Otemon Economic Studies, 26（1993） 53

ConsistentEstimators Associated with Output

　Processesof Fuzzy Moving Average Models

TOKUO　FUKUDA

　　　　　　　　　　　　　　　　　　　　　　　Abstract

　　In this paper, we investigate the consistent property of the estimators associated

with the output process of fuzzy stochastic systems (FSSs).

　　First, the fuzzy random variables are reviewed briefly in order to construct

fuzzy stochastic systems with fuzzy system parameters. Secondly, the estimators of

the mean and the second statisticalmoments of the fuzzy random output processes

of FSSs are derived heuristically. Finally, the consistent properties of the estimators

are investigated theoretically when the FSSs are described by those called fuzzy

moving average models (FMAs).

Keywords : fuzzy stochastic systems, fuzzy moving average models, fuzzy set theory,

consistency

1

Introduction

The purpose of this paper is to investigate the consistency of the heuristic esti-

mators associated with output processes of fuzzy stochastic systems （FSSs）with

fuzzy system parameters ［1],[2],［3］,[4].

　　Motivated by the importance of treating data exhibiting both vagueness and

randomness, fuzzy random variables（FRVs）axe investigated intensively. Such

phases are observed in e.g.　Klement, Puri and Ralescu [5］,[6], Puri and Ralescu

［7], Kruse and Meyer ［8], Kwakernaak［9],［10], Miyakoshi and Simbo ［11],

Boswell and Taylor ［12］and Inoue ［13]. For example, Kwakernaak［9],［10］pro-

posed FRVs considering that the fuzzy random data is obtained by perceiving

vaguely outcomes of ordinary random variables. Using the concept of random sets

（see e.g.［14],［151),Klement, Puri and Ralescu ［5］,[6], and Puri and Ralescu ［7］

investigated another type of FR Vs.

　　The basic properties of fuzzy sets associated with arithmetic operations and

topological properties are summarized in Sections 2 and 3,respectively. In section

　　　　　　　　　　　　　　　　　　　　　　（1）

54 TOKUO FUKUDA

4, the concept of FR Vs is reviewed as a basis for deriving .FSSs with fuzzy system

parameters in Section 5, where the definition of FR Vs is given by using set repre-

sentation concept [8] and hence it is the extended one of those given by Puri and

Ralescu [7]. The heuristic estimators of the mean and second statisticalmoments

of output processes of FSSs are also given in Section 5. Furthermore, restricting

the class of FSSs to a particular one of them called fuzzy moving average models,

the consistent property of the estimators is investigated mathematically in Section

6.

Throughout this paper, a fuzzy set is given by the triple U―(R", (£/),s) [4],

where R" is the ordinary w-dimensional Euclidean space, called the basic space;

the membership function (U) is a mapping jB"-*[O, 1] ; and s: Rn^&> with SP the

"universe of discourse" defined by a set of statements, assigns a proposition six) to

each element x&R". The corresponding value iU)ix) of the membership function

is the truth value of the proposition six), i.e.,(U)(x)=((s(x)) where *(*) is the

truth function of *.

The strong a-cut LJU and the a-levelset LaU of U are defined respectively by

LaU={x

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT　PROCESSES OF FUZZY MOVING AVERAGE MODELS

55

exists a set representation such that its each element びαis compact for ∀αeE(O, 1).

The set of all fuzzy sets in the basic space £" is denoted by 夕(Rり.The family of

all normal fuzzy sets in R ”is denoted by よ沢勺.The family of fuzzy sets satisfy-

ing that (i)each fuzzy set in this family is normal and (ii)it and its support are

compact is denoted by 'S(R"). Furthermore, a fuzzy set びeぎ(R”)belongs to

ぎじ(刄りif there eχists a set representation for U, which is conveχ and compact.

2　Arithmetic Operations on Fuzzy Sets

The extension principle introduced by Zadeh ［16］is one of the most basic idea of

fuzzy set theory. It provides a general method for eχtending nom-fuzzy mathe-

matical concepts to fuzzy ones.　Let consider that the acceptability of the state-

ment s（x）＝{x巳U}is given by its truth function, and it coincides with the mem-

bership function of U. Let

　　　　　　　　　　　　　φ：XX‥･XX >Y(X^R", Y

56 TOKUO FUKUDA

is a setrepresentationfor 4>(JJ＼,･･･,Um).

(ii) For VaG[0, 1),

LjbWi, -, Um)=^{LaUu -,LaUm) (2.3)

is valid.

(iii) If

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT

PROCESSES OF FUZZY MOVING AVERAGE MODELS57

dH(A, 5)= max (sup inf ＼＼a-bII,sup inf ||a-6|||, (3.2)la

58 TOKUO　FUKUDA

4　Fuzzy Random Variables

Let (a ぶ,巧be a complete probability space, where sd is the (7-algebra generated

by the subsets of Q, and F is a nonatomic probability measure. Then, the fuzzy

random variables (FRVs)is defined as follows [3,4]:

Definition 4. 1　The FRV X is defined as a ma)ping.

　　　　　　　　　　　　　X：口→(ぎ(i?勺，d 。，　　　　　　　　(4.1)

if there exists a set representation {X。(ω)|ω巳Q, αG(O，1)} of X, which is a compact

random set [15]and is also integrably bounded[14]for each a£E(O, 1)and co^Q、

The family {X。(ω)|ω∈£,α∈(O, 1)} satisfying the above conditions in Definition

4.1 is called a measurable set representation of χ in this paper. In the following,

{X。ω)|ω庄砲αG(O，1)} is abbreviated to {Z。α(E(O, 1)} when no confusion will

occur.

Theorem 4.1　　Let X be a FRV. Then, L^X，L.X and supp χare random sets.

（Proof）

　　Let {X。α(E（Oバ）}be a measurable set representation of X. Applying Theo-

rem 2.2 m［19], we have

and

玩X＝U X。　　for　α6E(0, 1)

LaX　∞　∩

r＝I

xら　　for゛゛∈[0, 1]･

(4.2)

(4.3)

where ｛αよEiV｝and {βrIr∈菌are strictly decreasing and increasing sequences

respectively, such that

α＝lim 恥＝limβ。r－゛∞ r~゛∞

Hence, for every Borel subset j ixi刄”we have

and

し^ぺ^ムズ{功⊂J

I t白駒。{功⊂^}

　　　　　　　　　r=l

ｼr⊂^}eぶ

（6）

(4.4)

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT　PROCESSES OF FUZZY MOVING AVERAGE MODELS

ﾚG£|≒X(功∩Å≠oトド^（n 右府)n^≠≫)

　　　　　　　　　　　ﾆﾊﾞﾚE叫亀{功nj≠o}eぶ　　　(4.5)

59

which imply L,X and £/iχare measurable. The measurability of supp χ is

derivedfrom that oflevelsets£gX forαe（O，1）［51.

Remark :　The definition of F?Vs may be a general version of that given by

Klement, Puri and Ralescu (see e.g.圓，[6]and[7]), wkere the level sets of FRVs

are used instead of their set representation.

　　By £'(P,R"), we denote the space of P-integrable point-valued function f:Q

→R″. Furthermore, we denote by S(F)the set of all£'(P,R") selection of F,i.e.，

S(F) ＝{/£EL'(P, J?")ﾚ(ω)^F圃)w. p. 11 (4.6)

Then, the integral of the seレvalued function F is defined by Aumann ［14], i.e

£㈲= (Aり'ﾆ几fdPﾚes(約} (4.7)

Let F(ω)be an integrably bounded random set in R”. Then, it can be shown that

the Aumann integral of F(ω)is a non-empty convex subset of R". Furthermore,

if F(ω)is closed for every ω∈以its Aumann integral is compact [14].Therefore,

for the measurable set representation {X。αGE(O，1)} of the FRV X, we know that

the set几几∈£”definedby

訂。＝(AりX。

is non-empty, convex and compact, and clearly

訂．⊆訂β⊂R”for Oくa

60 TOKUO　FUKUDA

Furthermore, we can see that E{X]∈影R")because L^(E{X))≠O and the com-

pact property of supp E {X] are easily verified.

Theorem 4.2　　For each FRV χ',there exists a unique expectation £収}∈影(召”)

sa匈fying

　　　　　　　　　　　　　乙E{X))＝E{玩X} for　αe(O, 1)｡　　　　　　　(4.10)

In order to prove Theorem 4.2,we need the following lemma [7]:

Lemma 4.1　　Let F,：ひ→(jr(≪"), dH)(ん＝＼，2,…)be a series of random sets. If

thereexists h^L＼P,R") such that

　　　　　　　　　　　||ム(ω)||

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT　PROCESSES OF FUZZY MOVING AVERAGE MODELS

Then, from (4.14)and (4.18)we have

乱£ぼ}={Aり玩X＝E伍aX}

5　Fuzzy stochastic Systems

(4. 19)

61

In this paper, we assume that the fuzziness in the system lies only in the sysem

parameters, and hence, it may be reasonable to consider that the set representation

of the fuzzy output process of the underlying FSS is given by

(Fム＝{ぬぬ=几(y←＼，…, yい。;ら，…, e *う心5j}　　　(5.1)

for each α∈(O，1),where几is the function describing the system mechanism ；and

{ら}is the non-fuzzy unobservable random disturbances. Furthermore, ら＝(ら,＼,

….らブin (5. 1)is the parameter vector given by 肌^^吼，where {吼αG(O, 1)} is

a set representation of the fuzzy system parameter a

Definition 5.1 ([3])　The Fuzzy stochasticsystem (FSS)is described formally by

　　　　　　　　　　　　　Yた＝ｱた(Yいi，…，7←,,;≒,…，召ト,; 0)　　　　　　　　　　　　l

Yo＝Y-,＝…＝Y-,,,,＝0　(non-fuzzy).　　　　　　　　(5,2)

　　The meaning of (5. 2)is that its set representation is given by (5.1). It should

be noted that if the extension principle is simply applied in order to define FSSs,

the set representation of Yだwill be given by the following equation :

where

(i"J。=長((F,,-,)α，…，(Yい,X。；ら，…, e k一-祖;Θ≫)

　　　　　　/((^←,)a，…，(Yみ｡。)話e*，…,ら｡,,;θ。

　　　　　　　　={yが三刄ﾚた一丿^(Yト几forf＝l,…, n and d．^Θ。

　　　　　　　　　　　　　with几＝ム(yト1，…,yトn', ら／‥,らー；臨)}

Then, we can easily find by comparing (5. 1)with (5.3)that

(5.3)

(5.4)

　　（setrepresentation of Yb given by （5.1)）⊆（setrepresentation of Y,,given by（5.3））.

Furthermore, it may be fair to say that the fuzziness of {八｝given by （5.3）will

increase eχplosively in a few time steps, and it will have no meaning as a system

　　　　　　　　　　　　　　　　　　　　　　　（9）

62　　　　　　　　　　　　　　　　　　TOKUO　FUKUDA

model, whereas that given by（5.1）will never become so. This is the reason why

we adopt（5.1）as set representations of the FSSs.

Definition 5.2　　The mean process {M^} of the system output Y斤り。£)is defined

妙

　　　　　　　　　　　　　　　　　　　訂｡＝£{Yよ　　　　　　　　　　　　(5.5)

Furthermore, the second moments of Y i,and Ybぺare defined by

　　　　　　　　　　　　Φ(ん；ん＋z)＝£{几几｝　(r＝0，土1,…),　　　　　　　(5.6)

for any k ＝l, 2,…and for any fixed time interval r, where the multiplicationbetween

Yi,and Yk^^ is given bバ＼hesense of Definiti爪2. 1.

Definition 5.3　　1ハ＼hemean process 収^i is time inde)endent and the second mo-

meritsΦ(か：八十z)depend only on the time step intervals妖0,土1,…), we callthat the

output process {Y,} is wide-sense stationary･

　　The estimators of the mean and the second moments of the output process of

the i^SS are heuristicallygiven by

　　　　　　　　　　　　　　ル,v=万倉

ly゛　　　　　　　　　　　　　（5.7）

　　　　　　　　　　　　　町か）ニ‾ﾌしﾒj

lyj≒台（7゛O，1，…,m）,　　　（5.8）

when {FJ is wide-sense stationary.

6　Fuzzy Moving Average Models

Hereafter, for simplicity of discussions, we focus on the class of FSSs described by

those called fuzzy moving average models of order m(FMA(m)models). FMA(m)

models are the eχtended version of ordinary moving average models of order m

(MA(m) models)described by

　　　　　　　　　　　　　　　　几＝/（ら，･･･.e ■,ｰ,; 0)｡　　　　　　　　（6.1）

The true meaning of（6.1）is that the set representation of the output process Yた

is given by

　　　　　　　　　　　　　　　　　　　　　（↓O）

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT　PROCESSES OF FUZZY MOVING AVERAGE MODELS

(yム＝{几八＝ﾃﾞら,…, らｰ。;θ);θ^凪} (6.2)

where ca is the set representation of the 刑-dimensional fuzzy parameter 0, and

/(ら，…- らｰm ；θa)＝ら十臥ノい＼十‥･十θ。,a召ト。1

ら＝(θiα，…. θ..J'eΘ.、(6.3)

63

If the system parameter θえa（j＝l，2，…，m）axemutually independent, then the set

representation of Y,is given by

(几)。＝{ぬ|ぬ＝ら十∂yae卜卜…十θ。.,e^－。いdi,。∈咸,。(E刄}，　(6.4)

where 亀a is the element of the set representation of the scalar (one-dimensional)

fuzzy parameter 吠^‰£).Therefore, it may be natural that the system model

with independent fuzzy parameters is described formally by

　　　　　　　　　　　　　　几＝ら十θ1 ら＿,十…十Θ}>} ^い m .　　　　　　　　　（6,5）

　　We assume here that the underlying FMA（m）model satisfies the following

conditions :

（C-1）The random disturbance ｛ら｝is the sequence of i.i.d. random variables

　　　with the mean me and the variance a}.

（C一2）The fuzzy system parameter e is the element of 汐ズ刄町

Then, from the definition of FMA（m），and the conditions（C-1）and （C-2）, it is

immediately shown that the output process {几｝of FMA（m）is wide-sense sta-

tionary and the elements of悟(皿）.

　　Applying Theorems 2.1 to 2.3, the following theorem is derived easily:

Theorem 6.1　　Assumed that the conditions (C-1)and (C-2)hold. Then，we have

　　　　　　　　　　　　　　訂頌≡影(淘)，ら心)

64 TOKUO　FUKUDA

　　　　　　　　　　　　　　　　　1 ^　　　　　　　　　　　　総覧(幻ニズ三(総几)(総几⊇)

arelevelsetsof MχandΦ,v(t), respectively.

Theorem 6.2　　Assumt日hat the conditions ((ト1)and (C-2)hold. Then,

　　　　　　　　　　　　め(M,v，釧F,})→O w. p. 1 as N→y

and

　　　　　　　　　　山(ら心), E{Y,Y,。J}→0 w, p. 1 as 八八

In order to prove Theorem 6.2,we need following lemma [5]:

(6.9)

(6. 10)

(6. 11)

Lemma 6.1　　Let{X,,＼k^N] and X be fuzzy random 斑面lables with values in

め(i2勺and such thatE{ IIsuppXj II}く十CO and £{llsuppX||}＜十∽，S駄卸lose that

　　　　　　　　　　　　　尚(X,,X)→O w. p.1 as ん→∽　　　　　　　　(6. 12)

and

　　　　　　　　　　　　尚{匙バ0})

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT　　PROCESSES OF FUZZY MOVING AVERAGE MODELS

ニﾄ{友S^(^“’゛)゛)十_^E(ﾀ i(F“゛,)゛)‾(7“(’゛)゛))}

1しし

゜(jﾍﾞﾚ上)九[･訂V〒*=0]十(器)jﾗ丿言で宍0⊇]

(6. 16)

65

where 5^0＝0.　Since (彫R), d,)is separable as shown in Theorem 3. 1, it is easy

to show that (x,|・||)is also separable, which implies that (χ,||・‖)is a Banach

space. Furthermore, from the definition of FMA(m)given by (6.1), we know that

{ﾌﾟ(Y,(,,,。,)づ)}(ん＝1，2，…)is the sequence of the i.i.d. χ-valued random elements.

Then, applying the standard theorem for strong law of large numbers in Banach

space, we have

万V

Tﾒ1

)

ﾌﾞ(y白う,-,.,■)一万ぴ(y,)}||→Ow p l as 訂→呵

and applying (6.17)to (6.16), we have

|し(訂,)-£び(Y)}||く(ﾍﾟﾄﾞう九||

　　　　　　　　　　べ打豆

万VTjyﾌﾞ(Y尚＋l卜)‾£ぴ(yl)}

M

嵩)し(几尚丿づ)－五口(r,)}

　　　　　　　　　　　　　　　　　->o　　w.p. 1 as　が→∽.

Therefore, if we can show E{j(Y)}＝ﾌﾞ(E{Y,}), we can conclude that

　　　尚(訂,v,£{F,})＝＼＼ﾌﾞ(Ms)-j(E{Y,})|I→O　w. p. 1　as　/V-≫oo,

So, the remainder is devoted to show

　　　　　　　　　　　　　　　　　Eij(Y)}づ(£{F,}).

Assume first that Y, is a simple function, i. e.,

　　　　　　　　　　　　　　　　　　　　　　　　　　瓦　　　　　　　　　　　　　　　　　　　　　　　Y,＝Σ防ら，

　　　　　　　　　　　　　　　　　　　　　　　　　　卜l

(6. 17)

(6. 18)

(6. 19)

(6. 20)

(6. 21)

where び,∈タズ-R)'Afi≡y and I A-is the characteristic function of A;.It is easy to

check that E{j(Y))づ(£{7,})because of £{|lsupp7, !i}く十CO from the condi-

tion (C-1). Since Y, is measurable, there eχistsa sequence of simple functions

{SJ(F?Vs)with the property

d^(S。，Y)→O　w. p. 1　as　謂→∽

　　　　　　　　　∩3)

(6. 22)

66　　　　　　　　　　　　　　　　　　TOKUO　FUKUDA

because of the separability of (影田)，d).Also,it follows

　　　　　　　　　　d1(S , バOD→山(Fi，{0})　w. p. 1　as　m→∽　　　　(6. 23)

from the continuity of 山，Consider here the truncated FR Vs{Tj(m＝l，2，…)

given by

斗，ifdi{S,,,，{0})

CONSISTENT ESTIMATORS ASSOCIATED WITH OUTPUT　PROCESSES OF FUZZY MOVING AVERAGE MODELS

7　Conclusions

67

In this paper, the consistency of the heuristically derived estimators associated

with FSSs are investigated theoretically. The FSSs in this paper have been as-

sumed to be described by FMA models with fuzzy parameters, which is an ex-

tended version of ordinary non-fuzzy MA ones｡

　　　Thekey aspects for proving the consistency of the estimators are to embed

the fuzzy random variables into the normed linear space by using Theorem 3.2 in

Section 3, and to use the law of large numbers in Banach Space.

　　　　　　　　　　　　　　　　　　　　　　　　　References

[1]T. Fukuda and Y. Sunahara. An identification method of fuzzy parameters for a class

　　　　offuzzy stochastic systems.　In nth World Congress of IFAC, volume 2, pages Ill －

　　　　116,Tallinn, Estonia, USSR, 1990.

[2]Y. Sunahara T. Fukuda and H, Hayashi. Asymptotic properties of fuzzy estimators for

　　　　fuzzymoving average models. In 22th ISCIE Sym卸'sium on Stochastic Systems Theory

　　　　andits Applications,卸:s-eA-59－62,1990.

[3]T. Fukuda and Y. Sunahara. Identification of vaguely dependent parameters for a

　　　　classof fuzzy stochastic systems. In IEEE International Conference on Fuzzy Systems

　　　　1992,pages 1419 －1426，1992.

[4]T. Fukuda and Y. Sunahara. Parameter identification of fuzzy autoregressive models.

　　　　In12th World Congress oﾁﾞlFAC, Sydney, Australia, 1993.

[5]E. P. Klement, M. L. Puri, and D. A. Ralescu. Law of large numbers and central limit

　　　　theorem for fuzzy random variables. In R. Trappl, editor, Cybernetics and Systems Re-

　　　　search,pages 525－529. North Holland， 1984.

[6]E. P. Klement, M. L. Puri, and D. A. Ralescu. Limit theorems for fuzzy random varia-

　　　　bles.Proceedings of Royal Society of London，A 407 : 171 －182,!986.

[7]M. L. Puri and D.Λ. Ralescu. Fuzzy random variables. Journal of Mathematical Analy-

　　　　sisand its Applications,χ14:409 －422,1986.

[8]R. Kruse and K. D. Meyer. Statisticswith Vague Data. D. Reidel Publishing Company,

　　　　Dordecht,1987.

[9]H. Kwakernaak. Fuzzy random variables-I: definitions and theorems. Information Sci-

　　　　ences,15: 1－29,1978.

[10]H. Kwakernaak. Fuzzy random variables- H : algorithms and eχamples for the discrete

　　　　case.Information Sciences, 17:253 －278, 1979.

[11]M. Miyakoshi and M. Simbo. A strong law of large numbers for fuzzy random varia-

　　　　bles.Fuzzy Sets and Systems, U: 133－ 142，1984.

[12]S. B. Boswell and M. S. Taylor. A central limit theorem for fuzzy random variables.

　　　　FuzzySets and Systems, 24 : 331 －344，1987.

[13]H. Inoue. A strong law of large numbers for fuzzy random sets. Fuzzy Sets and Sys-

　　　　terns,41:285－291, 1991.

[14]R. J. Aumann. Integrals of set-valued functions. Journal of Mathematical Analysis and

　　　　itsApplications, 12: 1－12, 1965.

∩5 ）

68 TOKUO　FUKUDA

[15]G. Debreu. Integration of correspondences. In Proc. of fifth Berkeley Sym）. on Math.

　　　Statist,and Probability pages 351 - 372, 1965.

[16]L. A. Zadeh. The concept of a linguistic variables and its application in approximate

　　　reasoning, parts 1, 2 and 3. Information Sciences, 8 and 9 : 199 －249，301－357,43－

　　　80,respectively, 1975,76.

[17]R. E. Bellman and M. Giertz. On the analytic formalism of the theory of fuzzy sets. In-

　　　formation Sciences, 5 : 149－156, 1973.

[18]T. Fukuda. Statisticalproperties of fuzzy stochastic processes. In F. Kozin and T. Ono,

　　　editors,Sytems and Control-Topics in Theory and A卸lications, pages 61 －72. Mita Press,

　　　Tokyo, Japan, 1991.

[19]T, Fukuda. Basic properties of fuzzy stes. The Kyoto Gakuen University Review，18: 125

　　　－164,1990.

[20]D. G. Kendall. Random sets and integral geometry. In E. F. Harding and D. G. Kendall，

　　　editors,Stochastic Gemetry, pages 322－376. Jhon-Wiley, 1974.

∩6 ）

page1page2page3page4page5page6page7page8page9page10page11page12page13page14page15page16