TAIEX Forecasting Based on Fuzzy Time Series

and Fuzzy Variation Groups

Shyi-Ming Chen, Fellow, IEEE, and Chao-Dian Chen

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 1, FEBRUARY 2011

Abstract

• 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 andCalculate the weighted variations of SF

Secondary factor process

Preliminaries

Fuzzy logical relationship

• Let F(t − 1) = Ai and let F(t) = Aj .• The relationship between F(t − 1) and F(t)

can be denoted by the fuzzy logical relationship

• “Ai → Aj ”

Lefthand side (LHS)Righthand side (RHS)

• Assume that the following fuzzy logical relationships exist:

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

Method and Realization

• [Step 1]: Define the universe of discourse U, U = [Dmin −D1,Dmax + D2 ].

• [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 u1, u2, . . . , and u18, which are defined as follows:

• ui = [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 Vart 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 “A8 →A9 ”, and from Table V, we can see that the fuzzy variation of trading day 2004/1/5 is B9 .

• Therefore, we can group the fuzzy logical relationship “A8 → A9” into the FLRG “GroupB9 ”.

• [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 DifferSFi between the variation of the main factor MF of trading day t and the variation of the elementary secondary factor SFi of trading day t − 1, where 1 ≤ i ≤ n is shown as follows:

• Calculate the weighted variations WVSF1 ,WVSF2 ,WVSF3 ,. . . , andWVSFn of the elementary secondary factors SF1 , SF2 ,SF3, . . . , and SFn , 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 Vars of the secondary factor “the Dow Jones and the NASDAQ” of trading day 2004/1/5,

• Fuzzify the variation Vart 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 Bx and the fuzzy variation of the secondary factor at trading day t − 1 is Bz , then put the fuzzy variation “Bx” of the main factor of trading day t into the fuzzy variation group “Group Bz ”.

Group Bz | Bx

• 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 B9 and that the fuzzy variation of the main factor of trading day 2004/1/6 is B8 . 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 B9 | B8

• Analyze the fuzzy variations appearing in each fuzzy variation group obtained in last step for

different situations.• For example, from Group B6 shown in Table IX,

we can see that the fuzzy variations appearing in the fuzzy variation group “GroupB6”

• We can see that S = 6.• The fuzzy variations BM when M <S are

B1,B5,B5,B3 , and B4 , where the total number of fuzzy variations is 5 (i.e., B6,1 = 5)

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

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

• [Step 7] Calculate the weights WB6,1,WB6,2 , and WB6,3 of Bs,1,Bs,2 , and Bs,3 , respectively.

• [Step 8] Perform forecasting.• 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 B8 .

• 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 B8,1 = 0.493671,B8,2 = 0.316456, andB8,3 = 0.189873.

• Because the minimum value, the middle point, and the maximum value of the interval u5 obtained in Step 1 are 5700, 5750, and 5800,

• that the weighted value u5* is calculated as

follows: 5700 × 0.493671 + 5750 × 0.316456 +5800 × 0.189873 = 5734.81 (i.e., u5

* = 5734.81).

• In the same way, the interval u4 obtained in Step 1 are 5600,5650, and 5700,

• Weighted value u4* is calculated as follows:

5600 × 0.493671 +5650 × 0.316456 + 5700 × 0.189873 = 5634.81 (i.e., u4

* =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:

EXPERIMENTAL RESULTS