水文时间序列的混沌性分析及预测研究（毕业设计）(8)

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This paper presents a PRBF network for one-step iterative prediction of Lorenz and hydraulic pump chaotic time series. The proposed prediction model, based on Camastra’s approach [6] and Yang’ method [7], can reduce the cumulative error effect and improve the prediction stability of the RBF. Besides, one-step iterative prediction ismore useful than one-step prediction in practical.There are mainly three sections in this paper as follows:Section 2 describes the theory of phase space reconstruction to obtain the estimation of correlation dimension; Section 3 is to propose a new PRBF based method for chaotic time series prediction; and then two study cases are given to verify the proposed model using Lorenz and actual hydraulic pump signals in Section 4.

II.PHASE-SPACE RECONSTRUCTION OF CHAOTIC TIME SERIES An important issue in the study of dynamic systems is dynamic phase-space reconstruction theory discovered by Packard in 1980[8]. It regards a one-dimensional chaotic time series as the compressed information of high-dimensional space. The time series x(t), t=1, 2, 3, …, N can be represented as a series of points X(t)in a m–dimensional space.

Where, m is called as system embedding dimension. In particular, Takens’ embedding theory [9] states that, in order to obtain a dependable phase-space reconstruction of dynamic system, it must be

Where, D is the dimension of system attractor. In order to obtain a correct system embedding dimension, starting from the time series, it is necessary to estimate the attractor dimension D.

Among different dimension definitions, correlation dimension discovered by Grassberger and Procaccia in 1983[10], is the most popular one due to its calculation simplicity. It is defined as followings. If the correlation integral Cm(r) is defined as

水文时间序列的混沌特性及预测研究 33

Where, H is the Heaviside function, m is the embedding dimension and N is the

number of vectors in reconstructed phase space. It is proved that if r is sufficiently small, and N would be sufficiently large, the correlation dimension D is equal to

The algorithm plots a cluster of lnCm(r)-ln(r) curves through increasing m until the slope of the curve’s linear part is almost constant. Then, the correlation dimension estimation D can be attained using least square regression.

III. MODEL OF PREDICTION

In practice, it is difficult to get the exact estimation value of minimum embedding dimension through G-P algorithm. Furthermore, a single RBF network uses the estimation value of minimum embedding dimension as the number of its input, usually resulting in inaccurate output due to the inaccurate estimation of embedding dimension from human factor.

Therefore, a PRBF network consisting of multiple RBF subnets is proposed to increase system performance with decreased error.

A. Structure of PRBF

The PRBF is constituted of multiple RBF networks connected in parallel for time series prediction. The structure of a PRBF is shown in Fig. 1.

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. B. Input Nodes of Subnet

Estimation value of the minimum embedding dimension is regarded as the number of input nodes in the central subnet, and each of other subnets uses a different numbers (calculated based on m) as its input size. Once the correlation dimension D is obtained by G-P algorithm and least square regression, the number of input nodes in the center subnet can be determined as

Then, the number of input nodes of each subnet can be defined as

C. Calculation of Weighted Factors

It is necessary to employ weighted factor ω to gain proper prediction result, because each RBF subnet has different influence on the whole system. Here, the optimal weighted value of the each subnet is determined according to the minimum predicted absolute percent error (APE) of in each study case. The output of PRBF net is the weighted combination of each individual RBF subnet and the final predicted result can be represented by the following equation.

D. One-step iterative prediction

Considering the lack of practicability from a common one-step prediction method, one-step iterative prediction should be adopted to predict chaotic time series instead. The detail of the one-step iterative prediction based on PRBF is described as following:

Step 1.Normalize the original chaotic time series;

Step 2.Determine the number of input nodes of each subnet in PRBF according to G-P algorithm and Takens’ theory;

Step 3.Obtain weighted factor ω on the basis of one-step prediction value of each subnet;

Step 4.Calculate one-step prediction value of PRBF;

Step 5.Regard the predicted result at Step 4 as the next input data to achieve one-step prediction;

Step 6.The future trend of actual case (Lorenz’s attractor, hydraulic pump) can be obtained gradually with the repetition of Step 3 and 4, and the loop times depends on is the length of actual expected data.

IV. CASES STUDY A. Simulation of Lorenz’s Attractor In this section, the simulation results of Lorenz’s attractor data is given to verify the performance of the proposed method. Equation (8) is

水文时间序列的混沌特性及预测研究 35

employed to generate the Lorenz’s time series data.

Where, σ=16 r=45.95 b=4. 1,000 points of X-

component Lorenz time series data were firstly normalized and used for the following

prediction. According to G-P algorithm, a cluster of lnCm(r)-ln(r) curves is plotted with the increase of the embedding dimension m, and the minimum embedding dimension was found to be 5, as shown in Fig.2. As a whole, 1,000 points were divided into two groups (training set and testing dataset). The first 800 samples were used for RBF …… 此处隐藏：5780字，全部文档内容请下载后查看。喜欢就下载吧 ……