![Page 1: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/1.jpg)
1
A framework for type 2 fuzzy time series models
K. Huarng and H.-K. YuFeng Chia University, Taiwan
![Page 2: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/2.jpg)
2
Outlines
Literature ReviewChen’s ModelType 2 fuzzy setsA FrameworkEmpirical analysisConclusion
![Page 3: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/3.jpg)
3
Literature Review
![Page 4: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/4.jpg)
4
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.
![Page 5: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/5.jpg)
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.
![Page 6: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/6.jpg)
6
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)
![Page 7: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/7.jpg)
7
Applications
Enrollment Stock index Temperature Some were shown to outperform their traditional counterpart models
![Page 8: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/8.jpg)
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)
![Page 9: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/9.jpg)
9
Applications of Type 2 Fuzzy Sets
Decision makingData processingSurvey processingTime series modelingFuzzy relation equations
![Page 10: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/10.jpg)
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 onlyWhy Type 2 fuzzy sets are not so popular
![Page 11: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/11.jpg)
11
Chen’s Model
![Page 12: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/12.jpg)
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.
![Page 13: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/13.jpg)
13
Two major processes
Steps 1 – 3: fuzzification, lengths of intervals
Steps 4 – 5: fuzzy relationships
![Page 14: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/14.jpg)
14
Enrollment forecasting
University of Alabama
Data from 1979 to 1991
![Page 15: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/15.jpg)
15
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].
![Page 16: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/16.jpg)
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/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
![Page 17: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/17.jpg)
17
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.
![Page 18: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/18.jpg)
18
Year Enrollment Fuzzy Enrollment Ai
1971 13055 A1
1972 13563 A1
1973 13867 A1
1974 14696 A2
1975 15460 A3
…
![Page 19: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/19.jpg)
19
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
![Page 20: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/20.jpg)
20
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
![Page 21: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/21.jpg)
21
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)
![Page 22: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/22.jpg)
22
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.
![Page 23: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/23.jpg)
23
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
![Page 24: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/24.jpg)
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 u1 and u2:
(13500+14500)/2=14000
![Page 25: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/25.jpg)
25
Type 2 fuzzy sets
![Page 26: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/26.jpg)
26
1.0
0.5
0.0
2
μ
x
![Page 27: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/27.jpg)
27
1.0
0.5
0.0
2
μ
x
0.6
0.5
0.4
![Page 28: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/28.jpg)
28
1.0
0.5
0.0
2
μ
x
0.6
0.5
0.4
![Page 29: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/29.jpg)
29
1.0
0.5
0.0
2
μ
x
![Page 30: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/30.jpg)
30
George J. Klir, Bo Yuan (1995)
0
1
0.5
α1
α2
a x
A(x)
![Page 31: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/31.jpg)
31
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)
![Page 32: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/32.jpg)
32
A Framework
![Page 33: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/33.jpg)
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.
![Page 34: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/34.jpg)
34
Rationale (2)
Lower bound - conservativeUpper bound - optimistic
![Page 35: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/35.jpg)
35
Conservative
1.0
0.5
0.0
2
μ
x
![Page 36: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/36.jpg)
36
Optimistic1.0
0.5
0.0
2
μ
x
![Page 37: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/37.jpg)
37
Rationale (3)
Conservative – Intersection operation
Optimistic – Union operation
![Page 38: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/38.jpg)
38
Rationale (4)
Conservative - To refine Type 1 fuzzy relationships
Optimistic - To include more information in the Type 1 fuzzy relationships
![Page 39: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/39.jpg)
39
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
![Page 40: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/40.jpg)
40
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
![Page 41: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/41.jpg)
41
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
![Page 42: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/42.jpg)
42
Empirical Analysis
![Page 43: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/43.jpg)
43
Data
TAIEX from 2000 to 2003.Jan – Oct.: estimationNov. – Dec.: forecastingDaily closing, high, low prices
![Page 44: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/44.jpg)
44
Setup
Lengths of intervals is set to 100.
Root mean squared errors (RMSEs) are used to evaluate forecasting results.
![Page 45: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/45.jpg)
45
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
![Page 46: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/46.jpg)
46
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
![Page 47: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/47.jpg)
47
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
![Page 48: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/48.jpg)
48
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
… … … …
![Page 49: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/49.jpg)
49
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
![Page 50: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/50.jpg)
50
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
![Page 51: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/51.jpg)
51
Forecasts (intersection)
The forecast for 11/8 is A13The forecast for 11/9 is A13, A14, and A16The forecast for 11/10 is A13.
![Page 52: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/52.jpg)
52
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
![Page 53: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/53.jpg)
53
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
![Page 54: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/54.jpg)
54
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
![Page 55: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/55.jpg)
55
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
![Page 56: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/56.jpg)
56
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
![Page 57: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/57.jpg)
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
![Page 58: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/58.jpg)
58
Calculations
Average 1 = (Intersection+Union)/2
Average 2 = (Type 1 + Intersection + Union)/3
![Page 59: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/59.jpg)
59
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
![Page 60: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/60.jpg)
60
Conclusion
![Page 61: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/61.jpg)
61
Conclusion
Applying Type 2 fuzzy sets to utilize extra information
A framework for applying Type 2 fuzzy time series models
![Page 62: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/62.jpg)
62
Conclusion
Lower and upper boundsConservative and optimisticIntersection and union operations
![Page 63: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/63.jpg)
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.
![Page 64: A framework for type 2 fuzzy time series models K. Huarng ... · (George J. Klir and Bo Yuan, 1995) Type 2 fuzzy sets possess a great expressive power zMotivation 1: Apply Type 2](https://reader031.vdocuments.us/reader031/viewer/2022020413/5b3be02f7f8b9a986e8c9dfb/html5/thumbnails/64.jpg)
64
Discussion