Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi: 10.1016/S2095-3119(13)60610-3

The Prediction Model of Weekly Retail Price of Eggs Based on Chaotic Neural Network LI Zhe-min1,2, CUI Li-guo3, XU Shi-wei1,2* , WENG Ling-yun1,2 , DONG Xiao-xia1,2, LI Gan-qiong 1,2 and YU Hai-peng1,2 1

Agricultural Information Institute, the Chinese Academy of Agricultural Sciences, Beijing 100081, China

2

Key Laboratory of Agri-information Service Technology, Ministry of Agriculture, Beijing 100081, China

3

Bei jing Research Center for Information Technology in Agriculture, Beijing Academy of Agriculture and

Forestry Science, Beijing 100081,China

Abstract Based on the weekly retail prices of eggs from Jan. 2008 to Dec. 2012, this paper establishes a short-term prediction model of weekly retail prices of eggs based on chaotic neural network. In the process of determining the structure of the chaotic neural network, the number of input layer nodes of the network is calculated by reconstructing phase space and computing its saturated embedding dimension, and then the number of hidden layer nodes is estimated by trial and error. Finally, this model is applied to predict the retail prices of eggs and compared with ARIMA. The result shows that the chaotic neural network has better nonlinear fitting ability and higher precision in the prediction of weekly retail price of eggs. The empirical result also shows that the chaotic neural network can be widely used in the field of short-term prediction of agricultural prices.

Keywords: chaos theory, chaotic neural network, neural network technology, short-term prediction, weekly retail price of eggs

INTRODUCTION China is the country with the largest production of eggs in the world. Since 1985, China has maintained the No. 1 spot worldwide in egg production for 28 years, and eggs have always taken an important ratio in the food consumption structure of our residents. The price of eggs fluctuates frequently since 2008, the fluctuation range of price was more than 4% month on month in 17 months of 50 months and more than 2% month on month in 31 months of 50 months by the end of February 2012. The lowest price of eggs was US$ 1.51/kg in February 2012, while the highest was US$ 1.80/kg in September 2011, and the fluctuation between the highest and the lowest prices exceeds 18%. Frequent and abnormal fluctuation of egg prices had a serious impact on residents’ daily life and poultry farmers’ income increase, so it has becoming the focus of public concern. Therefore, to conduct research on short-term forecasting of egg prices, to predict the future trend of *

Corresponding author: XU Shi-wei, Tel: +86-10-82109902, E-mail address: [email protected]

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

egg prices accurately, to find the change rules of egg prices and to develop appropriate policies and measures in order to avoid acute fluctuation of egg prices, there is great significance for stabilizing the whole industry of eggs, guaranteeing the interests of farmers and protecting the normal living of residents. Some research achievements have been made for short-term forecasts of egg prices by international and domestic academics. S. N. Kulshreshtha (1971) constructed a short-term forecasting model and predicted the production of eggs in Canada. Based on the data of egg prices from 1965 to 1969, the model predicted the production of the next 5 months, and the results showed that the forecast error was about 5%. Based on the data of monthly retail egg prices from 1980 to 1990, Oguri et al. (1992) designed a unitary regression model and an improved multiple regression model to predict the egg prices, they drew some important conclusions and discussed the aspects which the model needed to be further improved. H. A. Ahmad et al. (2001) adopted the general neural network method to forecast egg prices. The results showed that the fitting effect and the forecast accuracy of neural network method were better than the traditional prediction methods. They believed that reasonable data collection methods and processing techniques were very important for any neural network. Based on analyzing the variety of egg market prices systematically and the main influence factors, Li and Li (2009, 2010) filtered 4 correlation factors which can access data timely and comprehensively to construct the regression model and forecast short-term egg prices, making a valuable exploration to the short-term price prediction of agricultural product. According to the historical data of the egg prices of Huai’an City in Jiangsu province, Wang (2008) established a grey wave forecasting model to predict the egg prices over the next few months and analyzed the reasons of price fluctuation. According to the existing documents, the forecasting of egg price was studied through the traditional econometrics method which cannot express the nonlinear characteristics of egg prices accurately, so the accuracy of prediction needed to be improved. With the rapid development of computer technology and artificial intelligence technology (Mehdi Khashei and Mehdi Bijari 2010, 2011), international and domestic scholars try to introduce neural network methods into the research of forecasting short-term egg prices. However, this is still in its infancy. The chaotic neural network technology is a new technology which joins the chaos theory in neural network algorithm (Chen 2010; Muhammad and Saeed 2010). The performance of neural network and the speed of convergence have been optimized and improved. The powerful nonlinear mapping ability of chaotic neural network technology overcomes the shortcomings of traditional forecasting techniques. It also improves the accuracy of prediction and practicability significantly. In recent years, chaotic neural network technology has been applied maturely in flow forecasting (Dian et al. 2011; Zhang et al. 2011), electrical load forecasting (Niu 2009; Amjady and Keynia 2011; Sousa et al. 2012), and the prediction of stock and futures price (Yang et al. 2001; Xu et al. 2011; Soloviev et al. 2010). However, the application in the fields of short-term forecasting for 2

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

agricultural products (e.g., eggs) is still in the blank. The chaotic neural network technology is applied in the short-term prediction of egg prices in this paper. The applicability of this model in the short-term prediction field of egg prices is verified by the establishment of the model and empirical analysis. We hope this study will provide support and reference for in-depth studies of egg price trends and fluctuation rules, stabilizing the market prices of eggs, and promoting the healthy development of the entire egg industry.

OVERVIEW OF CHAOTIC NEURAL NETWORK MODEL The chaotic neural network determines its number of input layer nodes by using the reconstruction technique of phase space for the time series which has chaotic property, and then the network topology of this model will be confirmed (Yu 2011; Li 2013). Accordingly the fitting capacity and prediction accuracy of the model are improved greatly in the short-term prediction field. Fig. 1 shows the structure of chaotic neural network model. The main steps of model construction are as follows: To determine whether the time series is the chaotic sequence Input data

Determine the network topology Chaotic neural network construction

Reconstruct phase space

Determine the delay

Saturated embedding

time τ

dimension m

Input layer

Output layer

Trial and error

Prediction of chaotic neural network

Predict the outcome

Hidden layer

Training the chaotic neural network

End practice

Test data

Training

Initialization

Optimization The additional momentum method

Set learning rate

Fig. 1. The structure of chaotic neural network model

Deciding the number of input layer nodes of the chaotic neural network by chaos theory According to the chaos theory, we put the saturated embedding dimension of phase space as the

3

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

optimal combination of system input variables to modeling chaotic time series. This model can effectively reflect the dynamic characteristics of the system, and has good extrapolation ability, i.e., generalization ability (Guo et al. 2000; Dou 2003). Besides, the problem that previous neural network methods in choosing input layer node number only depend on experience and lack theoretical support can be solved well by making the saturated embedding dimension of phase space as the number of the input layer nodes. For the limited length of time sequence, characteristics of phase space depend on the choice of the optimal delay time. So in the process of determining the number of input layer nodes by using the reconstruction technique of phase space, the first step is to decide the optimal delay time of the system, then to calculate the saturated embedding dimension. Both of the parameters have a considerable impact on reconstructing the phase space (Wang and Xu 2006; Li 2011). This paper decides the optimal delay time by using depolarization complex self-correlation method, confirmed the saturated embedding dimension by using classical G-P algorithm (Grassberger and Procaccia 1983; Cui and Li 2013).

Deciding the optimal delay time using depolarization complex self-correlation method There are lots of methods to decide the optimal delay time, e.g., autocorrelation function method, the average displacement method, complex self-correlation method, depolarization complex self-correlation method, C-C method and mutual information method. This paper adopted the depolarization complex self-correlation method to confirm the optimal delay time after comprehensive comparison of these methods. 1

τ R 𝑥𝑥 = 𝑁 ∑𝑁− 𝑖=1 (𝑥𝑖 − 𝑥̅ ) (𝑥𝑖+τ − 𝑥̅ )

(1)

Where N is the number of samples, 𝑥𝑖 is the sample, 𝑥̅ is the mean of sequence, and τ is the lag time, also known as time delay. With the increment of τ, the value of R 𝑥𝑥 becomes smaller. When it goes down to 1 − 1/𝑒 times of initial value, we call the corresponding values of τ as the optimal delay time.

Deciding the saturated embedding dimension by using classical G-P algorithm According to Takens’s embedding theory (Takens 1981) , for a single variable time sequence at regular intervals x1, x2, x3 …, it can brace an embedded vector space by constructing a number of m-dimensional vectors. Generally, the system can recover the original dynamics shape in the topological equivalence sense if embedding dimension is high enough (generally requires m≥ 2D+1, D is the dimension of attractor, namely correlation dimension). Supposing the original time

4

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

series is 𝑥(t 𝑖 ), 𝑖 = 1,2, … 𝑁. We reconstruct the following phase space after determining the system’s optimal delay time τ and the saturated embedding dimension m: Y(t 𝑖 ) = [𝑥(t 𝑖 ), 𝑥(t 𝑖 + τ), 𝑥(t 𝑖 + 2t), … , 𝑥(t 𝑖 + (𝑚 − 1)τ)]

(2)

There are also many methods to calculate the saturated embedding dimension m, this paper chooses the classical G-P algorithm to study it. The main steps of G-P algorithm include the following aspects: Supposing that the reconstructed phase space has n vectors, we calculate the logarithmic of associated vector (namely the distance between two vectors is less than the given positive number r). Let proportion which accounts for n2 vector be the correlation integral: C(𝑟) =

1 𝑛2

𝑛

∑𝑖,𝑗=1 θ(r − |𝑦𝑖 − 𝑦𝑗 |)

(3)

where C(𝑟) is cross correlation integral, n is the number of vectors in phase space, and θ(𝑥) is the unit function of Heaviside. θ(𝑥) = 0 when x≤0 and θ(𝑥) = 1 when x＞0; r is a smaller positive number which has fixed; |𝑦𝑖 − 𝑦𝑗 | is the Euclidean distance between any two vectors in phase space. The cross correlation integral could meet the following equation through choosing a proper value of r: C(𝑟) = 𝑟 𝐷

(4)

Both sides were taken logarithm: 𝐷=

ln C(𝑟)⁄ ln 𝑟

(5)

In a practical application, the value m will be given. We just change the size of the positive number r within a certain range, and then a curve of lnC(𝑟) − ln𝑟 can be obtained according to (3). We remove the straight line whose slope is 0 and ∞ in the curve. The slope of the best fitting straight line is the correlation dimension D. Keep increasing the value of embedding dimension m constantly, we can draw the D-m curve. Now we observe the curve. If the value D increases with the growth of value m and does not converge to a stable value, it indicates that the selected time series is a random sequence. If the D value no longer increases with the value m and remains unchanged within a certain error range, it indicates that the sequence is chaotic time series, and then the corresponding value of m is the best embedding dimension of reconstructed phase space.

Deciding the number of hidden layer nodes of the chaotic neural network by trial and error For the hidden layer of the neural network, the design of hidden layer nodes has an important influence on the entire network performance and its prediction effect. The common method for deciding the number of best hidden layer nodes is the trial and error. The first step is to determine 5

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

the value ranges of hidden layer nodes based on our experience. Then we train the neural network with different numbers of hidden layer nodes respectively. Finally, we can find the best one when the network error achieves the minimum value.

Deciding the number of output layer nodes of the chaotic neural network based on the actual situation The number of output layer nodes can be determined according to the actual need. In terms of time series forecasting problem, single output is generally chosen for the output layer. In this paper we select the first m weeks of egg prices as the independent variables, the m+1 week of egg prices as the dependent variable and so on. Therefore, the number of output layer nodes of chaotic neural network in this paper is 1.

Deciding the training method of the chaotic neural network In this paper we train the chaotic neural network using the standard BP algorithm. Besides, in order to enhance the speed of training, improve the prediction accuracy and avoid getting into local minimum, we introduced the learning rate and appending momentum factors in the practical operation. The selection of learning rate is very important. Big value may cause system unstable and small value will lead to a long training time, slower convergence rate and fall short of requirements errors. The general data range of learning rate is 0.01~0.1. The appending momentum factors are parameters of the additional momentum method. The function of appending momentum factors resembles a low-pass filter which allows the network to ignore the network tiny change characteristic, as far as possible slip the part of network in minimum value. The method is on the basis of the back-propagation algorithm, and combines change amount of per one weight and one proportional to the amount of change of the previous weight value to produce a new weight change. The essence of this method is to deliver the influence of weights change in the last time through a momentum factor. Due to space constraints, verbose description of the specific function of BP algorithm and additional momentum method are omitted.

APPLYING THE CHAOTIC NEURAL NETWORK MODEL TO THE SHORT-TERM PREDICTION OF WEEKLY RETAIL PRICES OF EGGS IN CHINA Data selection and processing In this section, we firstly propose to introduce the chaotic neural network model to the short-term

6

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

prediction of egg prices and explore the prospects of artificial intelligence forecasting methods in the field of agricultural prices’ short-term forecasting. Annual or monthly samples were chosen mostly in previous papers. However, for the prediction of short-term price, the shorter the interval of sample points is and the greater the size of samples is, the higher the prediction accuracy of the model can be. Therefore, the weekly retail prices of eggs are firstly selected in this paper. We establish and train the chaotic neural network model using 260 weekly price data from January 2008 to December 2012, then predict the latter 5 weekly prices and compare the prediction accuracy with actual price and the output of ARIMA model. All of the price data above were retail prices which came from the Livestock Department, Ministry of Agriculture. The trend of weekly retail price of eggs is shown in Fig. 2 (January 2008—December 2012). It can be seen from the figure that the weekly retail prices of eggs have the characteristics of periodicity, seasonality and long-term volatility. In terms of the whole year, the price change rule can be concluded that the lower price was distributed from February to April. Time from May to September was the rise period of the egg retail prices. Before the National Day (1st October), the prices reached their highest point, and then it began to decline slowly. The prices also had a short callback around the Spring Festival. However, due to the increased complexity of eggs market in recent years, the domino effect appeared among the agricultural prices, the fluctuations of egg prices had been greatly increased from both amplitude and frequency.

Fig. 2. The trend of weekly retail prices of eggs (January 2008—December 2012)

The establishment of the chaotic neural network topology structure According to the chaotic neural network theory, we reconstruct phase space by the time series of China’s weekly retail prices of eggs. According to (1), the optimal delay time τ of Phase Space Reconstruction is 17. On this basis, we further adopt the G-P algorithm to determine the saturated embedding dimension m. Firstly, we used (3) to get different embedding dimension of the corresponding lnC(𝑟) − ln𝑟 curve as shown in Fig. 3. The slope of each curve is the corresponding

7

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

correlation dimension, and then we get the D-m curve. It can be seen from Fig. 4 that the corresponding correlation dimension reaches the maximum value D=1.9098 when the embedding dimension m is 8, which indicates that the saturated embedding dimension of the reconstruction phase space is also 8. According to the principle of chaos theory, when the value of correlation dimension D no longer increases with the embedding dimension m and tends to be stable within a certain error range, it indicates that the time series of sample is the chaotic time series. So the weekly retail prices series of eggs in China belongs to the chaotic time series, we can have deep analysis by the chaotic neural network theory.

Fig. 3. The lnC(r) − lnr curve of weekly retail prices of eggs in China

Fig. 4. The D-m curve of weekly retail prices of eggs in China As the saturated embedding dimension of the reconstructed phase space above is 8, the number of input layer nodes of chaotic neural network is 8, and the number of output layer nodes is 1. For the determination of the number of hidden layer nodes, we determine that the number range of hidden layer nodes is between 21 and 30 by experience, and then we get the corresponding network error of different hidden layer nodes by trial and error, which is shown in Table 1. As seen from the table, we find that when the number of hidden layer nodes is chosen as 26, the corresponding network error reaches its minimum value. So the number of hidden layer nodes of the chaotic neural network model in this paper is 26, and the whole structure of the chaotic neural network model is 8

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

8-26-1. Table 1. Different numbers of hidden layer nodes and the corresponding network error Hidden layer nodes Network error

21

22

23

24

25

26

27

28

29

30

0.5018 0.5016 0.5006 0.5015 0.4977 0.4904 0.5015 0.5011 0.5006 0.4990

Training of the chaotic neural network model After determining the topological structure of the chaotic neural network model, we use the MATLAB neural network toolbox to program and train it. Let “tansig” be the transfer function of network hidden layer neuron; let “purelin” be the transfer function of output layer neurons, thus we can get any value from entire network output. The training function uses “trainlm”, which has a faster training speed. The network training parameters which have been set are shown in Table 2. Table 2. Training parameters of the chaotic neural network Step size

Iterations

Learning rate

Minimum error

Momentum factor

100

2000

0.1

0.001

0.9

The training process of the chaotic neural network model costs 3 seconds to run and it satisfies the error requirements after 110 iterations. The changing curve of error in the process of network training is shown in Fig. 5. The changing process of network gradient and mean is shown in Fig. 6. All of the above indicates that the chaotic neural network model satisfies the setting requirement of relevant parameters by training, the model’s fitting effect is good, and it can be applied to predict values.

Fig. 5. Error training curves of the chaotic neural network model

9

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

Fig. 6. Training gradient and the mean curve of the chaotic neural network model

Prediction of the chaotic neural network model According to the trained chaotic neural network model, we predict the weekly retail prices of eggs in the next five weeks. The simulation results of the chaotic neural network model and the actual results of the weekly retail prices of eggs are shown in Fig. 7. As the output of the neural network model starts from the 9th week, the time span in Fig. 7 is from week 9 to 260. It can be seen from the figure that the fitting effect of the chaotic neural network model is very satisfied, and the stability and error of model all meet actual needs.

Fig. 7. Actual results and simulation results of the chaotic neural network The prediction results comparison between the chaotic neural network model and the traditional time series model (ARMA model) are shown in the Table 3. It can be seen obviously from the table that the relative errors are all less than 1.0% in the prediction of the weekly retail prices of eggs in the next five weeks using the chaotic neural network model. In terms of the error range, the stability of the predicted results is ideal. Whereas the relative predicted errors of the ARMA model are much larger, and the error range are even enlarging. Table 3. The prediction error comparison between chaotic neural network model and ARMA model (unit: USD/kg) 10

Journal of Integrative Agriculture Advanced Online Publication: 2013 Date

Doi：

Real Prediction by Absolute Relative Prediction by Absolute Relative price neural network error error (%) ARMA model error error (%)

1th week, 2013.01 1.74

1.7478

-0.0074

-0.43

1.7868

-0.0465

-2.67

2th week, 2013.01 1.75

1.7535

-0.0050

-0.28

1.7991

-0.0506

-2.90

3th week, 2013.01 1.76

1.7468

0.0115

0.65

1.8063

-0.0481

-2.73

4th week, 2013.01 1.77

1.7609

0.0054

0.31

1.8189

-0.0525

-2.97

5th week, 2013.01 1.77

1.7581

0.0083

0.47

1.8206

-0.0542

-3.07

CONCLUSIONS As for design of the chaotic neural network model, we use the reconstruction technique of phase space to determine the input layer nodes of the neural network, which providing a theoretical support to the establishment of the neural network structure and improving the model’s prediction precision. In terms of determining the number of hidden layer nodes, the way of comparing the network error can greatly reduce the training time and improve network performance. Through empirical analysis and comparison with the traditional time series model, the chaotic neural network model which is constructed in this paper has much higher prediction accuracy and ideal fitting effect in short-term forecast of nonlinear time series data (e.g., the weekly retail prices of eggs). It also indicates that the chaotic neural network model has a broad prospect of application in the field of short-term price prediction of agriculture.

Acknowledgements This research was financially supported by Key Projects of National Key Technology R&D Program during the Twelfth Five-Year Plan Period（No. 2012BAH20B04), 948 program of MoA (No.2013-Z1)

Reference 1.

Ahmad H A, Dozier G V and Roland D A. 2001. Egg price forecasting using neural networks. The Journal of Applied Poultry Research, 10, 162-171.

2.

Amjady N, Keynia F. 2011. A new neural network approach to short term load forecasting of electrical power systems. Energies, 4, 488-503.

3.

Ardalani-Farsa M, Zolfaghari S. 2010. Chaotic time series prediction with residual analysis method using hybrid Elman–NARX neural networks. Neurocomputing, 73, 2540-2553.

11

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

4.

Chen Z, Lu C, Zhang W J, Du X W. 2010. A chaotic time series prediction method based on fuzzy neural network and its application. In: Chaos-Fractals Theories and Applications (IWCFTA), 2010 International Workshop on. IEEE, Kunming, pp. 355-359.

5.

Cui L G, Li Z M. 2013. Comparing with the results of short-term price forecasting for cabbage by different optimization algorithm of chaos-RBF neural network model. Journal of System Science and Mathematical, 33, 45-54. (in Chinese)

6.

Dian S M, He R, Fu S W. 2011. Approach of network flow prediction based on chaotic time series and neural network. Modern Electronics Technique, 3, 65-71. (in Chinese)

7.

Dou C X. 2003. Design of fuzzy neural network controller based on chaos neural network forecast model and application. Systems Engineering —Theory& Practice, 8, 48-52. (in Chinese)

8.

Guo G, Shi Z K, Dai G Z. 2000. Select optimal number of variable to nonlinear modeling with chaotic theory. Control and Decision, 2, 233-235. (in Chinese)

9.

Grassberger P, Procaccia I. 1983. Measuring the strangeness of strange attractors. Physica D: Nonlinear Phenomena, 9, 189-208.

10. Li J. 2011. Applications and research on chaotic time series prediction model. Computer Simpulation, 4, 100-102. (in Chinese) 11. Li Z M, Li G Q. 2010. The short-term forecasting of the market price of eggs. Food and Nutrition in China, 6, 36-40. (in Chinese) 12. Li Z M, Xu S W, Cui L G, Li G Q, Dong X X, Wu J Z. 2013. The short-term forecast model of pork price based on CNN-GA. Advanced Materials Research, 628, 350-358. 13. Mehdi K, Mehdi B. 2010. An artificial neural network (p, d, q) model for time series forecasting. Expert Systems with Applications, 37, 479–489. 14. Mehdi K, Mehdi B. 2011. A novel hybridization of artificial neural networks and ARIMA models for time series forecasting. Applied Soft Computing. 11, 2664–2675. 15. Niu D X, Wang Y L, Duan C M, et al. 2009. A new short-term power load forecasting model based on chaotic time series and SVM. Journal of Universal Computer Science, 15, 2726-2745. 16. Oguri K, Adachi H, Yi C H, Sugiyama M. 1992. Study on egg price forecasting in Japan. Research Bulletin of the Faculty of Agriculture - Gifu University, 57, 157-164. 17. Soloviev V, Saptsin V, Chabanenko D. 2010. Financial time series prediction with the technology of complex markov chains. Computer Modelling and New Technologies, 14, 63– 67.

12

Journal of Integrative Agriculture Advanced Online Publication: 2013

Doi：

18. Sousa J C, Neves L P, Jorge H M. 2012. Assessing the relevance of load profiling information in electrical load forecasting based on neural network models. International Journal of Electrical Power & Energy Systems, 40, 85-93. 19. Surendra N. Kulshreshtha. 1971. A short-run model for forecasting monthly egg production in Canada. Canadian Journal of Agricultural Economics, 19, 36-46. 20. Takens F. 1981. Detecting strange attractor in turbulence. Lecture Notes in Math, 898, 366-381. 21. Wang Y, Xu W. 2006. The methods and performance of phase space reconstruction for the time series in Lorenz system. Journal of Vibration Engineering, 2, 277-282. (in Chinese) 22. Wang S H. 2008. Gray predict model application in egg prices forecast. Guide to Chinese Poultry, 15, 48-50. (in Chinese) 23. Xu N, Liao S Y, Deng G L. 2011. Prediction of petroleum futures price based on PSO-BP network. Computer Engineering and Applications, 47, 234-236. (in Chinese) 24. Yang Y W, Liu G Z, Zhang Z P. 2001. Chaotic data prediction and its applications in stock market based on embedding theory and neural networks. Systems Engineering —Theory & Practice, 6, 52-58. (in Chinese) 25. Yu G R, Yang J R, Xia Z Q. 2011. The prediction model of chaotic series based on support vector machine and its application to Runoff. Advanced Materials Research, 255, 3594-3599. 26. Zhang G M, Yuan Y H, Gong S J. 2011. A predictive model of short-term wind speed based on improved least squares support vector machine algorithm. Journal of shanghai Jiaotong University (Science), 45, 1125-1129.

13