For all the historical knowledge, the two-step- ahead fuzzy relational functions are found and union operator is applied to obtained the two-step-ahead model:

_R22—Union of R22(i) ⁽²²⁾

(7*) Obtained the second possibility Ai2 for the forecasted output data using r22.

As the two-step-ahead relational function is ^determined, historical data can be

used to obtained the predicted values by

F,2—F(i-2) _R22 ⁽²³⁾

(8*) Take the intersection of Fj and f to form the output f(I).

Based on the ^{proposed method,} the essence of high-order time series models can

be obtained while the simplicity of first-order calculation is retained. ^Furthermore, the formulation can be easily extended to higher-order models in the same fashion. The following presents the ^application and comparisons of the results for the Chen’s ^method, those for the first-order model and the second­ order modeL

20W0

igojo

18W0

:!

::

15W0

140J0

13W0

12W0

1965 ¹⁹⁷¹⁾ 1975 1980 1985 1990 15

Year

Fig.4 Forescasted enrollments and actual enrollments

^p 4 t I J 1i^I J 1

[

rIrt I4 —I-’ TT1

_‘- Iu

20000

19000

18000

17000

16000

15000

14000

13000

12000

^{1965 1970 1975} 1980 ^{1985 1990 1995}

Year

Fig.7 Effect of partition nLtmber on the forecasted enrollments by the proposed model

As another evidence for the advantages of the new ^scheme, the same example of the historical records of the enrollments of the university of Alabama are analyzed using three different models ^(the Chen’s ^method, the Song and Chissom method and high-order ^model). In Fig.4 and Fig.5, the results obtained from the three models are compared with the actual data for n=7. From these figures, the advantages of the proposed approach are clearly shown for the third-order case. Better forecasting results are presented. Fig.6 shows predicted results for higher order model. It is found that better forecasts can be obtained from the higher order model. Fig.7 shows forecasts for different partitions(include ^{n=7, 11, 15, 19).} It demonstrates that the reasonable larger value partition have the effect of improving the forecasting. These could be also evidenced from Fig.8,9, which shows the root mean square error obtained from forecasting models. The RMSE reduces from 630.86 for the Chen method for n=7 to199.32 for the second-order model using n=19 in Fig. 8. And in Fig.9, the RMSE improved from 624.23 for the Song-Chissom method for n=7 to 421.32 for the fifth- order method using n=7 also. These results show a great improvement in forecasting. The feasibility of the current formulation of the high-order fuzzy time series model is confirmed.

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2. Forecasting population

As another ^case, forecasting popLilation of Taiwan area is carried out ^with the proposed high-order model. The results ^will be beneficial to conform of the improvement by the new scheme. The historical data are yearly records from Jan. 1973 to1991. Since these data are not stationary, which is increasing with the time and making the universe can not be defined and the meaningful partition can not be ^taken, thus they must be differentiated to the third difference at first. Then the time series are implemented for these stationary data. And the procedure follows:

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

Usually, the minimum value _X1,1 and the maximum value Xjnax are found and ^{used to define the universe of discourse U. In this case,} X,,1 ^{=-145 and} Xjnax =

105. For simplicity, the universe is chosen to be the interval of U=[-145,105].

2. Determine the number of fLizzy partition.

In this ^case, if n=5 is used then the universe is segmented into five intervals ^with eqLial range. ^Thus, _{u1, U2, U3,} u, and _u5 are applied to denote the five

intervals.

3. Define fuzzy set on the universe U. _A1 would be the linguistic variable of ^{records of population, which is corresponding to each} u, ^{i=1,2, ,5.}

4. Transfer the historical data into fuzzy set.

In this step, the equivalent fuzzy set to every year’s record is found. The memberships of each record to _A1 will be determined. The fuzzy sets to each year’s record are shown in Table 7 and each fuzzy set has five ^elements, which can be interpreted as the degree of each record belonging to each _4.

5. Setup the fuzzy relations based on historical knowledge.

Suppose that the maximum membership on one year’s record is located at

then the year’s record is viewed as _A,,. For example, the first record in Table ⁷ represents for the population in Taiwan area at ^1973, and the maximum membership is under _A4. ^Thus, we can view the first record as the linguistic _A4. Hence all the historical data can be transferred into linguistic expressions.

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