A framework for type 2 fuzzy time series models. K. Huarng and H.-K. Yu Feng Chia University, Taiwan

Transcription

1 A framework for type 2 fuzzy time series models K. Huarng and H.-K. Yu Feng Chia University, Taiwan 1

2 Outlines Literature Review Chen s Model Type 2 fuzzy sets A Framework Empirical analysis Conclusion 2

3 Literature Review 3

4 Why Fuzzy Time Series Time series Stock index (open, close, high, low, average) Temperature (high, low, average) A need to model multiple values for any time t. 4

5 Fuzzy Time Series Models Tanaka et al. - linear programming to solve problems in fuzzy regression. Watada - fuzzy regression to solve the problems of fuzzy time series. Tseng et al. - fuzzy regression for autoregressive integrated moving average (ARIMA) analyses. 5

6 Fuzzy Time Series Models Song and Chissom (1993a, b, 1994) - defined fuzzy time series and proposed methods to model fuzzy relationships among observations. S.-M. Chen (1996) S.-M. Chen, and J.R. Hwang (2000) K. Huarng (2001a, 2001b) K. Huarng and H.-K. Yu (2003, 2004) R. Hwang, S.-M. Chen, and C.-H. Lee (1998) H.T. Nguyen, B. Wu (2000) 6

7 Applications Enrollment Stock index Temperature Some were shown to outperform their traditional counterpart models 7

8 Type 2 Fuzzy Set Models R.I. John, P.R. Innocent, M.R. Barnes (1998) N.N. Karnik, J.M. Mendel (1999) J.M. Mendel (2000) M. Wagenknecht, K. Hartmann (1988) R.R. Yager (1980) 8

9 Applications of Type 2 Fuzzy Sets Decision making Data processing Survey processing Time series modeling Fuzzy relation equations 9

10 Characteristics (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power Motivation 1: Apply Type 2 to improve fuzzy time series forecasting Type 2 fuzzy sets require complicated calculations Motivation 2: Apply Type 2 concept only Why Type 2 fuzzy sets are not so popular 10

11 Chen s Model 11

12 Chen, 1996 (1) Define the universe of discourse and the intervals, (2) Define the fuzzy sets, (3) Fuzzify the data, (4) Establish fuzzy logical relationships, (5) Establish fuzzy logical relationship groups, (6) Forecast, (7) Defuzzify the forecasting results. 12

13 Two major processes Steps 1 3: fuzzification, lengths of intervals Steps 4 5: fuzzy relationships 13

14 Enrollment forecasting University of Alabama Data from 1979 to

15 Step 1. Defining the universe of discourse and the intervals As in [1], U [13000, 20000]; the length of the intervals is Hence, there are intervals u1, u2, u3, u4, u5, u6, u7, where u1 =[13000, 14000], u2 [14000, 15000], u3 [15000, 16000], u4 [16000, 17000], u5 [17000, 18000], u6 [18000, 19000], u7 [19000, 20000]. 15

16 Step 2. Defining the fuzzy sets Ai The linguistic variable is enrollment; Ai(i=1, 2,...) as possible linguistic values of enrollment. Each Ai is defined by the intervals u1, u2, u3,..., u7. A1=1/u1+0.5/u2+0/u3+0/u4+0/u5+0/u6+0/u7 A2=0.5/u1+1/u2+0.5/u3+0/u4+0/u5+0/u6+0/u7 A3=0/u1+0.5/u2+1/u3+0.5/u4+0/u5+0/u6+0/u7 A4=0/u1+0/u2+0.5/u3+1/u4+0.5/u5+0/u6+0/u7 A5=0/u1+0/u2+0/u3+0.5/u4+1/u5+0.5/u6+0/u7 A6=0/u1+0/u2+0/u3+0/u4+0.5/u5+1/u6+0.5/u7 A7=0/u1+0/u2+0/u3+0/u4+0/u5+0.5/u6+1/u7 16

17 Year Enrollment A1 A2 A3 A4 A5 A6 A F(1971) = (1, 0.5, 0, 0, 0, 0, 0) F(1972) = (1, 0.8, 0.3, 0, 0, 0, 0) F(1973) = (1, 0.9, 0.4, 0, 0, 0, 0), etc. 17

18 Year Enrollment Fuzzy Enrollment Ai A A A A A3 18

19 Step 4. Establishing the fuzzy logical relationships (FLRs) A1 A1 A2 A3 A3 A4 A4 A3 A6 A6 A7 A7 A1 A2 A3 A3 A4 A4 A4 A6 A6 A7 A7 A6 19

20 Step 5. Establish fuzzy logic relationship groups (FLRGs) An FLRG is established by FLRs with the same LHSs. For example, there are FLRs A 1 A 1, A 1 A 2 These FLRs can be grouped together as an FLRG: A 1 A 1, A 2 20

21 Step 6. Forecast If A i s FLRG is empty (A i ), the forecast for the next observation, F(t) = A i. (1) If A i s FLRG is A i A j1, A j2,, A jk, the forecast for F(t) = A j1, A j2,, A jk. (2) 21

22 Step 7. Defuzzifying Suppose F(t-1) = A j. The defuzzified forecast of F(t) is calculated as follows. Rule 1. If the FLRG of A j is empty; i.e., A j, the defuzzified forecast of F(t) is m j, the midpoint of u j. 22

23 Step 7. Defuzzifying Rule 2. If the FLRG of A j is one-to-many; i.e., A j A p1, A p2,..., A pk, the forecast of F(t) is equal to the average of m p1, m p2,..., m pk, the midpoints of u p1, u p2,..., u pk, respectively. Forecast = k i =1 m k pi 23

24 Step 7. Defuzzifying [1972, 1973, 1974]: the forecasts of 1972, 1973, and 1974 are all equal to the arithmetic average of the mid points of u 1 and u 2 : ( )/2=

25 Type 2 fuzzy sets 25

26 x 26

27 x 27

28 x 28

29 x 29

30 George J. Klir, Bo Yuan (1995) A(x) a x 30

31 George J. Klir, Bo Yuan (1995) y 1 A(x) y I a (y) 0 a b x I b (y) 31

32 A Framework 32

33 Rationale (1) Apply Type 2 s expressive power to utilize extra information to improve forecasting For example, in Type 1 fuzzy time series forecasting of TAIEX, only closing prices are considered. However, in Type 2 fuzzy time series models, we may utilize high and low prices. 33

34 Rationale (2) Lower bound - conservative Upper bound - optimistic 34

35 Conservative x 35

36 Optimistic x 36

37 Rationale (3) Conservative Intersection operation Optimistic Union operation 37

38 Rationale (4) Conservative - To refine Type 1 fuzzy relationships Optimistic - To include more information in the Type 1 fuzzy relationships 38

39 Premise Suppose at t -1, HIGH=A j, LOW=A k Suppose F(t-1) = A i Type 1 FLRGs A i A x1, A x2, A x3,, A xp A j A y1, A y2, A y3,, A yq A k A z1, A z2, A z3,, A zr 39

40 Intersection Conservative = Intersection = {A x1, A x2, A x3,, A xp } {A y1, A y2, A y3,, A yq } {A z1, A z2, A z3,, A zr } = forecast If Intersection = Then the forecast is set to A i 40

41 Union Optimistic = Union = {A x1, A x2, A x3,, A xp } {A y1, A y2, A y3,, A yq } {A z1, A z2, A z3,, A zr } = forecast If upper bound = Then the forecast is set to A i 41

42 Empirical Analysis 42

43 Data TAIEX from 2000 to Jan Oct.: estimation Nov. Dec.: forecasting Daily closing, high, low prices 43

44 Setup Lengths of intervals is set to 100. Root mean squared errors (RMSEs) are used to evaluate forecasting results. 44

45 Date TAIEX Fuzzy Sets 2000/10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A /10/ A 10 45

46 Fuzzy Logic Relationships A 15 A 16, A 16 A 14, A 14 A 15, A 15 A 18 A 18 A 18, A 18 A 17, A 17 A 15, A 15 A 13 A 13 A 13, A 13 A 11, A 11 A 12, A 12 A 9 A 9 A 5, A 5 A 9, A 9 A 10, A 10 A 11 A 11 A 14, A 15 A 14, A 14 A 13, A 11 A 10 46

47 FLRGs A 5 A 9 A 9 A 5, A 10 A 10 A 11 A 11 A 12, A 14, A 10 A 12 A 9 A 13 A 13, A 11 A 14 A 13, A 15 A 15 A 16, A 18, A 13, A 14 A 16 A 14 A 17 A 15 A 18 A 18, A 17 47

48 Data for Forecasting Date Closing High Low 11/ /A /A /A12 11/ /A /A /A13 11/ /A /A /A14 48

49 Intersection Date FLRG Intersection 11/8 A13 --> A11, A13 A13 A13 --> A11, A13 A12 --> A9 11/9 A15 -->A13, A14, A16, A18 A13, A14, A16 A16 --> A15, A14 A13 --> A11, A13 11/10 A15 -->A13, A14, A16, A18 A13 A15 -->A13, A14, A16, A18 A14 --> A15, A13 49

50 Union Date FLRG Union 11/8 A13 --> A11, A13 A9, A11, A13 A13 --> A11, A13 A12 --> A9 11/9 A15 -->A13, A14, A16, A18 A11, A13, A14, A15, A16, A18 A16 --> A15, A14 A13 --> A11, A13 11/10 A15 -->A13, A14, A16, A18 A13, A14, A15, A16, A18 A15 -->A13, A14, A16, A18 A14 --> A15, A13 50

51 Forecasts (intersection) The forecast for 11/8 is A13 The forecast for 11/9 is A13, A14, and A16 The forecast for 11/10 is A13. 51

52 Forecasts (union) The forecast for 11/8 is A9, A11, and A13 The forecast for 11/9 is A11, A13, A14, A15, A16, and A18 The forecast for 11/10 is A13, A14, A15, A16, and A18 52

53 Date Actual Type 1 Intersection Union 11/ / / / / / / /

54 11/ / / / / / / / /

55 11/ / / / / /

56 56

57 Findout The forecast from the Intersection may not necessarily be lower than that of the Union Type 1 forecasts may not fall between those of the Intersection and the Union 57

58 Calculations Average 1 = (Intersection+Union)/2 Average 2 = (Type 1 + Intersection + Union)/3 58

59 Type 1 Intersection Union Average 1 Average

60 Conclusion 60

61 Conclusion Applying Type 2 fuzzy sets to utilize extra information A framework for applying Type 2 fuzzy time series models 61

62 Conclusion Lower and upper bounds Conservative and optimistic Intersection and union operations 62

63 Conclusion TAIEX used as the forecasting target Based on RMSEs, type 2 fuzzy time series models perform better than their type 1 counterparts (Chen model) in most cases. 63

64 Discussion 64