3. Theory for model equation and forecasting procedures

1. Theory for model equation

For the theory of the new ^scheme, the following theorems are proposed to set the element of model equation

Theorem 1(Chao-Chih Tsai)

Let D and B be two ^row vectors ^of dimension ^m. And let ^matrix

_C=(c ^{)=DT X} B ⁽⁹⁾

Assume that for any fixed n, there exists m, such that _{dm b} then

D C=B ⁽¹⁰⁾

Proof:

^{Let the row vector V= D} _° ^{C and be} represented by (Vk). ^Where vk=max(dI, clk).

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^And _Cik = min(d1, bk). ^{ThLls, if for each fixed x there exists m such that} _{dm b} ^{then the} mth-component of ^the column vector(c1) is _b. ^{Or, the} component of ^the column vector ^(cij, is less than _b. ^Thus, vk=maxl(dl, clk)= _bk. That ^{is, V=B. Q.E.D.} It is also noted that if the assumption is not ^satisfied, then we have _{v=max d1} which is taken as near as possible to _b. The above theorem help to clarify the operation suggested by Mamdani[6] used to calculate the fuzzy relational matrix. ^But,

to obtain the fuzzy model equation, all the historical data should be used. The next theorem indicates the change of the relational matrix due to this consideration.

Theorem 2(Chao-Chih Tsai)

Let _A1 ^, _A ^{,--- ,} _A be ^row vectors ^of dimension ^m. And

_{Rq= AqT} ^X _{Aq+j .} q=1,2,---p ⁽¹¹⁾

are matrice. And let

RP=Uq1

_Rq (12)

‘

_D=A1 ,B=A2 ^and

(v1) D R. ⁽¹³⁾

Assume ^for any ^fixed E ^, there exists ^, such that _d2>b then _{v= b} ^or _{d> v>b}

Proof:

When ^n=l, it is proven in theorem 1.

Assume ^n=k, it is also valid. For ^n=k+1, consider for a fixed value ^{t ,} let _{ur= max} minr(dj, ^{rk ii). By assumption, we have} _u5 = _b5 ^or _u>b. That ^{is to say, the maximum value of the} components ^{for the vector} (minr(dj, ^{rk ii)) has two} possibilities.

One is equal to _b. The other is between _d and _b.

Consider a fixed component ^ye of D

_° S. For _v5=

max1

minr(dj, ^where

_s = max

^{( A} jt

_{, nv ) with A = Rk÷1} ^Thus, _{s r} , min(di, ) min(di, ) But it is

apparently that ^{the value} of ^minr(di, 5) ^can not ^be greater than _d1. ^Hence, _{v= B} or _d1

Q.E.D.

In view of the change of the relationship ^matrix, every component of forecasted

vector D

S either maintains the origin correct value or become larger. ^Hence, we

°

must use some mechanism to improve it.

The first-order model assumes that ^F(t) is affected by ^F(t-l) only. ^However, in order to infer correctly, all the historical data must be used. Because of the addition of the historical ^data, represented by some implications, the inference form of forecasting becomes valid. ^Hence, we use this mechanism to improve it. Before improving, let us examine the first-order model.

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2. CALCULATION OF FIRST-ORDER MODEL

Based on fuzzy time ^series, to introdLice the forecasting procedLire, we begin with the first-order model then propose the high-order fuzzy time series.

The forecasting model can be constructed by the following step ^[9]:

1. Define the universes of discourse on which the fuzzy sets will be defined.

2. Collect historical data ^(can be linguistic ^values).

3. Define fuzzy sets on the universe of discourse based on the historical fuzzy data.

4. If historical ^{data are} not linguistic, a ^fuzzy transformation is used to get ^fuzzy data.

5. Setup fuzzy logical relationships(for Chen ^method) or relational matrix ^(for Song ^{Chissom method)} Lising ^{historical data.}

6. Construct the fuzzy forecasting model by the fuzzy logical relationships (for Chen

^method) or the model equations(for Song-Chissom ^method).

7. Use the historical data as inputs to the forecasting model and compute the output, which will be the forecasted values.

8. Transfer fuzzy outputs of the model into real values if needed.

The advantage of this model includes:

1. In every step of this procedure, one’s experience knowledge can be added.

2. While we use fuzzy sets to represent the correspondent ^variables, the uncertainty can be handled.

^But, the first-order ^model, like mentioned ^above, has some limitation. The assumption, each term of the fLizzy time series only caused by consecutive ^term, makes forecasts incorrect. ^Hence, the high-order models are proposed.