2. Review of fuzzy set theory and fuzzy time series

1. Fuzzy Set Theory

In the fuzzy set concept, the membership of an individual in a fuzzy set is a matter of degree. A ^function, called a membership ^function, assigns to each element a number in the closed ^unit interval ^[0,1] that characterizes the degree of membership of the element. In the work of H.J. Zimmermann ^[19], the fuzzy set is defined as follows.

Definition I.

^If X ^{is a} collection ^of objects denoted generically by ^x then ^a fuzzy set A ⁱⁿ X ^{is a} set ^of ordered ^{pairs: (x,} u(x)) ^, u(x) ^is called the membership ^{function or} degree ^of membership.

In the same ^fashion, the ^truth or falsity of fuzzy propositions is also a matter of degree. Due to the fuzziness of a fuzzy proposition may arise from a combination of different linguistic components.

While classical relations describe solely the presence of association between

elements of two ^sets, fuzzy relations are capable of capturing the strength of association. The definition of the binary relations is given in Zimmmermann work as follows.

Definition II

Let X,Y are two universal sets, then

^{R =} {x,y), uR(x,y)) _\ (x,y) ^{is an} element ^{of X X} ₁₇ ^is called ^a fuzzy relation ^{on X X Y}

Fuzzy relations are obviously fuzzy sets in product spaces. ^Thus, union is defined by HI. Zimmermann as follows

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Definition III

Let R and Z be two fuzzy relations in the ^same product ^space; the membership ^{of union of R with Z is} then ^defined by

^{uR U} Z(x,y)=max (uR(x,y), uZ(x,y)) ⁽¹⁾

2. Fuzzy Time Series

In this study, the fuzzy logical relationship will be employed to model fuzzy time series. In this approach, the values of fuzzy time series are fuzzy sets. ^And, there is a relationship between the observations at time t and those at previous times. For ^convenience, the following definitions are introduced by Song and Chissom ^[4].

Definition ^IV Fuzzy time series

Let ^{Y(t)(t=O,1,2,...) , a} subset ^{of Ri,} be the ^{universe of} discourse ^on which fuzzy set fi(t)(i=1,2,...) are defined. F(t) is ^a collection of fl(t), f2(t),...,then F(t) is called ^a fuzzy ^{time series defined on Y(t).}

In definition ^{IV, F(t)} could be viewed as a linguistic variable. This represents for

the major differences between fuzzy time series and traditional time ^series, whose values must be real numbers. Note that conventional time series models fail to work when its values are linguistic ones.

^{Definition V.} First-order model ^(Song-Chissom model)

Suppose F(t) is ^affected by F(t-1) only, then the fuzzy relation can be expressed by F(t)=F(t-1) R(t,t-1) (2)

where ^{R(t,t-1) is} the fuzzy relationship between ^F(t-1) and ^F(t). And the model F(t)

=F(t-1) ^o R(t,t-1) is called the first order model of F(t).

Note that the fuzzy relationship defined by R(t,t-1) can be dependent or independent of time. ^However, if R(t,t-1) is independent of t, then F(t) is called a time- invariant fuzzy time series; otherwise it is called a time-variant fuzzy time series. In case of time-invariant time ^series, the fuzzy relationship can be rewritten as:

R(t,t-1)=R ⁽³⁾

where R contains only constant elements. The algorithm proposed by Mamdani[6] for deriving the fuzzy relational matrix using an operator ^X of two ^vectors, It can be defined as:

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Definition VI

The operator ^)< of two vectors is defined as follows; Let D and B be two row vectors of dimension ^m and let

_C=(c1 ^)=DT>< B ⁽⁴⁾

then the entry cj ^{of matrix C at row i} and column j ^{is defined as}

min(d1, _b1), i,j ^{=1,2,---m (5)}

iv here _d1 and ^are the ith and the jth elements ^of vectors D and ^B, respectively. And the column vector DT is the transpose of the row vector D.

In the model equation, we use a compositional rule of inference to relate a vector and a matrix. We call it the max-mm operator.

Definition VII

The ^max-mm operator is defined as follows

Let D be a row vector of dimension in and _R=(r ) be a matrix. The composition of D and R is a row vector

D R= (maximinj(di, _r ) ) ⁽⁶⁾

To develop fuzzy relations among the observations at different times of ^interests, Song and Chissom [4] define the mth-order model of fLizzy time series in general as follows

^{Definition IIX.} mth-order model ^of fuzzy ^{time series}

^{Suppose F(t) is} caused by F(t-1) ^F(t-2), and ^F(t-m) (m>O) simultaneously ^, then the fuzzy relation ^can be expressed by

^{F(t)=(F(t-1) X F(t-2)} >< >< F(t-m)) R(t, ^{t-m) (7)}

where >< is the ^Cartesian product . Then the model is called the m-th order model

There might be more than one approach to modeling fuzzy time series. ^But, in this

paper, we propose that if F(t) is caused by F(t-1) and F(t-2), ^{- - -} F(t-m) simultaneously. Then take

F(t)=min(F(t-1) R(t,t-1), F(t-2) R(t,t-2), ---, F(t-m) R(t,t-m)) ⁽⁸⁾

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The model is called the proposed mth order model , the new scheme embedded into Song-Chissom method

Definition IX. Chen model

^{Suppose F(t) is affected} by ^F(t-1) only, then the fuzzy logical relationship ^can be

expressed by ^F(t-1)

F(t), which means that if the data of time t -1 is F(t-1) then

—

that of time t is F(t). And the model is called Chen model of F(t).

In the above ^definition, the fuzzy logical relationship, i.e. the fuzzy implication, contains linguistic variables as antecedent and consequent. The consideration that ^F(t) is affected by ^F(t-1) only qualify the fuzzy proposition. For the sake of the validity of the reasoning, we try to consider that ^F(t) is caused by ^{F(t-1) F(t-2),} and ^{F(t-m) (m>}

⁰⁾ simultaneously.

^Definition X. mth-order model ^{(the new} scheme embedded ^{into Chen} model) ^Suppose F(t) ^is caused by ^{F(t-1) F(t-2),} and F(t-m) ^(m>O) simultaneously ^, then the fuzzy logical relation ^can be expressed by

(F(t-1) ^X F(t-2) ^{X ..• X} F(t-m))

F(t),

—

Then the model is called the mth order model of F(t) embedded with the new

scheme.