approximation of stochastic processes by t–s fuzzy systems

Fuzzy Sets and Systems155 (2005) 215–235www.elsevier.com/locate/fss

Approximation of stochastic processes by T–S fuzzy systems�

Puyin Liua,b,∗, Hongxing LibaDepartment of Mathematics, National University of Defense Technology, Changsha 410073, China

bDepartment of Mathematics, Beijing Normal University, Beijing 100875, China

Received 8 May 2004; received in revised form 14 March 2005; accepted 5 April 2005Available online 25 April 2005

Abstract

Fuzzy systems can provide us with universal approximation models of deterministic input–output relationships,but in the stochastic environment few achievements related to the subject have so far achieved. In the paper a novelstochastic Takagi–Sugeno (T–S) fuzzy system is introduced to represent approximately existing randomness inmany real-world systems. By recapitulating the general architecture of the stochastic T–S fuzzy rule-based system,weanalyze systematically approximating capability of the stochastic system to a class of stochastic processes. By thecanonical representation of the stochastic processes, the stochastic fuzzy system is capable of with arbitrary accuracyproviding the approximation to the stochastic processes in mean square sense. Finally, an efficient algorithm for thestochastic T–S fuzzy system is developed. A simulation example demonstrates how a stochastic T–S fuzzy systemcan be constructed to realize the given stochastic process, approximately.© 2005 Elsevier B.V. All rights reserved.

Keywords:Stochastic T–S fuzzy system; Universal approximator; Stochastic process; Canonical representation; Stochasticintegral

1. Introduction

The fuzzy systems including T–S systems and Mamdani systems are capable of approximating a wideclass of functions, such as continuous functions and integrable functions[3,12,20,22–25], and so on.Like artificial neural networks [4], the approximation research is of much theoretic importance as it can

� Project supported by grants from National Natural Science Foundation of China (No. 60375023, No. 60474023,No. 69974041).∗ Corresponding author.

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

0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2005.04.002

http://www.elsevier.com/locate/fss

mailto:[email protected]

216 P. Liu, H. Li / Fuzzy Sets and Systems 155 (2005) 215–235

enrich approximation theory[3,12,24], also of practical importance as it has found very useful in manyapplied areas, e.g., system identification [17,20], automatic control [3,20], system modelling [17,25],pattern recognition [20] and telecommunication [9], etc.Although above universal approximation property of fuzzy systems can guarantee their ability for

modeling deterministic complex anduncertain systems, sucha superior traitmaybedegradedbyexistenceof randomness—a statistical uncertainty. It should be noted that many practical systems, such as financialmarkets [2], weather forecast models and control models [14] etc., operate in environment in whichincludes different types of uncertainty, especially randomness and fuzziness.Since the mid-1980s, research on the properties of articial neural networks in a stochastic environ-

ment has attracted attention from many scholars. Hinton et al. [7] built the Boltzmann machine in 1985based on Statistical Physics, in which the neuron states evolute with given probability distributions. Thenetwork model works like a simulated annealing procedure, and it serves as a simulator of probabil-ity distributions [7]. In 1989, Gelenbe [8] put forward a random neural network to simulate biophys-ical neural behavior, in which the neurons exchange positive and negative impulse signals with givenprobability distribution. To build efficient data classifying techniques Specht [16] developed in 1990another stochastic neural network model—probabilistic neural network (PNN). It operates by defininga probability density function to constitute a nonlinear, nonparametric pattern recognition algorithm.Based on Bayes’ classifying method PNN puts the statistical kernel estimator into the framework ofradial basis function networks. The performance of a PNN is often superior to other state-of-the-artclassifier. However, all those stochastic neural networks, Boltzmann machine-like networks, Gelenbe’smodels and PNN’s cannot be used as direct tools to approximate or simulate stochastic processes withsome metric senses. This problem begins to attract some scholars’ attention, for example, authors in[1,19] have proven that the approximation identity neural networks [5,18] can with mean square senseapproximate a class of stochastic processes to arbitrary degree of accuracy. But the fuzziness in real-world systems leads also to insufficiency of the neural network approach to model real input–outputprocesses.Above facts bring about the motivation of integrating randomness and fuzziness as a useful sys-

tem model to deal with different types of uncertainty. The stochastic fuzzy system can bridge the gapbetween randomness and fuzziness, and thus it can simultaneously handle data information andlinguistic information, in which statistical uncertainty can arise [11,14,15]. An important andmeaningful problem related is to analyze the approximating capability of stochastic fuzzy systems andemploy them to model real stochastic processes. In this paper, we concentrate on stochastic T–S fuzzysystems for this purpose and use the stochastic T–S fuzzy systems directly to simulate stochasticprocesses.The paper is organized as follows. In Section 2, we present a novel reasoning scheme—stochastic

T–S fuzzy rule incorporating randomness with fuzziness in an inference rule as ‘IF...THEN...’, and ageneral architecture of T–S stochastic fuzzy rule–based system, i.e. a stochastic T–S fuzzy system isrecapitulated, also some useful properties of the system is summarized. Section 3 recalls the theoryof stochastic processes, and the ‘canonical representation’ of a class of processes, which is of centralimportance to the approximation analysis. Section 4 deals with the approximation of the stochastic T–S fuzzy system to stochastic processes. If the processes are uniformly continuous with mean squaresense, the approximation with arbitrary degree of accuracy is analyzed by the processes themselves; Ifthe processes possess the regular representation form, the covariance functions of the processes and thestochastic integrals are the key tools for the discussions of the approximation procedures. In Section 5,

P. Liu, H. Li / Fuzzy Sets and Systems 155 (2005) 215–235 217

an efficient learning algorithm for the stochastic T–S fuzzy system is reported, and a simulation exampleis employed to demonstrated how a stochastic T–S fuzzy system for approximating the given process isconstructed. Finally, some further problems related the subject are presented.

2. Stochastic T–S fuzzy system

To account for T–S fuzzy logic with stochastic sense, let us now first consider the following sentencein natural language:

IF it is cloudy in the west at nightfall, THEN it will rain with great rainfall next day.

Although here ‘cloudy’ ‘great rainfall’ are fuzzy concepts and can be described with fuzzy sets, it is nolong enough to report the weather conditions by using fuzziness only. Obviously some random elementsexist in the weather prediction. Let us consider another fact: although fuzzy systems can utilize humanexpert knowledge to model many real systems, there is not a systematic way to ensure optimality ofthe models related, as expert opinions may be different, even conflicting. For instance, choose randomlysome experts to judge a design scheme synthetically. This is a subjective matter, and we will get dif-ferent answers from different experts. An efficient method to model the real systems with randomnessand fuzziness is to design a novel inference system—stochastic fuzzy logic, and construct the corre-sponding stochastic fuzzy systems. In this paper, we concentrate a simple case—stochastic T–S fuzzyinference rules, that is, the antecedents of the rules contain one type of uncertainty—fuzziness, and theconsequents are linear functions of antecedent variables, but the coefficients are random variables (r.v.s)[17]. To this end let us next recall some necessary preliminaries and notations. If we choose the valuesof r.v.s in a discrete set, our stochastic T–S fuzzy rules may degenerate as the probabilistic inferencerules in [2,14].

Wedenote the natural number set byN.AndRd means thed-dimensional Euclidean space, andR1 $= R.If X is the universe,F(X) stands for the collection of all fuzzy sets onX. Considering the fact that inpractice the varying index of the stochastic process related is time parameter wemay discuss our subjectsin R+, the nonnegative real number set. Since the problems related are inR2+, let d = 2.For adjustable parametersm, n ∈ N, partition the interval[0,m] into n equal parts: 0< m/n <

2m/n < · · · < m. Assume∼Aij∈ F([0,m]) (i = 1,2; j = 0,1,2, . . . , n) to be the antecedent fuzzy

sets of the fuzzy rules related a fuzzy system. For eachi,∼Aij may describe ‘large’, ‘medium’, ‘small’,

etc. Fuzzy phenomena, such as ‘positively large’, ‘positively small’, ‘approximate zero’ and so on may

be described by∼Aij ’s.

In the paper the following conditions always hold for the fuzzy sets∼Aij (i = 1,2; j = 0,1,2, . . . , n):

(i)∼Aij (·) is an integrable function onR+, andt /∈ [0,m] ⇒

∼Aij(t) = 0;

(ii)∼Aij is a fuzzy number, the kernal ker(

∼Aij ) includesmj/n. And for eachj : j − 1, j, j + 1 ∈

{0,1, . . . , n}, t ∈ [0,m], if ∼Ai(j−1)(t) ∧∼Ai(j+1)(t) > 0, then

∼Aij(t) > 0;

(iii) ∀t ∈ [0,m], i = 1,2,∑n

j=0∼Aij(t) ≡ 1;


Fig. 1. The curves of antecedent fuzzy sets.

(iv) There isc0 ∈ N independent ofn, such that∀t ∈ [0,m], we have

∀i ∈ {1, . . . , d}, card({j | ∼Aij(t) > 0})�c0, (1)

where card(A) means the cardinal ofA.

Obviously by condition (iv), card({j | ∼Aij(t) > 0})�1 for eacht ∈ [0,m]. Fig. 1 gives themembershipcurve family the fuzzy sets satisfying (i)–(iv) whenc0 = 2.Forp1, p2, p ∈ {0,1, . . . , n}, (t1, t2) ∈ [0,m]2, t ∈ [0,m], let

Hp1p2(t1, t2) =∼A1p1(t1) T

∼A2p2(t2), Hp(t) =

∼A1p(t).

In this paper we letT be the continuous t-norm ‘×’ which is also denoted by ‘·’. Write

N(t1, t2) = {(p1, p2)|∼A1p1(t1) ·

∼A2p2(t2) > 0}, N(t) = {p| ∼A1p(t) > 0}.

Proposition 1. For (t1, t2) ∈ [0,m]2, t ∈ [0,m], it follows that

(p1, p2) ∈ N(t1, t2) ⇒ m(pi − c0)

n� ti�

m(pi + c0)

n(i = 1,2),

p ∈ N(t) ⇒ m(p − c0)

n� t�

m(p + c0)

n.

The proof is reported in Appendix I. By Proposition 1, we may easily show that

(p1, p2) ∈ N(t1, t2) ⇒∣∣∣mpin− ti

∣∣∣ �mc0

n(i = 1,2), p ∈ N(t) ⇒

∣∣∣mpn− t

∣∣∣ �mc0

n. (2)

With the antecedent fuzzy set∼Aij (i = 1,2; j = 0,1, . . . , n)we express a simple stochastic T–S fuzzy

rule as following form, which is identical in form to one of a conventional T–S rule[12,20]:

Rule(p1, p2) : IF t1 is∼A1p1 andt2 is

∼A2p2 THENu isa0;p1p2 + a1;p1p2t1+ a2;p1p2t2, (3)

wherea0;p1p2, a1;p1p2, a2;p1p2 (p1, p2 = 0,1, ..., n) are r.v.s. Ifai;p1p2 (i = 0,1,2) is a discrete r.v.,the stochastic T–S fuzzy rule as Eq. (3) can be rewritten as the form of probabilistic fuzzy rules defined in[2,14]. Corresponding to above stochastic T–S fuzzy rules, we can construct the following input–output


(I/O) relationship:

Tnm(t1, t2) =∑n

p1,p2=0(Hp1p2(t1, t2) · (a0;p1p2 + a1;p1p2t1+ a2;p1p2t2))∑np1,p2=0Hp1p2(t1, t2)

, t1, t2 ∈ R+, (4)

where we always assume 0/0≡ 0. Eq. (4) is called a stochastic T–S fuzzy system, which is of the same

form as one of the T–S fuzzy systems[12,22,23]. By the assumption that∑n

p=0∼Aij(t) ≡ 1 (i = 1,2),

we may easily obtain

n∑p1,p2=0

Hp1p2(t1, t2)=n∑

p1,p2=0[∼A1p1(t1)] · [

∼A2p2(t2)]=

(n∑

p1=0∼A1p1(t1)

)·(

n∑p2=0

∼A2p2(t2)

)=1.

Thus, Eq. (4) may be represented as follows:

Tnm(t1, t2) =n∑

p1,p2=0Hp1p2(t1, t2)(a0;p1p2 + a1;p1p2t1+ a2;p1p2t2), t1, t2 ∈ R+. (5)

In the case of one-dimension, the corresponding stochastic T–S fuzzy system is expressed as follows:

Tnm(t) =n∑

p=0Hp(t)(a0;p + a1;pt) =

n∑p=0

∼A1p(t)(a0;p + a1;pt), t ∈ R+. (6)

Suppose(R+,B,G) to be a finite measure space, whereB is a �-ring generating by semi-intervalsuch asI = [�1, �2). For convenience from now on we always assumeG(dt) = exp(−10t)dt. LetL2(R+,B,G) =

{f : R+ −→ R|

∫R+|f (t)|2G(dt) < +∞

}.Forf ∈ L2(R+,B,G),wedenote‖f ‖ $={∫

R+|f (t)|2G(dt)

}1/2. Furthermore we letF also be a finite measure onB which may change corre-

sponding to the stochastic process related.L2(R+ ×R+,B×B, F ×G) is a product measure space. By[12] we have

Proposition 2. For arbitrary g ∈ L2(R+×R+,B×B, F ×G), and for eachε > 0, there is a T–S fuzzysystem defined as Eq. (5),whose adjustable parametersa0;p1p2, a1;p1p2, a2;p1p2 (p1, p2 = 0,1, . . . , n)

are real numbers, so that

{∫R+

∫R+|g(t, �)− Tnm(t, �)|2F(d�)G(dt)

}1/2< ε.

The details of the proof of above conclusion see[12]. By the proof we obtain

Remark 1 (Liu and Li [12] ). If let m ∈ N :∫t>m,�>m

|g(t, �)|2F(d�)G(dt) < ε2/4. Then there is

h > 0, if let

D =∨

t,�,t+h,�+h∈[0,m]

{ |g(t, �+ h)− g(t, �)|h

∨ |g(t + h, �)− g(t, �|)h

},


then

n>4mDc0 ·√(F ×G)([0,m]2)

/ε,⇒

{∫R+

∫R+|g(t, �)− Tnm(t, �)|2F(d�)G(dt)

}1/2< ε.

3. Stochastic process and stochastic integral

Let (�,A, P ) be a probability space, andu : � −→ R be a r.v. with following conditions:E(u) =0, E(u2) < +∞. All the r.v.s constitute a Hilbert space written byL2(�) with the following innerproduct〈·, ·〉 and norm which also denoted by‖ · ‖ [10,21]:

∀u, v ∈ L2(�), 〈u, v〉 = E(u · v), ‖u‖ = {E(u2)}1/2.Let {un, n ∈ N} ⊂ L2(�) be a r.v. sequence,u ∈ L2(�). If lim n→+∞E(|un − u|2) = 0, {un, n ∈ N} iscall to converge touwith mean square sense, which is written byun

m.s.−→ u (n→+∞).

Assume thatx = {x(t), t ∈ R+} is a stochastic process with the condition that∀t ∈ R+, x(t) ∈ L2(�).For s, t ∈ R+, letBx(s, t) = E(x(t) · x(s)), the functionBx(·, ·) is called the covariance function ofx.Write

C(�)={x∣∣ ∃�(·, ·) : R+×R+−→R,�(t, ·) ∈ L2(R+,B, F ), Bx(s, t)

=∫ +∞

0�(s, �)�(t, �)F (d�)

}.

If x ∈ C(�), andBx(t, t + �t) is the function of�t, independent oft, x is called a weakly stationaryprocess. ObviouslyC(�) can include some weakly stationary processes. In order to obtain the canonicalrepresentation of each process inC(�), let us now recall the framework of random measure, for detailssee[6,10,21].Assume that� : B −→ L2(�) satisfies the following conditions:

(i) �(∅) = 0, and if{B0, B1, B2, . . .} ⊂ B :⋃+∞i=1 Bi = B0, Bi∩Bj = ∅ (i �= j) ⇒∑m

i=1 �(Bi)m.s.−→

�(B0) (m→+∞);(ii) ∀C1, C2 ∈ B, E(�(C1)�(C2)) = F(C1

⋂C2).

We call� the random measure based onF.By Karhunen Theorem[6,10,21],∀x ∈ C(�), there is a random measure� on (R+,B, F ), so that

∀t ∈ R+, x(t) =∫

R+�(t, �)�(d�), (7)

moreover,E(|�(I )|2) = F(I).Let� = {�(�), � ∈ R+}beastochastic processwith orthogonal increments,i.e.

∀�1, �2, �3, �4 ∈ R+, �1 < �2��3 < �4 ⇒ E((�(�2)− �(�1))(�(�4)− �(�3))) = 0.

It is worth to note that a random measure� may be resulted in from an orthogonal increment process� = {�(�), � ∈ R+}. In fact, let�(I ) = �(�2) − �(�1) for I = [�1, �2). Thus, the representation Eq. (7)


may also be rewritten as follows:

∀t ∈ R+, x(t) =∫

R+�(t, �)d�(�). (8)

We can construct the isometric correspondence betweenL2(R+,B, F ) andL2(�) for a measureF onB.In fact, letf (�) = E(|�(�)− �(0)|2). The fact that� = {�(�), � ∈ R+} is an orthogonal increment processimplies f as monotonically increasing and continuous from the left onR+. For arbitrary semi-closedinterval I = [�1, �2), defineF(I) = f (�2) − f (�1). By the measure extension theorem[21], we canestablish a measureF onB.For arbitraryg ∈ L2(R+,B, F ), by the stochastic integralJ (g) =

∫R+g(�)d�(�) we can define a r.v.

J (g) ∈ L2(�). Forg1, g2 ∈ L2(R+,B, F ), by [6,10,21] we have

E

([∫R+g1(�)d�(�)

]·[∫

R+g2(�)d�(�)

])=

∫R+g1(�)g2(�)F (d�). (9)

For f1, f2 ∈ L2(R+,B, F ), putuk = J (fk) (k = 1,2). Then by lettingg1 = g2 = f1 − f2 in Eq. (9)we obtain

‖f1− f2‖2=∫

R+|f1(�)− f2(�)|2F(d�) = E

([∫R+(f1(�)− f2(�))d�(�)

]2)

=E([J (f1)− J (f2)]2) = E([u1− u2]2) = ‖u1− u2‖2. (10)

SoJ : L2(R+,B, F ) −→ L2(�) is an isometric mapping.By [6,10] for eachx ∈ C(�), Eqs. (7) and (8) can be extended to ones in the vector case, i.e., the

covariance function ofx is given by

Bx(s, t) =∫

R+〈�T(s, �),�(t, �)〉F(d�),

where〈·, ·〉 also means the inner product of the vector valued function�T = (�1,�2, ... ), and�T is thetranspose of the vector�. Thus the canonical representation ofx ∈ C(�) is expressed as follows:

∀t ∈ R+, x(t) =∫

R+〈�T(t, �), Υ (d�)〉, (11)

whereΥ (·) = (�1(·), �2(·), ... )T is a vector valued random measure satisfying

∀I ∈ B, E(|�1(I )|2) = E(|�2(I )|2) = · · · = F(I), E(�i(I )�j (I )) = 0 (i �= j).

If � = (�1, �2, ... )T, and�i = {�i(�), � ∈ R+} (i = 1,2, ...) is an orthogonal increment process, such

that

∀I = [�1, �2) ∈ B, �i(I ) = �i(�2)− �i(�1) (i = 1,2, ... ).


Eq. (11) can also be written as follows:

∀t ∈ R+, x(t) =∫

R+〈�T(t, �),d�(�)〉, (12)

where d�(�) = (d�1(�),d�2(�), ... )T satisfying

E(|d�1(�)|2) = E(|d�2(�)|2) = · · · = F(d�), E(d�i(�)d�j (�)) = 0. (i �= j).

4. Universal approximation to stochastic process

In this section, the discussion about the approximation of T–S fuzzy systems to the deterministic I/Orelationships will be extended to one in stochastic environment.Assume thatx = {x(t), t ∈ R+} is a stochastic process. For the finite measure space(R+,B, F ), if

∀ε > 0, there is an one-dimensional stochastic T–S fuzzy systemTnm(·) defined as Eq. (6), such that{∫R+E(|x(t)− Tnm(t)|2)G(dt)

}1/2< ε.

Then we call stochastic T–S fuzzy systems the universal approximators tox. Provided that the sampletrajectory of the processx is almost everywhere (a.e.) uniformly continuous onR+, we can analyze theapproximation by the process itself.

Theorem 1. Letx = {x(t), t ∈ R+} be a stochastic process belonging to the classC(�), and the sampletrajectory of x is a.e. uniformly continuous onR+. Define following finite sum

Fnm(t) =∑n

p=0[∼A1p(t)](a0;p + a1;pt)∑np=0 [

∼A1p(t)]

, (13)

where0��+∞. Then there area0;p, a1;p ∈ L2(�) (p = 0,1, . . . , n), so thatFnm(t)m.s.−→ x(t) (n→

+∞) uniformly holds for eacht ∈ R+.

The theorem is proven in Appendix II.

In Eq. (13) if = 1, then by∑n

p=0∼A1p(t) ≡ 1,we haveFnm(t) = Tnm(t) (t ∈ R+),which is defined

by Eq. (6). So the following corollary is obvious.

Corollary 1. Letx = {x(t), t ∈ R+} be a process whose sample trajectory is a.e. uniformly continuouson R+. Then stochastic T–S fuzzy systems are universal approximators tox, i.e. ∀ε > 0, there is a

stochastic T–S fuzzy systemTnm, so that∫

R+E(|x(t)− Tnm(t)|2)G(dt) < ε.

If the sample trajectory ofx is not a.e. uniformly continuous, we cannot directly discuss the approxima-tion by the process itself.We will use the covariance function and canonical representation of the process


x to study this problem.At first we give a conclusion about the stochastic integral of the two-dimensionalstochastic T–S fuzzy systemTnm(·, ·).

Proposition 3. For eachm, n ∈ N, Define a stochastic T–S fuzzy systemTnm(·, ·) as Eq. (5). Let � ={�(�), � ∈ R+}beanorthogonal increment process.Then the stochastic integral

∫R+Tnm(t, �)d�(�)exists

for eacht ∈ R+. Furthermore for arbitraryε > 0, there are�1, . . . , �k : 0 = �0 < �1 < · · · < �k,independent oft, such that

E

∣∣∣∣∣∫

R+Tnm(t, �)d�(�)−

k∑j=1

Tnm(t, �j )��j

∣∣∣∣∣21/2

< ε,

hold for eacht ∈ R+, where��j = �(�j )− �(�j−1) (j = 1, . . . , k).

A proof of the proposition is reported in Appendix I.

Let x = {x(t), t ∈ R+} be a stochastic process inC(�), by Eq. (12)x(t) =∫

R+〈�T(t, �),d�(�)〉 for

eacht ∈ R+. For constant vectorsA0;p1p2, A1;p1p2, A2;p1p2 : Ai;p1p2 = (a1i;p1p2, a

2i;p1p2, . . . )

T

(i = 0,1,2; p1, p2 = 0,1, . . . , n), define a two-dimensional T–S systemsT 1nm(t, �), T

2nm(t, �), . . . , so

that if letGn(t, �) = (T 1nm(t, �), T

2nm(t, �), . . . ), it follows that

GTn(t, �) =

n∑p1,p2=0

Hp1p2(t, �)(AT0;p1p2 + AT

1;p1p2t + AT2;p1p2�).

ByProposition 3, the stochastic integral�n(t)$=

∫ +∞

0〈GT

n(t, �),d�(�)〉 exists. By Eq. (10) the followingfact holds for eacht ∈ R+ :

E(|x(t)− �n(t)|2) =∫

R+‖�T(t, �)−GT

n(t, �)‖2F(d�).

Thus, we obtain

∫R+E

(∣∣∣∣∫

R+〈�T(t, �)−GT

n(t, �), d�(�)〉∣∣∣∣2)G(dt)

=∫

R+

∫R+‖�T(t, �)−GT

n(�)‖2F(d�)G(dt). (14)

Define for eacht ∈ R+ :

Tnm(t) =k∑

j=1〈GT

n(t, �j ), ��j 〉 (��j = (�1(�j )− �1(�j−1), �2(�j )− �2(�j−1), . . . )T.


Then by Proposition 3, we can easily prove thatE

(∣∣∣∣∫

R+〈GT

n(t, �), d�(�)〉 − Tnm(t)

∣∣∣∣2)may be suffi-

ciently small for eacht ∈ R+ whenj$= �j − �j−1→ 0. RewriteTnm(t) as follows:

Tnm(t)=n∑

p=0∼A1p(t)

(m∑j=1

n∑p2=0

∼A2p2(�j )(〈AT

0;pp2 + �jAT2;pp2, ��j 〉 + t · 〈AT

1;pp1, ��j 〉))

$=n∑

p=0∼A1p(t)(b0;p + b1;pt), (15)

b0;p, b1;p can be determined by the following learning algorithm, which is of an analytic form:

b0;p =k∑

j=1

n∑p2=0

(∼A2p2(�j )

((a10;pp2 + �j a

12;pp2)(�1(�j )− �1(�j−1))

+(a20;pp2 + �j a22;pp2)(�2(�j )− �2(�j−1))+ · · ·

))b1;p =

k∑j=1

n∑p2=0

∼A2p2(�j )

(a11;pp2(�1(�j )− �1(�j−1))+ a21;pp2(�2(�j )− �2(�j−1))+ · · ·

), (16)

wherep = 0,1, . . . , n.

Theorem 2. Let x ∈ C(B). Then for arbitraryε > 0, there is a one-dimensional stochastic T–S fuzzy

systemTnm(·) defined as Eq. (15),such that{∫

R+E(|x(t)− Tnm(t)|2)G(dt)

}1/2< ε.

We will prove the theorem in Appendix II.

5. Simulation example

As the application of previous results we will establish an approximation of a stochastic process bythe stochastic T–S fuzzy systems. At first we transform the process� = {�(�), � ∈ R+} into the standardBrown motionb = {b(�), � ∈ R+} for convenience of application. To this end letg ∈ L2(R+,B, F ), sothat∀� ∈ R+, g(�) > 0. Define a random measure� : B −→ L2(�) as follows:

∀I ∈ B, �(I ) =∫I

g(�)�(d�),

where�(·) is generated by the standard Brown motionb = {b(�), � ∈ R+} (for details see[6,10,21]).HenceE(|db(�)|2) = E(|�(d�)|2) = d�. If let M(I) = E(|b(�2) − b(�1)|2) for I = [�1, �2), wecan establish a measureM on B, i.e.,M(I) =

∫I

[g(�)]2F(d�) for I ∈ B. After extending a standard

Brown motion to vector one, we writeB = (b1, b2, . . . ) as a vector-valued Brown motion, wherebi (i = 1,2, . . . ) is a standard Brown motion, and� = (�1, �2, . . . ) is a vector-valued randommeasure,


�i (i = 1,2, . . . ) is generated bybi. Thus, the canonical representation of processxbecomes as follows:

x(t) =∫

R+

⟨1

g(�)�T(t, �), �(d�)

⟩=

∫R+

⟨1

g(�)�T(t, �),dB(�)

⟩. (17)

If write T(t, �) = 1

g(�)�T(t, �) (t, � ∈ R+), thenBx(t, s) =

∫R+〈 T(t, �), (s, �)〉M(d�). So let

∼� (t) =

∫ +∞

0〈GT

n(t, �),dB(�)〉, corresponding to Eq. (14) we obtain∫

R+E(|x(t)− ∼

�n (t)|2)G(dt) =∫

R+

∫R+‖ T(t, �)−GT

n(t, �)‖2M(d�)G(dt).

Thus, the corresponding one-dimensional stochastic T–S fuzzy system can be expressed as follows:

Tnm(t) =n∑

p=0∼A1p(t)(b0;p + b1;pt) (t ∈ R+),

by Eq. (16)b0;p, b1;p can be established by the following learning algorithm:

b0;p =k∑

j=1

n∑p2=0

(∼A2p2(�j )((a

10;pp2 + �j a

12;pp2)(b1(�j )− b1(�j−1))

+(a20;pp2 + �j a22;pp2)(b2(�j )− b2(�j−1))+ · · · );

b1;p =k∑

j=1

n∑p2=0

∼A2p2(�j )(a

11;pp2(b1(�j )− b1(�j−1))+ a21;pp2(b2(�j )− b2(�j−1))+ · · · ).

AndE(|bi(�j )− bi(�j−1)|2) = |�j − �j−1| (i = 1,2, . . . ; j = 1, . . . , k).

From now on letc0 = 2 in Eq. (1), the errorε = 0.2, and∼Aij (i = 1,2; j = 0,1, . . . , n) be the

translation of the triangular fuzzy number∼A (·).

∼A(t) =

n

mt + 1, −m

n� t�0,

n

mt − 1, 0< t�

m

n,

0, otherwise;

∼A10(t) =

{ ∼A(t), t�0,0, otherwise;

∼A1n(t) =

{ ∼A(t −m), t�m,0, t > m.

Let∼A1j≡

∼A2j :

∼A1j=

∼A (t −mj/n) (j = 1, ..., n−1). In the following example, we define the stochastic

processx = {x(t), t ∈ R+} as follows:x(t) = �(t) · (sin(w1t)+ cos(w2t)) (t ∈ R+),

wherew1 > 0, w2 > 0 are constants and�(·) is a weakly stationary process with zeromean value, whosecovariance function is defined as follows:

B�(t, s)$= B�(�) = |�| · exp{−|�|}, � = t − s.


In practicex = {x(t), t ∈ R+} may stand for the telegraph signal process. By[21] we may obtain therelationship between the covariance functionB�(·, ·) of � and its spectral density functionf (·) :

f (�) = 1

�

∫ +∞

−∞exp(−I��)B�(�)d�, B�(�) = 1

2

∫ +∞

−∞exp(I��)f (�)d�, (18)

whereI2 = −1. By Eq. (18) we obtain

f (�) = 2

�· 1− �2

(�2+ 1)2,⇒ F(d�) = f (�)d� = 2

�· 1− �2

(�2+ 1)2d�.

SinceBx(t, s) = E(�(t) · �(s)) · (sin(w1t)+ cos(w2t)) · (sin(w1s)+ cos(w2s)), it follows that

Bx(t, s)= 1

2

∫ +∞

−∞exp{I��} · 2

�

1− �2

(�2+ 1)2d� · (sin(w1t)+ cos(w2t))(sin(w1s)+ cos(w2s))

= 2 (sin(w1t)+ cos(w2t)) (sin(w1s)+ cos(w2s))

�

×∫ +∞

0

cos(�t) cos(�s)+ sin(�t) sin(�s)

(�2+ 1)2(1− �2)−1d�

$=∫ +∞

0〈�T(t, �), �(s, �)〉F(d�),

where�T(t, �) = ((sin(w1t)+ cos(w2t)) cos(�t), (sin(w1t)+ cos(w2t)) sin(�t)) . Consequently bythe facts T(t, �) = �T(t, �)/g(�); M(d�) = d� = (g(�))2 · f (�)d� implies

g(�) =√

�

2(1− �2)(�2+ 1),

we have

T(t, �)=√2(1−�2)

�· (sin(w1t)+cos(w2t)) cos(�t)

�2+ 1,

√2(1−�2)

�· (sin(w1t)+cos(w2t)) sin(�t)

�2+ 1

.

Let T(t, �)$= (�1(t, �), �2(t, �)). i.e.,

�1(t, �)=√2(1− �2)

�· (sin(w1t)+ cos(w2t)) cos(�t)

�2+ 1,

�2(t, �)=√2(1− �2)

�· (sin(w1t)+ cos(w2t)) sin(�t)

�2+ 1.


We may easily prove‖ (·, ·)‖ ∈ L2(R+ × R+,B × B,M × G). Furthermore if letw1 = w2 = 35,we can easily prove that the following fact holds:∫

t>1,�>1

(|�1(t, �)|2+ |�2(t, �)|2)M(d�)G(dt) < 0.005= ε2

8.

So in the examplem = 1. By Remark 1, by calculation we may establishD�40. Since�([0,m]2) =(M ×G)([0,1]2) < 1/10. By remark 1 let

n >4× 1×Dc0 ·

√(M ×G)([0,1]2)ε

⇒ n�4× 40× 2

0.2× 1√

10,⇒ n = 506.

So letn = 506, andAT1;p1p2 = AT

2;p1p2 = (0,0), moreover

a10;p1p2 = �1

( p1506

,p2

506

), a20;p1p2 = �2

( p1506

,p2

506

)(p1, p2 = 0,1, . . . ,506).

We can easily demonstrate the following fact:

∫R+

∫R+

∥∥∥∥∥ T(t, �)−506∑

p1,p2=0Hp1p2(t, �) · AT

0;p1p2

∥∥∥∥∥2

M(d�)G(dt) <ε2

4= 0.01.

If let �j = j/10 (j = 0,1, . . . ,10) inProposition3,E

(∣∣∣∣∫

R+Tnm(t, �)db(�)−∑10

j=1 Tnm(t, �j )�bj∣∣∣∣2)

< ε2/4= 0.01. Therefore by Eq. (16) we have forp = 0,1, . . . ,506 thatb1;p = 0 and

b0;p =10∑j=1

506∑p2=0

∼A2p2

(j

10

){a10;pp2

[b1

(j

10

)− b1

(j − 1

10

)]

+a20;pp2[b2

(j

10

)− b2

(j − 1

10

)]},

a10;pp2 =�1

( p

506,p2

506

), a20;pp2 = �2

( p

506,p2

506

).

Thus,

Tnm(t)=506∑

p,p2=0

10∑j=1

∼A1p(t)·

∼A2p2

(j

10

){�1

( p

506,p2

506

) [b1

(j

10

)− b1

(j − 1

10

)]

+�2

( p

506,p2

506

) [b2

(j

10

)− b2

(j − 1

10

)]}.

Hence by the fact

E

([bp

(j1

10

)− bp

(j1− 1

10

)][bq

(j2

10

)− bq

(j2− 1

10

)])

=

∣∣∣∣ j10− j − 1

10

∣∣∣∣ = 1

10, j1 = j2, p = q;

0, otherwise,


Fig. 2. The curve surface ofBx .

Fig. 3. The curve surface ofBTnm .

we obtain byBTnm(t, s) = E(Tnm(t)Tnm(s)) the following hold:

BTnm(t, s)=1

10·

506∑p1,p2,p3,p4=0

10∑j=1

∼A1p1(t)·

∼A1p2(s)·

∼A2p3

(j

10

)· ∼A2p4

(j

10

)

·(�1

( p1506

,p2

506

)· �1

( p3506

,p4

506

)+ �2

( p1506

,p2

506

)· �2

( p3506

,p4

506

)).

The figures ofBx(·, ·) andBTnm(·, ·) are shown in Figs. 2 and 3, respectively. We can easily find thatabove steps can guarantee the approximating accuracyε = 0.2.

6. Conclusion remarks

In this paper, we generalize the approximation analysis related toT–S fuzzy systems from the determin-istic I/O relationships to the stochastic ones. That is, the stochastic T–S fuzzy systems with multiplication‘ ·’ norm can with arbitrary degree of accuracy approximate a class of stochastic processes. Thus, theapplication fields of T–S fuzzy systems will be extended, strikingly. A few of important and meaningfulproblems related the subject for the future research are as follows. First, if triangular normT is not ‘·’,how may we study the approximation capability of the corresponding T–S fuzzy system? Second, if thefuzzy systems related are the Mamdani systems[12], whether may such systems be the universal approx-


imators to a class of stochastic processes? How can some adaptive learning algorithms for fuzzy systemsapproximating the stochastic processes be designed?

Appendix I

Proof of Proposition 1. The fact that(p1, p2) ∈ N(t1, t2) implies∼A1p1(t1) ·

∼A2p2(t2) > 0. Hence for

i = 1,2,∼Aipi (ti) > 0. Since

∼Aipi (i = 1,2) is a fuzzy number, and

∼Aipi (mpi/n) = 1, if we assume

ti > m(pi + c0)/n, thenti > mpi/n, andpi + c0�n. Moreover it follows that[13]:∼Aipi

(m(pi + c0)

n

)�∼Aipi(ti) > 0.

Since∼Ai(pi+c0)(m(pi + c0)/n) = 1> 0, we obtain by condition (ii) for

∼Aij that

∼Aij (m(pi + c0)/n) >

0 for eachj : pi�j�pi + c0. Thus, card({j |∼Aij (m(pi + c0)/n) > 0})�c0 + 1, which contradicts

the definition ofc0. So ti�m(pi + c0)/n. Similarly we may proveti�m(pi − c0)/n. Hencem(pi −c0)/n� ti�m(pi + c0)/n (i = 1,2).With the same reason we may show the conclusion related toN(t). �

Proof of Proposition 3. By Eq. (5)� > m ⇒ Tnm(t, �) ≡ 0, and whent�m,

Tnm(t, �) =n∑

p1=0∼A1p1(t)

(n∑

p2=0(a0;p1p2 + a1;p1p2t)

∼A2p2(�)+

n∑p2=0

∼A2p2(�)a2;p1p2�

). (A.1)

By [6,10,21], the stochastic integral∫ m

0Tnm(t, �)d�(�) exists, which implies

∫R+Tnm(t, �)d�(�) exists.

Furthermore∫R+Tnm(t, �)d�(�)=

n∑p1=0

∼A1p1(t)

(n∑

p2=0(a0;p1p2 + a1;p1p2t

) ∫ m

0

∼A2p2(�)d�(�)

+n∑

p2=0a2;p1p2

∫ m

0�· ∼A2p2(�)d�(�)

). (A.2)

So simultaneously use the definition of stochastic integral in[6,10,21] to∫ m

0

∼A2p2 (�)d�(�) and∫ m

0�· ∼A2p2 (�)d�(�), we may obtain�1, . . . , �k ∈ R+, obviously independent oft, moreover, 0=

�0 < �1 < · · · < �k, so that

E

∣∣∣∣∣∫ m

0

∼A2p2(�)d�(�)−

k∑j=1

∼A2p2(�j )��j

∣∣∣∣∣21/2

<ε

2(n+4)/2 ·∑np1,p2=0(a

20;p1p2 +m2 · a21;p1p2)

,


E

∣∣∣∣∣∫ m

0�· ∼A2p2(�)d�(�)−

k∑j=1

�j ·∼A2p2(�j )��j

∣∣∣∣∣21/2

<ε

2(n+4)/2 ·∑np1,p2=0 a

22;p1p2

. (A.3)

We can easily prove that∀t > m, Tnm(t, �) ≡ 0; t�m the following facts hold:

∫R+Tnm(t, �)d�(�)−

k∑j=1

Tnm(t, �)��j =n∑

p1=0∼A1p1(t)

(n∑

p2=0(a0;p1p2 + a1;p1p2t)

×[∫ m

0

∼A2p2(�)d�(�)−

k∑j=1

∼A2p2(�j )��j

]

+n∑

p2=0a2;p1p2

[∫ m

0�· ∼A2p2(�)d�(�)

−k∑

j=1�j∼A2p2(�j )��j

]). (A.4)

Since∑n

p1=0 [∼A1p1(t)]2�

∑np1=0

∼A1p1(t) = 1 for eacht ∈ [0,m], and by Eqs. (A.1)–(A.4) we obtain

E

∣∣∣∣∣∫

R+Tnm(t, �)d�(�)−

k∑j=1

Tnm(t, �j )��j

∣∣∣∣∣21/2

�

E

(n∑

p1=0

∣∣∣∣∣n∑

p2=0(a0;p1p2 + a1;p1p2t)

{∫ m

0

∼A2p2(�)d�(�)−

k∑j=1

∼A2p2(�j )��j

}

+n∑

p2=0a2;p1p2

{∫ m

0�· ∼A2p2(�)d�(�) −

k∑j=1

�j ·∼A2p2(�j )��j

}∣∣∣∣∣21/2

�√2 ·

n∑p1=0

E

∣∣∣∣∣

n∑p2=0

(a0;p1p2 + a1;p1p2t){∫ m

0

∼A2,p2 (�)d�(�)−

k∑j=1

∼A2p2(�j )��j

}∣∣∣∣∣2

+E∣∣∣∣∣

n∑p2=0

a2;p1p2

{∫ m

0�· ∼A2p2(�)d�(�)−

k∑j=1

�j ·∼A2p2(�j )��j

}∣∣∣∣∣21/2

�2(n+2)/2 ·

n∑p1,p2=0

(a20;p1p2+a21;p1p2m2)·E

∣∣∣∣∣∫ m

0

∼A2p2(�)d�(�)−

k∑j=1

∼A2p2(�j )��j

∣∣∣∣∣2


+a22;p1p2 · E∣∣∣∣∣∫ m

0�· ∼A2p2(�)d�(�)−

k∑j=1

�j ·∼A2p2(�j )��j

∣∣∣∣∣21/2

< 2(n+2)/2 ·(

ε

2n+42

+ ε

2n+42

)= ε,

which implies the proposition.�

Appendix II

Proof of Theorem 1. For givenm, and arbitraryε > 0, by the uniform continuity of the sample trajectoryof the processx and Eq. (2) there is sufficiently largen ∈ N, such that

∀t ∈ R+, p ∈ N(t),∣∣∣x(t)− x

(mpn

)∣∣∣ < ε, a.e..

Define random variablesa0;p, a1;p (p = 0,1, . . . , n), respectively, as follows:

a0;p(w) = x(mpn,w

), a1;p(w) = 0.

Obviouslya0;p, a1;p ∈ L2(�) (p = 0,1, . . . , n). Then∀t ∈ R+, we obtain

E(|x(t)− Fnm(t)|2

) = E

∣∣∣∣∣∣x(t)−

∑np=0 [

∼A1p(t)](a0;p + a1;pt)∑np=0 [

∼A1p(t)]

∣∣∣∣∣∣2

= E

∣∣∣∣∣∣∑n

p=0 [∼A1p(t)](x(t)− a0;p − a1;pt)∑n

p=0 [∼A1p(t)]

∣∣∣∣∣∣2

= E

∣∣∣∣∣∣∑n

p=0[∼A1p(t)](x(t)− x(mp/n))∑n

p=0[∼A1p(t)]

∣∣∣∣∣∣2

� E

∣∣∣∣∣∣∑n

p=0 [∼A1p(t)]|x(t)− x(mp/n)|∑n

p=0[∼A1p(t)]

∣∣∣∣∣∣2

= E

∣∣∣∣∣∣∑

p∈N(t) [∼A1p(t)]|x(t)− x(mp/n)|∑p∈N(t) [

∼A1p(t)]

∣∣∣∣∣∣2

< E

∣∣∣∣∣∣∣∣

∑p∈N(t)

[∼A1,p (t)] · ε∑

p∈N(t) [∼A1p(t)]

∣∣∣∣∣∣∣∣

2 = ε2 · E

∣∣∣∣∣∣∣∣∑

p∈N(t) [∼A1p(t)]∑

p∈N(t)[∼A1p(t)]

∣∣∣∣∣∣∣∣

2 = ε2.


Therefore limn→+∞E(|Fnm(t) − x(t)|2) = 0 holds uniformly for t ∈ R+, which proves thetheorem. �

Proof of Corollary 1. For arbitraryε > 0, since 0< G(R+)�1. Let ε′ = ε/(2G(R+)), by Theorem 1,there is an ∈ N, satisfyingE(|Tnm(t)− x(t)|2) < ε′ for eacht ∈ R+. Consequently∫

R+E(|Tnm(t)− x(t)|2)G(dt)�

∫R+ε′G(dt) = G(R+) · ε′� ε

2< ε,

which implies the conclusion.�

Proof of Theorem 2. By the assumption and Eq. (12) we can establish the following spectral represen-tation of the processx:

∀t ∈ R+, x(t) =∫ +∞

0�1(t, �)d�1(�)+ �2(t, �)d�2(�)+ · · · $=

∫ +∞

0〈�T(t, �),d�(�)〉,

where

�T(t, �) = (�1(t, �),�2(t, �), . . .), d�(�) = (d�1(�),d�2(�), . . .)T.

Furthermore,�1 = {�1(�), � ∈ R+}, �2 = {�2(�), � ∈ R+}, . . . are the orthogonal increment processeswith the following conditions:

E(|d�1(�)|2) = E(|d�2(�)|2) = · · · = F(d�),E(d�i(�) · d�j (�)) = 0 (i �= j).

Since Proposition 2 implies that the T–S fuzzy systems are universal approximators to each function inL2(R+ × R+,B × B, F × F), there are

AT0;p1p2 = (a10;p1p2, a

20;p1p2, . . .), AT

1;p1p2 = (a11;p1p2, a21;p1p2, . . .),

AT2;p1p2 = (a12;p1p2, a

22;p1p2, . . .),

wherep1, p2 = 0,1, . . . , n, such that if‖ · ‖ also means the Euclidean norm, we have∫R+

∫R+

∥∥∥∥∥�T(t, �)−n∑

p1,p2=0Hp1p2(t, �)(A

T0;p1p2 + AT

1;p1p2t + AT2;p1p2�)

∥∥∥∥∥2

F(d�)G(dt)

<ε2

2. (A.5)

Moreover,∫t>m

∫�>m

‖�T(t, �)‖2G(dt)F (�) < ε2/4.Write

GTn(t, �) =

n∑p1,p2=0

Hp1p2(t, �)(AT0;p1p2 + AT

1;p1p2t + AT2;p1p2�),

which implies the following hold:∫R+

∫R+‖�T(t, �)−GT(t, �)‖2F(d�)G(dt) < ε2

2. (A.6)


Obviously by Proposition 2 the stochastic integral�n(t)$=

∫ +∞

0〈GT

n(t, �),d�(�)〉 exists. Our aim is to

define an one-dimensional stochastic T–S fuzzy systemTnm(·) to satisfy the condition of the theorem.Considering the triangular inequality for metric we have(∫

R+E(|x(t)− Tnm(t)|2)G(dt)

)1/2

=(∫

R+E(|x(t)− �n(t)+ �n(t)− Tnm(t)|2G(dt)

)1/2

�(∫

R+E(|x(t)− �n(t)|2G(dt)

)1/2

+ (E(|�n(t)− Tnm(t)|2G(dt)

)1/2. (A.7)

Thus, we obtain by Eqs. (10) and (A.6)∫R+E

∣∣∣∣∫

R+〈(�T(t, �)−GT

n(t, �)),d�(�)〉∣∣∣∣2

G(dt)=∫

R+×R+‖�T(t, �)−GT

n(t, �)‖2F(d�)G(dt)

<ε2

2. (A.8)

Hence by the canonical representation ofx and Eq. (A.8) the following fact holds:∫R+E(|x(t)− �n(t)|2)G(dt) =

∫R+E

(∣∣∣∣∫

R+〈(�T(t, �)−GT

n(t, �)),d�(�)〉∣∣∣∣2)G(dt) <

ε2

2.

By Proposition 3, forε > 0 there are�1, �2, . . . , �k : �1 < �2 < · · · < �k independent oft, so that

E

∣∣∣∣∣∫ +∞

0〈GT

n(t, �),d�(�)〉 −k∑

j=1〈GT

n(t, �j ),��j 〉∣∣∣∣∣2 <

ε2

2,

where��j = (�1(�j )− �1(�j−1), �2(�j )− �2(�j−1), . . . )T (j = 1, . . . , k; �0 = 0). Let

Tnm(t) =k∑

j=1〈GT

n(t, �j ),��j 〉 (t ∈ R+).

We may easily prove

Tnm(t)=m∑j=1

⟨n∑

p1,p2=0Hp1p2(t, �j )

(AT0;p1p2 + AT

1;p1p2t + AT2;p1p2

), ��j

⟩

=n∑

p=0∼A1p(t)

(m∑j=1

n∑p2=0

∼A2p2(�j )

(⟨AT0;pp2 + �jA

T2;pp2, ��j

⟩+ t ·

⟨AT1;pp1, ��j

⟩)).

Forp = 0,1, . . . , n, let

b0;p =m∑j=1

n∑p2=0

(∼A2p2(�j )

((a10;pp2 + �j a

12;pp2)(�1(�j )− �1(�j−1))

+(a20;pp2 + �j a22;pp2)(�2(�j )− �2(�j−1))+ · · ·

)),


b1;p =m∑j=1

n∑p2=0

∼A2p2(�j )

(a11;pp2(�1(�j )− �1(�j−1))+ a21;pp2(�2(�j )− �2(�j−1))+ · · ·

)

it follows thatTnm(t) =∑np=0

∼A1p(t)(b0;p + b1;pt) (t ∈ R+). By Eq. (A.8) we have

∫R+E

∣∣∣∣∣�n(t)−

n∑p=0

∼A1p(t)(b0;p + b1;pt)

∣∣∣∣∣2G(dt) =

∫R+E(|�n(t)− Tnm(t)|2)G(dt) < ε2

4.

Synthesizing Eqs. (A.6)–(A.8), we obtain(∫R+E(|x(t)− tnm(t)|2)G(dt)

)1/2

<

{ε2

2+ ε2

2

}1/2= ε,

which implies{E(|x(t)− Tnm(t)|2)}1/2 < ε. �

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approximation of stochastic processes by t–s fuzzy systems

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