IAENG International Journal of Computer Science, 42:2, IJCS_42_2_06. Approximation Capabilities of Interpretable Fuzzy Inference Systems

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1 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 Approxiation Capabilities of Interpretable Fuzzy Inference Systes Hirofui Miyajia, Noritaka Shigei, and Hiroi Miyajia 3 Abstract Many studies on odeling of fuzzy inference systes have been ade. The issue of these studies is to construct autoatically fuzzy inference systes with interpretability and accuracy fro learning data based on etaheuristic ethods. Since accuracy and interpretability are contradicting issues, there are soe disadvantages for self-tuning ethod by etaheuristic ethods. Obvious drawbacks of the ethod are lack of interpretability and getting stuck in a shallow local iniu. Therefore, the conventional learning ethods with ulti-objective fuzzy odeling and fuzzy odeling with constrained paraeters of the ranges have becoe popular. However, there are little studies on effective learning ethods of fuzzy inference systes dealing with interpretability and accuracy. In this paper, we will propose a fuzzy inference syste with interpretability. Firstly, it is proved theoretically that the proposed ethod is a universal approxiator of continuous functions. Further, the capability of the proposed odel learned by the steepest descend ethod is copared with the conventional odels using function approxiation and pattern classification probles in nuerical siulation. Lastly, the proposed odel is applied to obstacle avoidance and the capability of interpretability is shown. Index Ters approxiation capability, fuzzy inference, interpretability, universal approxiator, obstacle avoidance. I. INTRODUCTION FUZZY inference systes are widely used in syste odeling for the fields of classification, regression, decision support syste and control [], []. Therefore, any studies on odeling of fuzzy inference systes have been ade. The issue of these studies is to construct autoatically fuzzy systes with interpretability and accuracy fro learning data based on etaheuristic ethods [3] [7]. Since accuracy and interpretability are contradicting issues, there are soe disadvantages for self-tuning ethod by etaheuristic ethods. Obvious drawbacks of the ethod are lack of interpretability and getting stuck in a shallow local iniu [8]. As etaheuristic ethods, soe novel ethods have been developed which ) use GA and PSO to deterine the structure of fuzzy systes [7], [9], ) use fuzzy inference systes coposed of sall nuber of input rule odules, such as SIRMs and DIRMs ethods [], [], 3) use a self-organization or a vector quantization technique to deterine the initial assignent and 4) use cobined ethods of the [4], [7], [9]. Since accuracy and interpretability are conflicting goals, the conventional learning ethods with ulti-objective fuzzy odeling and fuzzy odeling with constrained paraeters of the ranges have Affiliation: Graduate School of Science and Engineering, Kagoshia University, --4 Korioto, Kagoshia 89-65, Japan corresponding auther to provide eail: iya@eee.kagoshia-u-ac.jp eail: k376885@kadai.jp eail: shigei@eee.kagoshia-u-ac.jp 3 eail: iya@eee.kagoshia-u-ac.jp becoe popular [7], []. However, there are little studies on effective learning ethods of fuzzy inference systes satisfying with interpretability and accuracy. It is desired to develop effective fuzzy systes with interpretability to realize the linguistic processing of inforation which is the original issue of fuzzy logic. Shi has proposed a fuzzy inference syste with interpretability and accuracy [3]. Further, Miyajia has proposed a generalized odel of it [4]. However, there are no studies on the detailed capability of this type of odel. In this paper, we will propose a fuzzy inference syste with interpretability and accuracy. Firstly, it is proved theoretically that the proposed ethod is a universal approxiator of continuous functions. Further, the capability of the proposed odel learned by the steepest descend ethod is copared with the conventional odels using function approxiation and pattern classification probles in nuerical siulation. Lastly, the proposed odel is applied to obstacle avoidance and the capability of interpretability is shown. II. PRELIMINARIES A. The conventional fuzzy inference odel The conventional fuzzy inference odel is described []. Let Z j {,, j} for the positive integer j. Let R be the set of real nubers. Let x (x,, x ) and y r be input and output data, respectively, where x j R for j Z and y r R. Then the rule of fuzzy inference odel is expressed as R i : if x is M i and and x is M i then y is f i (x,, x ) (), where i Z n is a rule nuber, j Z is a variable nuber, M ij is a ebership function of the antecedent part, and f i (x,, x ) is the function of the consequent part. A ebership value of the antecedent part µ j for input x is expressed as µ i M ij (x j ). () j If Gaussian ebership function is used, then M ij is expressed as follow (See Fig.): ( M ij a ij exp ( ) ) xj c ij. (3), where a ij, c ij and b ij are the aplitude, the center and the width values of M ij, respectively. b ij (Advance online publication: 4 April 5)

2 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 Fig.. ¹ ¹ ¹ The Gaussian ebership function The output y of fuzzy inference is calculated by the following equation: n y i µ i f i n i µ. (4) i Specifically, siplified fuzzy inference odel is known as one with f i (x,, x ) w i for i Z n, where w i R is a real nuber. The siplified fuzzy inference odel is called Model. B. Learning algorith for the conventional odel In order to construct the effective odel, the conventional learning ethod is introduced. The objective function E is defined to evaluate the inference error between the desirable output y r and the inference output y. In this section, we describe the conventional learning algorith. Let D {(x p,, xp, yp) p r Z P } be the set of learning data. The objective of learning is to iniize the following ean square error(mse): E P P (yp yp) r. (5) p In order to iniize the objective function E, each paraeter α {a ij, c ij, b ij, w i } is updated based on the descent ethod as follows []: α(t + ) α(t) K α (6) α where t is iteration tie and K α is a constant. When Gaussian ebership function with a ij for i Z n and j Z are used, the following relation holds [6]. w i c ij b ij µ i n i µ (y y r ) (7) i µ j n i µ (y y r ) (w i y ) xj c ij i b (8) ij µ i n i µ (y y r ) (w i y ) (x j c ij ) i b 3 ij (9) Then, the conventional learning algorith is shown as below [], [], [6]. Learning Algorith A Step A : The threshold θ of inference error and the axiu nuber of learning tie T ax are given. The initial assignent of fuzzy rules is set to equally intervals. Let n be the nuber of rules and n d for an integer d. Let t.» Step A : The paraeters b ij, c ij and w i are set to the initial values. Step A3 : Let p. Step A4 : A data (x p,, xp, yp) D r is given. Step A5 : Fro Eqs.() and (4), µ i and y are coputed. Step A6 : Paraeters c ij, b ij and w i are updated by Eqs.(8), (9) and (7). Step A7: If p P then go to Step A8 and if p < P then go to Step A4 with p p +. Step A8: Let E(t) be inference error at step t calculated by Eq.(5). If E(t) > θ and t < T ax then go to Step A3 with t t+ else if E(t) θ and t T ax then the algorith terinates. Step A9: If t > T ax and E(t) > θ then go to Step A with n d as d d + and t. C. The proposed odel It is known that Model is effective, because all the paraeters are adjusted by learning. On the other hand, all the paraeters ove freely, so interpretability capability is low. Therefore, we propose the following odel. R i i : if x is M i and and x is M i then y is f i i (x,, x ) (), where i j l j, M ij j(x j ) is the ebership function for x j and j Z. A ebership value of the antecedent part µ i i for input x is expressed as µ i i M ij j(x j ) M i (x ) M i (x ) () j Then, the output y is calculated by the following equation. y i µ i i f i i (x,, x ) () µ i i In this case, the odel with f i,,i (x, x ) w i,,i and triangular ebership function with a ijj has already proposed in the Ref. [3]. We will consider a odel with f i i (x,, x ) w i i and Gaussian ebership functions. The odel is called Model. [Exaple ] Let us show an exaple of the proposed odel with and i, i as follows: R : if x is M and x is M then y is w R : if x is M and x is M then y is w R : if x is M and x is M then y is w R : if x is M and x is M then y is w Fig. show figures for Exaple. Reark that Model is one that the paraeters of ebership function for each input are adjusted by learning. (Advance online publication: 4 April 5)

3 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 d de de dd dd d dd d dd d dd» (a)before learning. (a)mebership functions of the variable x d de de dd dd (b)after learning. Fig.. The figure to explain Model with and i i. The assignent (a) of fuzzy rules for Model is transfored into the assignent (b) after learning. Fig. 3. Eq.(9). d» dd d dd d dd (b)mebership functions of the variable x The assignents of fuzzy rules of before and after learning for Learning equation for Model is obtained as follows: M ij j i i j i j+ i µ i i (y y r ) (y w i i µ ) i i (3) µ i i w i i (y y r ) (4) µ i i When Gaussian ebership function of Eq.(5) is used, the following equations for a ijj, c ijj and b ijj are obtained; M ijj(x j ) a ijj exp (x j c ij j) (5) M ijj c ijj M ij j b ijj M ij j a ij j exp (x j c ijj) b i jj (x j c ij j) b 3 i j j j b i j j (x j c ij j) (6) j exp exp b i jj (x j c ijj) (7) j b i jj (x j c ij j) (8) j b i j j Learning algorith for the proposed odel with a ijj is also defined using Eqs.(4), (7) and (8). Fig. explains the assignents of the antecedent parts of ebership functions for before and after learning. Let us show an exaple for the practical oveent of ebership functions. [Exaple ] Let us consider the following function: y (x + 4x )(x + 4x +.) (9) 37., where x, x. In order to approxiate Eq.(9), learning of the proposed odel with and i, i 3 is perfored, where the nuber of learning data is. Fig.3 shows the assignents of ebership functions after learning, where waving and solid lines ean the assignents of ebership functions before and after learning. Madani type odel is special case of Model [7]. It is the odel with the fixed paraeters of antecedent part of fuzzy rule and ebership function assigned to equally intervals. The odel is called Model 3. It has good interpretable capability, but the accuracy capability is law. Therefore, TSK odel with the weight of linear function f i (x,, x ) is introduced as a generalized odel of Model 3 [3]. The odel is called Model 4. III. FUZZY INFERENCE SYSTEM AS UNIVERSAL APPROXIMATOR In this section, the universal approxiation capabilities of Model,, 3 and 4 are discussed using the wellknown Stone-Weierstrass Theore. See Ref. [] about the atheatical ters. [Stone-Weierstrass Theore] [] Let S be a copact set with diensions, and C(S) be (Advance online publication: 4 April 5)

4 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 a set of all continuous real-valued functions on S. Let Φ be the set of continuous real-valued functions satisfying the conditions: (i) Identity function : The constant function f(x) is in Φ. (ii) Separability : For any two points x, x S and x x, there exists a f Φ such that f(x ) f(x ). (iii) Algebraic closure : For any f, g Φ and α, β R, the functions f g and αf + βg are in Φ. Then, Φ is dense in C(S). In other words, for any ε > and any function g C(S), there is a function f Φ such that for all x S. g(x) f(x) < ε It eans that the set Φ satisfying the above conditions can approxiate any continuous function with any accuracy. Since the sets of RBF and Model are satisfied with the conditions of Stone-Weierstrass Theore, they hold for universal approxiation capabilities [], [6], [7]. Further, we can show the result about Model in the following. [Theore] Let Φ be the set of all functions that can be coputed by Model on a copact set S R as follows: Let Φ l l { f(x) j M i j j(x j )w i i i i M ijj(x j ), w i w, a ij j, c ij j, b ij j R, x S} ( ( for M ijj(x j ) a ijj exp xj ) ) c ij j and Φ l l b ij j Φ l l () Then Φ is dense in C(S). [Proof] In order to prove the result, it is shown that three conditions of Stone-Weierstrass Theore are satisfied. (i)the function f(x) for x S is in Φ, because it can be considered as a Gaussian function with infinite variable b. (ii)the set Φ satisfies with the separability since the exponential function is onotonic. (iii)let f and g be two functions in Φ and be represented as f(x) j M f i (x jj j)w f i i i j M f i (x () jj j) l g(x) l j M g l (x jj j)w g l l l l j M g l (x () jj j) for ( M f i j j (x j) a f i j j exp xj c f i j j M g l j j (x j) a g l j j exp b f i j j ( xj c g l jj b g l jj ) ) Then, we show αf + βg Φ and f g Φ for α, β R. (3) (4) We define M fg i (x jl jj j) M f i (x jj j) M g l j (x j ) ( a fg i j l j j exp xj c fg ) i jl jj (5) b fg i jl jj w fg i i l l w f i i + w g l l (6) w fg i i l l w f i i w g l l (7), where a fg i j l j j, cfg By using these values, i j l j j, bfg i j l j j, wfg i w i l l, w fg i i l l R. αf + βg l l j M fg i j l j j (x j)w fg i i l l i l l j M fg i j l j j (x j) f g l l j M fg i j l j j (x j)w fg i i l l i l l j M fg i j l j j (x j) Further, Eqs.(8) and (9) have the sae for as Model. Therefore, (αf + βg) and f g Φ hold. [Rearks] Reark that the results using Stone-Weierstrass Theore hold only for Model and with f i (x x ) w i and Gaussian ebership function. On the other hand, Stone- Weierstrass Theore does not always hold for Model 3, 4 and the odels with triangular ebership function, because the ultiplicative condition fails. Further, it is an existence theore and there is another proble whether we can get the syste with high accuracy. Therefore, we need effective learning algorith. Learning Algorith A is a learning algorith based on the steepest descend ethod. IV. NUMERICAL SIMULATIONS In this section, three kinds of siulations are perfored to copare the capability of the proposed odel with the conventional odel for learning ethod based on steepest descend ethod. In the siulations, let a ij and a ijj for i Z n and j Z. A. Function approxiation This siulation uses four systes specified by the following functions with [, ] [, ]. y sin(πx 3 ) x (3) y sin(πx3 ) cos(πx ) + (3) y.9(.35 + ex sin(3(x.6) ) e x sin(7x )) 7 (3) y sin((x.5) + (x.5) ) +. (33) The condition of the siulation is shown in Table I. The value θ is. 5 and the nubers of learning and test (8) (9) (Advance online publication: 4 April 5)

5 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 TABLE I INITIAL CONDITION FOR SIMULATION OF FUNCTION APPROXIMATION. Model Model Model 3 Model 4 T ax K c.... K b.... K w. d Initial c ij equal intervals Initial b ij (the doain of input) (d ) Initial w ij rando on [, ] TABLE II RESULTS FOR SIMULATION OF FUNCTION APPROXIMATION. Eq.(3) Eq.(3) Eq.(3) Eq.(33) learning Model test paraeter learning Model test paraeter learning Model 3 test paraeter learning Model 4 test paraeter data selected randoly are and 5, respectively. Table II shows the results on coparison aong four odels. In Table II, Mean Square Error(MSE) of learning( 4 ) and MSE of test( 4 ) are shown. The result of siulation is the average value fro twenty trials. Table II shows that Model and have alost the sae capability in this siulation, where paraeter eans the nuber of paraeters..8 z z x.8 (a) Sphere..4.6 x.8 Sphere (r.3) y. Sphere (r.) Sphere (r.4) (b) Double-Sphere y B. Pattern classification In the next, we perfor two-category classification probles as in Fig. 4 to investigate another approxiation capability of Model and. In the classification probles, points on [, ] [, ] [, ] are classified into two classes: class and class. The class boundaries are given as spheres centered at (.5,.5,.5). For Sphere, the inside of sphere is associated with class and the outside with class. For Double-Sphere, the area between Spheres and is associated with class and the other area with class. For triple-sphere, the inside of Sphere and the area between Sphere and Sphere 3 is associated with class and the other area with class. The desired output y r p is set as follows: if x p belongs to class, then y r p.. Otherwise y r p.. The nubers of learning and test data selected randoly are 5 and 64, respectively. The siulation condition is shown in Table III. The results on the rate of isclassification are shown in Table IV. The result of siulation is the average fro twenty trials. Table IV shows that Model and have alost the sae capability. C. Obstacle avoidance and arriving at designated point In order to show interpretability, let us perfor siulation of control proble for Model and Model []. As shown in Fig.5, the distance r and the angle θ between obile object and obstacle and the distance r and the angle θ.8 z.6.4. Sphere (r.) Sphere (r.) Sphere3 (r.4) y x.8 (c) Triple-Sphere Fig. 4. Two-category Classification Probles [] TABLE III INITIAL CONDITION FOR SIMULATION OF TWO-CATEGORY CLASSIFICATION PROBLEMS. Model Model T ax 5 5 K w.. K c.. K b.. d 3 5 Initial c ij equal intervals Initial b ij (the doain of input) (d ) Initial w ij rando on [, ] (Advance online publication: 4 April 5)

6 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 TABLE IV SIMULATION RESULT FOR TWO-CATEGORY CLASSIFICATION PROBLEM. Sphere Double-Sphere Triple-Sphere learning Model test paraeter learning.9.8. Model test paraeter » TABLE V INITIAL CONDITION FOR SIMULATION OF OBSTACLE AVOIDANCE. Model Model T ax 5 5 K c.. K b.. K w. d 3 3 Initial c ij equal intervals Initial b ij (the doain of input) (d ) Initial w ij. paraeters 79 5 K É É»».8.6 Fig. 5. place. K Siulation on obstacle avoidance and arriving at the designated between obile object and the designated place are selected as input variables, where θ and θ are noralized. The proble is to construct fuzzy inference syste that obile object avoids obstacle and arrives at the designated place. Fuzzy inference rules for Model and Model are constructed fro learning for data of 4 points shown in Fig. 6. An obstacle is placed at (.5,.5) and a designated place is placed at (.,.5). The nuber of rules for each odel is 8 and the nuber of attributes is 3. Then, the nubers of paraeters for Model and are 79 and 93, respectively. The obile object oves with the vector A at each step, where A x of A is constant and A y of A is output variable. Learning for two odels are successful and the following tests are perfored. ()Test is siulation for obstacle avoidance and arriving at the designated place when the obile object stars fro various places (See Fig.7). Fig.7 shows the results of oves of obile object for starting places» (a)model (b)model Fig. 7. Siulation for obstacle avoidance and arriving at the designated place starting fro various places after learning. at (., ), (., ),, (.8, ), (.9, ) after learning. As shown in Fig.7, the test siulations are successful for both odels. ()Test is siulation for the case where the obile object avoids obstacle placed at different place and arrives at the different designated place. Siulations with obstacle placed at the place (.4,.4) and arriving at the designated place (,.6) are perfored for two odels. The results are successful as shown in Fig.8. Fig. 6. Learning data to avoid obstacle and arrive at the designated place (.,.5). (3)Test 3 is siulation for the case where obstacle oves with the fixed speed. Siulations with obstacle oving with the speed (.,.) fro the place (.3,.) to the place (.8,.) and object arriving at the place (,.6) are (Advance online publication: 4 April 5)

7 IAENG International Journal of Coputer Science, 4:, IJCS_ (a)model (i)model (step t 3) (b)model Fig. 8. Siulation for obstacle placed at the different place (.4,.4) and arriving at the different place (.,.6) (ii)model (step t 4) Fig. 9. Siulation for oving obstacle avoidance with fixed speed and the different designated place (.,.6)...8 perfored. Fig.9 shows the results of the steps 3 and 4 for test siulation. Since the object does not collide in the obstacle at t 3, thereafter it does not collide in the obstacle. The final results for both odels are successful as shown in Fig.. (4)Test 4 is siulation for the case where obstacle oves with the speed B randoly as shown in Fig., where B is constant, and the angle θ b is deterined randoly at each step. Siulations with obstacle starting fro the place (.5,.) are perfored for two odels. The results for both odels are successful as shown in Fig.. Lastly, let us consider interpretability for the proposed odel. Let us consider fuzzy rules constructed for Model by learning. Assue that three attributes are short, iddle and long for r and r, inus, central and plus for θ and θ and left, center and right for the direction of A y, respectively. Then, ain fuzzy rules for Model are constructed as shown in TableVI. Fro TableVI, we can get the rules : If the object approaches to the obstacle, ove in the direction away fro the object. and If the object approach to the goal, ove toward to the goal. They are siilar to huan activity to solve the proble. On the other hand, fuzzy rules for Model are not so clear (a)model (b)model Fig.. Siulation for oving obstacle avoidance with fixed speed and the different designated place (.,.6). (Advance online publication: 4 April 5)

8 IAENG International Journal of Coputer Science, 4:, IJCS_4 6 Fig.. The obstacle oves with the vector B, where B is constant and θ b is selected randoly.. V. CONCLUSION In this paper, a theoretical result and soe nuerical siulations including obstacle avoidance are presented in order to copare the capability of the proposed odel with the conventional odels. It is shown that Model and Model with Gaussian ebership function and f i (x,, x ) w i are satisfied with the conditions of Stone-Weierstrass Theore, so both odels are universal approxiators of continuous functions. Further, in order to copare the capability of learning algoriths for odels, nuerical siulation of function approxiation and pattern classification probles are perfored. Lastly, soe siulations on obstacle avoidance are perfored. In the siulations, it is shown that both odels are successful in all trials. Specifically it is shown that Model with the sall nuber of paraeters and interpretability is constructed. In the future work, we will find an effective learning ethod for Model (a)model (b)model Fig.. Siulation for oving obstacle randoly and the different designated place (.,.6). TABLE VI MAIN FUZZY RULES OBTAINED BY LEARNING FOR MODEL. r r θ θ A y Rule short long plus center right Rule inus left Rule 3 iddle iddle right Rule 4 plus plus left Rule 5 iddle left Rule 6 inus inus right REFERENCES [] C. Lin and C. Lee, Neural Fuzzy Systes, Prentice Hall, PTR, 996. [] M.M. Gupta, L. Jin and N. Hoa, Static and Dynaic Neural Networks, IEEE Press, 3. [3] J. Casillas, O. Cordon and F. Herrera, L. Magdalena, Accuracy Iproveents in Linguistic Fuzzy Modeling, Studies in Fuzziness and Soft Coputing, Vol. 9, Springer, 3. [4] S. M. Zhoua, J. Q. Ganb, Low-level Interpretability and High-Level Interpretability: A Unified View of Data-driven Interpretable Fuzzy Syste Modeling, Fuzzy Sets and Systes 59, pp.39-33, 8. [5] J. M. Alonso, L. Magdalena and G. Gonza, Looking for a Good Fuzzy Syste Interpretability Index: An Experiental Approach, Journal of Approxiate Reasoning 5, pp.5-34, 9. [6] H. Noura, I. Hayashi and N. Wakai, A Learning Method of Siplified Fuzzy Reasoning by Genetic Algorith, Proc. of the Int. Fuzzy Systes and Intelligent Control Conference, pp.36-45, 99. [7] O. Cordon, A Historical Review of Evolutionary Learning Methods for Madani-type Fuzzy Rule-based Systes, Designing interpretable genetic fuzzy systes, Journal of Approxiate Reasoning, 5, pp ,. [8] S. Fukuoto and H. Miyajia, Learning Algoriths with Regularization Criteria for Fuzzy Reasoning Model, Journal of Innovative Coputing Inforation and Control,,, pp.49-63, 6. [9] K. Kishida, H. Miya jia, M. Maeda and S. Murashia, A Self-tuning Method of Fuzzy Modeling using Vector Quantization, Proceedings of FUZZ-IEEE 97, pp397-4, 997. [] N. Yubazaki, J. Yi and K. Hirota, SIRMS(Single Input Rule Modules) Connected Fuzzy Inference Model, J. Advanced Coputational Intelligence,,, pp.3-3, 997. [] H. Miyajia, N. Shigei and H. Miyajia, Fuzzy Inference Systes Coposed of Double-Input Rule Modules for Obstacle Avoidance Probles, IAENG International Journal of Coputer Science, Vol. 4, Issue 4, pp.-3, 4. [] M. J. Gacto, R. Alcala and F. Herrera, Interpretability of Linguistic Fuzzy Rule-based Systes:An Overview of Interpretability Measures, Inf. Sciences 8, pp ,. [3] Y. Shi, M. Mizuoto, N. Yubazaki and M. Otani, A Self-Tuning Method of Fuzzy Rules Based on Gradient Descent Method, Japan Journal of Fuzzy Set and Syste 8, 4, pp , 996. [4] H. Miyajia, N. Shigei and H. Miyajia, On the Capability of a Fuzzy Inference Syste with Iproved Interpretability, Lecture Notes in Engineering and Coputer Science, Proceedings of The International MultiConference of Engineers and Coputer Scientists 5, IMECS 5, 8- March, 5, Hong Kong, pp [5] W. L. Tung and C. Quek, A Madani-takagi-sugeno based Linguistic Neural-fuzzy Inference Syste for Iproved Interpretability-accuracy Representation, Proceedings of the 8th International Conference on Fuzzy Systes(FUZZ-IEEE 9), pp , 9. [6] L. X. Wang and J. M. Mendel, Fuzzy Basis Functions, Universal Approxiation and Orthogonal Least Square Learning, IEEE Trans. Neural Networks, Vol.3, No.3, pp.87-84, 99. [7] D. Tikk, L. T. Koczy and T. D. Gedeon, A Survey on Universal Approxiation and its Liits in Soft Coputing Techniques, Journal of Approxiate Reasoning 33, pp.85-, 3. (Advance online publication: 4 April 5)