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Fuzzy basis functions for triangle-shaped membership functions: Universal approximation - MISO case Fernando di Sciascio , Ricardo Carelli INSTITUTO DE AUTOMATICA, UNIV. NAC. DE SAN JUAN Av. San Martin 1109(O), 5400 San Juan, ARGENTINA Tel.+54-64-213303, Fax: +54-64-213672 EMAIL: [email protected]

Abstract- In this paper, the universal approximation property of one of the most frequently used type of fuzzy systems is proved. The type of fuzzy system in the present work employs triangle-shaped fuzzy membership functions (TSMF) for its input variables. The proof of the universal approximation property does not use the Stone-Weierstrass theorem because the TSMFs are not closed for the product; instead, it is based on the " Density lemma for linear subspaces of C (X ) (the space of all bounded continuous real-valued functions on X 0 U n )". This lemma is a powerful tool to prove the universal approximation property for other classes of fuzzy systems. I- Introduction The approximation problem of multi-input-single-output (MISO) fuzzy systems is discussed in this work. A multi-output system can always be separated into several systems of a single output. From a mathematical point of view, fuzzy inference systems are functions which map inputs to outputs and can be represented by a linear combination of fuzzy basis functions (FBFs), [6]. The function approximation problem using fuzzy systems has recently been addressed in [1], [5], [6], [7], [8] and [9], and was motivated by publications on the approximation capacity of neural networks and by the fact that problems found in designing fuzzy systems can be regarded as approximation problems as well. The authors considered the universal approximation property for some types of fuzzy systems, that is, any continuous function on a compact set can be uniformly approximated by fuzzy systems with an arbitrary degree of accuracy. Based on the Stone-Weierstrass theorem, Wang and Mendel [7], proved the approximation property by considering the fuzzy systems class with singleton fuzzifier, product inference, centroid defuzzifier, and scaled Gaussian membership function. They have defined FBFs and expressed the above class fuzzy systems by means of a linear combination of FBFs. It is clear that the approximation property of fuzzy systems is closely related to FBFs properties. Nguyen and Kreinovich [6], proved via Stone-Weierstrass, theorem the universal approximation property of the linear combination of FBFs corresponding to the fuzzy systems class, with singleton fuzzifier, min-inference, centroid defuzzifier, and Gaussian membership function. Zeng and Singh, [9], considered the SISO case for a broad class of membership functions (pseudo trapezoid-shaped (PTS) membership function). In their paper, they have carried out a detailed analysis of the FBFs properties, and they also have proven important approximation properties of fuzzy systems. Scaled Gaussian membership functions are not normal, hence they do not have the consistency property, [9]. On the other hand, it is desirable to use triangle-shaped membership

functions (TSMF) in fuzzy systems, because they are normal and they also have the consistency property. Besides, TSMF are easier to generate and probably they are the most frequently used functions in applications. The proof of the universal approximation property based on the Stone-Weierstrass theorem, limits the type of membership functions to be used merely to Gaussian function forms, because the latter ones are closed for the product. The present paper analyzes the MISO case and considers the fuzzy systems class with singleton fuzzifier, product inference, centroid defuzzifier, and triangle-shaped membership functions. The proof of universal approximation property for the fuzzy systems class considered here, does not use the Stone-Weierstrass theorem because the TSMFs are not closed for the product. The proof is based on the " Density lemma for linear subspaces of C ( X ) ", [3], where C ( X ) is the space of all bounded continuous real-valued functions on X . This lemma is a powerful tool to prove the universal approximation property for other classes of fuzzy systems. II- Fuzzy system specifications In this paper a multi-input, single-output (MISO) fuzzy inference system (FIS) is considered. Multi-output systems (MIMO) can always be separated into single-output systems. A MISO fuzzy system performs a static mapping FS : S dUn 6 U , where S is a compact set. The main elements of fuzzy systems in this work are: singleton fuzzifier, Mamdani's fuzzy rule base, fuzzy product inference engine and centroid defuzzifier. The former performs a mapping from the crisp input space S d Un to the fuzzy sets defined in the compact set X ' [0 , 1] n d U n usually employed in the approximation theory ( the spaces C [ 0 , 1 ] n and C ( U ) are homeomorphic, where U d Un is a dense set ). The fuzzy set A defined in [0 , 1] is characterized by a membership function µ A , with µ A : [ 0 , 1 ] 6 [ 0 , 1] , and it is labelled by a linguistic term defined in the term set of the considered linguistic variable. Next, a summary of the characteristics of the fuzzy inference systems used in this paper is presented.

c1- The fuzzy system inputs are considered after the normalization process, being the n components xi of the vector x ' ( x1 x2 ... x n )T 0 [ 0 , 1 ] n , and y ' F( x ) 0 U being its output. c2- The number p of fuzzy sets is the same for the n input variables. c3- Every possible rule is considered, being m ' number of rules ' p n . c4- The fuzzy sets for the output variable y are singletons, placed at any point aj 0 U . c5- The fuzzy membership functions of the input variable are triangle-shaped. They are shown in Fig.1

The entire static mapping performed by the fuzzy inference systems with singleton fuzzifier, product inference, centroid defuzzifier, and triangle-shaped membership functions, has the form, j ... j a j ( µ A ki i ( x i )

j p

p

p

k n '1 k n & 1 '1

F ( a,x ) '

j

n

k 1 '1

j ... j

p

k n '1

p

p

i'1

(µ

k1 '1 i '1

k n &1 '1

(5)

n

k

Ai i

(x i )

with j ' j ( k1 , k2 , ...,kn ) . Eq. (5) is obtained after setting up a similar order to the one of the basis p numeration algorithm. This order is fixed between the j & esim rule and the fuzzy sets regarded to evaluate the j & esim rule. The subindex i of ki indicates the variable x i that is being considered and ki indicates the fuzzy sets order of the variable x i from left to right. A one-to-one correspondence between index set ki and the number or index j of each rule is established. The following expression for the j index is obtained, j & 1 ' ' j (ki &1 ) p i&1 ' j $i p i&1 ' b T $ ( j )

Fig.1 Equally spaced triangles

n

n

i '1

i'1

(6)

with $ i ' k i &1 ' 0 , 1 ,...,( p&1) , $ ( j ) ' ( $1 $2 ...$n )T 0 U n, b ' ( p 0 p 1 ... p n&1 )T 0 U n . The components of vector $ ( j) are the basis p digits of the number j&1 , ( $i corresponds to the digit i ) Proposition 1: Denominator of Eq. (5) is equal to 1. j p

j ... j ( µ A iki (x i ) ' p

p

' j µA p

kn '1

n

(7)

k1 '1 i '1

k n '1 kn & 1 '1

Fig.2 Parameters definitions

n

( x n ) j µ k n& 1 (x n&1 )... j µ k1 (x 1 ) ' 1 kn A A p

k n & 1 '1

p

n& 1

k1 '1

$

1

The triangle-shaped membership functions of each ki

variable x i are determined by two parameters, ) and c i they are indicated in the Fig.2. k i% 1

) ' ci k

ci i '

k

k

& c i i ' c i i & ci

ki & 1

1 p&1

'

(1) (2)

it is verified that: µ

ki

Ai

(

k ci i

p

p

n

k n& 1 '1

k 1 '1

i '1

i

' j aj nj ( x ) ' a T n (x ) m'p n

(8)

where a ' ( a1 a2 ... am )T 0 U m isis the parameter vector and nj ( x) nj ( x) '

)'1

j µ A ki (x i ) ' µ Ai1 (x i ) % ...% µ Aip ( x i )' 1

k µ A iki ( x i ) n

(j )

(9)

i'1

(3) and n (x ) ' ( n 1 (x ) n 2 ( x) ... n m (x ) )T

For each x i and every value of the interval [ 0 , 1 ] the following equality holds, p

i

j ... j aj ( µ A k i ( x i ) '

is,

i ' 1 , 2 , ..., n ; k i ' 1 , 2 ,..., p

k i '1

p

j'1

p& 1

for x i '

F ( a,x ) ' j

k n '1

ki & 1

k ci i

By substituting Eq. (7) into Eq. (5), it yields,

(4)

The j exponent in Eq. (9) is symbolic and implies that in the product it must be taken into account only the fuzzy sets corresponding to the j rule.

Proposition 2: For the m ' p n points of the space X ' [0 ,1] n given by x ' ) $( j ) , the j rule equals to 1 and all the others equals to 0, then n j ( ) $( k ) ) ' * j , k (10) therefore Eq. (8) becomes, F( a ,) $( j ) ) ' j aj nj () $( j ) )' a T n ( ) $( j ) ) ' a j . m'2 n

(11)

j '1

Density lemma for linear subspaces of C ( X ) : Let X be a topological space, let C ( X ) be the space of all bounded continuous real-valued functions on X , and let C ( X ) have the topology of uniform convergence ( equivalently, norm C ( X ) by 2 f 2 ' sup * f (x )* : x 0 X ). A subset L of C ( X ) is said to have the two-set property iff for closed disjoint subsets A and B of X and for each closed real interval [ " , $ ] there is a member f of L such that f maps X into [ " , $ ] , f is " on A , and f is $ on B . Each linear subspace of C ( X ) which has the two-set property is dense in C ( X ) .

III- Universal approximation The set 6 nj ( x ) > is the fuzzy basis functions, where n are the input variables, and m is the number of fuzzy rules. The span of the function set 6 nj ( x) > generates Fm which is a linear subspace of C(X) , where C(X ) is the space of all continuous real-valued functions on X ' [ 0 , 1 ] n (every function of Fm is continuous because it is a linear combination of continuous functions, then Fm d C(X ) ), that is, F m ' span 6 n j ( x) > ' F (a ,x ) ' j a j n j (x ) : a ' ( a 1 ... a m ) T 0 U m

Proof: It must be shown that F m complies with the two-set property. s1- A ^ B … X ,for all closed disjoint subsets A and B in X ' [ 0 , 1 ] n ( if A ^ B ' X 6 B ' A c 6 A and B cannot be simultaneously closed). s2- A metric space X is a normal topological space. Then, for all closed disjoint subsets A and B in X ' [ 0 , 1 ] n exists open disjoint subsets G and H such that A d G and B d H . Besides, G ^ H … X by same reason as for s1.

m'p n

'

j '1

Each and every member of F m is a fuzzy system and the linear space Fm has the following properties: p1- The set of functions 6 nj (x ) > is linearly independent, so for different aj parameters, different functions are obtained; p2- The constant functions belong to Fm ; let aj ' k ; with j ' 1 , 2 ,..., m , and through Eq. (7),

s3- Closed disjoint sets A and B are compact because they are closed and bounded subsets of a compact space X d U n . Then every open covering of A and B has a finite open subcovering. Open subsets G and H may be depicted as a joining of finite numbers of open sets. Let S j :' x 0 X : n j ( x) > 0 be open subsets ( S j is the support of n j (x ) ), then the diameter of the sets S j are, D( S j ) :' sup d (s 1 , s2 ) : s 1 , s2 0 S j the maximum diameter of the sets S j are,

j nj ( x) ' 1

m 'p n

(12) D :' max D(S j ) '

j '1

F( a ,x ) ' j aj nj (x ) ' j k nj (x ) ' k j nj ( x ) ' k m

m

m

j '1

j '1

j '1

j

$

n p&1

(14)

The sets S j may depict G and H as a finite cover of A and B . p3- Linear functions in C(X ) are members of Fm [3]. p4- Each member of Fm is a fuzzy system and the linear space Fm has the universal approximation property. These property can be proved by means of the following theorem: Theorem ( universal approximation ): Let C (X) be the space of all continuous real-valued function on the compact set X d U n , then for any given f 0 C (X) and arbitrary g > 0 , there exists a number m 0 ' m 0 (g ) 0 ù such that,

s4- The space X ' [ 0 , 1 ] n is a totally bounded subset of U n ; then, for each , > 0 there exists a break down of X in a finite number of sets so that the each set diameter be less than , . This is equal to say that X has a , & net for each , > 0 . Let the set C be a , & net of X whose elements are the m ( n , p ) ' p n points of X , where the functions n j (x ) ' 1 C ' 6 x 0 X : n j (x )' 1 > ' c j ' ) $ ( j ) : j ' 1, 2,..., m

(15)

œ m > m 0 there exists F (a ,x ) 0 F m ; such that : sup * f( x ) & F(a ,x ) * < g

(13)

x0 X

The theorem proof consists of demonstrating that Fm is dense in C ( X ) , and this is based upon the following technical lemma, [4]:

, (n , p) >

1 n 1 ) n' ' D 2 2(p& 1) 2

(16)

Let the following subsets of C be defined: C A ' 6 cj 0 C : d ( cj , A ) # )

n ' D >

(17)

C B ' 6 c j 0 C : d ( cj , B ) # )

n ' D >

C C ' C & ( C A ^ CB )

(18) (19)

By choosing the value m ' p n adequate for C A , C B y C C be disjoint sets: CA _ C B ' CA _ C C ' CB _ C C ' i

(20)

This can always be performed on the basis of the separation axiom and due to the fact that A and B are compact sets. The following sets are defined as a function of sets C , C A , C B y CC : I ( 6 cj > ) ' 6 j >

I( @ ) : 6 U n > 6 6 ù >

,

(21)

I( C A ) ' 6 j of cj 0 C A >

(22)

s5 Let's regard any one set S j ) definition of c j , d ( x , cj ) ) #

D ( sj ) )

œ x 0 X : d ( x, c j ) ) >

œ x 0 Sj )

,

2

from (14) and from the

D( Sj ) )

then,

6 x ó S j ) 6 n j ) ( x )' 0

2

(23)

s6- Every a j associated to the indexes j 0 I ( C A ) are given the value a j ' " ; every a j associated to the indexes j 0 I ( C B ) are assigned the value a j ' $ . Finally, every a j associated to the indexes j 0 I ( C C ) are given the values " # a j # $ . With this parameter assignation F ( a , x ) is: F( a , x ) ' j

m 'p n

%

j '1

j

a j n j ( x) %

j 0 I ( CB )

j

a j n j ( x) ' j

j 0 I ( CA )

j 0 I ( CC )

a j n j ( x) %

a j n j ( x)

(25)

for x 0 A and by considering Eq. (23), it follows: n j ( x ) ' 0 œ j 0 I ( C B ^ CC )

(26)

by regarding Eqs.(26) and (25) turns to be: F( a , x ) ' j

m 'p n

'"

j

j '1

j 0 I ( CA )

a j n j ( x)

'

n j (x )

j 0 I ( CA )

j

j 0 I ( CC )

" n j (x ) ' (27)

through Eq. (26) and p2, Eq. (27) becomes: F( a , x ) ' " j n j (x ) % " % "

j

j 0 I ( CA )

n j( x ) ' " j

m' p n j'1

j

n j( x ) %

n j (x ) ' " ,

œx0 A

j 0 I ( CB )

By following the same procedure for x 0 B and x 0 ( A ^ B ) c , it is obtained a function F ( a , x) 0 F m such that it verifies:

i) ii) iii)

F (a , x ) ' " F (a , x ) ' $ " # F( a , x ) # $

, , ,

œx0 A œx0 B œx0 X

Since the closed sets A , B and the interval [ " , $ ] are arbitrary, F m verifies the two-set property. Then, by regarding the lemma, F m is dense in C( X ) $ VI- Conclusions It has been shown that a specific class of fuzzy systems with triangle-shaped membership functions, presents the universal approximation property. The methodology used in proving this property is applicable also to prove the universal approximation property for other classes of fuzzy systems. References [1] Buckley, J.J. (1992). "Universal Fuzzy Controllers". Automatica, vol.28, Nº6, pp. 1240-1245. [2] di Sciascio, F; Carelli, R.O. (1994). "Modelling of Dynamic Systems by means of Fuzzy Inference Systems", in Spanish. AADECA'94, Buenos Aires. Argentina, pp.415-420. [3] di Sciascio, F; Carelli, R.O. (1995). "Fuzzy representation theorem of linear functions: Automatic control applications", submited to IFSA'95. [4] Kelley, J.L. (1955). General Topology. D. Van Nostrad Company, Inc, pp. 243. [5] Kosko, B. (1992). "Fuzzy systems as universal approximators". Proceeding of the IEEE International Conference on Fuzzy Systems, San Diego, CA, pp. 11531162. [6] Nguyen, H.T; Kreinovich, V. (1992). "On Approximation of Controls by Fuzzy Systems". LIFE, Technical Report TR92-93/302. [7] Wang, L.-X.; Mendel, J.M. (1992). "Fuzzy Basis Functions, Universal Approximation, and Orthogonal Least-Square Learning". IEEE Trans. on Neural Network. vol 3 no 5. [8] Wang, L.-X.; Mendel, J.M. (1992). "Generating Fuzzy Rules by Learning from Examples". IEEE T.S.M.C. vol 22 no 6. [9] Zeng, X.-J.; Singh, M.G. (1994). "Approximation Theory of Fuzzy Systems-SISO Case". IEEE Trans. on Fuzzy Systems, vol.2, NO. 2, pp. 162-176 LLL