Figures 4 through to 8 show the observed and forecast streamflow by RTRL in 1992 to 1996, respectively. According to these
results the network provides stable and precise predictions. The mean absolute error (MAE) is smaller than 0Ð8 cms and the RMAE is less than 2Ð5%. The result of one-step ahead prediction by RTRL in the case of typhoon Herb, which hit Taiwan in 1996 and was the largest typhoon in last 20 years, is presented in Figure 9. Although it has a slight time shift problem, the forecast, in general, is quite adequate.
CONCLUSIONS
The RTRL is an algorithm for training a completely recurrent, continually updated network to learn tasks. This feature is especially important for the extraordinary time variate characteristics of hydrological time-series. We
1200 1000 ESTIMATION (CMS) 800 600 400 200 0 0 200 400 600 OBSERVATION (CMS) 800 1000 1200
Figure 6. Comparison of observed and forecast stream flow at Son-Mou gauge in 1994 120 100 ESTIMATION (CMS) 80 60 40 20 0 0 20 40 60 OBSERVATION (CMS) 80 100 120
1200 1000 ESTIMATION (CMS) 800 600 400 200 0 0 200 400 600 OBSERVATION (CMS) 800 1000 1200
Figure 8. Comparison of observed and forecast stream flow in Son-Mou gauge in 1996
0 200 400 600 800 1000 1200 RTRL OBSERVATION 1996/7/31 1996/8/1 1996/8/2 1996/8/3 1996/8/4 1996/7/30 STREAMFLOW (CMS)
Figure 9. The observed and forecast stream flow at Son-Mou gauge during passage of Typhoon Herb in 1996
first use five different ARIMA time-series of simulation experiments to investigate the power and properties of this algorithm. The results indicate that the RTRL networks have similar performance to those of the fitted ARIMA models for one-step ahead prediction, and the networks could persist with the main statistical properties of the generated ARIMA series. This reveals the excellent learning ability and good performance of RTRL. Overall, RTRL is a well-suited model for time series that possess autoregressive moving average components.
The rainfall and runoff data of Da-Chia river in Taiwan are used to demonstrate the practicability and applicability of RTRL for real-time stream-flow forecasting. A comparison of RTRL and ARMAX is also performed. The results show that (i) RTRL has better performance than ARMAX, and (ii) RTRL can be applied successfully to building a real-time stream-flow forecasting network with high accuracy. To deal with the one-step-ahead forecasting x t C 1, both on synthetic time-series and rainfall–runoff series, only current information x t is used as input. This provides evidence that the use of a dynamically driven feedback algorithm has the potential of reducing the memory requirement significantly.
ACKNOWLEDGEMENT
This paper is based on partial work supported by National Science Council, R.O.C. (Grant No. NSC89-2313-B-002-041).
REFERENCES
Box GEP, Jenkins GM. 1976. Time Series Analysis Forecasting and Control , 2nd edn. Holden-Day: San Francisco.
Chang FJ, Chen YC. 2001. A counterpropagation fuzzy-neural network modeling approach to real-time stream-flow prediction. Journal of
Hydrology 245: 153– 164.
Chang FJ, Hwang YY. 1999. A Self-organization algorithm for real-time flood forecast. Hydrological Processes 13(2): 123– 138. Chang FJ, Hu HF, Chen YC. 2001. Counterpropagation fuzzy neural network for stream-flow reconstruction. Hydrological Processes 15(2):
219– 232.
Haykin S. 1999. Neural Networks: a Comprehensive Foundation, 2nd edn. Prentice Hall: Upper Saddle River, NJ.
Hopfield JJ. 1984. Neuron with graded response have collective computational properties like those of two-state neurons. Proceedings of
the National Academy of Sciences, USA 81: 3088– 3092.
Hsu K-L, Gupta HV, Sorooshian S. 1995. Artificial neural network modeling of the rainfall– runoff process. Water Resources Research 31(10): 2517– 2530.
Hydrological Engineering Center. 1990. HEC-1 Flood Hydrograph Package. Program Users Manual . U.S. Army Corps of Engineers: Davis, CA.
McCulloch WS, Pitts W. 1943. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics 5: 115– 133. Nie J, Linkens DA. 1994. Fast self-learning multivariable fuzzy controllers constructed from a modified CPN network. International Journal
of Control 60(3): 369– 393.
Rumelhart DE, Hinton GE, Williams RJ. 1986. Learning internal representation by error propagation. Parallel Distributed Processing 1: 318– 362.
Sajikumar N, Thandaveswara BS. 1999. A non-linear rainfall– runoff model using an artificial neural network. Journal of Hydrology 216: 32– 55.
Salas JD, Delleur JW, Yevjevich V, Lane WL. 1985. Applied Modeling of Hydrologic Time Series. Water Resources Publications: Littleton, Colorado.
Shamseldin AY. 1997. Application of a neural network technique to rainfall– runoff modelling. Journal of Hydrology 199: 272– 294. Sherman LK. 1932. Stream-Flow from rainfall by the unit-graph method. Engineering News Record 108: 501– 505.
