Shyi-Ming Chen, Fellow, IEEE, and
Chao-Dian Chen
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2011

• First , the proposed method fuzzifies the historical training data of the TAIEX into fuzzy sets to form fuzzy logical relationships.
• Second , it groups the fuzzy logical relationships into fuzzy logical relationship groups (FLRGs) based on the fuzzy variations of the secondary factor.

• Third ,it evaluates the leverage of the fuzzy variations between the main factor and the secondary factor to construct fuzzy variation groups.
• Fourth ,it gets the statistics of the fuzzy variations appearing in each fuzzy variation group.

• Fifth , it calculates the weights of the statistics of the fuzzy variations appearing in each fuzzy variation group, respectively.
• Finally , based on the weights of the statistics of the fuzzy variations appearing in the fuzzy variation groups and the FLRGs, it performs the forecasting of the daily TAIEX.

Main factor process

Calculate the total difference variation and
Calculate the weighted variations of SF
Secondary factor process

• Let F(t − 1) = A i and let F(t) = A j
.
• The relationship between F(t − 1) and F(t) can be denoted by the fuzzy logical relationship
• “A i
→ A j
”
Righthand side (RHS)
Lefthand side (LHS)

• Assume that the following fuzzy logical relationships exist:

• Then, these fuzzy logical relationships can be grouped into a FLRG, shown as follows:

• [Step 1]:
Define the universe of discourse U,
U = [D min
−D
1
,D max
+ D
2
].
• [Step 2]:
Define the linguistic terms A1 ,A2 , . . .,An

• The minimum value and the maximum value of the training data of the TAIEX of the year
2004 are 5316.87
and 7034.1
, respectively.
• Let D1 = 16.87, and let D2 = 65.9.
• The universe of discourse U = [5300, 7100] .

• U can be divided into 18 intervals u
1
, u
2
, . . . , and u
18
, which are defined as follows:
• u i
= [5300 + (i − 1) × 100, 5300 + i × 100)] where i = 1, 2, . . . , 18.

• [Step 3] Fuzzify each historical datum of the main
factor into a fuzzy set defined in Step 2.
• [Step 4] Based on the results of Step 3, because
the fuzzified TAIEX of trading day 2004/1/2 is A8
and because the fuzzified TAIEX of trading day
2004/1/5 is A9 , we can construct the following fuzzy logical relationship :
A8 → A9 .

• [Step 5] Fuzzify the variations between the
adjacent historical data of the main factor, respectively, and then group the fuzzy logical relationships of the main factor into FLRGs.

• The variation Var t of trading day t is calculated as follows:

• Define the universe of discourse V and the
intervals in V .

• Based on Table IV, we can define the linguistic terms B1,B2, . . . , andB14 represented by fuzzy sets, which are shown as follows:

• Group the fuzzy logical relationships having the same fuzzy variation at the LHS into a
FLRG.
• For example,from Table I, we can see that the fuzzy logical relationship from trading day
2004/1/2 to trading day 2004/1/5 is “ A
8
→A
9
”, and from Table V, we can see that the fuzzy variation of trading day 2004/1/5 is B
9
.
• Therefore, we can group the fuzzy logical relationship “ A
8
→ A
9
” into the FLRG
“ GroupB
9
”.

• [Step 6] Fuzzify the variations of the secondary
factor of the training data and analyze the fuzzy variations appearing in the fuzzy variation groups obtained in Step 5.

• Calculate the total different variation Differ
SF i between the variation of the main factor MF of trading day t and the variation of the elementary secondary factor SF i
of trading day t − 1, where 1 ≤ i ≤ n is shown as follows:

• Calculate the weighted variations
WV
SF
1
,WV
SF
2
,WV
SF
3
,. . . , andWV
SF n of the elementary secondary factors SF
1
,
SF
2
,SF
3
, . . . , and SF n
, respectively, where

• Calculate the total difference variation between the main factor “TAIEX” and the elementary secondary factor “Dow Jones” from 2004/1/5 to 2004/10/29, which is shown as follows:

• Calculate the total difference variation between the main factor “TAIEX” and the elementary secondary factor “NASDAQ” from
2004/1/5 to 2004/10/29, which is shown as follows:

• Calculate the normalized weight of the elementary secondary factor “ Dow Jones ”
• 2.129032/(2.129032 + 1.885714) =
0.53030303
• Calculate the normalized weight of the secondary factor “ NASDAQ ”
• 1.885714/(2.129032 +1.885714) =
0.46969697

• Use “the Dow Jones and the NASDAQ” as the secondary factor to fuzzify the variations
• Calculate the variation Var s of the secondary
factor “the Dow Jones and the NASDAQ” of trading day 2004/1/5,

• Fuzzify the variation Var t of the secondary
factor of each trading date t obtained in Step 6
into a fuzzy variation represented by a fuzzy set defined in Step 5.

• Based on the fuzzy variations of the secondary factor obtained in last step, if the fuzzy variation of the main factor at trading day t is
B x and the fuzzy variation of the secondary factor at trading day t − 1 is B z
, then put the
fuzzy variation “ B x
” of the main factor of
trading day t into the fuzzy variation group
“ Group B z
”.
Group B z
| B x

• For example, from Tables V and VIII, we can see that the fuzzy variation of the secondary factor of trading day 2004/1/5 is B
9 and that the fuzzy variation of the main factor of trading day 2004/1/6 is B
8
. Therefore, we put
the fuzzy variation “ B8 ” of the main factor of
trading day 2004/1/6 into the fuzzy variation group “ Group B9 ”.
Group B
9
| B
8

• Analyze the fuzzy variations appearing in each fuzzy variation group obtained in last step for different situations.
• For example, from Group B
6 shown in Table IX,
we can see that the fuzzy variations appearing in the fuzzy variation group “GroupB
6
”
• We can see that S = 6.
• The fuzzy variations B
M when M <S are
B
1
,B
5
,B
5
,B
3
, and B
4
, where the total number
of fuzzy variations is 5 (i.e., B
6,1
= 5)

• The fuzzy variations B
M when M = S, the total number of fuzzy variations is 6
(i.e.,B6,2 = 6)
• The fuzzy variations B
M when M >S, the total number of fuzzy variations is 23
(i.e., B6,3 = 23)

• [Step 7] Calculate the weights W
B6,1
,W
B6,2
, and
W
B6,3 of Bs,1,Bs,2 , and Bs,3 , respectively.

.
• Assume that we want to forecast the TAIEX of trading day 2004/11/2 by using the first-order fuzzy logical relationships.
• We can see that the TAIEX of trading day
2004/11/1 is 5656.17.
• Then, the value “5656.17” is fuzzified into the fuzzy set A4 .

• From Table III, we can see that the variations of the Dow Jones and the NASDAQ of trading day 2004/11/1 are 0.268463% and 0.247090%, respectively. Based on Steps 6 , we can see that the variation of the secondary factor “the
Dow Jones and the NASDAQ” is equal to
0.268463% × 0.53030303
+ 0.247090% ×
0.46969697
= 0.258424%.

• We can see that the variation (i.e., 0.258424%) of the secondary factor “the Dow Jones and the NASDAQ” is fuzzified into the fuzzy variation represented by the fuzzy set B
8
.
• Therefore, we choose the fuzzy variation group “Group B8” shown in Table X and
choose the fuzzy logical relationship in
“Group B8” of Table VI , whose LHS is A4 , i.e., we choose the fuzzy logical relationship
“ A4 →A5,A4,A4,A4,A5 ” appearing in Group B8 of Table VI.

• See that the weights B
8,1
= 0.493671,
B
8,2
= 0.316456, andB
8,3
= 0.189873.
• Because the minimum value, the middle point, and the maximum value of the interval u
5 obtained in Step 1 are 5700, 5750 , and 5800 ,
• that the weighted value u
5 follows:
* is calculated as
5700 × 0.493671 + 5750 × 0.316456
+5800 × 0.189873 = 5734.81
(i.e., u
5
* = 5734.81).

• In the same way, the interval u
4 obtained in
Step 1 are 5600 , 5650 , and 5700 ,
• Weighted value u
4
*
5600 is calculated as follows:
× 0.493671 +5650 × 0.316456 +
5700 × 0.189873 = 5634.81
(i.e., u
4
* =5634.81)

• The fuzzy logical relationship
“ A4 →A5,A4,A4,A4,A5 ”
• That the forecasted TAIEX of trading day
2004/11/2 is calculated as follows: