3. Formulation of high-order models

Assume F(t) is caused by F(t-1) F(t-2), and F(t-m) (m>O) simultaneously , then How to ^{relate the} n-tuple ^F(t-1) >< ^{F(t-2) X} >< ^F(t-m) to ^F(t)

1. the new scheme embedded into Chen method

As the new scheme applied to Chen ^method, a high-order ^{model, F(t)} is caused by ^F(t-1) and ^F(t-2) and ^... and ^F(t-m), that ^is, we may treat this as

F(t-1) ^—, F(t) and

F(t-2)

F(t) and

—

347

F(t-3)

F(t-m)

F(t) and. F(t)

—

—

The fuzzy logical relationships groups between F(i-1) and F(i) can be obtained ^{following the processes by} putting ^{together the propositions having the same antecedent} throughout all time ^t. Note that the fuzzy logical relationships groups between ^F(i-k) and ^F(i)can be obtained in the same fashion. Once all groups for ^{k=1,2,3, - - -,} m are ^obtained, for each ^forecast, they are applied simultaneously. Since all the relations should be satisfied simultaneously. Then we take the intersection of the consequent of the correspondent groups for ^{k=1,2,3, - -} -, m. The intersection is the output of this inference. To defuzzify the output, the principals has been follow

1. For the output having only one fuzzy ^set, the midterm of the correspondent interval is given as the forecast.

2. For the output having two or more fuzzy ^set, the average value of the midterm

of the correspondent intervals is given as the forecast.

About the computation amount of this proposed ^model, only inference and intersection process must be added once for each order. The total computation remains simple.

2. The new scheme embedded into Song-Chissom method

On the other ^hand, the high-order calculation procedure is also achieved by applied the new scheme to Song-Chissom method as follows. For the high-order ^{model, F(t)} is caused by ^{F(t-1), F(t-2), - - -} and ^F(t-m), we may treat this as

F(t-1)

—

F(t-2)

—

F(t-3)

—

F(t-rn)

—

^F(t) and ^F(t) and ^F(t) and. F(t)

In each projection of the above expression ^{F(t-k) —‘ F(t),} we relate them by the relationship ^matrice _Rmk. ^The first-order ^fuzzy relational function Rk(t) ^{can be} obtained by following the processes described in the above section. In the same ^fashion, we have

^{Rmk(t). By taking the union of Rmk(t) throughout all t, we can calculate} Rmk, ^which represents one of the fuzzy relational function of order m. Note that this relational function shows the relationship between F(i-k)and F(i). Once all _Rmk for ^{k=1,2,3,...,m}

are ^obtained, they are applied simultaneously to produce m possible values

=F1

o

r Jr

ik’ i-k ^“- k

‘

Y

D

C

Xi

349

4. RESULTS AND DISCUSSIONS

1. Forecasting Enrollments

In this paper, the historical records of enrollments of the University of Alabama shown in Song and ^{Chissom(1993, 1994)} are modeled and forecasted. The details will be beneficial to the understanding of the fuzzy calculation. The historical data are yearly records from 1971 to1992. The forecasting enrollments are mainly carried out with two time series as follows

1. Chen method ^/proposed high-order model ^(a new scheme embedded into Chen ^method)

The procedLires follows:

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

^{UsLially, the minimum valLie} X,,, ^{and the maximum valLie} X,,, ^{are found and used to define the universe of discourse U. In this case,} X,,, ^{=13000 and} _Xmax

=20000. For simplicity, the universe is chosen to be the interval of U=[

13^000,20000].

2. Determine the number of fuzzy partition.

In this ^case, for example, if n=7 is used and the universe is segmented into seven intervals with equal range then _{u1, u2, u3, u4, u5, u6} and _u7 are applied to denote the seven intervals.

3. Define fuzzy set on the universe U.

_A, would be the linguistic variables of records of ^enrollments, which is corresponding to each _u1

4. Transfer the historical data into fuzzy set.

In this step, the equivalent fuzzy set to the yearly record is found out. If the Maximum membership of one year’s enrollment is under _A , then the fuzzified enrollment for this year is treated as _A. Hence all the historical data can be

transferred into linguistic expressions, that is fuzzy sets.

5. Setup the fuzzy logical relations based on historical knowledge.

For all the historical ^data, representing by fuzzy ^set, the fuzzy logical relationships are found out. In this ^case, the consecutive yearly fuzzy sets served as antecedent and consequent of the fuzzy propositions for the first-order model. The two-step ahead yearly fuzzy sets served as antecedent and consequent of the additional fuzzy propositions for the second-order model. In the same fashion, the fuzzy propositions must include the three-step ahead consideration for the

350 ^Iflgfl

third-order model and the foLir-step ahead consideration for the fourth-order model and so on. Then the fuzzy logical relationship groups for the first-order model ^and high-order model ^are obtained by taking together ^{the fuzzy} propositions with the same antecedent.

6. Obtained forecasted output by approximate reasoning.

Once the correspondent fuzzy proposition, found in the fuzzy logical relationship group is ^determined, historical data can be used to obtain the output by inference. And the forecasted outputs are given in tables 1-3. Note that the fuzzy output is the intersection of the consequent in the corresponding fuzzy propositions

Table.1 Fuzzified historical enrollments and outputs of the models ⁽ⁿ⁼⁷⁾

Year	Actual enrollment	FLizzified enrollment	Ouipuis 1-order	Outputs 2-order	Ouiputs 3-order	Outpuis 4-order
1971	13055	Al
1972	13563	Al	A1A2
1973	13867	Al	A1A2	A1A2
1974	14696	A2	A1A2	A1A2	A2
1975	15460	A3	A3	A3	A3	A3
1976	15311	A3	A3A4	A3	A3	A3
1977	15603	A3	A3A4	A3A4	A3	A3
1978	15861	A3	A3A4	A3A4	A3A4	A3
1979	16807	A4	A3A4	A3A4	A3A4	A3A4
1980	16919	A4	A3A4A6	A3A4A6	A3A4A6	A3A4A6
1981	16388	A4	A3A4A6	A3A4A6	A3A4A6	A3A4A6
1982	15433	A3	A3A4A6	A3A4A6	A3	A3
1983	15497	A3	A3A4	A3A4	A3	A3
1984	15145	A3	A3A4	A3A4	A3	A3
1985	15163	A3	A3A4	A3A4	A3A4	A3
1986	15984	A3	A3A4	A3A4	A3A4	A3A4
1987	16859	A4	A3A4	A3A4	A3A4	A3A4
1988	18150	A6	A3A4A6	A3A4A6	A3A4A6	A3A4A6
1989	18970	A6	A6A7	A6	A6	A6
1990	19328	A7	A6A7	A7	A?	A7
1991	19337	A7	A6A7	A7	A?	A7
1992	18876	A6	A6A7	A6	A6	A6

4 kI41 rIrt I

^[P IJ1i

YTI

2. IL

141t

^I J 1

11t

-

Table.2 Fuzzified historical enrollments and outputs of the models ⁽ⁿ⁼¹⁴⁾

Year	Actual enrollment	Fuzzified enrollment	Outputs 1-order	Outputs 2-order	Outputs 3-order	Outputs 4-order
1971	13055	Al
1972	13563	A2	A2
1973	13867	A2	A2A4	A2
1974	14696	A4	A2A4	A4	A4
1975	15460	A5	A5	A5	A5	A5
1976	15311	A5	A5A6	A5	A5	A5
1977	15603	A6	A5A6	A5A6	A6	A6
1978	15861	A6	A6A8	A6A8	A6A8	A6
1979	16807	A8	A6A8	A8	A8	A8
1980	16919	A8	A7A8A11	A8A11	A8	A8
1981	16388	A7	A7A8A11	A7	A7	A7
1982	15433	A5	AS	A5	A5	AS
1983	15497	AS	ASA6	AS	A5	AS
1984	15145	A5	A5A6	A5A6	A5	A5
1985	15163	AS	ASA6	A5A6	ASA6	AS
1986	15984	A6	A5A6	A5A6	A5A6	A6
1987	16859	A8	A6A8	A6A8	A6A8	A6A8
1988	18150	All	A7A8A11	A8A11	A8A11	A8A11
1989	18970	A12	A12	A12	A12	A12
1990	19328	A13	A13	A13	A13	A13
1991	19337	A13	A12A13	A13	A13	A13
1992	18876	A12	A12A13	A12	A12	A12

r)

^J)L 119 lIIJ IY-

Table.3 Fuzzified historical enrollments and outputs of the models ⁽ⁿ⁼²⁸⁾

Year	Actual enrollment	Fuzzified enrollment	Outputs 1-order	Outputs 2-order	Outputs 3-order	Outputs 4-order
1971	13055	Al
1972	13563	A3	A3
1973	13867	A4	A4	A4
1974	14696	A7	A7	A7	A7
1975	15460	AlO	AlO	AlO	AlO	AlO
1976	15311	AlO	A9A1OA11	AlO	AlO	AlO
1977	15603	All	A9A1OA11	A9A11	All	All
1978	15861	A12	A12	A12	A12	A12
1979	16807	A16	A16	A16	A16	A16
1980	16919	A16	A14A16A21	A16A21	A16	A16
1981	16388	A14	A14A16A21	A14	A14	Al4
1982	15433	AlO	AlO	AlO	AlO	AlO
1983	15497	AlO	A9A1OA11	AlO	AlO	AlO
1984	15145	A9	A9A1OA11	A9A11	A9	A9
1985	15163	A9	A9A12	A9A12	A9A12	A9
1986	15984	A12	A9A12	A12	A12	A12
1987	16859	A16	A16	A16	A16	A16
1988	18150	A21	A14A16A21	A16A21	A16A21	A21
1989	18970	A24	A24	A24	A24	A24
1990	19328	A26	A26	A26	A26	A26
1991	19337	A26	A24A26	A26	A26	A26
1992	18876	A24	A24A26	A24	A24	A24

7. Transfer the output fuzzy set into forecasted records.

Since the output fuzzy ^results, they should be transfer into real numbers. The transformation presented by Chen ^[9] is used in this study. In table ^4-6, it is found that the forecasts of higher-order is closed to actual enrollments. And the more close the larger n. ^However,the optimization for n is studied by ^Tsai,

CC. using an adaptive partition[16]

4 kI41 rIrt I

^[P IJ1i

YTI

2. IL

141t

^I J 1

11t

-

Table.4 Fuzzified historical enrollments and forecasts of the models ⁽ⁿ⁼⁷⁾

Year	Actual enrollment	Forecasts 1-order	Forecasts 2-order	Forecasts 3-order	Forecasts 4-order
1971	13055
1972	13563	14000
1973	13867	14000	14000
1974	14696	14000	14000	14500
1975	15460	15500	15500	15500	15500
1976	15311	16000	15500	15500	15500
1977	15603	16000	16000	15500	15500
1978	15861	16000	16000	16000	15500
1979	16807	16000	16000	16000	16000
1980	16919	16833	16833	16833	16833
1981	16388	16833	16833	16833	16833
1982	15433	16833	16833	15500	15500
1983	15497	16000	16000	15500	15500
1984	15145	16000	16000	15500	15500
1985	15163	16000	16000	16000	15500
1986	15984	16000	16000	16000	16000
1987	16859	16000	16000	16000	16000
1988	18150	16833	16833	16833	16833
1989	18970	19000	18500	18500	18500
1990	19328	19000	19500	19500	19500
1991	19337	19000	19500	19500	19500
1992	18876	19000	18500	18500	18500