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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