Fig. 1. Time Series c(t) and the partitions

Fig. 2. Construction of flat-top IT2FS for partition A3: (a) the close price, (b) Type-1 MFs for regions R1 and R2, c)IT2FS representation of (b), (d) Flat-top IT2FS obtained by joining the peaks of two lobes.

Fig. 3. Construction of Fuzzy Logical implication and their grouping under MFVS , VMF(t): (a) If

M i kc (t ) A and 1M i jc (t ) A ; then the rule is jk AA , (b) If 1M i sV (t + )= B , s ,then grouping is done as

skjAAB jks ,,,: in Fig. (b)

Fig. 4. Grouping of Fuzzy logical implications (FLI)

Fig. 5 Computation and partitioning of Composite Secondary Variation Series (CSVS)

Fig.7. Secondary to Main factor variation mapping: If ( )M kV t B ,then the mapping is written as kj BB :

Fig. 6 Composite Secondary Variation (CSV) to Main Factor Variation Series (MFV) mapping

Fig. 8. The main steps in the prediction algorithm of a stock index time-series

Fig. 9 Inference Generation with T1/ IT2 antecedent-consequent pairs

Fig. 10. Optimal Selection of MFs to minimize RMSE

Fig.11. Forecasted TAIEX of the months November and December 2003 using Gaussian MFs and Triangular MFs, (a) Actual TAIEX, forecasted data using proposed method1 without adopting the Gaussian MFs and forecasted data

using proposed method2 adopting the Gaussian MFs (b) Actual TAIEX, forecasted data using proposed method3 without adopting the triangular MFs and forecasted data using proposed method4 adopting the triangular MFs

The prediction was done for each trading day during the month of November and December. The

results of the prediction in comparison to those of Chen et al. [10] are shown in Fig 12, Fig. 13

and Fig. 14. Fig. 12 shows the prediction results (November 1 to December 31, 2004) of the

proposed approach designed by triangular MF without adaptation, considering the influence of

secondary index following Chen [10]. Fig. 13 shows the effect of secondary factor as introduced

in the proposed method. In Fig. 14, we report the forecasting result employing adaptation cycle

to the MFs.

Fig. 12. Three close price of TAIEX data of the months November and December are plotted, (a) actual TAIEX 2004, forecasted data using proposed method without adopting the triangular MFs (influence of secondary factor is considered as Chen[10]) and forecasted data using Chen et al.[10], b) RMSE and errors for proposed method, (c)

RMSE and errors for Chen et al.

Fig. 13. Three close price of TAIEX data of the months November and December (2004) are plotted, (a) actual TAIEX 2004, forecasted data using proposed method3 without adopting the triangular MFs and forecasted data using Chen [10], (b) RMSE and errors for proposed method3.The error plot for Chen method [10] is same as in

Fig. 11(c)

Fig. 14. Three close price of TAIEX data of the months November and December (2004) are plotted, (a)actual

TAIEX 2004, forecasted data using proposed method4 with adopting the triangular MFs and forecasted data using Chen [10], (b) RMSE and errors for proposed method4. The error plot for Chen method [10] is same as Fig. 11(c)

TABLE-I(A) MAIN FACTOR FUZZY LOGICAL IMPLICATION (FLI)

TABLE-I(B)

MAIN FACTOR FUZZY LOGICAL IMPLICATION (FLI) UNDER VARIATION GROUPS

Group Antecedent of Main Factor

A1 A2 A3… A5… … A18

B1 ... ... ... ... ...

. ... … ... ... ...

B5

22 AA 12 AA

23 AA 45 AA

. ... ... ... ... ...

B14 ... ... ... ... ...

Group

Time points

t1 ... ti+1… … tj+1 ... … … tk+1 … … … tl+1 ... tn

B1 .

… … … …

… … … … … … … … … … … … … … … … … … … …

B5 22 AA 23 AA 45 AA 12 AA

. B14

… … … …

… … … … … … … … … … … … … … … … … … … …

TABLE-II

FREQUENCY OF OCCURRENCE OF MAIN FACTOR VARIATION IN EACH GROUP OF SECONDARY FACTOR VARIATION

TABLE-III

Years Methods

1999 2000 2001 2002 2003 2004 MEAN

1.Huarng et al.[24 ](Using NASDAQ) Nil 158.7 136.49 95.15 65.51 73.57 105.88 2.Huarng et al.[24 ](Using Dow Jones) Nil 165.8 138.25 93.73 72.95 73.49 108.84 3.Huarng et al.[ 24](Using M1b) Nil 160.19 133.26 97.1 75.23 82.01 111.36 4.Huarng et al.[24](Using NASDAQ and M1b)

Nil 157.64 131.98 93.48 65.51 73.49 104.42

5.Huarng et al.[ 24](Using Dow Jones and M1b)

Nil 155.51 128.44 97.15 70.76 73.48 105.07

6.Huarng et al.[24 ] (Using NASDAQ,Dow Jones and M1b)

Nil 154.42 124.02 95.73 70.76 72.35 103.46

7.Chen et al.[ 6],[12],[18] 120 176 148 101 74 84 117.4 8.U_R Model[12 ],[18] 164 420 1070 116 329 146 374.2

COMPARISON OF RMSEs OBTAINED BY THE PROPOSED TECHNIQUE WITH EXISTING TECHNIQUES

TABLE-IV RESULTS OF STATISTICAL SIGNIFICANCE WITH THE PROPOSED METHOD 1 TO 4

AS THE REFERENCE, ONE AT A TIME Statistical significance

Existing Methods

Reference Methods Proposed method1

Proposed method2

Proposed method3

Proposed method4

1 + + + 2 + + + + 3 + + + + 4 + + + + 5 + + + + 6 + + + 7 + + + + 8 + + 9 10 + 11 + + + + 12 + + + + 13 14 15 + + + + 16 + + + + 17 + + +

9.U_NN Model[ 12],[18] 107 309 259 78 57 60 145.0 10.U_NN_FTS Model [12 ],[18],[23] 109 255 130 84 56 116 125.0 11.U_NN_FTS_S Model [12 ],[18],[23] 109 152 130 84 56 116 107.8 12.B_R Model[12 ],[18] 103 154 120 77 54 85 98.8 13.B_NN Model[12],[18] 112 274 131 69 52 61 116.4 14.B_NN_FTS Model [12],[18] 108 259 133 85 58 67 118.3 15.B_NN_FTS_S Model [ 12],[18] 112 131 130 80 58 67 96.4 16.Chen et al.[10] (Using Dow Jones) 115.47 127.51 121.98 74.65 66.02 58.89 94.09 17.Chen et al.[ 10] (Using NASDAQ) 119.32 129.87 123.12 71.01 65.14 61.94 95.07 18.Chen et al.[10 ] (Using M1b) 120.01 129.87 117.61 85.85 63.1 67.29 97.29 19.Chen et al.[10 ] ( Using NASDAQ and Dow Jones)

116.64 123.62 123.85 71.98 58.06 57.73 91.98

20.Chen et al.[ 10] (Using Dow Jones and M1b)

116.59 127.71 115.33 77.96 60.32 65.86 93.96

21.Chen et al.[10 ] (Using NASDAQ and M1b)

114.87 128.37 123.15 74.05 67.83 65.09 95.56

22.Chen et al.[10 ] (Using NASDAQ, Dow Jones and M1b)

112.47 131.04 117.86 77.38 60.65 65.09 94.08

23. Karnik-Mendel [33] induced stock prediction

116.60 128.46 120.62 78.60 66.80 68.48 96.59

24. Proposed Metod0 114.20 127.12 110.50 70.56 52.10 48.40 87.18 24. Proposed Method1 101.842 125.876 111.604 71.6678 52.771 50.1667 85.654 25. Proposed Method2 92.6657 108.183 105.510 66.9913 47.1108 43.83 77.382 26. Proposed Method3 94.6108 113.041 110.814 66.0441 48.7705 46.1796 79.910 27. Proposed Method4 89.0214 99.7658 101.714 58.3214 37.123 36.6006 70.424

18 + + + + 19 + + 20 + + + 21 + + + 22 + + 23 + + + +