hierarchical ts fuzzy system and its universal approximation

Information Sciences 169 (2005) 279–303

www.elsevier.com/locate/ins

Hierarchical TS fuzzy system and itsuniversal approximation

Puyin Liu a,b,*, Hongxing Li a

a Department of Mathematics, Beijing Normal University, Beijing 100875, Chinab Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received 23 March 2004; accepted 10 April 2004

Abstract

An efficient tool to deal with the �rule explosion� problem is the hierarchical system

by which a fuzzy system can be decomposed into a number of hierarchically connected

low-dimensional systems. In this paper a generalized hierarchical Tagaki–Sugeno (TS)

system is built. It is shown that the input–output (I/O) relationship of this generalized

hierarchical system can be represented as one of a standard TS fuzzy system. And the

system approximation capability is analyzed by taking piecewise linear functions as a

bridge. By constructive method it is proven that the hierarchical fuzzy systems (HFS�s)can be universal approximators. For the given approximation accuracy, an estimation

formula about the number of the rules needed in the HFS is established. Finally some

simulation examples confirm that the HFS�s with smaller size rule base can approximate

the given functions with high accuracy. The results obtained here provide us with the

theoretical basis for various applications of HFS�s.� 2004 Elsevier Inc. All rights reserved.

Keywords: TS fuzzy system; Hierarchical system; Universal approximator; Piecewise linear function

0020-0255/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2004.04.008

* Corresponding author. Address: Department of Mathematics, National University of Defense

Technology, Changsha 410073, China.

E-mail address: [email protected] (P. Liu).

mailto:[email protected]

280 P. Liu, H. Li / Information Sciences 169 (2005) 279–303

1. Introduction

Fuzzy systems have emerged as one of the most active and fruitful research

fields [17]. In many real areas, such as pattern recognition [17], automatic con-

trol [1,17], system identification [16] etc., we can find many successful applica-

tions of fuzzy systems. Also the research on fuzzy systems is of theoreticimportance [17,20,23,24]. For instance, through studying approximation capa-

bility of fuzzy systems systematically we can build the approximation theory of

systems [23,24].

In most rule based fuzzy systems, the fuzzy rule base consisting of a number

of inference rules defined as �IF. . .THEN. . .� is a key part. In the paper the

fuzzy rule base is assumed to be complete, that is, the rule base is valid for

all possible conditions. As the number of the system input variables increases

the number of rules in the complete fuzzy rule base increases exponentially.That is the �Rule explosion� problem, which is in nature the �curse of dimensio-

nality� which exists in many fields [6,7]. That will not only generate the compli-

cated system structures, but also cause long computational time, even memory

overload of the computer.

To make fuzzy systems usable in real complex systems, we have to deal with

the �rule explosion� problem. A few of methods for the problem have so far

been put forward [2,3,6,7,13]. In the fuzzy systems or fuzzy controllers, two

classes of such methods are significant. One is based on the equivalence of�intersection rule configuration� and �union rule configuration� [2–4,13]. Thatis, if P and Q are two antecedents, and R is a consequent, then

½ðP ^ QÞ ) R� () ½ðP ) RÞ _ ðQ ) RÞ�:Another one is to introduce a hierarchical system configuration [8,14,15,18,19],

i.e. instead of applying a fuzzy system with higher-dimensional input, a number

of lower-dimensional fuzzy systems are linked in a hierarchical fashion. Bysuch a hierarchy, the number of the fuzzy rules will increase linearly with the

number of the input variables. So the HFS�s can be efficiently used in some

large scale systems. And the application fields of fuzzy systems are undoubtedly

extended and expanded. The paper focuses on the HFS�s mainly in input–out-

put (I/O) relationship representation and approximating properties.

Naturally we may put forward an important problem, that is, how can the

approximating capability of a HFS be analyzed? Kikuchi et al. in [8] show that

it is impossible to utilize a HFS to build the precise expression of an arbitrarilygiven function. So we have to analyze the approximate representation of a

function by the HFS�s. Whether can the HFS�s be universal approximators

or not? If a function is continuously differentiable on the whole space, Wang

in [18] shows the arbitrarily close approximation of a function by the HFS�s;and he also in [19] gives the sensitivity properties of HFS�s and designs a

suitable system structure. For each compact set U and an arbitrarily given

P. Liu, H. Li / Information Sciences 169 (2005) 279–303 281

continuous function f on U, how can we find a HFS to approximate f uni-

formly with arbitrary error bounds e? In order to analyze this problem, in

the paper we show that the I/O relationship of a HFS can be represented as

one of a standard fuzzy system. Based on the fact the universal approximation

of HFS�s are studied systematically. Comparing with the approximation meth-

ods suggested by Wang [17–19], Buckley [1], Ying [20–22], Zeng and Singh[23,24], we may easily find that our methods are directly based on the I/O rela-

tionship information of the functions to be approximated, no intermediate step

needed, consequently more applicable in practice.

The paper is organized as follows. After some notations are introduced, Sec-

tion 2 defines a class of piecewise linear functions as a bridge for the study of

approximation. The TS fuzzy inference and generalized TS fuzzy system are re-

ported in Section 3. In Section 4, a simply generalized HFS is built, and the

corresponding I/O relationship is represented as one of a standard fuzzy sys-tem. With the conclusions in Sections 2–4, the universal approximation of

the hierarchical systems is shown in Section 5, and the number of the fuzzy

rules in the rule base of the hierarchical system is estimated. Finally we give

some simulation examples to confirm that the HFS�s can uniformly approxi-

mate continuous functions defined in a high dimensional space, and the fuzzy

rules in their rule base are strikingly less than those in the standard fuzzy sys-

tems to ensure the same approximating accuracy. Some further topics related

the subject for the future research are put forward.

2. Piecewise linear function

Suppose N to be the natural number set, and Rd to be the d dimensional Eu-

clidean space, R , R1. By eA; eB; eC ; . . . etc. we denote the fuzzy set defined on R.

In the paper, we refer to the approximation with the maximum norm. Let

U � Rd , by k � k1;U we denote the maximum norm on U, i.e. if f : Rd ! R,kf k1;U ¼ supx2Ufjf ðxÞjg ,

Wx2Ufjf ðxÞjg, i.e. �� is the supremum operator.

Obviously if U1, U 2 � Rd are compact sets, then U 1 � U 2 ) kf k1;U16

kf k1;U2. FðRÞ means the collection of all fuzzy numbers [10].

To analyze the approximation capability of HFS�s, let us now define the

piecewise linear functions as a intermediate tool. The function S : Rd ! R is

called a piecewise linear function, if there is M>0, such that the support

supp(S) satisfies the condition: supp(S)�[�M, M]d, and the following facts

hold:

(i) S is a continuous function on [�M, M]d;

(ii) There is a l(SÞ 2 N, and the polyhedrons D1,. . .,Dl(S)�[�M, M]d, satisfyingSlðSÞk¼1Dk ¼ ½�M ;M �d ; so that for each k=1,. . .,l(S), S is linear on Dk, that is,

there are kk0; kk1; . . . ; kkd 2 R,


ðx1; . . . ; xdÞ 2 Dk;) Sðx1; . . . ; xdÞ ¼Xd

i¼0

kki � xi;

where x0 ” 1. Considering the linearity of S, we may easily prove that the one-

side partial derivatives of S on [�M, M]d exist. Furthermore, if let V(Dk) be the

set of all vertices of Dk, and V ðSÞ ,SlðSÞ

k¼1V ðDkÞ: Then for arbitraryðx01; . . . ; x0dÞ 2 ð�M ;MÞd ; we have for i=1,. . .,d:

oSþðx01; . . . ; x0dÞoxi

�� _ oS�ðx01; . . . ; x0dÞoxi

�� 6

_ðx1;...;xd Þ2V ðSÞ

oSþðx1; . . . ; xdÞoxi

�� _ oS�ðx1; . . . ; xdÞoxi

�� : ð1Þ

If let

DiðSÞ ¼_

ðx1;...;xd Þ2V ðSÞ

oSþðx1; . . . ; xdÞoxi

�� _ oS�ðx1; . . . ; xdÞoxi

�� ;

then for h1; . . . ; hd 2 R we can easily obtain by (1) the following fact [12]: If

(x1+h1,. . .,xd+hd), (x1,. . ., xd)2[�M, M]d, it follows that

Sðx1 þ h1; . . . ; xd þ hdÞ � Sðx1; . . . ; xdÞj j6Xd

i¼1

DiðSÞ � jhij: ð2Þ

Proposition 1. Let f : Rd ! R be continuous, and U � Rd be an arbitrarycompact set. Then there is M>0, such that U�[�M, M]d, moreover for

arbitrary e>0, there is a piecewise linear function S : Rd ! R, satisfying

supp(S)�[�M, M]d, further kf � Sk1;U < e.

The proof of the proposition is reported in Appendix A.

By Proposition 1, for a continuous function f : Rd ! R and an arbitrary

compact set U � Rd , let M>0:U�[�M, M]d. Then there is a sufficiently large

n 2 N, satisfying "(p1, . . .,pd) 2 {�n,

� nþ 1; . . . ; n� 1; ngd ;)

fMp1n

; . . . ;Mpdn

� �¼ S

Mp1n

; . . . ;Mpdn

� �: ð3Þ


3. Generalized TS fuzzy system

In the following we assume that there are d input variables x1,. . .,xd relatedto the fuzzy systems. Since the universal approximations of fuzzy systems are

analyzed in a given compact set, we can always scale the input variables so that

they all fall in [�1, 1]. Thus, without loss of generality, we suppose �16 xi6 1for i=1,. . ., d.

For given n 2 N, partition [�1, 1] into 2n equal parts: [(j�1)/n, j/n]

(j=�n+1, �n+2, . . ., n�1, n). For each i2{1,. . .,d}, we define 2n+1 fuzzy

numbers eAijðj ¼ �n;�nþ 1; . . . ; n� 1; nÞ on [�1, 1] with the following condi-

tions:

(i) The kernel kerðeAijÞ , fx 2 ½�1; 1�jeAijðxÞ ¼ 1g includes {j/n}.

(ii) "j1, j2, j2{�n, �n+1, . . ., n�1, n}:j1< j< j2, then 8x 2 ½�1; 1�, eAij1ðxÞ > 0,eAij2ðxÞ > 0, ) eAijðxÞ > 0.

(iii) There is a c0 2 N, independent of n, such that 16 cardðfjjeAijðxÞ > 0gÞ6 c0for each x 2 ½�1; 1�, where card(A) means the cardinal number of the

set A.

Fig. 1 gives a family of membership curves of such a class of fuzzy numbers.eAij�s will serve as the antecedents of the following TS fuzzy inference [16]:

IF x1 IS eA1p1 AND . . .AND IF xd IS eAdpd THEN y IS a0;p1;...;pd

þXd

i¼1ai;p1;...;pd xi; ð4Þ

where p1,. . .,pd 2 {�n,�n+1,. . .,n�1,n}, ai;p1,. . .,pd (i=0, 1,. . .,d) is an adjustable

parameter, and y is an output variable.

For arbitrary (x1,. . .,xd) 2 [�1, 1]d, and (p1,. . .,pd)2{�n,�n+1,. . .,n�1,n}d,let

Hp1;...;pd ðx1; . . . ; xdÞ ¼ eA1p1ðx1ÞT � � � T eAdpd ðxdÞ;

Fig. 1. The curves of antecedent fuzzy sets.


where �T � is a t-norm to evaluate �and� in the TS inference rule. In the rest of the

paper, we assume �T � to be the product �·�, that is, T(a,b)=a·b(a,b2[0, 1]). For(x1,. . .,xd)2[�1, 1]d. Let

Nðx1; . . . ; xdÞ ¼ ðp1; . . . ; pdÞ 2 f�n;�nþ 1; . . . ; n� 1; ngd eA1p1ðx1Þ��n

� � � � � eAdpd ðxdÞ > 0o:

Proposition 2. Let (x1,. . .,xd)2[�1, 1]d. Then

ðp1; . . . ; pdÞ 2 Nðx1; . . . ; xdÞ;)pi � c0

n6 xi �

pi þ c0n

ði ¼ 1; . . . ; dÞ:

The proof of the proposition is reported in Appendix A.We use a generalized defuzzifier introduced in [5] to calculate the output of

a fuzzy system determined by TS inference, that is, the output Tn(x1,. . .,xd)is determined as follows [21,22]:

T nðx1; . . . ; xdÞ ¼Pn

p1;...;pd¼�nHp1;...;pd ðx1; . . . ; xdÞa a0;p1;...;pd þ

Pdi¼1ai;p1;...;pd xi

� �Pn

p1;...;pd¼�nHp1;...;pd ðx1; . . . ; xdÞa ;

ð5Þwhere a : 06 a6 þ1 is an adjustable parameter. The fuzzy system whose I/O

relationship is determined by (5) is called a generalized TS fuzzy system.By the fact suggested by [5], if a=1, (5) is a fuzzy system with the centroid

defuzzifizer (see also [1,16,17,21–24]; If a ¼ þ1, (5) is one with the mean of

maximum defuzzification method [17]; and if a=0, (5) is a fuzzy system with

weighted sum [9]; If a=1�c+c/d(c2[0, 1]), then (5) becomes one corresponding

to the compensatory neuro-fuzzy system [25].

In (5) if the parameters a1;p1;...;pd ; . . . ; ad;p1;...;pd satisfy the condition:

a1;p1;...;pd ¼ � � � ¼ ad;p1;...;pd ¼ 0, the system (5) is called a simply generalized TS

fuzzy system [22].

4. Hierarchical TS fuzzy systems

In the complete fuzzy rule base of a fuzzy system, the fuzzy rules cover all

possible combinations of the antecedent fuzzy sets eAij�s [14,21]. The number

of the fuzzy rules is (2n+1)d. Thus the total rule number in the complete fuzzy

rule base is an exponential function of the number of the system variables. Fora large d, it may become unproductive to realize a d-dimensional fuzzy system.

To solve such a problem, a hierarchical structure is presented in the section. In

this hierarchy, the first level fuzzy system gives an approximate output yn1,


which is modified by the second level fuzzy system as an input variable; the

third level system will modify the output yn2 of the second level fuzzy system. . .;and so on. This process is repeated in succeeding sub-systems of the hierarchy.

Such a hierarchical structure is shown in Fig. 2.

In Fig. 2, yn1; . . . ; ynL�1 are also employed as the intermediate input variables

of the corresponding sub-system. For each k=1,. . .,L�1, we introduce fuzzynumber eBkj 2 FðRÞ ðj ¼ 0;�1; . . . ;�nÞ as the antecedent fuzzy set for the

input variable ynk , such that 8y 2 R; 8k ¼ 1; . . . ; L� 1;

card j 2 f�n;�nþ 1; . . . ; n� 1; ng eBkjðyÞ > 0�� on �

P 1:�

ð6Þ

In the hierarchy as Fig. 2, the first level and kth level are the generalized TS

fuzzy systems for k=2,. . .,L. The first level with d1 input variables x1,. . .,xd1,

and kth level with dk+1 inputs xdk�1 + 1,. . .,xdk�1 + dk and the intermediate varia-

ble ynk�1, the output of the (k�1)th level, are constructed by the following TS

fuzzy inference rules:

First: IF x1 IS eA1p1 AND � � �AND xd1 IS eAd1pd1THEN y1 is a10;p1;...;pd1

þPd1i¼1a

1i;p1;...;pd1

xi;

kth: IF xdk�1 + 1 IS eAðdk�1þ1Þp1 AND� � �AND xdk�1 + dkIS eAðdk�1þdkÞpdk

AND ynk�1 ISeBðk�1Þik�1THEN ynk is a

k0;p1;...;pdk

þ ckik�1ynk�1 þ

Pdki¼1a

ki;p1;...;pdk

xdk�1þi,

where k ¼ 2; . . . ; L;PL

k¼1dk ¼ d; ik�1; p1; p2; . . . 2 f�n;�nþ 1; . . . ; n� 1; ng,and aki;p1;...;pk ; c

kik�1

are adjustable parameters.

Fig. 2. Hierarchical fuzzy system.


The following conclusions show the linear relation of the fuzzy rule number

with respect to the input variable number d.

Proposition 3. [18] Let there be L levels in the HFS as Fig. 2, and d1 be the input

variable number of the first level system. And the kth (k=2,. . .,L) level system has

dk+1 input variables, in which the intermediate variable ynk�1 is included. Ifd1=dk+1=c, then the total number of the fuzzy rules in the rule base of the HFS

is (2n+1)c(d�1)/(c�1).

By the fact suggested by Raju et al. [14], the value of (2n+1)c(d�1)/(c�1) in

Proposition 3 is minimum when c=2, that is, the number of rules in the fuzzy

rule base of the HFS is (2n+1)2(d�1), a linear function of the input variable

number d.

For arbitrary ðx1; . . . ; xdÞ 2 Rd , nonnegative integer i, and k 2 N, so that06 i< i+k6 d. From now on we write for p1,. . .,pk2{�n,�n+1,. . .,n�1,n}:

Hp1;...;pk ðxiþ1; . . . ; xiþkÞ ¼ eAðiþ1Þp1ðxiþ1Þ � � � � � eAðiþkÞpk ðxiþkÞ: ð7Þ

In the hierarchical system shown as Fig. 2, the I/O relationships of the first

level and kth level fuzzy systems are determined by the generalized TS fuzzy

system:

yn1 ¼ T n1ðx1; . . . ; xd1Þ ¼

Pn

p1 ;...;pd1¼�n

Hp1 ;...;pd1ðx1;...;xd1 Þ

a a10;p1 ;...;pd1

þPd1

i¼1a1i;p1 ;...;pd1

xi

� �Pn

p1 ;...;pd1¼�n

Hp1 ;...;pd1ðx1;...;xd1 Þ

a ;

ynk ¼ T nkðxlkþ1; . . . ; xlkþdk ; y

nk�1Þ

¼Pn

ik�1 ;plkþ1 ;...;plkþdk¼�n

Hplkþ1 ;...;plkþdkðxlkþ1;...;xlkþdk

Þ�eBðk�1Þik�1ðynk�1

Þ� �a

Y ðIk ;PkÞPn


Hplkþ1 ;...;plkþdkðxlkþ1;...;xlkþdk

Þ�eBðk�1Þik�1ðyn

k�1Þ

� �a ;

8>>>>>>>>>><>>>>>>>>>>:ð8Þ

where k ¼ 2; . . . ; L; lk ¼Pk�1

p¼1dp and

Y ðIk; PkÞ ¼ ak0;plkþ1;...;plkþdkþ ckik�1

ynk�1 þXdki¼1

aki;plkþ1;...;plkþdkxlkþi:

The I/O relationship determined by (8) is called the generalized hierarchical TS

fuzzy system.

Lemma 1. Let ðx1; . . . ; xdÞ 2 Rd , and i; j; k1; k2 2 N, such that 16 i< j<

i+k1+k26 d, i+k1=j. Then the following hold for each a : 06 a6 þ1:


Xn

p1;...;pk1¼�n

ðHp1;...;pk1ðxiþ1; . . . ; xiþk1ÞÞ

a

0@ 1A�

Xn

q1;...;qk2¼�n

ðHq1;...;qk2ðxjþ1; . . . ; xjþk2ÞÞ

a

0@ 1A¼

Xn

p1;...;pk1þk2¼�n

Hp1;...;pk1þk2ðxiþ1; . . . ; xiþk1þk2Þ

a:

The proof of the lemma is reported in Appendix B.For simplicity, in the rest of the paper we choose the parameters in (8) for

k=2,. . .,L as follows:

a1i;p1;...;pd1¼ 0ði ¼ 1; . . . ; d1Þ; aki;plkþ1;...;plkþdk

¼ 0ði ¼ 1; . . . ; dkÞ

consequently rewriting (8) we obtain

yn1 ¼ T n1ðx1; . . . ; xd1Þ ¼

Pn

p1 ;...;pd1¼�n

Hp1 ;...;pd1ðx1 ;...;xd1 Þ

aa10;p1 ;...;pd1Pn

p1 ;...;pd1¼�n

Hp1 ;...;pd1ðx1 ;...;xd1 Þ

a ;

ynk ¼ T nkðxlkþ1; . . . ; xlkþdk ; y

nk�1Þ

¼Pn


Hplkþ1 ...plkþdkðxlkþ1 ;...;xlkþdk

Þ�eBðk�1Þik�1ðynk�1

Þh ia

ak0;plkþ1 ;...;plkþdk

þckik�1ynk�1Pn


Hplkþ1 ;...;plkþdkðxlkþ1 ;...;xlkþdk

Þ�eBðk�1Þik�1ðyn

k�1Þ

h ia ;

8>>>>>>>>>><>>>>>>>>>>:ð9Þ

where k ¼ 2; . . . ; L, lk ¼Pk�1

p¼1dp. (9) is called a simply generalized HFS.

For k=2,. . .,L, we introduce the integer vectors Ik, Pk and the function

J(Ik;y1,. . .,yk�1) respectively as follows:

Ik ¼ ði1; . . . ; ik�1Þ; Pk ¼ ðp1; . . . ; plkþdk Þ;

JðIk; y1; . . . ; yk�1Þ ¼ eB1i1ðy1Þ � � � � � eBðk�1Þik�1ðyk�1Þ:

Theorem 1. Let yn1; . . . ; ynL be a simply generalized HFS defined as (9). Then for

each k=2,. . .,L, and Ik=(i1,. . .,ik�1), Pk ¼ ðp1; . . . ; plkþdk Þ: i1; . . . ; ik�1;

p1; . . . ; plkþdk 2 f�n;�nþ 1; . . . ; n� 1; ng, there is the coefficient a0(Ik, Pk), so

that the following fact holds:

PnH ðx ;...;x ÞJðIk;yn;...;yn Þ

� �aa ðIk;PkÞ

ynk¼i1;...;ik�1;p1;...;plkþdk

¼�np1;...;pdkþlk

1 dkþlk 1 k�1 0

Pni1;...;ik�1;p1;...;pdkþlk

¼�nHp1;...;pdkþlk

ðx1;...;xdkþlk ÞJðIk;yn1;...;ynk�1Þ� �a :

ð10Þ(10) is a simply generalized TS system. So by Theorem 1 the output of the

HFS defined as (9) can be represented as one of a simply generalized TS fuzzysystem. Further the adjustable parameters a 0(I

2, P2),. . .,a0(IL, PL) may be

determined by following iterative scheme:

a0ðI2; P 2Þ ¼ a10;p1;...;pd1c2i1 þ a20;pd1þ1;...;pd1þd2

;

a0ðIk; PkÞ ¼ a0ðIk�1; Pk�1Þckik�1þ ak0;plkþ1;...;plkþdk

;

8<: ð11Þ

where k=3,. . .,L. (11) may be implied by the proof of Theorem 1, which isreported in Appendix B.

On the other hand, by (11) if a0(I2, P2),. . .,a0(I

L, PL) are some known

parameters, we may obtain a10;p1;...;pd1, ckik�1; . . . ; a

k0;plkþ1;...;plkþdk

ðk ¼ 2; . . . ; LÞ suchthat (11) holds, that is, by a known simply generalized TS fuzzy system, we may

obtain a corresponding HFS [11,12].


5. Universal approximation

By HFS�s we can successfully solve the �rule explosion� problem. Undoubt-

edly such a hierarchy is of theoretical and practical importance. Naturally a

meaningful and important problem related to the subject is to analyze the

approximating capability of the HFS�s. Wang in [18] firstly put forward this

question, and the approximation of the HFS�s to continuously differentiable

functions is established by solving a large scale linear system of equations.

The antecedent fuzzy sets eAij�s suggested by Wang [18] must be the triangular

fuzzy number. In the section, the universal approximation of the generalizedHFS�s is presented, that is, the simply generalized HFS�s defined as (9) can

arbitrarily closely approximate any continuous function defined on each com-

pact set.

The HFS�s proposed in the paper is general. The shapes of the fuzzy numbers

are not especially restricted, also the defuzzification methods are general [5].

Theorem 2. Let S : ½�1; 1�d ! R be an arbitrary piecewise linear function. Then

for each e>0, there are n 2 N and a simply generalized HFS yn1; . . . ; ynL defined as

(9), so that kynL � Sk1;½�1;1�d < e.


The proof of Theorem 2 is reported in Appendix B.

By Theorem 1, The HFS defined as (9) can be represented as a generalized

TS fuzzy system (see also [11,12,20–22]), so for arbitrary e>0, there are n 2 N,

and the parameter a0;p1;...;pd ðp1; . . . ; pd ¼ 0;�1; . . . ;�nÞ:

a0;p1;...;pd ¼ Sp1n; . . . ;

pdn

� �:

The adjustable parameters a10;p1;...;pd1, ak0;plkþ1;...;plkþdk

and ckik�1ðik�1; p1; . . . ; pd1 ;

plkþ1; . . . ; plkþdk 2 f�n;�nþ 1; . . . ; n� 1; ng; i 2 f1; . . . ; dg; k=2,. . .,L) can be

determined by the backward iteration method:

a0ðIL; PLÞ ¼ a0;p1;...;pd ¼ S p1n ; . . . ;

pdn

� ;

a0ðIk; PkÞ ¼ a0ðIk�1; Pk�1Þckik�1þ ak0;plkþ1;...;plkþdk

ðk ¼ L; . . . ; 3Þ;

a0ðI2; P 2Þ ¼ a10;p1;...;pd1c2i1 þ a20;pd1þ1;...;pd1þd2

:

8>>><>>>: ð12Þ

By Proposition 1 and Theorem 2, we may easily obtain

Theorem 3. Let f : ½�1; 1�d ! R be a continuous function. Then for arbitrary

e>0, there is n 2 N, if yn1; . . . ; ynL are defined by (9), kynL � f k1;½�1;1�d < e.

By Theorem 3, the simply generalized HFS�s can be the universal approxi-

mator. So the HFS�s can be extensively utilized as controllers and modeling

tools.

For given error bounds e>0, we can estimate n, consequently the fuzzy rule

number in the rule base of a hierarchical system can be determined.

Theorem 4. Let f : ½�1; 1�d ! R be a continuous function. Then for each e>0,

there is h>0. Writing

D ¼_di¼1

_x1;...;xd ;xiþh2½�1;1�

f ðx1; . . . ; xi�1; xi þ h; xiþ1; . . . ; xdÞ � f ðx1; . . . ; xdÞh

�� ;

ð13Þ

we have, kf � ynLk1;½�1;1�d < e holds provided n>2Dc0d/e.

The proof of Theorem 4 is reported in Appendix B.

If let c0=2,d=3, then n>12D/e implies kf � ynLk1;½�1;1�d < e. For a given con-

tinuous function f and an error bound e>0, we may conveniently establish the

approximation representation of f by a simply generalized HFS defined as (9)

with following steps:


Step 1. By (13) the supremum D can be approximately calculated, and the

minimum value of n can be estimated.

Step 2. To a given practical problem, define the suitable antecedent fuzzy setseAijði ¼ 1; . . . ; d; j ¼ 0;�1; . . . ;�nÞ, and eBðk�1Þik�1ðk ¼ 2; . . . ; L; ik�1=0,

±1, . . ., ±n). Usually for convenience eAij�s are defined as the triangular or tra-

pezoidal fuzzy numbers.Step 3. In (10) we define the parameter a0(I

L, PL) by (3) and (12) as follows:

a0ðIL; PLÞ ¼ fp1n; . . . ;

pdn

� �:

Step 4. By (12), the parameters a10;p1;...;pd1, ak0;plkþ1;...;plkþdk

and ckik�1of the simply

generalized HFS can be backwards determined.

Step 5. By (9) we can establish the simply generalized HFS yn1; . . . ; ynL.

6. Simulation examples

In the section we present two examples to realize the approximation of the

HFS�s to the given functions with the algorithm developed in Section 5. From

these simulation examples we can obtain the fact that using HFS�s with much

less fuzzy rules than standard fuzzy systems we can realize the high dimensionalreal systems with high accuracy.

Example 1. Let a=1, d=3, c0=2 and d1=2,d2=1. Thus the number of the

input variables of the fuzzy system is three, and the hierarchy possesses two

levels. Define the continuous function f : ½�1; 1�3 ! R as follows:

f ðx1; x2; x3Þ ¼ exp � x21 þ x22 þ x2325

� �jx1j6 1; jx2j6 1; jx3j6 1ð Þ:

For given error bound e=0.1, considering

exp � x21 þ y21 þ z2125

� �� exp � x22 þ y22 þ z22

25

� �� 6

2

25jx1 � x2j þ jy1 � y2j þ jz1 � z2jð Þ; ð14Þ

and from (13) and (14), we can obtain D 6 2/25<1/12. So let n=12/(12e)=1/

0.1=10. That is, if we partition [�1, 1] into 20 equal parts, then by Theorem

4, it follows that

ynL � f

1;½�1;1�d¼ y102 � f

1;½�1;1�3< e:


By Proposition 3, the total number of fuzzy rules in the rule base of the hier-

archical system is (2n+1)2(d�1), i.e. 2(2· 10+1)2=882.

Define the triangular fuzzy number eA as follows for m 2 N:

eAðxÞ ¼ m 1m � x�

; 06 x6 1m ;

m 1m þ x�

; � 1m 6 x < 0;

0; otherwise:

8><>:Let eA1j ¼ eA2j ¼ eA3jðj ¼ 0;�1; . . . ;�mÞ; and eA1jðj ¼ 0;�1; . . . ;�ðm� 1ÞÞ

be obtained by the parallel translation of eA; that is, eA1jðxÞ ¼ eA x� j=mð Þ. And

eA1ðmÞðxÞ ¼m x� m�1

m

� ; m�1

m 6 x6 1;

0; otherwise;

(

eA1ð�mÞðxÞ ¼�m xþ m�1

m

� ; �16 x6 � m�1

m ;

0; otherwise:

(

Let B1j be defined as follows for j=0,±1,. . .,±m:

eB1jðyÞ ¼ exp � 1

2y � j

m

� �� : ð15Þ

Let m=n=10. Considering Hp1p2p3ðx1; x2; x3Þ ¼ eA1p1ðx1Þ � eA1p2ðx2Þ � eA1p3ðx3Þ,and the fact:

JðI2; yn1Þ ¼ Jði1; yn1Þ ¼ eB1i1ðyn1Þ ¼ exp � yn1 � i1=102

� �;

and by (9), we obtain the simply generalized HFS with two level as follows:

yn1 ¼P10

p1 ;p2¼�10eA1p1

ðx1Þ�eA2p2ðx2Þ�a10;p1p2P10

p1 ;p2¼�10eA1p1

ðx1Þ�eA2p2ðx2Þ

;

yn2 ¼P10

i1 ;p3¼�10eA3p3

ðx3Þ�eB1i1ðyn

1Þ� a2

0;p3þc2i1

yn1

� �P10

i1 ;p3¼�10eA3p3

ðx3Þ�eB1i1ðyn

1Þ

:

8>>>>>>><>>>>>>>:ð16Þ

By (12) and the fact f ðp1=10; p2=10; p3=10Þ ¼ Sðp1=10; p2=10, p3=10Þ, we may

define the parameters a10;p1p2 ; c2i1and a20;p3 in (16) such that

a1 � c2 þ a2 ¼ fp1 ;

p2 ;p3

� �:
0;p1p2 i1 0;p3 10 10 10


Letting

c2i1 ¼ 1; a10;p1p2 þ a20;p3 ¼ fp110

;p210

;p310

� �;

we may obtain a hierarchy defined as (9). By Theorem 1, the I/O relationship

of the hierarchical system may be represented as follows:

yn2 ¼

P10i1;p1;p2;p3¼�10

eA1p1ðx1Þ � eA2p2ðx2Þ � eA3p3ðx3Þ � eB1i1ðyn1Þ� �

� f p110; p210; p310

� P10

i1;p1;p2;p3¼�10

eA1p1ðx1Þ � eA2p2ðx2Þ � eA3p3ðx3Þ � eB1i1ðyn1Þ� � :

ð17ÞGiving x3=0 and x3=1, we may respectively obtain the surfaces of f and yn2

respectively as Figs. 3–6.From Figs. 3–6, with the approximate sense ��e� the given continuous

function may be directly realized by the HFS�s with the high accuracy. Further

Fig. 3. x3=0, surface of f.

Fig. 4. x3=0, surface of yn2.

Fig. 6. x3=1, surface of yn2.

Fig. 5. x3=1, surface of f.


the approximation is convenient to realize with the simple system. If we employ

the standard fuzzy system to realize the same accuracy, the number of the fuzzy

rules in rule base is (2n+1)3=213=9621. So the rule number of the HFS is

decreased strikingly.

Let us now proceed to show the approximation of HFS�s by a simulation

example. We handle the problem in a higher dimensional space.

Example 2. Suppose a=1, d=6, c0=2 and d1=2, d2= � � �=d6=1. Thus theinput variable number of the fuzzy system is six and the hierarchy possesses five

levels. Let the function f : ½�1; 1�6 ! R define as follows:

f ðx1; . . . ; x6Þ ¼

sinx41þx4

2ffiffiffiffiffiffiffiffiffix21þx2

2

p� �

þ lnQ6i¼3

1þ cos p=2� x3i� � �

;

x1; . . . ; x6 2 ½�1; 1�; x21 þ x22 6¼ 0;

x1; . . . ; x6 2 ½�1; 1�; x1 ¼ x2 ¼ 0:

8>>>><>>>>:


Similarly we choose error bound e=0.1. Using (13), we can estimate D, and

therefore n can be determined. Here we let n=20, that is, if we partition [�1, 1]

into 40 equal parts to define the antecedent fuzzy sets eAij and eBkj for i=1,. . .,6,j=0,±1,. . .,±m and k=1,. . .,4. We choose these fuzzy sets as in Example 1:eA1j ¼ � � � ¼ eA6j; eB1j ¼ � � � ¼ eB4j for j=0,±1,. . .,±m, and eB1j is defined as (15),

where m=20. Using (9) and (12) we obtain a simple HFS as follows for n=20:

yn1 ¼ T n1ðx1; x2Þ ¼

Pn

p1 ;p2¼�neA1p1

ðx1Þ�eA2p2ðx2Þ�a10;p1p2Pn

p1 ;p2¼�nHp1p2 ðx1;x2Þ

;

yn2 ¼ T nkðx3; yn1Þ ¼

Pn

i1 ;p3¼�neA3p3

ðx3Þ�eB1i1ðyn

1Þ a2

0;p3þc2i1

yn1

� �Pn

i1 ;p3¼�neA3p3

ðx3Þ�eB1i1ðyn

1Þ

;


Pn

i2 ;p4¼�neA4p4

ðx4Þ�eB2i2ðyn

2Þ a3

0;p4þc3i2

yn2

� �Pn

i2 ;p4¼�neA4p4

ðx4Þ�eB2i2ðyn

2Þ

;


Pn

i3 ;p5¼�neA5p5

ðx5Þ�eB3i3ðyn

3Þ a4

0;p5þc4i3

yn3

� �Pn

i3 ;p5¼�neA5p5

ðx5Þ�eB3i3ðyn

3Þ

;


Pn

i4 ;p6¼�neA6p6

ðx6Þ�eB4i4ðyn

4Þ a5

0;p6þc5i4

yn4

� �Pn

i4 ;p6¼�neA6p6

ðx6Þ�eB4i4ðyn

4Þ

;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð18Þ

where a10;p1p2 , ak�10;pk

, ck�1ik�2

for k=3,. . .,6 can be determined by (12) as follows: Letck�1ik�2

¼ 1, and

f p1n ; . . . ;

p6n

� ¼ a0ðI4; P 4Þ þ a50;p6 ;

a0ðI4; P 4Þ ¼ a0ðI3; P 3Þ þ a40;p5 ;

a0ðI3; P 3Þ ¼ a0ðI2; P 2Þ þ a30;p4 ;

a0ðI2; P 2Þ ¼ a10;p1p2 þ a20;p3 :

8>>>>><>>>>>:We choose some sample points randomly in [�1, 1]6 as in Table 1, and the

corresponding approximating errors of the sample HFS (18) at these points are

also demonstrated in Table 1, from which we can see the high approximating

accuracy of the hierarchical system (18).

By Proposition 3, the total number of fuzzy rules in the rule base of the

hierarchical system may be (2n+1)2(d�1), i.e. (6�1) Æ (2· 20+1)2=8405. If weutilize a standard fuzzy system to realize the same accuracy, the number of the

fuzzy rules in the rule base is (2n+1)6=416�4.8·109. The fuzzy system with

such large size fuzzy rule base is impossible to realize. So the rule number of the

HFS can be decreased strikingly.

Table 1

Approximating errors at randomly chosen sample points

No. Sample points Original values Approximate values Error

1 (�1.0, 0.4, �0.2, �0.7, 0.6, 0.5) 0.63172947647394 0.63172947647394 4.440892098500626e�016

2 (0.8, �0.6, 0.1, 0.0, 1.0, �0.7) 0.71506795219737 0.71506795219738 1.554312234475219e�015

3 (�0.7, �0.7, 0.2, �0.9, 0.1, 0.2) �0.61377539192791 �0.61377539192791 9.992007221626409e�016

4 (�0.9, 0.8, 0.7, 0.3, �0.1, �0.4) 1.02338498389300 1.02338498389301 1.998401444325282e�015

5 (0.5, 0.2, �0.5, �1.0, 0.2, �0.5) �1.98141917964883 �1.98141917964883 2.220446049250313e�016

6 (0.1, 0.8, �0.3, �0.4, �1.0, 0.1) �1.44769952494765 �1.44769952494765 4.440892098500626e�016

7 (�0.9, �0.4, 0.4,0.9, 0.1, �0.5) 1.07853688212088 1.07853688212088 6.661338147750939e�016

8 (0.7, �0.9, 0.3, �0.7, �0.1, 0.3) 0.34987698784961 0.34987698784961 1.887379141862766e�015

P.Liu,H.Li/Inform

atio

nScien

ces169(2005)279–303

295


7. Conclusions

The paper firstly gives the systematic analysis to the approximating capabil-

ity of HFS�s, and shows the fact that HFS�s can be the universal approxima-

tors. The results derived here provides us with the solid theoretical basis of

the applications of HFS. So our contribution is three-fold:

(1) In the discussions to HFS�s (for example, Raju and Zhou et al. [14,15] and

Wang [18,19]) the fuzzy sets must be triangular fuzzy numbers, and the de-

fuzzification is the centroid defuzzifier. The HFS�s presented in this paper

are very general. It use any type of fuzzy sets, and the generalized defuzz-

ifier containing the popular centroid method as a special case.

(2) Using the piecewise linear functions as the bridge, we constructively prove

that the HFS�s can arbitrarily closely approximate any continuous func-tion defined on each compact set. The data of the function to be approx-

imated are directly employed, so the approximation may be conveniently

realized.

(3) We establish a formula for estimating the minimal upper bounds of the

number of fuzzy rules needed for a given accuracy. So for given continuous

system and approximating accuracy, we may easily constitute a HFS to

represent the system approximately.

The further topics related the subject for the future research are as follows:

If the t-norm is not the product �·�, how may the approximation capability of

the HFS�s be studied? If the function related is not continuous, but integrable,

whether the hierarchical systems can be the universal approximators with the

mean square sense?

Acknowledgement

Project supported by a grant from National Natural Science Foundation of

China (Nos. 60375023 and 69974041).

Appendix A

Proof of Proposition 1. For simplicity, we prove the conclusion in two-dimensional Euclidean space R2, i.e. d=2. As for the cases of d>2 and d=1 the

proofs are similar. Obviously there is M>0, so that U�[�M, M]d because U is

a compact set in R2. It is no harm to assume [�M, M]d=[�1, 1]2. The fact that

f is continuous implies f is uniformly continuous on [�1, 1]2. Hence for e>0,

there is a n0 2 N, such that "n>n0, if we partition [�1, 1] into 2n equal parts


[�1, (1�n)/n],. . .,[0, 1/n],. . .,[(n�1)/n, 1], consequently 4n2 squares Dnij�s

(i,j=�n+1,�n+2,. . .,n�1,n) are yielded:

Dnij ¼ ðx; yÞ 2 U

i� 1

n6 x6

in;j� 1

n6 y6

jn

�� ;

we have 8ðx1; y1Þ; ðx2; y2Þ 2 Dnij, jf ðx1; y1Þ � f ðx2; y2Þj < e=5. Therefore

dij ,_

ðx;yÞ2Dnij

ff ðx; yÞg;

dij ,^

ðx;yÞ2Dnij

ff ðx; yÞg ) jdij � dijj <e4; ðA:1Þ

where i, j=�n+1, �n+2, . . .,n�1, n. Respectively linking a pair of oppose

vertices of Dnij; we obtain 8n2 equicrurally rectangular triangles D1,. . .,D8n2.

For each k=1,. . .,8n2, the plane equation z=S(x,y) determined by

ðxk1; yk1; f ðxk1; yk1ÞÞ, ðxk2; yk2; f ðxk2; yk2ÞÞ, ðxk3; yk3; f ðxk3; yk3ÞÞ of the vertices of Dk, is

as follows:

x y Sðx; yÞ 1

xk1 yk1 f ðxk1; yk1Þ 1



��

�� ¼ 0: ðA:2Þ

Let h is the side length of Dnij; i.e. h=1/n. It is no harm to assume yk1 ¼ yk2,

xk1 ¼ xk3, xk2 ¼ xk1 þ h, yk3 ¼ yk1 þ h. Rewriting (18), we obtain

Sðx; yÞ ¼ f ðxk1; yk1Þ

þ ðx� xk1Þðf ðxk2; yk2Þ � f ðxk1; yk1ÞÞ þ ðy � yk1Þðf ðxk3; yk3Þ � f ðxk1; yk1ÞÞh

:

ðA:3ÞTherefore "(x,y)2U, there is a k2{1,. . .,8n2}, such that (x,y)2Dk. So by

(A.3) the following hold:

jf ðx; yÞ � Sðx; yÞj ¼ f ðx; yÞ � f ðxk1; yk1Þ��

�ðx� xk1Þðf ðxk2; yk2Þ � f ðxk1; yk1ÞÞ þ ðy � yk1Þðf ðxk3; yk3Þ � f ðxk1; yk1ÞÞh

��6 jf ðx; yÞ � f ðxk1; yk1Þj þ

jx� xk1jh

jf ðxk2; yk2Þ � f ðxk1; yk1Þj

þ jy � yk1jh

jf ðxk3; yk3Þ � f ðxk1; yk1Þj:

Considering jx� xk1j=h6 1, jy � yk1j=h6 1 and (A.1), we have jf(x,y)�S(x,y)j6 e/4+ e/4+ e/4=3e/4. Consequently kf � Sk1;U 6 3e=4 < e. h


Proof of Proposition 2. Considering (p1,. . .,pd)2N(x1,. . .,xd), we obtaineA1p1ðx1Þ � � � � � eAdpd ðxdÞ > 0, hence 8i ¼ 1; . . . ; d; eAipiðxiÞ > 0. Given arbitrar-

ily i 2 f1; . . . ; dg; eAipi is a fuzzy number and eAipi pi=nð Þ ¼ 1. If we assume

xi>(pi+c0)/n, then xi>pi/n. By the properties of the fuzzy numbers [10], we

obtain eAipi ðpi þ c0Þ=nð ÞP eAi;piðxiÞ > 0. Since eAipiþc0ððpi þ c0Þ=nÞ ¼ 1 > 0; the

definition of eAijimplies 8j : pi 6 j6 pi þ c0; eAij ðpi þ c0Þ=nð Þ > 0. Thus

cardðfjjeAijððpi þ c0Þ=nÞ > 0gÞP c0 þ 1;

which contradicts the definition of c0. So xi6 (pi+c0)/n. With the same reason,

xiP (pi�c0)/n. Consequently (pi�c0)/n6 xi6 (pi+c0)/n(i=1,. . .,d). h

Appendix B

Proof of Lemma 1. Considering (7), we obtain

Hp1;...;pk1ðxiþ1; . . . ; xiþk1Þ

a ¼ eAðiþ1Þp1ðxiþ1Þa � � � � � eAðiþk1Þpk1ðxiþk1Þ

a;

Hq1;...;qk2ðxjþ1; . . . ; xjþk2Þ

a ¼ eAðjþ1Þq1ðxjþ1Þa � � � � � eAðjþk2Þqk2ðxjþk2Þ

a:

Therefore by the assumption that i+k1= j the following facts hold:

Xn

p1;...;pk1¼�n

Hp1;...;pk1ðxiþ1; . . . ;xiþk1Þ

a

0@ 1A �Xn

q1;...;qk2¼�n

Hq1;...;qk2ðxjþ1; . . . ;xjþk2Þ

a

0@ 1A¼

Xn

p1;...;pk1¼�n

ðeAðiþ1Þp1ðxiþ1Þa� � � �� eAðiþk1Þpk1ðxiþk1Þ

aÞ

0@ 1A�

Xn

q1;...;qk2¼�n

eAðjþ1Þq1ðxjþ1Þa� �� eAðjþk2Þqk2ðxjþk2Þ

a

0@ 1A¼

Xn

p1;...;pk1 ;q1;...;qk2¼�n

eAðiþ1Þp1ðxiþ1Þa� �� eAðiþk1Þpk1ðxiþk1Þ

a�

�eAðiþk1þ1Þq1ðxiþk1þ1Þa� �� eAðiþk1þk2Þqk2ðxiþk1þk2Þ

a�

¼Xn

p1;...;pk1þk2¼�n

eAðiþ1Þp1ðxiþ1Þa� �� eAðiþk1þk2Þpk1þk2ðxiþk1þk2Þ

a� �

¼Xn

p1;...;pk1þk2¼�n

Hp1;...;pk1þk2ðxiþ1; . . . ;xiþk1þk2Þ

a;

which implies the conclusion. h


Proof of Theorem 1. Let us prove the conclusion by the inductive method.

At first when k=2, then lk= l2=d1. By (9) we obtain

yn2 ¼

Pni1;p1;...;pd2¼�n

Hp1;...;pd2ðxd1þ1; . . . ; xd1þd2Þ

a � eB1i1ðyn1Þa a20;p1;...;pd2

þ c2i1yn1

� �Pn

i1;p1;...;pd2¼�nHp1;...;pd2

ðxd1þ1; . . . ; xd1þd2Þa � eB1i1ðyn1Þ

a� � :

ðB:1Þ

Considering (9) and (B.1) we have

a20;p1;...;pd2þ c2i1y

n1 ¼

Pnq1;...;qd1¼�n

Hq1;...;qd1ðx1; . . . ; xd1Þ

a a10;q1;...;qd1c2i1 þ a20;p1;...;pd2

� �Pn

q1;...;qd1¼�nHq1;...;q1ðx1; . . . ; xd1Þ

a:

ðB:2Þ

By (B.1) and (B.2), and Lemma 1, we obtain

yn2 ¼

Pni1 ;p1 ;...;pd1þd2

¼�nHp1 ;...;pd1þd2

ðx1; . . . ; xd1þd2Þ � eB1i1ðyn1Þaa10;p1 ;...;pd1

c2i1 þ a20;pd1þ1 ;...;pd1þd2Pni1 ;p1;...;pd1þd2

¼�nHp1 ;...;pd1þd2

ðx1; . . . ; xd1þd2Þ � eB1i1ðyn1Þ� �a :

ðB:3Þ

If let

a0ðI2; P 2Þ , a10;p1;...;pd1c2i1 þ a20;pd1þ1;...;pd1þd2

;

we obtain

yn2 ¼

Pni1;p1;...;pl2þd2

¼�nHp1;...;pd2þl2

ðx1; . . . ; xd2þl2ÞJðI2; yn1Þ� �a

� a0ðI2; P 2Þ

Pni1;p1;...;pd2þl2

¼�nHp1;...;pl2þd2

ðx1; . . . ; xd2þl2Þ � JðI2; yn1Þ� �a : ðB:4Þ

(B.4) implies that (10) holds when k=2. Suppose (10) holds when

k=L�1, i.e. we can imply by the fact lL�1+dL�1= lL that the following factholds:


ynL�1 ¼

Pni1;...;iL�2 ;p1 ;...;plL¼�n

Hp1 ;...;plLðx1; . . . ; xlLÞJðIL�1; yn1; . . . ; y

nL�2Þ

� �a� a0ðIL�1; PL�1Þ

Pni1 ;...;iL�2 ;p1 ;...;plL¼�n

Hp1 ;...;plLðx1; . . . ; xlLÞ � JðIL�1; yn1; . . . ; y

nL�2Þ

� �a :

ðB:5Þ

And by (9), we have

ynL ¼ T LðxlLþ1; . . . ; xlLþdL ; ynL�1Þ

¼

PniL;p1 ;...;pdL¼�n

Hp1;...;pdLðxlLþ1; . . . ; xlLþdLÞ � eBLiLðynL�1Þ

� �aðaL0;p1 ;...;pdL þ cLiL�1

ynL�1Þ

PniL ;p1;...;pdL¼�n

Hp1;...;pdLðxlLþ1; . . . ; xlLþdLÞ � eBiLðynL�1Þ

� �a :

ðB:6Þ

So by (B.5) and (B.6), if let

a0ðIL; PLÞ ¼ a0ðIL�1; PL�1Þ � cLiL�1þ aL0;plLþ1;...;plLþdL

;

we obtain

ynL ¼

Pni1 ;...;iL�1 ;p1 ;...;plLþdL

¼�nHp1;...;pdLþlL

ðx1; . . . ; xdLþlLÞJðIL; yn1; . . . ; ynL�1Þ� �a

� a0ðIL; PLÞ

Pni1 ;...;iL�1;p1;...;pdLþlL

¼�nHp1 ;...;pdLþlL

ðx1; . . . ; xdLþlLÞJðIL; yn1; . . . ; ynL�1Þ� �a ;

which means (10) holds when k=L. Thus the theorem is completed. h

Proof of Theorem 2. Suppose D1,. . .,Dl(S) to be the polyhedrons corresponding

to S. Moreover,SlðSÞ

k¼1Dk ¼ ½�1; 1�d . Let

Sðx1; . . . ; xdÞ ¼

Pdi¼1s1i � xi; ðx1; . . . ; xdÞ 2 D1;

� � � � � � ;Pdi¼1slðSÞi � xi; ðx1; . . . ; xdÞ 2 DlðSÞ:

8><>:We show that there are n 2 N, and the coefficient a0(I

L, PL) fori1,. . .,iL�1=0,±1,. . .,±n, so that if ynL is defined by (10), then kynL � Sk1;U < e.

Define a0(IL, PL) for i1,. . .,iL�1; p1,. . ., plL+ dL 2 {0, ±1, . . ., ±n} respectively

as follows:

a0ðIL; PLÞ ¼ Sp1n; . . . ;

pdn

� �: ðB:7Þ


By Proposition 2, "(p1,. . .,pd)2N(x1,. . .,xd), let pi/n=xi+hpi/n(i=1,. . .,d). Thenjhpij6 c0, and by (2), it follows that

ðp1; . . . ; pdÞ 2 Nðx1; . . . ; xdÞ

) Sðx1; . . . ; xdÞ � Sp1n; . . . ;

pdn

� �� 6 c0n�Xd

i¼1

DiðSÞ: ðB:8Þ

Therefore considering pL+dL=d, and letting Hp1;...;pd ðx1; . . . ; xdÞ�JðIL; yn1; . . . ; ynL�1Þ , UðIL; PLÞ; we can conclude by (B.7) and (B.8) that the

following facts hold:

kynL � Sk1;½�1;1�d

¼_

ðx1 ;...;xd Þ2½�1;1�dfjynL � Sðx1; . . . ; xdÞjg

¼_

ðx1 ;...;xd Þ2½�1;1�d

Pni1 ;...;iL�1 ;p1 ;...;pd¼�n

UðIL; PLÞa � a0ðIL; PLÞ

Pni1 ;...;iL�1 ;p1 ;...;pd¼�n

UðIL; PLÞa� Sðx1; . . . ; xdÞ

��

8>>><>>>:9>>>=>>>;

¼_

ðx1 ;...;xd Þ2½�1;1�d

Pni1 ;...;iL�1 ;p1 ;...;pd¼�n

UðIL; PLÞa Sðp1n ; . . . ;pdn Þ � Sðx1; . . . ; xdÞ

� Pn

i1 ;...;iL�1 ;p1 ;...;pd¼�nUðIL; PLÞa

��

8>>><>>>:9>>>=>>>;

6

_ðx1 ;...;xd Þ2½�1;1�d

Pni1 ;...;iL�1¼�n

Pðp1 ;...;pd Þ2Nðx1 ;...;xd Þ

UðIL; PLÞa Sðp1n ; . . . ;pdn Þ � Sðx1; . . . ; xdÞ

�� Pn

i1 ;...;iL�1¼�n

Pðp1 ;...;pd Þ2Nðx1 ;...;xd Þ

UðIL; PLÞa

��

��

8>>><>>>:9>>>=>>>;

6

_ðx1 ;...;xd Þ2½�1;1�d

c0n�Xd

i¼1

DiðSÞ( )

¼ c0n�Xd

i¼1

DiðSÞ:

So if let n > c0 �Pd

i¼1DiðSÞ=e; we imply, kynL � Sk1;½�1;1�d < e.Also by the proof of Theorem 1, we have

a0ðIk; PkÞ ¼ a0ðIk�1; Pk�1Þ � ckik�1þ ak0;plkþ1;...;plkþdk

; ðB:9Þ

for 2 6 k 6 L. By (B.7) and (B.9), we may determine backwards the parame-ters ak0;plkþ1;...;plkþdk

, ckik�1ðk ¼ 2; . . . ; LÞ and a10;p1;...;pd1

. So we can obtain the simply

generalized HFS defined as (9). Thus by Theorem 1, the conclusion is

proven. h

Proof of Theorem 4. By the proof of Proposition 1, for e>0, if we partition

[�1, 1]d into identical cubes, and divide further the cubes into d-dimension

polyhedrons D1; . . . ;DN ðN 2 NÞ respectively, a piecewise linear function S may


be defined. Moreover, kf � Sk1;½�1;1�d < e=2. Assume the side lengths of the

cubes to be h>0. It may be easily prove for D determined by (13) that

D ¼Wd

i¼1fDiðSÞg. So by Theorem 2, if n>2D Æ c0d/e, then n > 2c0 �Pd

i¼1DiðSÞ=e. Therefore we have, kynL � Sk1;½�1;1�d < e=2. Thus

kf � ynLk1;½�1;1�d 6 kf � Sk1;½�1;1�d þ kS � ynLk1;½�1;1�d <e2þ e2¼ e;

by which the theorem is proven. h

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hierarchical ts fuzzy system and its universal approximation

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