A framework for type 2 fuzzy time series models K. Huarng and H.-K. Yu Feng Chia University, Taiwan 1
Outlines Literature Review Chen s Model Type 2 fuzzy sets A Framework Empirical analysis Conclusion 2
Literature Review 3
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
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
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
Applications Enrollment Stock index Temperature Some were shown to outperform their traditional counterpart models 7
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
Applications of Type 2 Fuzzy Sets Decision making Data processing Survey processing Time series modeling Fuzzy relation equations 9
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
Chen s Model 11
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
Two major processes Steps 1 3: fuzzification, lengths of intervals Steps 4 5: fuzzy relationships 13
Enrollment forecasting University of Alabama Data from 1979 to 1991 14
Step 1. Defining the universe of discourse and the intervals As in [1], U [13000, 20000]; the length of the intervals is 1000. 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
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
Year Enrollment A1 A2 A3 A4 A5 A6 A7 1971 13055 1 0.5 0 0 0 0 0 1972 13563 1 0.8 0.3 0 0 0 0 1973 13867 1 0.9 0.4 0 0 0 0 1974 14696 0.8 1 0.8 0.3 0 0 0 1975 15460 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
Year Enrollment Fuzzy Enrollment Ai 1971 13055 A1 1972 13563 A1 1973 13867 A1 1974 14696 A2 1975 15460 A3 18
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
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
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
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
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
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 : (13500+14500)/2=14000 24
Type 2 fuzzy sets 25
1.0 0.5 0.0 2 x 26
1.0 0.6 0.5 0.5 0.4 0.0 2 x 27
1.0 0.6 0.5 0.5 0.4 0.0 2 x 28
1.0 0.5 0.0 2 x 29
George J. Klir, Bo Yuan (1995) 1 0.5 A(x) 2 1 0 a x 30
George J. Klir, Bo Yuan (1995) y 1 A(x) 4 3 4 4 1 3 2 2 1 y I a (y) 0 a b x I b (y) 31
A Framework 32
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
Rationale (2) Lower bound - conservative Upper bound - optimistic 34
Conservative 1.0 0.5 0.0 2 x 35
Optimistic 1.0 0.5 0.0 2 x 36
Rationale (3) Conservative Intersection operation Optimistic Union operation 37
Rationale (4) Conservative - To refine Type 1 fuzzy relationships Optimistic - To include more information in the Type 1 fuzzy relationships 38
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
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
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
Empirical Analysis 42
Data TAIEX from 2000 to 2003. Jan Oct.: estimation Nov. Dec.: forecasting Daily closing, high, low prices 43
Setup Lengths of intervals is set to 100. Root mean squared errors (RMSEs) are used to evaluate forecasting results. 44
Date TAIEX Fuzzy Sets 2000/10/11 6040.55 A 15 2000/10/21 5599.74 A 10 2000/10/2 6024.07 A 15 2000/10/12 5805.01 A 13 2000/10/23 5680.95 A 11 2000/10/3 6143.44 A 16 2000/10/13 5876.11 A 13 2000/10/24 5918.63 A 14 2000/10/4 5997.92 A 14 2000/10/16 5630.95 A 11 2000/10/25 6023.78 A 15 2000/10/5 6029.65 A 15 2000/10/17 5702.36 A 12 2000/10/26 5941.85 A 14 2000/10/6 6353.67 A 18 2000/10/18 5432.23 A 9 2000/10/27 5805.17 A 13 2000/10/7 6352.03 A 18 2000/10/19 5081.28 A 5 2000/10/30 5659.08 A 11 2000/10/9 6209.42 A 17 2000/10/20 5404.78 A 9 2000/10/31 5544.18 A 10 45
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
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
Data for Forecasting Date Closing High Low 11/7 5877.77/A13 5877.77/A13 5720.89/A12 11/8 6067.94/A15 6164.62/A16 5889.01/A13 11/9 6089.55/A15 6089.55/A15 5926.64/A14 48
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
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
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
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
Date Actual Type 1 Intersection Union 11/2 5626.08 5300 5450 5416.67 11/3 5796.08 5750 5650 5750 11/4 5677.3 5450 5750 5700 11/6 5657.48 5750 5650 5750 11/7 5877.77 5750 5650 5675 11/8 6067.94 5750 5850 5650 11/9 6089.55 6075 5983.33 6000 11/10 6088.74 6075 5850 6070 53
11/13 5793.52 6075 5950 6070 11/14 5772.51 5450 5750 5650 11/15 5737.02 5450 5750 5650 11/16 5454.13 5450 5750 5766.67 11/17 5351.36 5300 5450 5416.67 11/18 5167.35 5350 5350 5350 11/20 4845.21 5150 5150 5150 11/21 5103 4850 4850 5450 11/22 5130.61 5150 5150 5150 54
11/23 5146.92 5150 5150 5150 11/24 5419.99 5150 5150 5150 11/27 5433.78 5300 5450 5300 11/28 5362.26 5300 5450 5416.67 11/29 5319.46 5350 5350 5300 11/30 5256.93 5350 5350 5350 55
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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
Calculations Average 1 = (Intersection+Union)/2 Average 2 = (Type 1 + Intersection + Union)/3 58
Type 1 Intersection Union Average 1 Average 2 2000 176.32 131.86 175.47 139.39 143.52 2001 147.84 159.68 138.37 144.15 141.05 2002 100.62 79.6 89.17 82.56 83.13 2003 74.46 73.03 76.65 73.26 70.92 59
Conclusion 60
Conclusion Applying Type 2 fuzzy sets to utilize extra information A framework for applying Type 2 fuzzy time series models 61
Conclusion Lower and upper bounds Conservative and optimistic Intersection and union operations 62
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
Discussion 64