a framework for type 2 fuzzy time series models k. huarng...
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A framework for type 2 fuzzy time series models
K. Huarng and H.-K. YuFeng Chia University, Taiwan
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Outlines
Literature ReviewChen’s ModelType 2 fuzzy setsA FrameworkEmpirical analysisConclusion
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Literature Review
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Why Fuzzy Time Series
Time seriesStock index (open, close, high, low, average)Temperature (high, low, average)A need to model multiple values for any time t.
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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.
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Fuzzy Time Series ModelsSong 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)J. Sullivan, and W.H. Woodall (1994)
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Applications
Enrollment Stock index Temperature Some were shown to outperform their traditional counterpart models
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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)
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Applications of Type 2 Fuzzy Sets
Decision makingData processingSurvey processingTime series modelingFuzzy relation equations
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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 onlyWhy Type 2 fuzzy sets are not so popular
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Chen’s Model
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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.
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Two major processes
Steps 1 – 3: fuzzification, lengths of intervals
Steps 4 – 5: fuzzy relationships
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Enrollment forecasting
University of Alabama
Data from 1979 to 1991
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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].
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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/u7A3=0/u1+0.5/u2+1/u3+0.5/u4+0/u5+0/u6+0/u7A4=0/u1+0/u2+0.5/u3+1/u4+0.5/u5+0/u6+0/u7A5=0/u1+0/u2+0/u3+0.5/u4+1/u5+0.5/u6+0/u7A6=0/u1+0/u2+0/u3+0/u4+0.5/u5+1/u6+0.5/u7A7=0/u1+0/u2+0/u3+0/u4+0/u5+0.5/u6+1/u7
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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.
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Year Enrollment Fuzzy Enrollment Ai
1971 13055 A1
1972 13563 A1
1973 13867 A1
1974 14696 A2
1975 15460 A3
…
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Step 4. Establishing the fuzzy logical relationships (FLRs)
A1→A1 A1→A2
A2→A3 A3→A3
A3→A4 A4→A4
A4→A3 A4→A6
A6→A6 A6→A7
A7→A7 A7→A6
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Step 5. Establish fuzzy logic relationship groups (FLRGs)
An FLRG is established by FLRs with the same LHSs. For example, there are FLRs
A1 → A1, A1 → A2
These FLRs can be grouped together as an FLRG:
A1 → A1, A2
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Step 6. Forecast
If Ai‘s FLRG is empty (Ai → ), the forecast for the next observation,F(t) = Ai. (1)
If Ai‘s FLRG is Ai → Aj1, Aj2, …, Ajk, the forecast for F(t) = Aj1, Aj2, …, Ajk. (2)
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Step 7. Defuzzifying
Suppose F(t-1) = Aj. The defuzzifiedforecast of F(t) is calculated as follows.
Rule 1. If the FLRG of Aj is empty; i.e., Aj →, the defuzzified forecast of F(t) is mj, the midpoint of uj.
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Step 7. Defuzzifying
Rule 2. If the FLRG of Aj is one-to-many; i.e., Aj→ Ap1, Ap2, ..., Apk, the forecast of F(t) is equal to the average of mp1, mp2, ..., mpk, the midpoints of up1, up2, ..., upk, respectively.
Forecast = k
mk
ipi∑
=1
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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 u1 and u2:
(13500+14500)/2=14000
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Type 2 fuzzy sets
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1.0
0.5
0.0
2
μ
x
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1.0
0.5
0.0
2
μ
x
0.6
0.5
0.4
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1.0
0.5
0.0
2
μ
x
0.6
0.5
0.4
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1.0
0.5
0.0
2
μ
x
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George J. Klir, Bo Yuan (1995)
0
1
0.5
α1
α2
a x
A(x)
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George J. Klir, Bo Yuan (1995)
a bx0
1
Α4
α4
α3
α2
α1
β1
β4
β3
β2
I a(y) I b(y)
y
y
A(x)
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A Framework
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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.
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Rationale (2)
Lower bound - conservativeUpper bound - optimistic
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Conservative
1.0
0.5
0.0
2
μ
x
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Optimistic1.0
0.5
0.0
2
μ
x
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Rationale (3)
Conservative – Intersection operation
Optimistic – Union operation
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Rationale (4)
Conservative - To refine Type 1 fuzzy relationships
Optimistic - To include more information in the Type 1 fuzzy relationships
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Premise
Suppose at t -1, HIGH=Aj , LOW=Ak
Suppose F(t-1) = Ai
Type 1 FLRGsAi→ Ax1, Ax2, Ax3, …, Axp
Aj→ Ay1, Ay2, Ay3, …, Ayq
Ak→ Az1, Az2, Az3, …, Azr
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Intersection
Conservative = Intersection = {Ax1, Ax2, Ax3, …, Axp} ∩ {Ay1, Ay2, Ay3, …, Ayq} ∩ {Az1,Az2, Az3, …, Azr} = forecast
If Intersection = ∅Then the forecast is set to Ai
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Union
Optimistic = Union = {Ax1, Ax2, Ax3, …, Axp} ∪ {Ay1, Ay2, Ay3, …, Ayq} ∪ {Az1,Az2, Az3, …, Azr} = forecast
If upper bound = ∅Then the forecast is set to Ai
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Empirical Analysis
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Data
TAIEX from 2000 to 2003.Jan – Oct.: estimationNov. – Dec.: forecastingDaily closing, high, low prices
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Setup
Lengths of intervals is set to 100.
Root mean squared errors (RMSEs) are used to evaluate forecasting results.
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Date TAIEX Fuzzy Sets
… … …
2000/10/2 6024.07 A15
2000/10/3 6143.44 A16
2000/10/4 5997.92 A14
2000/10/5 6029.65 A15
2000/10/6 6353.67 A18
2000/10/7 6352.03 A18
2000/10/9 6209.42 A17
2000/10/11 6040.55 A15
2000/10/12 5805.01 A13
2000/10/13 5876.11 A13
2000/10/16 5630.95 A11
2000/10/17 5702.36 A12
2000/10/18 5432.23 A9
2000/10/19 5081.28 A5
2000/10/20 5404.78 A9
2000/10/21 5599.74 A10
2000/10/23 5680.95 A11
2000/10/24 5918.63 A14
2000/10/25 6023.78 A15
2000/10/26 5941.85 A14
2000/10/27 5805.17 A13
2000/10/30 5659.08 A11
2000/10/31 5544.18 A10
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Fuzzy Logic RelationshipsA15→ A16, A16→ A14, A14 → A15, A15→ A18
A18→ A18, A18→ A17, A17→ A15, A15→ A13
A13→ A13, A13→ A11, A11→ A12, A12→ A9
A9→ A5, A5→ A9, A9→ A10, A10→ A11
A11→ A14, A15→A14, A14→ A13, A11→ A10
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FLRGsA5 → A9
A9 → A5, A10
A10→ A11
A11 → A12, A14, A10
A12 → A9
A13 → A13, A11
A14 → A13, A15
A15 → A16, A18, A13, A14
A16 → A14
A17 → A15
A18 → A18, A17
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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
… … … …
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IntersectionDate 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
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UnionDate 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
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Forecasts (intersection)
The forecast for 11/8 is A13The forecast for 11/9 is A13, A14, and A16The forecast for 11/10 is A13.
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Forecasts (union)
The forecast for 11/8 is A9, A11, and A13The forecast for 11/9 is A11, A13, A14, A15, A16, and A18The forecast for 11/10 is A13, A14, A15, A16, and A18
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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
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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
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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
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4500
4700
4900
5100
5300
5500
5700
5900
6100
6300
2000/11/2
2000/11/9
2000/11/16
2000/11/232000/11/302000/12/7
2000/12/14
2000/12/212000/12/28
Actual Type 1 Intersection Union
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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
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Calculations
Average 1 = (Intersection+Union)/2
Average 2 = (Type 1 + Intersection + Union)/3
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Type 1 Intersection Union Average 1 Average 2
2000176.32 131.86 175.47
139.39 143.52
2001147.84 159.68 138.37
144.15 141.05
2002100.62 79.6 89.17
82.56 83.13
200374.46 73.03 76.65
73.26 70.92
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Conclusion
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Conclusion
Applying Type 2 fuzzy sets to utilize extra information
A framework for applying Type 2 fuzzy time series models
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Conclusion
Lower and upper boundsConservative and optimisticIntersection and union operations
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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.
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Discussion