Abstract—In this paper, we proposed a functional-link-based

neural fuzzy network to improve the traditional TSK-type neural fuzzy network. Besides, an efficient evolutionary learning algorithm, called the Symbiotic Taguchi-based Modified Differential Evolution (STMDE), is proposed for the neural fuzzy networks design. Firstly, in order to avoid trapping in a local optimal solution and to ensure the searching capability of near global optimal solution, the STMDE adopts the Taguchi method to effectively search towards the best individual and employs an adaptive parameter control to adjust scaling factor which is called the Taguchi method. Moreover, the proposed STMDE introduces the concept of symbiotic evolution to improve the individual structure. Unlike the traditional individual that uses each one in a population as a full solution to a given problem, symbiotic evolution assumes that each individual in a population represents only a partial solution, while complex solutions combine several individuals in the population.

I. INTRODUCTION S inheriting the advantages of fuzzy system and neural network simultaneously, neural fuzzy networks (NFNs)

have been demonstrated their advantage in lots aspects of research. In other words, NFNs have the inference characteristic of the fuzzy system and with the learning ability of the neural network to adjust fuzzy rules automatically. Therefore, NFNs have become a popular research target progressively, and been applied to various applications, such as in the fields of control, prediction, classification and pattern recognition.

In NFNs, it is necessary to apply some learning algorithm for network parameter adjusting. Many NFN approaches were implemented by using traditional backpropagation (BP) learning algorithm which is based on gradient descents that are known to be easily trapped at local minima. The other drawback of applying the BP algorithm is it will increase the complexity of solving the problem. Recently, the approach of evolutionary computation has been designed to optimize parameters of NFNs and achieved great success. Many researches have been successfully utilized evolutionary algorithms to solve a lot of problems. Several evolutionary algorithms, such as genetic algorithm (GA) and Immune Algorithm (IA), are able to efficiently explore the desired

Manuscript received March 22, 2010. C.-J. Lin, C.-H. Hsu, S.-Y. Wu and C.-C. Peng are with the Department of

Computer Science and Information Engineering, National Chin-Yi University of Technology, Taichung County, Taiwan, R.O.C. (e-mail: cjlin@ncut.edu.tw, chhsu828@gmail.com, SiaoYin28@gmail.com, goudapeng@gmail.com)

global search space, but the drawbacks of local minimum and premature convergence remain the same. Therefore, technologies that can be used to train the system parameters and find the global solution while optimizing the overall structure are generally required. Simultaneously, a new optimization algorithm, called the differential evolution (DE), is an evolutionary computation approach that was developed by Rainer and Kenneth in 1995 [1]. Recently, DE has emerged as a robust numerical optimization algorithm and been successfully applied to tackle various difficult optimization problems [2]. Basically, DE is a fast and easy-to-use method, which is not only astonishingly simple, but also performs extremely well on a wide variety of applications. However, DE sometimes explores too many search points before locating the global optimum. In addition, though DE is particularly simple to work with, i.e., having only a few control parameters, proper choice of these parameters is often critical to the performance of DE [3].

In this paper, a learning algorithm, called the symbiotic Taguchi-based modified differential evolution (STMDE) is proposed for designing of a functional-link-based neural fuzzy network (FNFN). Firstly, the proposed STMDE has two crucial ideas to balance the exploration abilities. The proposed STMDE adopts a method to effectively search towards the best individual and employs an adaptive parameter control procedure to adjust the scaling factor of the traditional DE algorithm. Therefore, STMDE does not only explore the search space by randomly chosen individuals, but also exploits the search capability of a near global optimal solution by the best individual currently. In addition, we use the Taguchi method to obtain the better evolutionary direction. Taguchi method was developed by Taguchi and Konishi in 1950 [4], while these techniques have been utilized widely in engineering analysis to optimize the performance characteristics within the combination of design parameters [5]. Tsai et al. [5] proposed a hybrid Taguchi-genetic algorithm (HTGA) which inherits both the merits of powerful global exploration capability of the traditional GAs and exploiting the optimum offspring of the Taguchi method. Taguchi technique is a powerful tool for the design of high quality systems. It introduces an integrated approach which is simple and efficient to find the best range of designs for quality, performance, and computational cost. It is designed for simply value analysis via orthogonal arrays. The characteristic of Taguchi method is to utilize fewer experiment sets and to grant more important information. Furthermore, the Taguchi method can point out the trend of optimization. On the other hand, the STMDE introduces the

An Efficient Symbiotic Taguchi-based Differential Evolution for Neuro-Fuzzy Network Design

Cheng-Jian Lin, Chia-Hu Hsu, Siao-Yin Wu, and Chun-Cheng Peng

A

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Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

concept of symbiotic evolution [6] to improve the traditional structure of the DE. Different to the traditional DE that uses each individual in a population as a full solution to a specific problem, symbiotic evolution assumes that each individual in a population represents only a partial solution to a problem. As the result, complex solutions combine several individuals in the population.

II. STRUCTURE OF THE FUNCTIONAL-LINK-BASED NEURAL FUZZY NETWORK

This section describes the structure of the FNFN model. The FNFN model adopted the functional link neural network (FLNN) generating complex nonlinear combination of the input variables as the consequent part of the fuzzy rules. Each fuzzy rule corresponds to a sub-FLNN, comprising a functional link. Fig. 1 presents the structure of the proposed FNFN model. The FNFN model realizes a fuzzy if-then rule in the following form. Rule-j:

1 1 2 2IF is and is ... and is ... and is ,i ij N Njj jx A x A x A x A

1

1 1 2 2

ˆTHEN

... ,

M

j kj kk

Mj Mj j

y w

w w w

φ

φ φ φ=

=

= + + +

∑ (1)

where xi and ˆ jy are the input and local output variables, respectively, Aij is the linguistic term of the precondition part with Gaussian membership function, N the number of input variables, wkj is the link weight of the local output,

kφ the basis trigonometric function of the input variables, M the number of basis function, and Rule-j the j-th fuzzy rule.

1̂y 2̂y 3̂y

1φ 2φ Mφ

∑ ∑ ∑

Fig. 1. Structure of proposed FNFN model.

The operation functions of the nodes in each layer of the FNFN model are now described. In the following description, u(l) denotes the output of a node in the l-th layer.

Layer 1 (Input layer): No computation is performed in this layer. Each node in this layer is an input node, which corresponds to one input variable, and only transmits input values to the next layer directly:

(1)iiu x= (2)

Layer 2 (Membership function layer): Nodes in this layer correspond to a single linguistic label of the input variables in Layer 1. Therefore, the calculated membership

value specifies the degree to which an input value belongs to a fuzzy set in layer 2. The implemented Gaussian membership function in layer 2 is

(1) 2(2)

2 ,[ ]exp iji

ijij

u mu σ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

−= − (3)

where and ijσ are the mean and variance of the Gaussian membership function, respectively, of the j-th term of the i-th input variable xi.

Layer 3 (Rule layer): Nodes in this layer represent the preconditioned part of a fuzzy logic rule. Here, the product operator described above is adopted to perform the IF-condition matching of the fuzzy rules. As a result, the output function of each inference node is (3) (2) ,j ij

iu u= ∏ (4)

where (2)uiji∏ of a rule node represents the firing strength of

its corresponding rule. Layer 4 (Consequent layer): Nodes in this layer are

called consequent nodes. The input to a node in layer 4 is the output from layer 3, and the other inputs are nonlinear combinations of input variables from a FLNN, as depicted in Fig. 1. For such a node,

(4) (3)

1,

M

kj kj jk

u u w φ=

= ⋅∑ (5)

where wkj is the corresponding link weight of functional link neural network andψk is the functional expansion of input variables. The functional expansion uses a trigonometric polynomial basis function, given by

sin ( ) cos ( ) sin ( ) cos( )1 1 1 2 2 2x x x x x xπ π π π⎡ ⎤⎣ ⎦

for

two-dimensional input variables. Therefore, M is the number of basis functions, i.e., 3M N= × , where N is the number of input variables.

Layer 5 (Output layer): Each node in this layer corresponds to a single output variable. The node integrates all of the actions recommended by layers 3 and 4 and acts as a defuzzifier with

(3)(4) (3)

1 11 1(5)

(3) (3) (3)

1 1 1

ˆ,

R MR R

jkj kjj jj kj j

R R R

j j jj j j

u wu u yy u

u u u

φ= == =

= = =

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠= =

∑ ∑∑ ∑= =

∑ ∑ ∑

(6)

where R is the number of fuzzy rules, and y is the output of the FNFN model. III. THE PROPOSED SYMBIOTIC TAGUCHI-BASED MODIFIED

DIFFERENTIAL EVOLUTIONARY ALGORITHM In a general evolution algorithm, a single individual is

responsible for the overall performance, with a fitness value assigned to that individual according to its performance, it can be found in the traditional DE. It also applies each individual to represent a population as a full solution to a problem. As stated previously, symbiotic evolution assumes that each individual in a population represents only a partial solution to a problem; the goal of each individual is to form a partial solution that can be combined with other partial

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solutions currently in the population to build an effectively full solution. The general structure of the individuals in the symbiotic evolution is shown in Fig. 2.

j2σj1σ ijσ

Fig. 2. The representation of a fuzzy system by STMDE.

The learning process of the STMDE includes the coding, initialization, fitness evaluation, parameter learning and solution aging mechanisms. The flowchart of the proposed STMDE algorithm is shown in Fig. 3(a) and Fig. 3(b), while the whole learning process is described phase-by- phase as follows.

A. Initialization Phase The coding step is related with the membership functions

and fuzzy rules of a fuzzy system that represent sub-individuals suitable for symbiotic evolution. The initialization step assigns the population values before the evolution process begins.

Coding Step The foremost step in the STMDE is the coding of the

individual (rule) into a fuzzy system. Fig. 4 shows an example of the coding of parameters of the neural fuzzy network into a fuzzy system where i and j represent the i-th input variable and the j-th rule, respectively. ijm and ijσ are the mean and

variance of a Gaussian membership function, respectively, and kjw represents the corresponding link weight of the

consequent part that is connected to the j-th rule node. In this study, a real number is used to indicate the element of each rule.

Create Initial Population Before the STMDE method is applied, every individual gix , must be created randomly within the range [0, 1], where

i=1, 2, …, PS, g is the generation index and PS the population size.

Fig. 3(a). Flowchart of the proposed STMDE designs method.

A

B

Create a mutated individual

Crossover

MutationBuilding a orthogonal arrays for the Taguchi

experiment

Set up two levels for three factors

The end of all experiments?

Compute the SNR of this experiment

Record the best levels

No

Yes

Fig. 3(b). Flowchart of the Taguchi method in the TMDE algorithm.

j1σ j2σ ijσ

Fig. 4. Coding FNFN model into a fuzzy system in the STMDE.

B. Evaluation Phase The fitness value of a fuzzy system is computed by the

fitness values of all the feasible combinations of that rule with all other randomly selected rules. The details of assigning the fitness value are described as follows. Step 1: Randomly select R fuzzy rules (individuals) from each subgroup with size PS for composing the fuzzy system. Step 2: Evaluate every fuzzy system that is generated from step1 to obtain a fitness value. The fitness function is defined as follows.

( )2

1

1 ,11

tNd

k kt k

fitness value

y yN =

=

+ −∑

(7)

where yk represents the model output of the k-th data, dky the

desired output of the k-th data, and Nt the number of the training data. Step 3: Compare fitness value of each current fuzzy system with the one of the best fuzzy system. If the current fitness value is higher than the best fuzzy system has, then the best

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fuzzy system will be replaced with the current fuzzy system. Step 4: Repeat the above steps until each rule (individual) in each subgroup has been selected a sufficient number of times.

C. Parameter Learning Phase In this phase, the STMDE performs the parameter learning

based on a sub-population, so all individuals are updated using the proposed TMDE method. Each sub-population allows the individual (rule) itself to evolve by evaluating the composed fuzzy system. Fig. 5 shows the structure of the individual in the rule-based symbiotic modified differential evolution.

Fig. 5. Structure of the individual in the STMDE.

The TMDE comprises of two major sub-phases, i.e., the procedures of parents choice and reproduction, which are summarized as follows.

1. Sub-phase of Parent Choice with Taguchi method In order to obtain the better evolutionary direction, we have

used the Taguchi method to determine the best parent combination according to some experimental results. The Taguchi method is based on orthogonal array experiments that provide many reduced variances and with optimal setting of control parameters. In this paper the two-level orthogonal array has been applied. The general symbol for two-level standard orthogonal arrays is defined as

( ),ca

L b (8)

where L represents the symbol of Latin square, a the number of the experimental, b the level of each factors, and c the number of factors.

Experimental Factors Determination and Orthogonal Array Construction In traditional DE algorithm, each individual in the current

generation is allowed to breed through mating with other randomly selected individuals from the population. Specifically, for each individual ,i gx , i=1, 2, …, PS, where g

denotes the current generation, three other random individuals

1 ,r gx , grx ,2 and grx ,3

are selected from the

population such as r1, r2, and r3 { }PS,...2,1 ∈ and

.1 2 3i r r r≠ ≠ ≠ Therefore, a mutated individual giv , can be

generated, according to the following equation

1 2 3, , , ,( ),i g r g r g r gv x F x x= + ⋅ − (9)

where F is commonly known as a scaling factor. In order to balance the evolutionary direction and improve the search capability, we have proposed a new update function of the mutated individual, while the function of an individual is stated as below

1 2 3

1

(1 ) ( )

( ),i,g r ,g r ,g r ,g

best,g r ,g

v x F x x

F x x

= + − ⋅ −

+ ⋅ − (10)

where F is defined as /g G to control the rate at which the population evolves, g denotes the current generation, G the maximum number of generations, and ,best gx the current best

individual. In this way, a parent pool of four individuals is formed to breed an offspring, while the evolutionary direction and the three parameters, i.e., r1, r2, and r3, exist in a close relation. Therefore, we adopted these parameters as the factors in the Taguchi method. Table I depicts a 3

4(2 )L

two-level orthogonal array used in our proposed method. In this table, the column of “Experiment #” represents the numbers of experiment which in this case, there are 4 numbers. For the )2( 3

4L orthogonal array, an experiment is

designed and proceeded in order to understand the influence of the three different independent factors (r1, r2, and r3), while each factor is with two levels. In Table I the column under “Performance value” represents a value of η which evaluates the performance of each experiment by signal-noise rate (SNR).

TABLE I: 34

2L ( ) TWO-LEVEL ORTHOGONAL ARRAY

Experiment # Factors Performance value (SNR) r1 r2 r3

1 1 1 1 η1 2 1 2 2 η2 3 2 1 2 η3 4 2 2 1 η4

The SNR has been originally used as the index of quality characteristic in communication engineering. Taguchi, whose background was communication and electronic engineering, introduced the same concept into the design of experiments. The homologous table of the factor to level is shown as in Table II.

TABLE II: THE HOMOLOGOUS TABLE OF THE FACTORS TO LEVEL.

Level # Factors

r1 r2 r3 1 r1,1 r2,1 r3,1

2 r1,2 r2,2 r3,2

Note that, in Table II, rf,l represents parental index (between 1 to PS) corresponding to the level l of each factor rf. In this paper, we have used tournament selection which chooses the

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best performing individual. The process was that we picked three individuals randomly and selected the one with the best fitness value. As the process was repeated six times, six different individuals from the population would be granted.

Experiment Analysis and Factor Effect Evaluation The performance value in Taguchi method is defined as the

quality characteristic. In general, we need to transfer the quality characteristic into the SNR, and the transformation way of SNR can be classified into three types, i.e., nominal-is-best, smaller-the-better, and larger-the-better. The transformation of the larger-the-better type has been applied in this paper, and it was calculated by

101

2

1 110 log .1

1 ( )

N

kd

k k

t

tN

y y

η=

⎛ ⎞⎜ ⎟⎜ ⎟

= − ⋅ ⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ −⎝ ⎠⎝ ⎠

∑ (11)

The effects of the various factors can be defined as , ,f,l iE η level l at factor f= ∀∑ 　 (12)

where i is the experiment number, f the factor name, and l the level number. Therefore, we can determinate which level in each factor can obtain the best performance, according to the effective value. Assume that all experiments are completed, and the performance is described as in Table III. The best level (BL) of this generation can be stated as following,

,1 ,2,1.

2 ,f f

fif E E

BLotherwise

>⎧= ⎨⎩

(13)

For example, E1,2 is bigger than E1,1 in Table III, so the best level of the factor 1 is equal to level 2. Finally we can obtain the best parent combination, i.e., r1,2, r2,1 and r3,1. TABLE III: 3

42L ( ) TWO-LEVEL ORTHOGONAL ARRAY (AFTER EVALUATION)

Experiment # Factors Performance

value (SNR) r1 r2 r3 1 1 1 1 -1.93 2 1 2 2 -3.06 3 2 1 2 -1.02 4 2 2 1 -1.59

Ef,1 -4.99 -2.95 -3.52

Ef,2 -2.61 -3.65 -4.08

Best level (BL) 2(r1,2) 1(r2,1) 1(r3,1)

2. Sub-phase of Reproduction

Offspring Generation After choosing the parents the TMDE approach applies a

differential operation to generate a mutated individual ,,i gv

according to (8). To complement the differential operation search strategy, then uses a crossover operation, often referred to as discrete recombination, the mutated individual

,i gv should be mated with gix , and generate the offspring

giu , . The element of individual ,i gu are inherited from gix ,

and ,,i gv and determined by a parameter called crossover

probability ( [0, 1]CR ∈ ), defined as follows:

,,

,

, if ,

, if ( )( )

id gid g

id g

v Rand d CRu x Rand d CR

⎧⎪= ⎨⎪⎩

≤>

(14)

where 1,2,...,d D= denotes the d-th element of individual vectors and ]1,0[)( ∈dRand is the d-th evaluation of a random number generator.

Mutation To prevent the STMDE from being trapped in local optima

of the search space (i.e., problems in which there are a number of points that are better than all their neighboring solutions, but do not have as good a fitness as the globally optimal one), we adopt a mutation scheme which maintains diversity in the population to increase the search capability. A one-point mutation operation is applied in this paper. For a given 1 2( , , ..., , ..., )d Dx x x x x= if the elements dx is randomly selected, then the resulted offspring is

1 2( , , ..., , ..., ).d Dx x x x x′= The new element dx′ is created randomly in the range [0, 1].

Survivor Selection The proposed STMDE approach applies selection pressure

only when selecting survivors. In order to compare the fitness values between the current composed fuzzy system, the trial composed fuzzy system and the best fuzzy system, there are several cases should be considered. Firstly, if the fitness value of the current composed fuzzy system exceeds those of the best fuzzy system, then the best fuzzy system should be replaced by the current composed fuzzy system. If the fitness value of the trial composed fuzzy system exceeds those of the best fuzzy system, and then the best fuzzy system should be replaced by the trial composed fuzzy system.

D. Solution Aging Mechanism In order to reinforce the performance of the proposed

algorithm, we have applied the solution aging mechanism in the STMDE algorithm to overcome bottlenecks, such as trapping into local optima. Here, we have used a counter cr,k to record the number of times if the fitness of each individual in the current generation is smaller than the previous generation. If the counter value exceeds a threshold ε, we will generate a neighborhood to maintain the solution variation. The following equation generates a new individual on feasible space, while the new individual is a neighborhood of the old one and stated as below,

, 1 , ,i g i gx x α+ = + (15)

where g denotes the current generation, and α is a parameter that controls the distance between the new individual and the old individual, which is within the interval [0.001,-0.001]. After generating the new individual, the counter will be reset to zero.

IV. ILLUSTRATIVE EXAMPLE In order to evaluate the performance of the proposed

183

STMDE approach, several experiments were conducted on a prediction problem, while the initial parameters are given in Table IV. The related programs were developed using Visual C++ 6.0 and the chosen problem was simulated for 10 runs on a Pentium IV 3.2GHz Windows desktop computer.

TABLE IV: THE INITIAL PARAMETERS BEFORE TRAINING

Parameters Value Number of fuzzy system (N) 50

Generation 500 Coding Type Real Number

CR 0.9 Mutation rate 0.3

Ε 3

Example: Forecast of the Sunspot Number The sunspot numbers (from 1700 to 2004) exhibit the

nonlinear, non-stationary, and non-Gaussian cycles, which is difficult to predict [7]. The inputs xi are defined as

( ) ( 1),1 1dx t y t= − ( ) ( 2)2 1

dx t y t= − and, ( ) ( 3),3 1dx t y t= − where

t represents the year and ( )1dy t is the sunspot numbers at the

t-th year. In this example, the first 180 years (from 1705 to 1884) of the sunspot numbers were used to train the proposed STMDE and TMDE method, while the remaining 121 years (from 1885 to 2004) of the sunspot numbers were for testing. The difference between the proposed STMDE and the TMDE is that the TMDE uses each individual to represent a population as a full solution.

Fig. 6. Learning curves of the STMDE and some baseline methods.

TABLE V: PERFORMANCE COMPARISON OF VARIOUS EXISTING METHODS

Methods RMSE (best)

RMSE (average)

Training error (best)

GA 14.90421 21.29034 11.94042 IA 14.58572 20.27216 10.75237 DE 13.44143 15.22812 10.07805

TMDE 11.83080 12.79109 9.022997 STMDE 11.07366 11.88599 8.563105

Methods Forecasting error (best)

Training error (average)

Forecasting error (average)

GA 14.48591 15.96930 20.70072 IA 13.67878 15.10594 19.31841 DE 12.62501 11.53510 14.36850

TMDE 11.49548 9.632608 12.33428 STMDE 10.69706 9.078579 11.69699

In this example, we set the numbers of fuzzy rules to three. After 500 generations of training, the final average RMSE (root mean squared error) of the prediction output of the

proposed STMDE approximates 11.88599. We also compared the performance with the TMDE, DE, IA and GA methods, while the generation numbers of the evolution learning was also set to 500. The best cases of their learning curves of the five compared approaches are depicted in Fig. 6. Table V summarizes the RMS error, the training error

(governed by 1884

1705

( ) ( )1 1

180t

dy t y t

=

−∑ ) and the forecasting error

(governed by 2004

1885

( ) ( )1 1

121t

dy t y t

=

−∑ ). Note that the average

error and minimum (best) error of each method were calculated from ten different simulations, respectively. As indicated in Table V, our proposed STMDE method outperforms other methods, in terms of lower RMSEs and forecasting errors.

V. CONCLUSION This paper proposes a functional-link-based neural fuzzy

network (FNFN) based on a symbiotic Taguchi-based modified differential evolutionary (STMDE) for a prediction problem. The proposed FNFN adopts a nonlinear combination of input variables to the consequent part of fuzzy rules and applies the STMDE to optimize the system parameters. The STMDE adopts the concept of symbiotic evolution to yield several variable fuzzy systems to find suitable rule combinations. In order to avoid trapping into the local optimal, we have also used the solution of aging mechanism. This STMDE method has succeeded in solving the problem of the convergence speed and learning accuracy. The experimental results demonstrate that the proposed STMDE can achieve better performances, comparing to the generally used TMDE, DE, IA and GA methods, for solving a extremely difficult prediction application.

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