Adaptive CMAC Filter for Chaotic Time Series Prediction

Chaotic signal is a natural phenomenon exhibiting in every condition of dynamical system. Chaotic signals are almost unpredictable, noise-like, uncertain and irregular behavior, yet they are very useful in numerous applications of signal processing. Due to their behaviors, the quest of a good method to model and analyze of the chaotic signal is very crucial. This paper present a novel strategy to analyze chaotic signal using the cerebellar model articulation controller (CMAC) network combined with evolutionary algorithms (EAs) such as biogeography-based optimization (BBO), genetic algorithm (GA) and particle swarm optimization (PSO). Mackey-glass chaotic signal time series is tested and demonstrated by the conventional and the proposed algorithms. They are compared with each other to determine the optimal filtering and prediction. The result demonstrated that the CMAC combined with EAs could filter, predict and estimate chaotic signal time series well rather than the conventional methods. The best result of the algorithms tested for chaotic signal time series is the CMAC combined with BBO algorithm.


Introduction
A chaotic signal is a natural phenomenon exhibiting in every condition of a dynamical system.The chaotic signal has become an attractive issue for many researchers in recent year.Some methods have been developed to obtain the meaning of the chaotic signal.One of the common techniques to predict the value of the chaotic signal is time delays and an embedding dimension [1].To realize this method, a predictor design involves a neural network to obtain a better result.A kind of neural networks imitating the human cerebellum is cerebellar articulation model controller (CMAC).The CMAC has advantages, such as low computation, good generalization capability and easy implementation [2].Due to some advantages, the CMAC can be applied as a filter to predict the chaotic signal series.A time windowing strategy called Time-division CMAC or CMAC filter is an appropriate to tackle the prediction problem.The CMAC filter is a modified classical CMAC networks for predicting chaotic signal series [3].To achieve a good prediction result, training process is an important part in the neural network.Some evolutionary algorithms (EAs) are employed to enhance the learning system capability.There are biogeography-based optimization (BBO), genetic algorithm (GA) and particle swarm optimization (PSO).

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The structure of the CMAC, comprises two input states, association cells (ai), memory cells, and an output state (y).In Figure 1(a), the simple example shows a memory structure of the CMAC, which each layer is mapped into three blocks.Each layer has nine areas labeled as hypercube Aa, Ab, Ac, …, and Cc.The hypercubes would be used as memory cell, which the active cell is addressed by an association cell.The actual CMAC output comes from the summation of the weights stored in the memory of each layer.The structure of the CMAC filter design involves some basic CMACs.The prediction output can be obtained by the equation (1): where Md is the memory size, w is the weight vector, and d is the number of delays.The equation for the learning phase can be written as follows: where ydes is the desired output, y(i) is the actual output, w(i) is the weight vector, β is the learning rate, and Ne is the number of activated memories.

Biogeography-based Optimization (BBO)
Biogeography is the study of biology as it relates to the geographical distribution of species over time and space.The emigration rate (μ) and the immigration rate (λ) determine the BBO algorithm.In Figure 2 ( The immigration and emigration rates can be written by equation ( 4) and ( 5): where I is the maximum immigration rate, E is the maximum emigration rate, k is the number of species, and a is the largest number of species.The mutation rate can be expressed as follows: P -55 where max m is the value defined by the user, s P is the mutation probability, and Pmax = argmax(Pn) for n=1,2,…,N.A species living in a habitat can be calculated using s P , which changes time t to time (t+∆t), and is given as follows: where Ps(t) is the probability that a habitat contains S species, S is the species at time t without migration, S-1 is the species at time t with one species immigrating, and S+1 is the species at time t with one species emigrating.

Genetic Algorithms (GAs)
Genetic algorithms are population-based modeled by biological evolution based on Charles Darwin's theory of natural selection.The principle of the GA is the movement from one population of chromosomes to a new population using crossover, mutation, and selection [4].To find a good solution, GA involves the chromosomes represented by strings of bits, the manipulation operation string, and the selection to their fitness.In one evolution cycle of the genetic algorithm, there are some steps such as evaluating the fitness of all the individuals, creating a new population, replacing the old population and iterating again using the new population.The crossover operation is swapping one segment of one chromosome with the segment on another chromosome at a random position.The mutation can be obtained by flipping the selected bits randomly and usually has small probability.The objective of mutation is to increase the various populations.

Particle Swarm Optimization (PSO)
The concept of PSO algorithm is the movement to search new space of swarm behavior toward an objective function by adjusting the trajectories of particles [5].Each particle moves to a new location determined by a velocity term, which represents the attraction of global best (g*) and its best location xi * in the history of the particle and random coefficient.If the location of the particle is better than any previously found location, updates such location as new current best for particle i.The purpose of this algorithm looks for the global best among all the current best solution for particles.A description of the movement of the particle is represented in Figure 2(d).The global best comes from the minimum value of function f(xi) for i=1,2,…,n.Let xi=(xi 1 , xi 2 ,…,xi d ) be the position vector and vi=(vi 1 ,vi 2 ,…,vi d ) be the velocity for the particle i. Particle updating to obtain the new velocity vector is specified by Equation ( 8) where 1 and 2 are two random vectors, 1 c and 2 c are the learning parameter, w is the inertia weight, and vi (t+1) is the new position.

Simulation Results
In this paper, the parameter values for the CMAC filter have two inputs, s1 and s2, in which the number of layers is five, the width is ten, and the number of delays is three.
The learning parameter values are configured as follows: the same parameters of BBO, GA and PSO have population: 15, chromosome: 10, mutation: 1%, and elitism: 2. The parameters for each EAs can be configured as follows: habitat probability of BBO is 1, upper is 1 and upper is 0; crossover probability and crossover type of GA is one; and initial w, c1, and c2 of PSO are 0.03, 1, and 1, respectively.The Mackey-glass is generated by a time-delay differential equation, such as that shown in Equation (10): where α = 0.2, β = 0.1, γ = 10, and = 17.In this paper, 400 and 100 data points are employed for training and testing.Figure 3 In Figure 3, the predictor employs the chaotic data series.The CMAC filter using BBO, GA or PSO learning system is employed to predict the chaotic signal data series.The comparative of Mackey-glass prediction results are presented in Table 1.  1, the learning process using BBO algorithm improves the performance of the CMAC filter.The CMAC filter using BBO learning process has the smallest MSE, MAPE, and ARV.They are 4.2610e-05 for MSE, 1.4106e-02 for MAPE, and 2.2484e-03 for ARV.Based on these errors and curves, the best prediction was made by the BBO algorithm.

Conclusion
This paper presents a novel filter using CMAC network to predict chaotic data time series, in which generated by Mackey-glass chaotic signal time series equations.The proposed CMAC filter is employed using EAs learning system to obtain a good performance.The EAs are BBO, GA, and PSO algorithm.The prediction results are compared with each other to determine the optimal filtering and prediction.Based on the result, the CMAC filter combined with EAs could filter, predict and estimate chaotic signal time series well.The best result of the algorithms tested for chaotic signal time series is the CMAC filter combined with BBO algorithm.

Figure 1 .
(a) memory structure of the CMAC, (b) memory structure of CMAC filter 3. Evolutionary Algorithms (EAs) (a), the BBO depicts emigration and immigration curves by straight lines.The equilibrium point of species S0 occurs when the immigration and emigration rates are equal.It occurs temporarily, as conditions continually vary.The equilibrium point of migration can be written by equation ( Fitness reproduction or elitism is achieved by the selection of an individual in a population.Diagram of crossover and mutation can be depicted in Figure 2(b) and 2(c).

©Figure 2 .
Figure 2. (a).Migration curve of species of a single habitat; (b).Diagram of crossover; c).Diagram of mutation; (d).Representation of the motion of particle i.

Table 1 .
The results of MSE, MAPE, and ARV MAPE, and ARV evaluate the output of the CMAC filter.The prediction results are compared with those of algorithms to identify the algorithm with the best performance.Based on the error results listed in Table