- Email: [email protected]
A new fuzzy time-series model of fuzzy number observations
sets and systems Fuzzy Sets and Systems 73 (1995) 341-348
ELSEVIER
A new fuzzy time-series model of fuzzy number observations Qiang Song a'*, Robert P. Leland a, Brad S. Chissom b aDepartment of Electrical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA bArea of Behavioral Studies, College of Education, The University of Alabama, Tuscaloosa, AL 35487, USA
Received October 1993;revisedAugust 1994
Abstract
Fuzzy time series [4] has been proposed to model dynamic processes whose observations are fuzzy sets or linguistic values, and some applications have been made with satisfactory results [5, 6]. In this paper, a new fuzzy time-series model is proposed by means of defining some new operations on fuzzy numbers. The proposed model is in the form of two theorems that relate the current value of a fuzzy time series with its past. Keywords: Fuzzy time series; Fuzzy numbers; Linguistic values and models
1. Introduction
The motivation for proposing the concept of fuzzy time series [4] was due to the need for modelling dynamic processes whose observations are linguistic values [8]. Since no traditional methodologies have been found to model these dynamic processes with satisfaction, fuzzy set theory, or specifically fuzzy logic [7], was applied and in I-4] three different models were proposed for fuzzy time series. Since those models were based upon the fuzzy logic reasoning, in this sense any fuzzy logic reasoning models can be applied to model fuzzy time series. Fuzzy numbers [1], a special form of fuzzy sets, have been studied extensively in the literature. Although being fuzzy sets, fuzzy numbers have been endued with different operations from the fuzzy set operations. If a fuzzy time series has fuzzy numbers as its observations, the proposed models in [4] may not work well. Hence new models need to be sought after. The simplest fuzzy time-series model might be the first-order time-invariant model which can be expressed as [4]
F(t) = F ( t -
1) o R(t, t - 1),
O)
*Correspondenceaddress: Schoolof Industrial and SystemsEngineering,Georgia Institute of Technology,Atlanta, GA 30332, USA. 0165-0114/95/$09.50 © 1995 - ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(94)00315-7
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
342
where F(t) is the collection of all possible values of a fuzzy time series at t, F(t - 1) the collection of all possible values at t - 1 and R(t, t - 1) the fuzzy relation between F ( t - 1) and F(t). In most cases, this relation is the union of all fuzzy relations between any possible values of F(t - 1) and F(t), as shown below
[4] R = R ( t , t - 1) . . . .
uf~,(t - 1) Xfjo(t)cJf~2(t - 2) xfj,(t - 1)~ ... uj~.(t - m)
×fJm ,(t - m + 1)c) .-.,
(2)
where m > 0 and each pair of fuzzy sets is different. One of the application areas of fuzzy time series is the forecasting problems under a fuzzy environment in which no numerical historical data are available but linguistic ones. In I-5, 6], two models were applied to forecast the enrollments of the University of Alabama by first fuzzifying the historic data. Yet, careful studies of those models lead to the conclusion that there still exists some information hidden within the historic linguistic data that could be included in the models. The ignored information, found at this stage, is the differences among the historical linguistic data. This means that our models should take those differences into account. A complete solution of this problem is never trivial, but under some conditions partial solutions may be arrived at. In this paper, with the assumption that the values of the fuzzy time series are fuzzy numbers I-1, 31, a new model is presented to utilize the differences. Our goal is to develop a relation between the current value and the values in the past. Since we assume that an infinite number of observations of the past on an infinite time interval could be obtained, this relation will be in the form of summation of a sequence of fuzzy numbers. In the following section, by defining some useful terms that will be applied in the discussion, the new model will be presented in the form of two theorems.
2. The new model
First, we will represent the definition of a fuzzy time series. Definition 1. Suppose Y(t), t ~ T, where T = { .... 0, 1, 2, ... }, is a subset of El and on it are defined some fuzzy setsf/(t) (i = 1, 2, ... ) and F(t) is the collection ofJ~(t) (i = 1, 2 . . . . ). Then F(t) is called a fuzzy time series on Y (t), t ~ T. It can be seen from the above definition that F(t) is a collection off,(t) (i = 1, 2 . . . . ) which are fuzzy sets defined on ~1 for a given t ~ T. The main difference between the traditional time series and the fuzzy time series is that the former has numerical values as its observations while the latter has fuzzy sets as its observations. Since time t is considered in the definition of the fuzzy time series, F(t) is a function of t and hence a kind of dynamic processes. In other words, there is a fuzzy one-to-many mapping [2] defined on T and this fuzzy mapping is a fuzzy time series. As is known, linguistic values are labels of fuzzy sets [8-1, that is to say, fuzzy sets can be defined on a predefined universe of discourse according to the linguistic values. Thus the observations of fuzzy time series are also said to be linguistic values. Since fuzzy numbers are also defined on ~1, under some conditions J] (t) can be treated as fuzzy numbers. Hence, in the following we will assume that all the observations of a fuzzy time series are fuzzy numbers. For argument's sake, we will also use F(t) for its observation and assume that at each time t ~ T, F(t) has only one observation. A fuzzy number N is defined as a fuzzy set in ~1 which satisfies I-1, 3-1: (1) the height of N is 1, i.e., hgt(N) = 1; (2) N is unimodal; (3) N is upper-semi-continuous; (4) N has a bounded support.
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341 348
343
O f extreme interest to us is the L - R fuzzy number whose membership function is defined as follows:
L(-~-~-~ N(x)=
if x~< m, x ~ ~l,
(3)
\/ / "~ _ \
where a > 0 and b > 0 are parameters that can be interpreted as the measure of the fuzziness of N and m is the "mean" value of N(x), and L(x) (or R(x)) is a real function satisfying the following conditions: (1) L( - x) = L(x); (2) L(O)= 1; (3) L is decreasing on [0, ~ ]. When a = b = 0, N(x) is a real number. Generally, the L - R fuzzy numbers are denoted as N(m, a, b). In this paper, we allow m = ~ , i.e., a fuzzy number can have a mean value being infinity. For two fuzzy numbers A1 = (ml, al, bl) and Az = (mz, a2, b2), their addition and subtraction are defined as follows [1, 3]: A1 + A 2 = ( m l + m2, al +a2, bl + b 2 ) ,
(4)
A1
(5)
-
A2 = (ml -
m2,
al + a2, bl + b2).
It can be shown that the + operation satisfies the commutative and the associative laws while the - operation does not. Obviously with the operations + and - , the resultant fuzzy numbers become more fuzzy after the operations. In addition, the summation of an infinite number of fuzzy numbers may not be a fuzzy number even if the summation of their means converges to a finite number. This is because in some cases the support of the resultant fuzzy number may not be bounded. For example, let us consider a sequence of fuzzy numbers {A.} = {(1/n 2, 1, 1)} for n = 1 to 00. Then Y~.~=1A. = ( y l / n 2, 00, ~ ) will not be a fuzzy number. This means that when dealing with a sequence of fuzzy numbers, the limit operation cannot be taken in general with the + and - operations. Thus, the + and - operations can only be applied for a finite number of fuzzy numbers, and when dealing with an infinite number of fuzzy numbers new operations are needed. For this reason, in the following we will tentatively introduce new addition and subtraction operations for fuzzy numbers, denoted as +f and - f , called linguisic addition and subtraction respectively, and defined below. 2. For fuzzy numbers A1 = (rnl, a l , bl) and subtraction are defined as follows:
Definition
A2 = (m2,
a2,
b2),
their linguistic addition and
A1 +fA2 = ( m l +m2,a,b),
(6)
A1 - f A 2
(7)
=
(ml -- m2, a, b),
where a = max {al, a2} and b = max {bx, b2}. In the case of an infinite number of fuzzy numbers, then in (6) and (7) we define a = sup {a;} and b = sup {b~} for i ~ I = {1, 2, 3 . . . . }. It can be shown that +f satisfies the commutative and the associative laws but - f does not. The difference between the operations + , - and + f and - f is quite obvious. With + f and - f the fuzziness of the resultant fuzzy number will be the m a x i m u m fuzziness of the two operands. The advantage of this definition is that it provides one the ease with which to manipulate multiple fuzzy numbers, and even sequences of fuzzy numbers. In addition, if A1 and A 2 a r e real numbers, -if and - r will yield the same results as + and - do. For the + f and - f operations, we have the following conclusion.
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Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
Proposition 1. L e t { N , } = {(mi, ai, bi)}iet be a sequence o f f u z z y numbers where I = {1, 2, 3, ... } with ~,i~=tmi < ~ . T h e n N = Nx +fN2 +f -.- +f Nk +f -'- is a f u z z y number if and only if {ai}i~1 and {bi}iel are uniformly bounded, i.e., there exists a constant c > 0 such that for any i ~ I ai <~ c and bi <. c. Proof. If {a~}~x and {b~}~t are uniformly bounded, then there exists a constant c > 0 such that a~ ~< c b~ <~ c for any i e I. Thus, there exist a > 0 and b > 0 such that a = supi~1{a~} <~ c and b = s u p i ~ { b i } <~ c. Denote N = (~,i~=lmi, a, b). Then N is a fuzzy number. If N = (y~ff=xm~, a, b)is a fuzzy number, then a = supi~{a~} and b = sup~{b~} for i ~ I. Therefore, {a~}i~ and {bi}i~ are uniformly bounded. This finishes the proof. [] If we allow m = ~ , Proposition 1 is naturally valid. Now, we will provide some useful definitions about the fuzzy time series.
Definition 3. Let F(t), t ~ T, be a fuzzy time series with the observation F(t) = (re(t), a(t), b(t)) being fuzzy numbers. Define DI F(t) = F(t) - f F(t - 1),
where DI F(t) = (m~b(t), alb(t), b~b(t)), mXab(t) = m(t) -- m(t -- 1),
a~b(t) = max {a(t), a(t -- 1)} and b~b(t) = max {b(t), b(t - 1)}; D~F(t) = D~ F(t)
--f
0 1 F ( t -- 1),
where D~F(t) = (m~b(t), a~b(t), blb(t)),
m2b(t) = mlb(t) -- mlb(t -- 1), a~b(t) = max {a~b(t), a~b(t - 1)} and blb(t) = max {b~b(t), b~b(t - 1)}. In general, D~,F(t) = D~,- 1F(t) - f D~,- t F(t - 1),
(8)
where
m~b(t) -- m~bl(t) -- mkabl(t -- 1),
(9) (lO)
akb(t) = max {a~b 1(0, akab- 1(t -- 1)},
(11)
bkb(t) = max {bk~- 1(0, bk~ l(t -- 1)}.
(12)
okbF(t) = (m~b(t), a~s(t), b~b(t)),
and
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
345
Then DkF(t) is called the kth order (k = 1, 2, ... ) b a c k w a r d linguistic difference of F(t) at t. Obviously, DkF(t), t ~ T, is also a fuzzy time series for any k > 0.
Definition 4. Let F(t), t E T, be a fuzzy time series and F(t) = (re(t), a(t), b(t)). Define
DX F(t) = F(t + 1) - f F ( t ) , where
D t F(t) = (m) (t), a~ (t), b~ (t)), m~(t) = m(t + 1 ) - m(t), a~(t) = m a x {a(t + 1), a(t)} and
b~(t) = m a x {b(t + 1), b(t)}; D 2 F ( t ) = DXF(t + 1) - - f D 1 F ( t ) ,
where
DZ F(t) = (m2(t), a2(t), b2(t)), m2(t) = m~(t + 1) - m~(t), a~(t) = m a x {a~(t + 1), adl(t)} and
bad(t) = m a x {bl(t + 1), b)(t)). In general,
DkF(t) = D k- 1F(t + 1) - - f D k- tF(t),
(13)
where
ak(t) = m a x {a k- l(t + 1), a k- l(t)},
(14) (15) (16)
b~(t) - m a x {b~- l(t + 1), bkd- 1(0}.
(17)
DkF(t) = (ink(t), akd(t), bkd(t)), m~(t) = mgd- l(t + 1) -- mk-l(t),
and
Then DkF(t) is called the kth order (k = 1, 2, ... ) forward linguistic difference of F(t) at t. Similarly, DkF(t), t ~ T, is also a fuzzy time series for any k > 0. O f all the different fuzzy numbers, the ones with equal a's and b's are extremely interesting to us, and so is the fuzzy time series with equal a's and b's.
Definition 5. Let F(t), t e T, be a fuzzy time series and F(t) = (re(t), a(t), b(t)). If a(t) = a(t - 1) and b(t) = b(t - 1) for any t ~ T, then F(t) is said to be a h o m o g e n e o u s fuzzy time series. It should be noted that Definition 5 is not arbitrarily proposed. As explained before, a(t) and b(t) can be taken as a measure of the fuzziness of a fuzzy number. T h e fuzziness can be u n d e r s t o o d as our incomplete understanding of the nature and usually a(t) and b(t) are related to the fuzzy m e m b e r s h i p functions which are
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
346
quite subjective. Therefore, if we argue that our understanding of the nature, although incomplete, is consistent with time, it can be acceptable. Another fact about Definition 5 is that we are focusing our discussion on a special class of fuzzy time series, i.e., the ones with equal a's and b's. F o r the h o m o g e n e o u s fuzzy time series, we have the following conclusion. L e m m a 1. L e t F(t) be a homogeneous f u z z y time series, t • T, and F(t) = (m(t), a(t), b(t)). T h e n a~b(t) = a(t), Proof. Obvious. Theorem
b~b(t) = b(t),
a~(t) = a(t),
b~(t) = b(t),
t • T ; k = 1, 2, ...
[]
1. I f F(t), t • T, is a homogeneous f u z z y time series, then
F(t + 1) = F(t) + f D 1 F ( t -
1) + f D 2 F ( t -
2) +f ... + f D k F ( t -
k) +f ...
(18)
i.e., F(t + 1) = ( E ioo : o m ai( t - i), a(t + 1), b(t + 1)).
Proof. Since F(t) is homogeneous, according to L e m m a 1 a(t + 1) = sup {a(t), a~(t), a~(t), a](t), ... ,a~(t) . . . . }, b(t + 1) = sup {b(t), b~(t), b~(t), b](t) . . . . , b~(t) . . . . }
for any t • T. In addition, m(t + 1) = m(t) + m ( t + 1) - m(t) = m(t) + m~(t) m~(t) = m~(t) + m~(t - 1) - m~(t - 1) = m ) ( t - 1) + m ] ( t - 1) m~(t - 1) = m~(t - 1) + m~(t - 2) - m~(t - 2) = m~(t - 2) + m](t - 2).
Generally, for k > 0, mk(t -- k + 1) = mk(t -- k) + m k + l ( t -- k).
Thus, m(t + 1) = re(t) + m~(t - l) + m2(t - 2) + ma(t - 3) + -.- + mk(t -- k) + ... = ~, m~(t -- i), i=O
where m°(t - O) = re(t). Therefore,
m~(t -
F(t + 1) = (m(t + 1), a(t + 1), b(t + 1)) =
i), a(t + 1), b(t + 1) .
\i=0
This finishes the proof.
[]
T h e o r e m 1 indicates that the value of F(t) at t + 1 can be related to its previous ones by means of forward linguistic differences of various orders. It can also be seen that the definition of the linguistic addition and the linguistic subtraction plays an important role in the derivation of the conclusion. With such a definition, it is convenient to manipulate multiple fuzzy numbers and even fuzzy n u m b e r sequences. N o t e that Eq. (18) in T h e o r e m 1 is the new model proposed in this paper.
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
347
Theorem 2. I f F(t), t ~ T, is a homogenous fuzzy time series, then F(t + 1) = F(t) +fO~F(t) +fO2F(t) +fDabF(t) ÷ f ... +fO[F(t) ÷ f ""
(19)
i.e., F(t + 1) = (~i=Omdb(t), oo i a(t + 1), b(t + 1)). Proof. Similar to that of Theorem 1.
[]
Note that Eq. (19) in Theorem 2 is another form of the new model. It indicates that the value of F(t) at t + 1 can be expressed as the linguistic summation of various order linguistic backward differences at t, which is analogous to the Taylor's expansion in calculus. Next, let us define an operator D F which satisfies: (1) D F : N ~ 1, where N is the fuzzy number and ~t is a real number; (2) DF(N~ ÷fN2) = DF(N1) + DF(N2); (3) DF(N~ - f N2) = DF(N~) - DF(N2); and (4) DF(N1 +fN2 +fN3 +f ... +fNk ÷f " - ) = DF(N~) 4- DF(N2) + DF(N3) + ... + DF(Nk) + ---, where Ni, i e I, are fuzzy numbers, + and - are the regular addition and subtraction. Then D F is called the defuzzification operator. With DF, we can easily obtain the following from Theorem 1: DF(F(t + 1)) = DF(F(t)) + DF(D1F(t - 1)) + DF(D2F(t - 2)) + ... + DF(DkF(t - k)) + ...
(20)
and from Theorem 2: D F t F ( t + 1)) = DF(F(t)) + DF(D~F(t)) + DF(D2F(t)) + ... + DF(Dk F(t)) + ...
(21)
F o r a fuzzy n u m b e r N = (m, a, b), if we define D F ( N ) = m, then for T h e o r e m 1, we have
m(t + 1) =
~
m~(t -
i),
where m ° = re(t),
i=0
and for T h e o r e m 2, we have
m(t + 1) = ~ m~b(t),
where mob(t) = re(t).
i=0
3. Concluding remarks In this paper, by defining linguistic addition and subtraction, we have derived a new model in two forms for the homogeneous fuzzy time series. The new model of Eq. (18) expresses the value of a homogeneous fuzzy time series at t + 1 as the linguistic summation of the previous values and linguistic differences of different orders, which indicates clearly the relationship of the current and the past. The new model of Eq. (19) relates the value at t + 1 as the linguistic summation of the value and various differences at t, which resembles the Taylor's expansion in calculus. Since fuzzy numbers are only special fuzzy sets, the conclusions drawn here are only valid for homogeneous fuzzy time series with fuzzy number observations. Further studies are needed to consider expanding the conclusions to the nonhomogeneous fuzzy time series and the general fuzzy time series. Applications of the new model will be pursued in a different paper.
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Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
Acknowledgement The authors wish to express their gratitude to the referees for their valuable comments on and suggestions for the original version of this paper.
References [1] D. Dubois and H. Prade, Addition of interactive fuzzy numbers, IEEE Trans. Automat. Control AC-26 (1981) 926-936. [2] D. Dubois and H. Prade, Towards fuzzy differential calculus Part 1: integration of fuzzy mappings, Fuzzy Sets and Systems 8 (1982) 1-17. I'3] W. Pedrycz, Fuzzy Control and Fuzzy Systems, Research Studies Press Ltd., Taunton, Somerset, England, 1989. 1'4] Q. Song and B.S. Chissom, Fuzzy time series and its models. Fuzzy Sets and Systems 54 (1993) 269-277. [5] Q. Song and B.S. Chissom, Forecasting enrollments with fuzzy time series: Part I, Fuzzy Sets and Systems 54 (1993) 1-9. I-6] Q. Song and B.S. Chissom, Forecasting enrollments with fuzzy time series: Part II, Fuzzy Sets and Systems 62 (1994) 1 8. 1-7] L.A. Zadeh, Fuzzy logic and approximate reasoning, Synthese 30 (1975) 407-428. 1-8] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning: Part 1, Inform. Sci. 8 (1975) 199-249; Part 2, Inform. Sci. 8 (1975) 301-357; Part 3, Inform. Sci. 9 (1975) 43-80.
ELSEVIER
A new fuzzy time-series model of fuzzy number observations Qiang Song a'*, Robert P. Leland a, Brad S. Chissom b aDepartment of Electrical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA bArea of Behavioral Studies, College of Education, The University of Alabama, Tuscaloosa, AL 35487, USA
Received October 1993;revisedAugust 1994
Abstract
Fuzzy time series [4] has been proposed to model dynamic processes whose observations are fuzzy sets or linguistic values, and some applications have been made with satisfactory results [5, 6]. In this paper, a new fuzzy time-series model is proposed by means of defining some new operations on fuzzy numbers. The proposed model is in the form of two theorems that relate the current value of a fuzzy time series with its past. Keywords: Fuzzy time series; Fuzzy numbers; Linguistic values and models
1. Introduction
The motivation for proposing the concept of fuzzy time series [4] was due to the need for modelling dynamic processes whose observations are linguistic values [8]. Since no traditional methodologies have been found to model these dynamic processes with satisfaction, fuzzy set theory, or specifically fuzzy logic [7], was applied and in I-4] three different models were proposed for fuzzy time series. Since those models were based upon the fuzzy logic reasoning, in this sense any fuzzy logic reasoning models can be applied to model fuzzy time series. Fuzzy numbers [1], a special form of fuzzy sets, have been studied extensively in the literature. Although being fuzzy sets, fuzzy numbers have been endued with different operations from the fuzzy set operations. If a fuzzy time series has fuzzy numbers as its observations, the proposed models in [4] may not work well. Hence new models need to be sought after. The simplest fuzzy time-series model might be the first-order time-invariant model which can be expressed as [4]
F(t) = F ( t -
1) o R(t, t - 1),
O)
*Correspondenceaddress: Schoolof Industrial and SystemsEngineering,Georgia Institute of Technology,Atlanta, GA 30332, USA. 0165-0114/95/$09.50 © 1995 - ElsevierScienceB.V. All rights reserved SSDI 0165-01 14(94)00315-7
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341-348
342
where F(t) is the collection of all possible values of a fuzzy time series at t, F(t - 1) the collection of all possible values at t - 1 and R(t, t - 1) the fuzzy relation between F ( t - 1) and F(t). In most cases, this relation is the union of all fuzzy relations between any possible values of F(t - 1) and F(t), as shown below
[4] R = R ( t , t - 1) . . . .
uf~,(t - 1) Xfjo(t)cJf~2(t - 2) xfj,(t - 1)~ ... uj~.(t - m)
×fJm ,(t - m + 1)c) .-.,
(2)
where m > 0 and each pair of fuzzy sets is different. One of the application areas of fuzzy time series is the forecasting problems under a fuzzy environment in which no numerical historical data are available but linguistic ones. In I-5, 6], two models were applied to forecast the enrollments of the University of Alabama by first fuzzifying the historic data. Yet, careful studies of those models lead to the conclusion that there still exists some information hidden within the historic linguistic data that could be included in the models. The ignored information, found at this stage, is the differences among the historical linguistic data. This means that our models should take those differences into account. A complete solution of this problem is never trivial, but under some conditions partial solutions may be arrived at. In this paper, with the assumption that the values of the fuzzy time series are fuzzy numbers I-1, 31, a new model is presented to utilize the differences. Our goal is to develop a relation between the current value and the values in the past. Since we assume that an infinite number of observations of the past on an infinite time interval could be obtained, this relation will be in the form of summation of a sequence of fuzzy numbers. In the following section, by defining some useful terms that will be applied in the discussion, the new model will be presented in the form of two theorems.
2. The new model
First, we will represent the definition of a fuzzy time series. Definition 1. Suppose Y(t), t ~ T, where T = { .... 0, 1, 2, ... }, is a subset of El and on it are defined some fuzzy setsf/(t) (i = 1, 2, ... ) and F(t) is the collection ofJ~(t) (i = 1, 2 . . . . ). Then F(t) is called a fuzzy time series on Y (t), t ~ T. It can be seen from the above definition that F(t) is a collection off,(t) (i = 1, 2 . . . . ) which are fuzzy sets defined on ~1 for a given t ~ T. The main difference between the traditional time series and the fuzzy time series is that the former has numerical values as its observations while the latter has fuzzy sets as its observations. Since time t is considered in the definition of the fuzzy time series, F(t) is a function of t and hence a kind of dynamic processes. In other words, there is a fuzzy one-to-many mapping [2] defined on T and this fuzzy mapping is a fuzzy time series. As is known, linguistic values are labels of fuzzy sets [8-1, that is to say, fuzzy sets can be defined on a predefined universe of discourse according to the linguistic values. Thus the observations of fuzzy time series are also said to be linguistic values. Since fuzzy numbers are also defined on ~1, under some conditions J] (t) can be treated as fuzzy numbers. Hence, in the following we will assume that all the observations of a fuzzy time series are fuzzy numbers. For argument's sake, we will also use F(t) for its observation and assume that at each time t ~ T, F(t) has only one observation. A fuzzy number N is defined as a fuzzy set in ~1 which satisfies I-1, 3-1: (1) the height of N is 1, i.e., hgt(N) = 1; (2) N is unimodal; (3) N is upper-semi-continuous; (4) N has a bounded support.
Q. Song et al. / Fuzzy Sets and Systems 73 (1995) 341 348
343
O f extreme interest to us is the L - R fuzzy number whose membership function is defined as follows:
L(-~-~-~ N(x)=
if x~< m, x ~ ~l,
(3)
\/ / "~ _ \
where a > 0 and b > 0 are parameters that can be interpreted as the measure of the fuzziness of N and m is the "mean" value of N(x), and L(x) (or R(x)) is a real function satisfying the following conditions: (1) L( - x) = L(x); (2) L(O)= 1; (3) L is decreasing on [0, ~ ]. When a = b = 0, N(x) is a real number. Generally, the L - R fuzzy numbers are denoted as N(m, a, b). In this paper, we allow m = ~ , i.e., a fuzzy number can have a mean value being infinity. For two fuzzy numbers A1 = (ml, al, bl) and Az = (mz, a2, b2), their addition and subtraction are defined as follows [1, 3]: A1 + A 2 = ( m l + m2, al +a2, bl + b 2 ) ,
(4)
A1
(5)
-
A2 = (ml -
m2,
al + a2, bl + b2).
It can be shown that the + operation satisfies the commutative and the associative laws while the - operation does not. Obviously with the operations + and - , the resultant fuzzy numbers become more fuzzy after the operations. In addition, the summation of an infinite number of fuzzy numbers may not be a fuzzy number even if the summation of their means converges to a finite number. This is because in some cases the support of the resultant fuzzy number may not be bounded. For example, let us consider a sequence of fuzzy numbers {A.} = {(1/n 2, 1, 1)} for n = 1 to 00. Then Y~.~=1A. = ( y l / n 2, 00, ~ ) will not be a fuzzy number. This means that when dealing with a sequence of fuzzy numbers, the limit operation cannot be taken in general with the + and - operations. Thus, the + and - operations can only be applied for a finite number of fuzzy numbers, and when dealing with an infinite number of fuzzy numbers new operations are needed. For this reason, in the following we will tentatively introduce new addition and subtraction operations for fuzzy numbers, denoted as +f and - f , called linguisic addition and subtraction respectively, and defined below. 2. For fuzzy numbers A1 = (rnl, a l , bl) and subtraction are defined as follows:
Definition
A2 = (m2,
a2,
b2),
their linguistic addition and
A1 +fA2 = ( m l +m2,a,b),
(6)
A1 - f A 2
(7)
=
(ml -- m2, a, b),
where a = max {al, a2} and b = max {bx, b2}. In the case of an infinite number of fuzzy numbers, then in (6) and (7) we define a = sup {a;} and b = sup {b~} for i ~ I = {1, 2, 3 . . . . }. It can be shown that +f satisfies the commutative and the associative laws but - f does not. The difference between the operations + , - and + f and - f is quite obvious. With + f and - f the fuzziness of the resultant fuzzy number will be the m a x i m u m fuzziness of the two operands. The advantage of this definition is that it provides one the ease with which to manipulate multiple fuzzy numbers, and even sequences of fuzzy numbers. In addition, if A1 and A 2 a r e real numbers, -if and - r will yield the same results as + and - do. For the + f and - f operations, we have the following conclusion.
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Proposition 1. L e t { N , } = {(mi, ai, bi)}iet be a sequence o f f u z z y numbers where I = {1, 2, 3, ... } with ~,i~=tmi < ~ . T h e n N = Nx +fN2 +f -.- +f Nk +f -'- is a f u z z y number if and only if {ai}i~1 and {bi}iel are uniformly bounded, i.e., there exists a constant c > 0 such that for any i ~ I ai <~ c and bi <. c. Proof. If {a~}~x and {b~}~t are uniformly bounded, then there exists a constant c > 0 such that a~ ~< c b~ <~ c for any i e I. Thus, there exist a > 0 and b > 0 such that a = supi~1{a~} <~ c and b = s u p i ~ { b i } <~ c. Denote N = (~,i~=lmi, a, b). Then N is a fuzzy number. If N = (y~ff=xm~, a, b)is a fuzzy number, then a = supi~{a~} and b = sup~{b~} for i ~ I. Therefore, {a~}i~ and {bi}i~ are uniformly bounded. This finishes the proof. [] If we allow m = ~ , Proposition 1 is naturally valid. Now, we will provide some useful definitions about the fuzzy time series.
Definition 3. Let F(t), t ~ T, be a fuzzy time series with the observation F(t) = (re(t), a(t), b(t)) being fuzzy numbers. Define DI F(t) = F(t) - f F(t - 1),
where DI F(t) = (m~b(t), alb(t), b~b(t)), mXab(t) = m(t) -- m(t -- 1),
a~b(t) = max {a(t), a(t -- 1)} and b~b(t) = max {b(t), b(t - 1)}; D~F(t) = D~ F(t)
--f
0 1 F ( t -- 1),
where D~F(t) = (m~b(t), a~b(t), blb(t)),
m2b(t) = mlb(t) -- mlb(t -- 1), a~b(t) = max {a~b(t), a~b(t - 1)} and blb(t) = max {b~b(t), b~b(t - 1)}. In general, D~,F(t) = D~,- 1F(t) - f D~,- t F(t - 1),
(8)
where
m~b(t) -- m~bl(t) -- mkabl(t -- 1),
(9) (lO)
akb(t) = max {a~b 1(0, akab- 1(t -- 1)},
(11)
bkb(t) = max {bk~- 1(0, bk~ l(t -- 1)}.
(12)
okbF(t) = (m~b(t), a~s(t), b~b(t)),
and
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345
Then DkF(t) is called the kth order (k = 1, 2, ... ) b a c k w a r d linguistic difference of F(t) at t. Obviously, DkF(t), t ~ T, is also a fuzzy time series for any k > 0.
Definition 4. Let F(t), t E T, be a fuzzy time series and F(t) = (re(t), a(t), b(t)). Define
DX F(t) = F(t + 1) - f F ( t ) , where
D t F(t) = (m) (t), a~ (t), b~ (t)), m~(t) = m(t + 1 ) - m(t), a~(t) = m a x {a(t + 1), a(t)} and
b~(t) = m a x {b(t + 1), b(t)}; D 2 F ( t ) = DXF(t + 1) - - f D 1 F ( t ) ,
where
DZ F(t) = (m2(t), a2(t), b2(t)), m2(t) = m~(t + 1) - m~(t), a~(t) = m a x {a~(t + 1), adl(t)} and
bad(t) = m a x {bl(t + 1), b)(t)). In general,
DkF(t) = D k- 1F(t + 1) - - f D k- tF(t),
(13)
where
ak(t) = m a x {a k- l(t + 1), a k- l(t)},
(14) (15) (16)
b~(t) - m a x {b~- l(t + 1), bkd- 1(0}.
(17)
DkF(t) = (ink(t), akd(t), bkd(t)), m~(t) = mgd- l(t + 1) -- mk-l(t),
and
Then DkF(t) is called the kth order (k = 1, 2, ... ) forward linguistic difference of F(t) at t. Similarly, DkF(t), t ~ T, is also a fuzzy time series for any k > 0. O f all the different fuzzy numbers, the ones with equal a's and b's are extremely interesting to us, and so is the fuzzy time series with equal a's and b's.
Definition 5. Let F(t), t e T, be a fuzzy time series and F(t) = (re(t), a(t), b(t)). If a(t) = a(t - 1) and b(t) = b(t - 1) for any t ~ T, then F(t) is said to be a h o m o g e n e o u s fuzzy time series. It should be noted that Definition 5 is not arbitrarily proposed. As explained before, a(t) and b(t) can be taken as a measure of the fuzziness of a fuzzy number. T h e fuzziness can be u n d e r s t o o d as our incomplete understanding of the nature and usually a(t) and b(t) are related to the fuzzy m e m b e r s h i p functions which are
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quite subjective. Therefore, if we argue that our understanding of the nature, although incomplete, is consistent with time, it can be acceptable. Another fact about Definition 5 is that we are focusing our discussion on a special class of fuzzy time series, i.e., the ones with equal a's and b's. F o r the h o m o g e n e o u s fuzzy time series, we have the following conclusion. L e m m a 1. L e t F(t) be a homogeneous f u z z y time series, t • T, and F(t) = (m(t), a(t), b(t)). T h e n a~b(t) = a(t), Proof. Obvious. Theorem
b~b(t) = b(t),
a~(t) = a(t),
b~(t) = b(t),
t • T ; k = 1, 2, ...
[]
1. I f F(t), t • T, is a homogeneous f u z z y time series, then
F(t + 1) = F(t) + f D 1 F ( t -
1) + f D 2 F ( t -
2) +f ... + f D k F ( t -
k) +f ...
(18)
i.e., F(t + 1) = ( E ioo : o m ai( t - i), a(t + 1), b(t + 1)).
Proof. Since F(t) is homogeneous, according to L e m m a 1 a(t + 1) = sup {a(t), a~(t), a~(t), a](t), ... ,a~(t) . . . . }, b(t + 1) = sup {b(t), b~(t), b~(t), b](t) . . . . , b~(t) . . . . }
for any t • T. In addition, m(t + 1) = m(t) + m ( t + 1) - m(t) = m(t) + m~(t) m~(t) = m~(t) + m~(t - 1) - m~(t - 1) = m ) ( t - 1) + m ] ( t - 1) m~(t - 1) = m~(t - 1) + m~(t - 2) - m~(t - 2) = m~(t - 2) + m](t - 2).
Generally, for k > 0, mk(t -- k + 1) = mk(t -- k) + m k + l ( t -- k).
Thus, m(t + 1) = re(t) + m~(t - l) + m2(t - 2) + ma(t - 3) + -.- + mk(t -- k) + ... = ~, m~(t -- i), i=O
where m°(t - O) = re(t). Therefore,
m~(t -
F(t + 1) = (m(t + 1), a(t + 1), b(t + 1)) =
i), a(t + 1), b(t + 1) .
\i=0
This finishes the proof.
[]
T h e o r e m 1 indicates that the value of F(t) at t + 1 can be related to its previous ones by means of forward linguistic differences of various orders. It can also be seen that the definition of the linguistic addition and the linguistic subtraction plays an important role in the derivation of the conclusion. With such a definition, it is convenient to manipulate multiple fuzzy numbers and even fuzzy n u m b e r sequences. N o t e that Eq. (18) in T h e o r e m 1 is the new model proposed in this paper.
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347
Theorem 2. I f F(t), t ~ T, is a homogenous fuzzy time series, then F(t + 1) = F(t) +fO~F(t) +fO2F(t) +fDabF(t) ÷ f ... +fO[F(t) ÷ f ""
(19)
i.e., F(t + 1) = (~i=Omdb(t), oo i a(t + 1), b(t + 1)). Proof. Similar to that of Theorem 1.
[]
Note that Eq. (19) in Theorem 2 is another form of the new model. It indicates that the value of F(t) at t + 1 can be expressed as the linguistic summation of various order linguistic backward differences at t, which is analogous to the Taylor's expansion in calculus. Next, let us define an operator D F which satisfies: (1) D F : N ~ 1, where N is the fuzzy number and ~t is a real number; (2) DF(N~ ÷fN2) = DF(N1) + DF(N2); (3) DF(N~ - f N2) = DF(N~) - DF(N2); and (4) DF(N1 +fN2 +fN3 +f ... +fNk ÷f " - ) = DF(N~) 4- DF(N2) + DF(N3) + ... + DF(Nk) + ---, where Ni, i e I, are fuzzy numbers, + and - are the regular addition and subtraction. Then D F is called the defuzzification operator. With DF, we can easily obtain the following from Theorem 1: DF(F(t + 1)) = DF(F(t)) + DF(D1F(t - 1)) + DF(D2F(t - 2)) + ... + DF(DkF(t - k)) + ...
(20)
and from Theorem 2: D F t F ( t + 1)) = DF(F(t)) + DF(D~F(t)) + DF(D2F(t)) + ... + DF(Dk F(t)) + ...
(21)
F o r a fuzzy n u m b e r N = (m, a, b), if we define D F ( N ) = m, then for T h e o r e m 1, we have
m(t + 1) =
~
m~(t -
i),
where m ° = re(t),
i=0
and for T h e o r e m 2, we have
m(t + 1) = ~ m~b(t),
where mob(t) = re(t).
i=0
3. Concluding remarks In this paper, by defining linguistic addition and subtraction, we have derived a new model in two forms for the homogeneous fuzzy time series. The new model of Eq. (18) expresses the value of a homogeneous fuzzy time series at t + 1 as the linguistic summation of the previous values and linguistic differences of different orders, which indicates clearly the relationship of the current and the past. The new model of Eq. (19) relates the value at t + 1 as the linguistic summation of the value and various differences at t, which resembles the Taylor's expansion in calculus. Since fuzzy numbers are only special fuzzy sets, the conclusions drawn here are only valid for homogeneous fuzzy time series with fuzzy number observations. Further studies are needed to consider expanding the conclusions to the nonhomogeneous fuzzy time series and the general fuzzy time series. Applications of the new model will be pursued in a different paper.
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Acknowledgement The authors wish to express their gratitude to the referees for their valuable comments on and suggestions for the original version of this paper.
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