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1450 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 τ includes autocorrelation function method, multiple correlation function method, mutual information method. Embedding dimension m is calculated by the methods of GP algorithm, pseudo-nearest-point method, correlation integral method and Cao method. The chaotic time series prediction is based on the Takens' delay-coordinate phase reconstruct theory. If the time series of one of the variables is available, based on the fact that the interaction between the variables is such that every component contains information on the complex dynamics of the system, a smooth function can be found to model the portraits of time series. If the chaotic time series are{ x() t }, then the reconstruct state vector is x( t) = ( x( t), x( t+ τ ), , x( t+ ( m−1) τ )) where m ( m = 2,3, ) is called the embedding dimension ( m= 2d + 1 , d is called the freedom of dynamics of the system), and τ is the delay time. The predictive reconstruct of chaotic series is a inverse problem to the dynamics of the system essentially. There exists a smooth function defined on the reconstructed m manifold in R to interpret the dynamics x( t+ T) = F( x( t)) , where T ( T > 0) is forward predictive step length, and F() ⋅ is the reconstructed predictive model. B. RBF Neural Network Function Approximation Theory Takens embedding theorem states that there is a smooth mapping F of the F makes: x( t+ τ ) = F( x( t)) (1) that is, xt ( + τ), xt (), , xt ( −( m− 2) τ) = F{[ xt (), xt ( −τ), , xt ( −( m−1) τ]} For purposes of calculation, equation (1) can be rewritten as: xt ( + τ ) = Fxt [ ( ), xt ( −τ), , xt ( −( m−1) τ] (2) where, f is the mapping from R M to R L . Chaos theory suggests that the chaotic time series is short-term forecast, and the essence of prediction is how to get a good approximation f on the function f . Chaotic time series determined by the internal regularity, this regularity comes from the non-linear, it exhibits the time series in the time delay state, this feature makes the system seem to have some kind of memory capacity. The same time, it is difficult to demonstrate such a regularity by using the analytic methods; this type of information processing happens to be the neural network, and the Kolmogorov continuity theorem in the neural networks theory provides a theoretical guarantee for the neural network nonlinear function approximation. Theorem (Kolmogorov continuity theorem) Let ϕ ( x) be a non-constant and bounded monotonically increasing a continuous function; M is a compact sub-set of R n , and f( x) = f( x1, x2, , x n ) is the continuous real value function on M , then for ∀ ε > 0 , exists a positive integer N and real numbers C , makes: N n f( x , x , , x ) = Cϕ( ϖ x −θ ) (3) 1 2 ∑ ∑ n i ij j j i= 1 j= 1 meet: max f( x , x , , x ) − f( x , x , , x ) < ε (4) M 1 2 n 1 2 By the above theorem, the nonlinear time series prediction process using neural network can be considered as dynamic reconfiguration, which is an inverse process. Namely, the existence of a three-layer network, the hidden unit output function, the network input and output function is linear, three-layer network input and output relation f can approximate p. Therefore, the theorem from mathematics is to ensure the feasibility of chaotic time series prediction by neural network. C. Realized Architecture of Adaptive RBF Neural Network Filtering Predictive Model After reconstructing the phase space, the RBF neural networks adopt three layers networks of Figure 1. Where the input layer has m nerve cells, the first layer feed to the second layer directly and it do not need the power processing. r i ( i = 1, 2, , L ) is the reference vector and ϖ k ( i = 1, 2, , L ) is the adjustable parameters in the adaptive RBF neural network filtering. Thus, the adaptive RBF neural network filtering is more flexible in studying the nonlinear functions. The differentiation between the networks and the traditional neural networks is that the activation function is a RBF function but not the Sigmoid function. The activation function usually choose the Gauss function, the spline function f ( di ( k )) , where d ( k) = x( k) − r( k) . In the adaptive RBF neural i network filtering, yk ( ) is expressed as i 2 L−1 yˆ( k) = f ( ∑ ϖ ( k) f( d ( k))) , i= 0 i = 0, 2, , L−1, where f () 2 ⋅ is the activation function of output signal. i x(k) −1 z −1 z −1 z r 0 r 1 r 2 r L i ϖ 0 ( k) ϖ 1 ( k) ϖ 2 ( k) (k) ϖ L n ∑ yˆ ( k) input hidden layer output Figure 1. Structure of adaptive RBF neural network filtering Generally, the learning of the RBF neural network filtering has three steps. If the gradient method and the Gauss activation function are adopted, the regulate formulas of RBF are shown as: © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1451 ⎧ϖi( k+ 1) = ϖi( k) + 2 μϖ e( k) f( di( k)) ⎪ 2 ⎪ di ( k) σi( k+ 1) = σi( k) + 2 μϖ e( k) f( di( k)) ϖi( k) ⎪ 3 σ i ( k) ⎪ ⎨ x( k) − ri ( k) (5) ⎪ri( k + 1) = ri( k) + 2 μre( k) f( di( k)) ϖi( k) 2 σ i ( k) ⎪ ⎪ x( k) − ri ( k) ⎪ri( k + 1) = ri( k) + 2 μre( k) f( di( k)) ϖi( k) 2 ⎩ σ i ( k) i = 0, 2, , L−1. The RBF neural network system can enhance the stabilization and associative memory of chaotic dynamics and generalization ability of predictive model even by imperfect and variation inputs by selecting the suitable nonlinear feedback term. The dynamics of network become chaotic one in the weight space. Thus, the regulate formula ϖ ( k) is shown as ϖ i( k + 1) = ϖi( k) + 2 μϖ e( k) f( di( k)) + g( ϖi( k) −ϖI( k−1)) (6) 2 where g( x) = tanh( ax)exp( − bx ), x = ϖ ( k) −ϖ ( k− 1) . That the feedback function g( x ) is chose is because that g( x) can get the difference feedback function corresponding to the dissimilar parameter, such as the staircase function, δ function and so on. If the feedback function is seen as the motion-promoting force, the different feedback parameters a and b corresponding to the amplitude and width of the motion-promoting force. The paper [18] was detailed to discuss the influences by selecting the suitable learning and predictive process. The simulation results indicated that the network system can enhance the stabilization and associative memory of chaotic dynamics and generalization ability of predictive model even by imperfect and variation inputs during the learning and prediction process by selecting the suitable nonlinear feedback term. III. DETERMINATION METHOD OF THE OPTIMAL DELAY TIME AND MINIMUM EMBEDDING DIMENSION A. Determination Method of the Optimal Delay Time τ During Phase Space Reconstruction in the Takens embedding theorem does not make limited to the delay time τ .In theory, when the observational data point is an infinitely long, the effect of embedded not too large. However, in actual operation, τ is caused a great impact. If τ is too small, the chaotic attractor cannot be fully expanded, redundant error is larger; if τ is too large, the no related error is larger. Therefore, in order for complex nonlinear systems, using the mutual information method to determine the optimal delay time τ , the mutual information method using a minimal value of the mutual information function to determine the optimal delay time τ , its expression is as follows: P, () r M( x , ) , ( )ln i j t xt− τ = ∑ Pi j r (7) PP i, j i j i i where, P i is the probability of point x t in the i time interval; Pi, j() r is the joint probability of the point x t in t moment fall into the i time interval and the t + τ moment fall into the j time intervals. B. Determination Method of the Minimum Embedding m In this paper, the commonly used pseudo-near-point method to calculate the minimum embedding dimension m , set the number of attractor dimension d , then m is just the minimum embedding dimension when the attractor is fully open. When m< d , the attractor in the phase space cannot be completely open, the attractor will produce some projection point in the embedded space, the projection point and the other points in the phase space will form the closest point. In the original system, the 2 points are not true nearest neighbors, so called pseudo adjacent points. Assume that any point yt () in the phase space, the criterion of false neighboring points are as follows: 1 2 2 D () () 2 m+ 1 t − Dm t xt ( + mτ) − xt ( ′ + mτ) = > ρm (8) D () t D () t m Where D () t is the Euclidean distance between the m N points of yt () with its nearest neighbor y () t in the phase space when the embedding dimension is m . According to this criterion, the calculation pseudo-nearest neighbor number N when m from small to large, and then calculate the change amount Δ N when the embedding dimension from m to m + 1. Draw the curve ΔN Δ from to m ; when Δ N = 0 , just N dropped to 0, N N the value m * of m is seeking the minimum embedding dimension. IV. ADAPTIVE RBF NEURAL NETWORK RAPID LEARNING ALGORITHM On the establishment of chaotic time series RBF, Network input the number of neurons, hidden layers and the number of neurons in the hidden layer are to be considered.The following chaotic time series used are from Lorenz chaotic sampling time series. The Lorenz chaotic sampling time series RBF neural network can be constructed: RBF neural network is designed to be three layers: input layer, single hidden layer and output layer; the number of hidden layer wavelet neural taken as 9 by Kolmogorov Theorem, the number of input layer neurons equal to the minimum embedding dimension, the number of output layer is 1, so that the 4-9-1 structure of Lorenz chaotic sampling time series RBF was obtained, specifically shown in Figure 1. Algorithm The steps of the chaotic time series learning and prediction of the adaptive RBF neural network filtering predictive model are showed: Step1) Based on the Takens' delay-coordinate phase reconstruct theory, the number of the input nerve cells m © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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