A Hybrid of Cooperative Particle Swarm Optimization and Cultural Algorithm for Neural Fuzzy Networks and Its Prediction Applications

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A Hybrid of Cooperative Particle Swarm

Optimization and Cultural Algorithm for Neural

Fuzzy Networks and Its Prediction Applications

Cheng-Jian Lin, Member, IEEE, Cheng-Hung Chen, Student Member, IEEE, and Chin-Teng Lin, Fellow, IEEE

Abstract—This study presents an evolutionary neural fuzzy work, designed using the functional-link-based neural fuzzy net-work (FLNFN) and a new evolutionary learning algorithm. This new evolutionary learning algorithm is based on a hybrid of cooper-ative particle swarm optimization and cultural algorithm. It is thus called cultural cooperative particle swarm optimization (CCPSO). The proposed CCPSO method, which uses cooperative behavior among multiple swarms, can increase the global search capacity using the belief space. Cooperative behavior involves a collection of multiple swarms that interact by exchanging information to solve a problem. The belief space is the information repository in which the individuals can store their experiences such that other individuals can learn from them indirectly. The proposed FLNFN model uses functional link neural networks as the consequent part of the fuzzy rules. This study uses orthogonal polynomials and linearly independent functions in a functional expansion of the functional link neural networks. The FLNFN model can generate the consequent part of a nonlinear combination of input variables. Finally, the proposed FLNFN with CCPSO (FLNFN-CCPSO) is adopted in several predictive applications. Experimental results have demonstrated that the proposed CCPSO method performs well in predicting the time series problems.

Index Terms—Chaotic time series, cultural algorithm, functional-link network, neural fuzzy network, particle swarm op-timization, prediction.

I. INTRODUCTION

P

REDICTION has been widely studied for many years as time series analysis [1], [2]. Traditionally, prediction is based on a statistical model that is either linear or nonlinear [3]. Recently, several studies have adopted neural fuzzy networks to predict time series [4]–[6]. Researchers have discussed that the network paradigm is a very useful model for predicting time series and especially for predicting nonlinear time series.

Neural fuzzy networks [7]–[13] have become a popular re-search topic. They bring the low-level learning and computa-tional power of neural networks into fuzzy systems and bring the high-level human-like thinking and reasoning of fuzzy sys-tems to neural networks. In the typical TSK-type neural fuzzy Manuscript received May 25, 2007; revised September 22, 2007. Current version published December 22, 2008. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 95-2221-E-324-028-MY2 and Grant NSC-96-2221-E-009-058, and in part by the Taiwan In-formation Security Center (TWISC), under the National Science Council Grant NSC96-2219-E-009-013. This paper was recommended by Associate Editor X. Guan.

C. J. Lin is with the Department of Computer Science and Information En-gineering, National Chin-Yi University of Technology, Taichung County 411, Taiwan, R.O.C. (e-mail: [email protected]).

C.-H. Chen and C.-T. Lin are with the Department of Electrical and Control Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C.

Digital Object Identifier 10.1109/TSMCC.2008.2002333

network [8]–[13], which is a linear polynomial of input vari-ables, the model output is approximated locally by the rule hyperplanes. However, the traditional TSK-type neural fuzzy network does not take full advantage of the mapping capabil-ities that may be offered by the consequent part. Introducing a nonlinear function, especially a neural structure, to the con-sequent part of the fuzzy rules has yielded the NARA [14] and the CANFIS [15] models. These models [14], [15] use multilayer neural networks in the consequent part of the fuzzy rules. Although the interpretability of the model is reduced, the representational capability of the model is significantly im-proved. However, the multilayer neural network has such dis-advantages as slower convergence and greater computational complexity. Therefore, we proposed the functional link neural fuzzy network (FLNFN), which uses the functional link neural network (FLNN) [16], [17] in the consequent part of the fuzzy rules [18]. The FLNN is a single-layer neural structure that is capable of forming arbitrarily complex decision regions by generating nonlinear decision boundaries. Additionally, using functional expansion effectively increases the dimensionality of the input vector and the hyperplanes that are generated by the FLNN provide a good discrimination capability in input data space.

Training of the parameters is the main problem in design-ing a neural fuzzy system. Backpropagation (BP) traindesign-ing is commonly adopted to solve this problem. It is a powerful train-ing technique that can be applied to networks with a forward structure. Since the steepest descent approach is used in BP training to minimize the error function, the algorithms may reach the local minima very quickly and never find the global solution.

The aforementioned disadvantages lead to suboptimal per-formance, even for a favorable neural fuzzy network topology. Therefore, technologies that can be used to train the system parameters and find the global solution while optimizing the overall structure, are required. Accordingly, a new optimization algorithm, called particle swarm optimization (PSO), appears to be better than the backpropagation algorithm. It is an evolu-tionary computation technique that was developed by Kennedy and Eberhart in 1995 [19], [20]. The underlying motivation for the development of PSO algorithm is the social behavior of animals, such as bird flocking, fish schooling and swarm the-ory. PSO has been successfully applied to many optimization problems, such as control problems [21]–[23] and feedforward neural network design [24]–[28]. However, PSO suffers from the burden of many dimensions, such that its performance falls 1094-6977/$25.00 © 2008 IEEE

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56 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 39, NO. 1, JANUARY 2009

as the dimensionality of the search space increases. Therefore, Bergh et al. [29] proposed a cooperative approach that em-ploys cooperative behavior, called CPSO, which uses multiple swarms to improve upon traditional PSO. However, the CPSO still uses the formula (the local best position of each particle and global best position in the swarm) of the traditional PSO to evolve. The trajectory of each particle in the search space is adjusted according to the local best position of the particle and the global best position in the same search space, but it is unable to yield high diversity of particles to increase search space. That is, it is lacking enough capability to satisfy the requirements of exploration [30], [31]. Therefore, the CPSO may find a sub-optimal solution. Additionally, the cultural algorithm [32], [33] can exploit the information of specific belief space to guide the feasible search space and it can also change the direction of each individual in solution space. Hence, the proposed cul-tural cooperative particle swarm optimization (CCPSO) learn-ing method, which combines the cooperative particle swarm optimization and cultural algorithm, to increase global search capacity, is proposed herein to avoid trapping in a suboptimal so-lution and to ensure that a nearby global optimal soso-lution can be found.

This study presents an efficient CCPSO for the FLNFN in several predictive applications. The proposed FLNFN model is based on our previous research [18]. The FLNFN model, which combines a neural fuzzy network with a functional link neu-ral network, is designed to improve the accuracy of functional approximation. The consequent part of the fuzzy rules that corre-sponds to an FLNN comprises the functional expansion of input variables. The orthogonal polynomials and linearly independent functions are adopted as functional link neural network bases. The proposed CCPSO is a hybrid method that combines co-operative particle swarm optimization and cultural algorithms. The CCPSO method with cooperative behavior among multiple swarms increases the global search capacity using the belief space. Cooperative behavior among multiple swarms involves interaction by exchanging information with each other to solve a problem. The belief space is the information repository in which the individuals can store their experiences for other individuals to learn from them indirectly. The advantages of the proposed FLNFN-CCPSO method are as follows: 1) the consequent of the fuzzy rules involves a nonlinear combination of input vari-ables. This study uses a functional link neural network to the consequent part of the fuzzy rules. The functional expansion in the FLNFN model can yield the consequent part of a nonlinear combination of input variables; 2) the proposed CCPSO with cooperative behavior among multiple swarms can accelerate the search and increase global search capacity using the belief space; and 3) as demonstrated in Section V, the FLNFN-CCPSO method is a more effective controller than the other methods.

The rest of this paper is organized as follows. Section II de-scribes the basic concept of particle swarm optimization and cultural algorithm. Section III presents the structure of the functional-link-based neural fuzzy network. Next, Section IV presents the cultural cooperative particle swarm optimization method. Section V presents the results of the simulation of sev-eral predictive applications. Section VI draws conclusions.

II. PARTICLESWARMOPTIMIZATION ANDCULTURALALGORITHM

This section describes basic concepts concerning particle swarm optimization and the cultural algorithm. The special-ization property of particle swarm optimspecial-ization and cultural algorithm is consistent with the learning property of the neural fuzzy network. Therefore, the development of a neural fuzzy network based on particle swarm optimization and the cultural algorithm is valuable.

A. Particle Swarm Optimization

In 1995, Kennedy and Eberhart introduced the particle swarm optimization algorithm (PSO) [19], [20] in the field of social and cognitive behavior. The PSO is a population-based optimization approach, in which the population is called a swarm. Further-more, each swarm consists of many particles. In the PSO, the trajectory of each particle in the search space is adjusted by dy-namically altering the velocity of each particle. Each particle has a velocity vector viand a position vector xi, which represents a

possible solution. Then, the particles move rapidly around and search the solution space using the moving velocity of each par-ticle. Each of these particle positions is scored to obtain a fitness value, based on how to define the solution of the problem. The local best position (Lbest) of each particle and the global best position (Gbest) in the swarm are used to yield a new velocity for each particle

vi(k + 1) = ω× vi(k) + φ1× rand() × (Lbest − xi(k))

+ φ2× rand() × (Gbest − xi(k)) (1)

where ω, φ1, and φ2 are called the coefficient of the inertia

term, the cognitive term, and the society term, respectively. The term vi is limited to the range±vm ax. If the velocity violates

this limit, then it is set to the actual limit.

Changing the velocity enables each particle to search around its individual best position and global best position. Based on the updated velocities, each particle changes its position according to

→

xi(k + 1) =x→i(k) +v→i(k + 1). (2)

Fig. 1 presents the concept of the updated velocity using (1) and (2).

B. Cultural Algorithm

Cultural algorithms [32], [33] involve acquiring the belief space from the evolving population space and then exploiting that information to guide the search. Fig. 2 presents the cultural algorithm components. Cultural algorithms can be described in terms of two basic components—belief space and the population space. The belief space is the information repository in which the individuals can store their experiences for other individuals to learn from them indirectly. In cultural algorithms, the infor-mation acquired by an individual can be shared with the entire population, unlike in most evolutionary techniques, in which the information can be shared only with the offspring of the

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Fig. 1. Diagram of the updated velocity in the PSO.

Fig. 2. Framework of cultural algorithm.

individual. The population space comprises a set of possi-ble solutions to the propossi-blem, and can be modeled using any population-based approach. The belief space and the population space are linked using a scheme that states rules that govern the individuals of the population space that can contribute to the be-lief space based on its experiences (according to the acceptance function), and the belief space can influence the new individuals of the population space (according to the influence function).

III. STRUCTURE OFFUNCTIONAL-LINK-BASED NEURALFUZZYNETWORK

This section describes the structure of functional link neural networks and the structure of the FLNFN model. In functional link neural networks, the input data usually incorporate high-order effects, and thus, artificially increase the dimensions of the input space. Accordingly, the input representation is enhanced and linear separability is achieved in the extended space. The FLNFN model adopted the functional link neural network gen-erating complex nonlinear combination of input variables as the consequent part of the fuzzy rules. The rest of this section details these structures.

A. Functional Link Neural Networks

The functional link neural network is a single-layer network in which the need for hidden layers is eliminated. While the input variables generated by the linear links of neural networks are linearly weighted, the functional link acts on an element of input variables by generating a set of linearly independent functions, which are suitable orthogonal polynomials for a

Fig. 3. Structure of the FLNN.

functional expansion, and then evaluating these functions with the variables as the arguments. Therefore, the FLNN structure considers trigonometric functions. For example, for a 2-D input X = [x1, x2]T, the enhanced data are obtained by

using trigonometric functions as functional expansion Φ = [1, x1, sin(π x1), cos(π x1), . . . , x2, sin(π x2), cos(π x2), . . .]T.

Thus, the input variables can be separated in the enhanced space [16]. In the FLNN structure with reference to Fig. 3, a set of basis functions Φ and a fixed number of weight parameters

W represent fW(x). The theory behind the FLNN for

multidimensional function approximation has been discussed elsewhere [17] and is analyzed next.

Consider a set of basis functions B ={φk ∈ Φ(A)}k∈K, K ={1, 2, . . .} with the following properties; 1) φ1 = 1; 2)

the subset Bj ={φk ∈ B}Mk = 1 is a linearly independent set,

meaning that if M_{k = 1}wkφk = 0, then wk = 0 for all k =

1, 2, . . . , M ; and 3) supj

j

k = 1φk2A

1/2

<∞.

Let B ={φk}Mk = 1 be a set of basis functions to be considered,

as shown in Fig. 3. The FLNN comprises M basis functions

{φ1, φ2, . . . , φM} ∈ B. The linear sum of the jth node is given

by ˆ yj = M k = 1 wk jφk(X) (3) where X∈ A ⊂ N_{, X = [x}

1, x2, . . . , xN]Tis the input vector,

and Wj = [w1j, w2j, . . . , wM j] is the weight vector associated

with the jth output of the FLNN. ˆyj denotes the local output of

the FLNN structure and the consequent part of the jth fuzzy rule in the FLNFN model. Thus, (3) can be expressed in matrix form as ˆyj = WjΦ, where Φ = [φ1(x), φ2(x), . . . , φM(x)]T is the

basis function vector, which is the output of the functional ex-pansion block. The m-dimensional linear output may be given by ˆy = WΦ, where ˆy = [ˆy1, ˆy2, . . . , ˆym]T, m denotes the

num-ber of functional link bases, which equals the numnum-ber of fuzzy rules in the FLNFN model, and W is a (m× M)-dimensional weight matrix of the FLNN given by W = [w1, w2, . . . , wm]T.

The jth output of the FLNN is given by ˆy_j = ρ(ˆyj), where

the nonlinear function ρ(·) = tanh(·). Thus, the m-dimensional output vector is given by

ˆ

Y = ρ(ˆy) = fW(x) (4)

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Fig. 4. Structure of the proposed FLNFN model.

B. Structure of the FLNFN Model

This subsection describes the FLNFN model, which uses a nonlinear combination of input variables (FLNN). Each fuzzy rule corresponds to a sub-FLNN, comprising a functional link. Fig. 4 presents the structure of the proposed FLNFN model. The FLNFN model realizes a fuzzy if-then rule in the following form.

Rulej: IF x1is A1jand x2 is A2j. . . and xiis Aij. . . and

xN]is AN j THEN ˆyj = M k = 1 wk jφk = w1jφ1+ w2jφ2+ . . . + wM jφM (5)

where xiand ˆyjare the input and local output variables,

respec-tively; Aij is the linguistic term of the precondition part with

Gaussian membership function; N is the number of input vari-ables; wk jis the link weight of the local output; φk is the basis

trigonometric function of input variables; M is the number of basis function, and Rulej is the jth fuzzy rule.

The operation functions of the nodes in each layer of the FLNFN model are now described. In the following description,

u(l)_{denotes the output of a node in the lth layer.}

No computation is performed in layer 1. Each node in this layer only transmits input values to the next layer directly

u(1)_i = xi. (6)

Each fuzzy set Aijis described here by a Gaussian

member-ship function. Therefore, the calculated membermember-ship value in layer 2 is u(2)_ij = exp −[u (1) i − mij]2 σ2 ij (7) where mij and σij are the mean and variance of the Gaussian

membership function, respectively, of the jth term of the ith input variable xi.

Nodes in layer 3 receive 1-D membership degrees of the associated rule from the nodes of a set in layer 2. Here, the product operator described earlier is adopted to perform the precondition part of the fuzzy rules. As a result, the output function of each inference node is

u(3)_j =

i

u(2)_ij (8) where the_iu(2)_ij of a rule node represents the firing strength of its corresponding rule.

Nodes in layer 4 are called consequent nodes. The input to a node in layer 4 is the output from layer 3, and the other inputs are calculated from a functional link neural network that has not used the function tanh(·), as shown in Fig. 4. For such a node

u(4)_j = u(3)_j ×

M

k = 1

(5)

where wk j is the corresponding link weight of

func-tional link neural network and φk is the functional

expansion of input variables. The functional expansion uses a trigonometric polynomial basis function, given by [x1sin(π x1) cos(π x1)x2sin(π x2) cos(π x2)] for 2-D input

variables. Therefore, M is the number of basis functions,

M = 3× N, where N is the number of input variables.

More-over, the output nodes of functional link neural network depend on the number of fuzzy rules of the FLNFN model.

The output node in layer 5 integrates all of the actions rec-ommended by layers 3 and 4 and acts as a defuzzifier with

y = u(5)= R j = 1 u(4)_j R j = 1 u(3)_j = R j = 1 u(3)_j _M k = 1 wk jφk R j = 1 u(3)_j = R j = 1 u(3)_j yˆj R j = 1 u(3)_j (10) where R is the number of fuzzy rules, and y is the output of the FLNFN model.

IV. LEARNINGALGORITHMS FOR THEFLNFN MODEL This section describes the proposed CCPSO method. Before the CCPSO method is designed, CPSO [29] that differs from the traditional PSO is introduced.

The traditional PSO uses one swarm of particles defined by the P -dimensional vectors to evolve. The CPSO method can change traditional PSO into P swarms of 1-D vectors, such that each swarm represents a dimension of the original problem. Fig. 5(a) and (b) shows the framework of the traditional PSO and CPSO method. The key point is that, instead of using one swarm (of I particles) to find the optimal P -dimensional vec-tor, the vector is split into its components so that P swarms (of

I particles each) optimize a 1-D vector. Notably, the function

that is being optimized still requires a P -dimension vector to be evaluated. However, if each swarm represents only a single dimension of the search space, it cannot directly compute the fitness of the individuals of a single population considered in isolation. A context vector is required to provide a suitable con-text in which the individuals of a population can be evaluated. To calculate the fitness for all particles in swarm, the other P -1 components in the context vector keep constant values, while the

pth component of the context vector is replaced, in turn, by each

particle from the pth swarm. Additionally, each swarm aims to optimize a single component of the solution vector essentially solving a 1-D optimization problem. Unfortunately, the CPSO still employs just the local best position and the global best po-sition of the traditional PSO to evolution process. Therefore, the CPSO may fall into a suboptimal solution. The CCPSO learning method, which combines the cooperative particle swarm opti-mization and the cultural algorithm to increase the global search capacity, is proposed to avoid trapping in a suboptimal solution and to ensure the ability to search for a near-global optimal solution.

The CCPSO method is characteristic of the cooperative par-ticle swarm optimization and cultural algorithm. Fig. 6 shows the framework of the proposed CCPSO learning method, which

Fig. 5. Framework of (a) PSO and (b) CPSO.

is based on a CPSO all of whose parameters are simultaneously tuned using the brief space of the culture algorithm (CA). The CCPSO method can strengthen the global search capability. If 50-dimensional vectors are used in the original PSO, then the vectors in CCPSO can be changed into 50 swarms of 1-D vec-tors. In the original PSO, the particle can exhibit 50 variations in each generation, whereas the CCPSO offers 50× 50 = 2500 different combinations in each generation. Additionally, each position of the CCPSO can be adjusted not only using the belief space that stores the paragons of each swarm, but also by search-ing around the local best solution and the global best solution. In the aforementioned scheme, the proposed CCPSO method can avoid falling into a suboptimal solution and ensure that the approximate global optimal solution can be found.

The detailed flowchart of the proposed CCPSO method is presented in Fig. 7. The foremost step in CCPSO is the coding of

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Fig. 6. Framework of the proposed CCPSO learning method.

Fig. 7. Flowchart of the proposed CCPSO learning method.

the neural fuzzy network into a particle. Fig. 8 shows an example of the coding of parameters of neural fuzzy network into a particle, where i and j represent the ith input variable and the jth rule, respectively. In this study, a Gaussian membership function

Fig. 8. Coding FLNFN model into a particle in the proposed CCPSO.

is adopted with variables that represent the mean and deviation of the membership function. Fig. 8 represents the neural fuzzy network given by (5), where mij and σij are the mean and

deviation of a Gaussian membership function, respectively, and

wk j represents the corresponding link weight of the consequent

part that is connected to the jth rule node. In this study, a real number represents the position of each particle.

The learning process is described step-by-step as follows.

Step 1: Create initial swarms

Before the CCPSO method is applied, every position xp,i(t)

must be created randomly in the range [0, 1], where p = 1, 2,

. . ., P represents the pth swarm, i = 1, 2, . . ., I represents the ith particle, and t denotes the tth generation.

Step 2: Create initial belief space

The belief space is the information repository in which the particles can store their experiences for other particles to learn from them indirectly. Create P belief space, Bp (p = 1, 2, . . .,

P ). Each initial Bp is defined as an empty set.

Step 3: Update every position

Step 3.1: Evaluate the performance function of each Particlei

The fitness function is used to evaluate the performance func-tion of each particle. The fitness funcfunc-tion is defined as follows

F = 1 D D d= 1 (yd − yd)2 (11)

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where yd represents the dth model output; ydrepresents the dth

desired output, and D represents the number of input data.

Step 3.2: Update local best position Lp,i and global best

position Gp

The local best position Lp,iis the best previous position that

yielded the best fitness value of the pth swarm of the ith particle, and the global best position Gp is generated by the whole local

best position. In Step 3.2, the first step updates the local best position. Compare the fitness value of each current particle with that of its local best position. If the fitness value of the current particle exceeds those of its local best position, then the local best position is replaced with the position of the current particle. The second step updates the global best position. Compare the fitness value of all particles in their local best positions with that of the particle in the global best position. If the fitness value of the particle in the local best position is better than those of the particles in the global best position, then the global best position is replaced with the current local best position

Lp,i(t + 1) =

xp,i(t), if F (xp,i(t)) < F (Lp,i(t))

Lp,i(t), if F (xp,i(t))≥ F (Lp,i(t))

Gp(t + 1) = arg min Lp , i

F (Lp,i(t + 1)), 1≤ i ≤ I. (12)

Step 3.3: Adjust each belief space Bp using an acceptance

function

The first part of Step 3.3 sorts these particles in each Swarmp

in the order of increasing fitness. Then, the paragon of each

Swarmp is put into the belief space Bp using an acceptance

function. This function yields the number of particles that are used to adjust each belief space, and is as follows. The number of accepted particles decreases as the number of generations increases

Naccepted = n%× I +

n%

t × I (13)

where n% is a parameter that is set by the user, and must specify the top performing 20% [34]; I is the number of particles, and

t represents the tth generation. The second step adjusts Bp.

The interval of belief space BIp is defined asBIp = [lp, up] =

{x|lp ≤ x ≤ up, x∈ }, where lp is the lower bound on belief

space Bpand upis the upper bound on belief space Bp. Then, the

position of each particle in Bpis compared with the lower bound

lp. If the position of the particle is smaller than the lower bound

lp, then the lower bound lpis replaced with the current position.

Furthermore, the position of each particle in the Bpis compared

with the upper bound up. If the position of the particle is greater

than the upper bound up, then the upper bound up is replaced

with the current position. These rules are given as follows:

lp = xp,i, if xp,i≤ lp lp, otherwise up = xp,i, if xp,i≥ up up, otherwise. (14)

Step 3.4: Generate each new Swarmp using

lp, up, Lp,i, and Gp

In Step 3.4, the first step adjusts every position of each Swarmp

using an influence function (15). This step can change the direc-tion of each particle in soludirec-tion space, not easily being trapped at a local optimum. Then, the second step updates the velocity and position of each particle to generate the each new Swarmp

using (16) and (17)

xp,i(t)=

xp,i(t) +|Rand() × (up − lp)| if xp,i < lp

xp,i(t)− |Rand() × (up − lp)| if xp,i > up

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vp,i(t + 1) = w× vp,i(t) + c1× Rand() × [Lp,i(t + 1)− xp,i(t)]

+ c2× Rand() × [Gp(t + 1)− xp,i(t)] (16)

xp,i(t + 1) = xp,i(t) + vp,i(t + 1) (17)

where c1 and c2 denote acceleration coefficients; Rand() is

generated from a uniform distribution in the range [0, 1], and w controls the magnitude of vp,i(t).

V. EXPERIMENTALRESULTS

This section discusses three examples that were considered to evaluate the FLNFN model with the CCPSO learning method. The first example involves predicting a chaotic signal that has been described in [7]; the second example involves predicting a chaotic time series [4], and the third example involves forecast-ing the number of sunspots [5].

A. Example 1: Prediction of a Chaotic Signal

In this example, an FLNFN model with a CCPSO learning method (FLNFN-CCPSO) was used to predict a chaotic signal. The classical time series prediction problem is a one-step-ahead prediction that has been described in [7]. The following equation describes the logistic function:

x(k + 1) = ax(k)(1− x(k)). (18) The behavior of the time series generated by this equation depends critically on parameter a. If a < 1, then the system has a single fixed point at the origin, and from a random ini-tial value between [0, 1], the time series collapses to a constant value. For a > 3, the system generates a periodic attractor. At

a≥ 3.6, the system becomes chaotic. In this example, a was

set to 3.8. The first 60 pairs (from x(1) to x(60)), with initial valuex(1) = 0.001, were the training dataset, while the remain-ing 100 pairs (from x(1) to x(100)), with initial value x(1) = 0.9, were the testing dataset used to validate the proposed method.

In this example, several particles will be found to minimize the fitness value using the proposed FLNFN-CCPSO method. The learning stage involved parameter learning using the CCPSO method. The coefficient ω was set to 0.4. The cognitive coef-ficient c1 1.6, and the society coefficient c2 was set to 2. The

swarm sizes were set to 50. The learning proceeded for 1000 generations, and was repeated 50 times. After 1000 genera-tions, the final average rms error of the predicted output is about 0.002285. In this example, three fuzzy rules are adopted. They

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Fig. 9. (a) Predictions of the proposed method. (b) Prediction errors of the proposed method. (c) Predictions of the PSO [19]. (d) Prediction errors of the PSO. (e) Predictions of the CPSO [29]. (f) Prediction errors of the CPSO.

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are shown as follows. Rule1: IF x is µ(0.763596, 19.0781) THEN ˆy1 =−0.846419 + 0.840237x + 0.0103279cos(π x) + 1.61874 sin(π x) − 0.364635x Rule2: IF x is µ(0.235112, 0.307009) THEN ˆy1= 0.145784− 0.961044x − 0.146496cos(π x) + 0.857966 sin(π x) + 6.62004x Rule3: IF x is µ(0.771367, 0.351594) THEN ˆy1 = 0.727383 + 0.625871x + 1.00717cos(π x) + 2.25003 sin(π x) + 0.178136x

where µ(mij, σij) represents a Gaussian membership function

with mean mij and deviation σij in the ith input variable and

the jth rule. Fig. 9(a) plots the predictions of the desired output and the model output in 1000 generations of learning. The solid line represents the desired output of the time series, and the no-tation “∗” represents the output of the FLNFN-CCPSO method. Fig. 9(b) presents the prediction errors of the proposed method. The experimental results demonstrate the perfect predictive ca-pability of theFLNFN-CCPSO method.

In this example, PSO [19] and CPSO [29] were applied to the same problem to show the effectiveness and efficiency of the FLNFN model with the CCPSO learning method. In the PSO and CPSO, the swarm sizes were set to 50. The coefficient

ω was set to 0.4. The cognitive coefficient c1was set to 1.6, and

the society coefficient c2was set to 2. Three rules were applied

to construct the fuzzy model. In the PSO [19] and CPSO [29], learning proceeded for 1000 generations, and was performed 50 times.

The performance of the FLNFN model with CCPSO learning was compared with the performance of other methods. First, the performance of the FLNFN-CCPSO method was compared with that of the PSO [19]. Fig. 9(c) plots the results predicted using PSO. Fig. 9(d) presents the prediction errors of the PSO. Second, CPSO [29] is adopted to solve the predictive problem. Fig. 9(e) and (f) plots the results and the errors of the CPSO. As presented in Fig. 9, the results predicted by the FLNFN model with the CCPSO learning method are better than those predicted by other methods.

Fig. 10 plots the learning curves of the best performance of the FLNFN model with the CCPSO learning method, PSO [19], and CPSO [29]. This figure indicates that the proposed method converges quickly and yields a lower rms error than the other methods. Computer simulations indicated that the proposed method outperforms other methods. The best performance of the CCPSO was compared with that of the PSO [19] and CPSO [29]. Table I compares the results. The comparison indi-cates that the rms error of training and predicting for the FLNFN-CCPSO method is better than those obtained using other methods.

Fig. 10. Learning curves of the proposed method, PSO [19] and CPSO [29]. TABLE I

COMPARISON OF THEBESTPERFORMANCE OF THE

CCPSO, PSO,ANDCPSOINEXAMPLE1

B. Example 2: Prediction of Chaotic Time Series

The Mackey–Glass chaotic time series x(t) was generated using the following delay differential equation

dx(t) dt =

0.2x(t− τ)

1 + x10_(t_{− τ)}− 0.1x(t). (19)

Crowder [4] extracted 1000 input–output data pairs{x, yd_}

using four past values of x(t)

[x(t− 18), x(t − 12), x(t − 6), x(t); x(t + 6)] (20) where τ = 17 and x(0) = 1.2. Four inputs to the FLNFN-CCPSO method, corresponded to these values of x(t), and one output was x(t + ∆t), where ∆t is a time interval into the future. The first 500 pairs (from x(1) to x(500)) were the training dataset, while the remaining 500 pairs (from x(501) to x(1000)) were the testing data used to validate the proposed method.

The learning stage entered parameter learning through the CCPSO method. The coefficient ω was set to 0.4. The cognitive coefficient c1 was set to 1.6, and the society coefficient c2 was

set to 2. The swarm sizes were set to 50. The learning proceeded for 1000 generations, and was performed 50 times. In this example, three fuzzy rules are applied. They are as follows

Rule1: IF x1 is µ(0.452959,−5.36833) and x2 is µ(−0.10799, 0.768855) and x3 is µ(−0.850613, −3.60999) and x4 is µ(1.09886, 0.495632) THEN ˆy1 = 2.20613 + 0.580829x1+ 0.391061 cos(πx1) + 0.332886 sin(π x1)− 4.68232x2 − 5.05388 cos(π x2) + 1.73753 sin(π x2)

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64 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 39, NO. 1, JANUARY 2009 − 0.656754x3+ 1.71626 cos(π x3) + 0.0923789 sin(π x3) + 4.93925x4 − 0.416084 cos(π x4) + 1.45935 sin(π x4) + 0.990628x1x2x3x4 Rule2 : IF x1 is µ(−0.596747, −0.896165) and x2 is µ(0.841226, 1.1499) and x3 is µ(0.20028, 0.310169) and x4 is µ(1.01531, 0.524704) THEN ˆy1 = 0.683119 + 0.649552x1 + 1.74121 cos(π x1)− 4.32156 sin(π x1) + 0.200504x2− 2.74432 cos(π x2) + 1.18918 sin(π x2) + 0.519391x3 + 0.641173 cos(π x3) + 3.17329 sin(π x3) − 0.22503x4+ 0.524293 cos(π x4) + 0.685239 sin(π x4) − 0.127742x1x2x3x4 Rule3: IF x1 is µ(1.03417, 0.919468) and x2 is µ(−0.115958, 1.69308) and x3 is µ(−0.114371, 1.1357) and x4 is µ(−0.152534, 0.74255) THEN ˆy1 = 0.58632− 1.28024x1 − 0.180169 cos(π x1)− 0.470873 sin(π x1) − 0.530146x2− 0.597328 cos(π x2) + 0.156929 sin(π x2) + 0.176057x3 + 0.0405789 cos(π x3) + 1.09262 sin(π x3) + 0.353992x4− 0.437468 cos(π x4) − 1.09654 sin(π x4) + 0.479358x1x2x3x4

the jth rule. The final rms error of the prediction output is about 0.008424. Fig. 11(a) plots the prediction outputs of the chaotic time series from x(501) to x(1000), when 500 training data from x(1) to x(500) were used. Fig. 11(b) plots the prediction errors between the proposed model and the desired output.

In this example, as in Example 1, the performance of the FLNFN model with the CCPSO learning method was compared to that of other methods. In the PSO [19] and CPSO [29], the parameters are the same as in Example 1. Three rules are set to construct the fuzzy model. The learning proceeded for 1000 generations, and was performed 50 times. Fig. 11(c) and (d) plots the predictions and the prediction errors of the PSO [19]. Fig. 11(e) and (f) plots the predictions and the prediction

er-Fig. 11. (a) Prediction results of the proposed method. (b) Prediction errors of the proposed method. (c) Prediction results of the PSO [19]. (d) Prediction errors of the PSO. (e) Prediction results of the CPSO [29]. (f) Prediction errors of the CPSO. (g) Prediction results of the DE [35]. (h) Prediction errors of the DE. (i) Prediction results of the GA [38]. (j) Prediction errors of the GA.

rors of the CPSO [29]. Fig. 11(g) and (h) plots the predictions and the prediction errors of differential evolution (DE) [35]. Fig. 11(i) and (j) plots the predictions and the prediction errors of the genetic algorithm (GA). Fig. 12 plots the learning curves of the best performance of the FLNFN model with CCPSO, PSO [19], CPSO [29], DE [35], and GA [38] learning methods. The proposed CCPSO method yields better prediction results than the other methods. Table II compares the best performance

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Fig. 12. Learning curves of the best performance of the proposed method, PSO [19], CPSO [29], DE [35], and GA [38].

TABLE II

COMPARISON OF THEBESTPERFORMANCE OF THECCPSO, PSO, CPSO, DE,ANDGAINEXAMPLE2

TABLE III

COMPARISON OF THEPERFORMANCE OFVARIOUSEXISTINGMODELS

of the CCPSO with those of the PSO [19], CPSO [29], DE [35], and GA [38]. Table III lists the generalization capabilities of the other methods [4], [36], [37]. The generalization capabili-ties were measured by using each model to predict 500 points immediately following the training dataset. The results show that the proposed FLNFN-CCPSO method offers a smaller rms error than the other methods.

C. Example 3: Forecast of the Number of Sunspots

The number of sunspots varied nonlinearly from 1700 to 2004, in nonstationary, and non-Gaussian cycles that are dif-ficult to predict [5]. In this example, the FLNFN model with the CCPSO learning method was used to forecast the number of sunspots The inputs xi of the FLNFN-CCPSO method are

defined as x1(t) = yd1(t− 1), x2(t) = yd1(t− 2) and x3(t) =

yd

1(t− 3), where t represents the year and y1d(t) represents the

number of sunspots in the year t. In this example, the num-ber of sunspots of the first 151 years (from 1703 to 1853) was used to train the FLNFN-CCPSO method while the number of

sunspots of all 302 years (from 1703 to 2004) was used to test the FLNFN-CCPSO method.

The learning stage involved parameter learning by the CCPSO method. The coefficient ω was set to 0.4. The cognitive coeffi-cient c1was set to 1.6, and the society coefficient c2 was set to

2. The swarm sizes were set to 50. The learning proceeded for 1000 generations, and was performed 50 times. In this example, three fuzzy rules are applied. They are as follows:

Rule 1: IF x1is µ(0.845312, 0.508771) and x2 is µ(0.90418, 0.451389) and x3 is µ(−0.0895866, 1.06449) THEN ˆy1 =−3.35896 − 0.436238x1+ 1.4272 cos(π x1) − 0.417788 sin(π x1) + 2.19244x2 − 0.32409 cos(π x2) + 0.2113 sin(π x2) − 1.36183x3− 0.480986 cos(π x3) + 2.59738 sin(π x3)− 0.361671(x1x2x3) Rule 2: IF x1is µ(1.44016, 0.616583) and x2 is µ(0.314697, 7.34735) and x3 is µ(2.38597, 1.31093) THEN ˆy1 = 1.88537 + 1.78931x1 + 1.73373 cos(π x1) + 2.86658 sin(π x1) + 4.53188x2− 3.75512 cos(π x2) − 7.18406 sin(π x2) + 0.868682x3 − 0.541793 cos(π x3) + 1.52449 sin(π x3) + 0.763891(x1x2x3) Rule 3: IF x1is µ(0.115385, 0.93777) and x2 is µ(0.326872, 1.02448) and x3 is µ(0.984958, 0.403378) THEN ˆy1 = 1.56458 + 0.703153x1 + 0.0115128 cos(π x1)− 0.119185 sin(π x1) − 0.0263568x2− 0.681762 cos(π x2) + 0.478785 sin(π x2) + 7.0577x3 − 0.808627 cos(π x3) + 0.462158 sin(π x3) + 10.6957(x1x2x3)

the jth rule. The final rms error of the forecast output is about 10.337347. Fig. 13(a) presents the forecast outputs for years 1703–2004, using 151 training data from years 1703 to 1853. Fig. 13(b) plots the forecast errors between the proposed model and the desired output.

In this example, as in Examples 1 and 2, the performance of the FLNFN model with the CCPSO learning method was

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Fig. 13. (a) Forecast results of the proposed method. (b) Forecast errors of the proposed method. (c) Forecast results of the PSO [19]. (d) Forecast errors of the PSO. (e) Forecast results of the CPSO [29]. (f) Forecast errors of the CPSO. (g) Forecast results of the DE [35]. (h) Forecast errors of the DE. (i) Forecast results of the GA. (j) Forecast errors of the GA [38].

compared with that of the other methods. In PSO [19] and CPSO [29], the parameters are the same as in Examples 1 and 2. Three rules are used to construct the fuzzy model. The learning proceeded for 1000 generations, and was performed 50 times. Fig. 13(c) and (d) plot the forecast results and the forecast errors of the PSO [19]. Fig. 13(e) and (f) plots the forecast results and the forecast errors of the CPSO [29]. Fig. 13(g) and (h) plots the forecast results and the forecast errors of the DE [35]. Fig. 13(i)

Fig. 14. Learning curves of the best performance of the proposed method, PSO [19], CPSO [29], DE [33], and GA [38].

TABLE IV

COMPARISON OF THEBESTPERFORMANCE OF

CCPSO, PSO,ANDCPSOINEXAMPLE3

TABLE V

COMPARISON OF THEPERFORMANCE OFVARIOUSEXISTINGMODELS

and (j) plots the forecast results and the forecast errors of the GA. Fig. 14 plots the learning curves of the best performance of the FLNFN model with CCPSO, PSO, CPSO, DE, and GA learning. The proposed CCPSO learning method yields better forecast results than the other methods. Table IV presents the best rms errors of training and forecasting for the CCPSO, PSO [19], CPSO [29], DE [35], and GA [38] learning methods. Table V lists the generalization capabilities of other methods [36], [37]. As presented in Tables IV and V, the proposed FLNFN-CCPSO method outperforms the other methods.

VI. CONCLUSION

This study proposes an efficient cultural cooperative par-ticle swarm optimization learning method for the functional-link-based neural fuzzy network in predictive applications. The FLNFN model can generate the consequent part of a nonlinear combination of input variables. The proposed CCPSO method with cooperative behavior among multiple swarms increases the global search capacity using the belief space. The advantages of the proposed FLNFN-CCPSO method are as follows. 1) The consequent of the fuzzy rules is a nonlinear combination of input variables. This study uses the functional link neural network to the consequent part of the fuzzy rules. The functional expansion in the FLNFN model can yield the consequent part of a nonlinear combination of input variables; 2) the proposed CCPSO with cooperative behavior among multiple swarms can accelerate the search and increase global search capacity using the belief

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space. The experimental results demonstrate that the CCPSO method can obtain a smaller rms error than the generally used PSO and CPSO for solving time series prediction problems.

Although the FLNFN-CCPSO method can perform better than the other methods, there is an advanced topic to the pro-posed FLNFN-CCPSO method. In this study, the number of rules is predefined. In future studies, it would be better if the proposed method has the ability to determine the number of fuzzy rules.

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Cheng-Jian Lin (S’93–M’95) received the B.S. de-gree in electrical engineering from Tatung University, Taipei, Taiwan, R.O.C., in 1986, and the M.S. and Ph.D. degrees in electrical and control engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 1991 and 1996, respectively.

From April 1996 to July 1999, he was an Associate Professor in the Department of Electronic Engineer-ing, Nan-Kai College, Nantou, Taiwan. From August 1999 to January 2005, he was an Associate Professor in the Department of Computer Science and Infor-mation Engineering, Chaoyang University of Technology, Taichung, Taiwan, where from February 2005 to July 2007, he was a full Professor. Currently, he is a full Professor in the Department of Electrical Engineering, National Uni-versity of Kaohsiung, Kaohsiung, Taiwan. From 2001 to 2005, he served as the Chairman of the Department of Computer Science and Information Engineer-ing, Chaoyang University of Technology, where, from 2005 to 2007, he served as the Library Director of Poding Memorial Library. He is the author or coauthor of more than 150 papers published in referred journals and conference proceed-ings. His current research interests include soft computing, pattern recognition, intelligent control, image processing, bioinformatics, and field-programmable gate array design.

Prof. Lin is a member of the Phi Tau Phi. He is also a member of the Chinese Fuzzy Systems Association, the Chinese Automation Association, the Taiwanese Association for Artificial Intelligence (TAAI), the Institute of Elec-tronics, Information, and Communication Engineers, and the IEEE Computa-tional Intelligence Society. He is an Executive Committee Member of the TAAI. He has served as the Associate Editor of the International Journal of Applied

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Cheng-Hung Chen (S’07) was born in Kaohsiung, Taiwan, R.O.C., in 1979. He received the B.S. and M.S. degrees in computer science and information engineering from the Chaoyang University of Tech-nology, Taichung, Taiwan, in 2002 and 2004, respec-tively. He is currently working toward the Ph.D. de-gree in electrical and control engineering at National Chiao-Tung University, Hsinchu, Taiwan.

His current research interests include fuzzy sys-tems, neural networks, evolutionary algorithms, in-telligent control, and pattern recognition.

Chin-Teng Lin (S’88–M’91–SM’99–F’05) received the B.S. degree in control engineering from National Chiao-Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1996, and the M.S.E.E. and Ph.D. de-grees in electrical engineering from Purdue Uni-versity, West Lafayette, IN, in 1989 and 1992, respectively.

Since August 1992, he has been with the Col-lege of Electrical Engineering and Computer Science, NCTU, where he is currently the Provost of Aca-demic Affairs and the Chair Professor of electrical and control engineering. He has served as the Founding Dean of the Computer Science College of NCTU from 2005 to 2007. He is the author or coauthor of more than 110 journal papers, including about 80 IEEE Transactions papers. He is the author of a textbook Neural Fuzzy Systems (Prentice-Hall, 1996) and

Neural Fuzzy Control Systems with Structure and Parameter Learning (World

Scientific, 1994). His current research interests include intelligent technology, soft computing, brain–computer interface, intelligent transportation systems, robotics and intelligent sensing, and nanobioinformation technologies and cog-nitive science.

Prof. Lin is a member of Tau Beta Pi, Eta Kappa Nu, and Phi Kappa Phi hon-orary societies. He was a Member of the Board of Governors BoG of the IEEE Systems, Man, Cybernetics Society (SMCS) from 2003 to 2005, and is the cur-rent BoG member of the IEEE Circuits and Systems Society (CASS). He was the IEEE Distinguished Lecturer from 2003 to 2005. He also serves as the Deputy Editor-in-Chief (EIC) of the IEEE TRANSACTIONS ONCIRCUITS ANDSYSTEMS, PARTII now. He was the Program Chair of the 2006 IEEE International Con-ference on Systems, Man, and Cybernetics held in Taipei. He was the President of the Board of Government of the Asia Pacific Neural Networks Assembly from 2004 to 2005. He has been the recipient of several awards including the Outstanding Research Award granted by the National Science Council (NSC), Taiwan, since 1997 to present, the Outstanding Professor Award granted by the Chinese Institute of Engineering in 2000, and the 2002 Taiwan Outstanding Information Technology Expert Award. He was also elected to be one of 38th Ten Outstanding Rising Stars in Taiwan in 2000.