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Junzo WatadaThe Concept and Progress of Fuzzy
Regression AnalysisThe Formation of
Fuzzy Variable Model and Its Applications
Junzo WatadaGraduate School of ISP
Waseda University
junzow@osb.att.ne.jp
Botho CollegeInterdisciplinary Research Conference 2012
Botho Education Park, Kgale, Gaborone18 th and 19 th October, 2012
Junzo Watada
Fuzzy Sets
Oct. 19, 2012 Botho Collage Conference 2012 2
Junzo Watada
Fuzzy Sets Lotfi A. Zadeh
June 27, 2012 at & CAIRO, UTM 3
Oil Palm Fruit Evaluation
Oct. 19, 2012 Botho Collage Conference 2012 44
Oil Palm Fruit Anatomy
DECISION MAKING
QUALITY INSPECTION
Graded Fruit
Condition
FruitletsColor
Surface
(MPOB, 2003)
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 5
Junzo Watada
Oct. 19, 2012 100th meetinhBotho Collage Conference 2012 6
Statistical Regression Analysis
-1950s Fransis Golton
- The solution can be obtained by Normal Equation
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Junzo WatadaDr. Hideo Tanaka
At Grand Hotel,
Taipei in 1983
June 27, 2012 at & CAIRO, UTM 7
Junzo Watada
June 27, 2012 at & CAIRO, UTM 8
Junzo Watada1. Fuzzy Regression Analysis
Oct. 19, 2012 Botho Collage Conference 2012 9
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Upper Bound
Lower Bound
Possibility
Junzo WatadaFuzzy Regression Model
Oct. 19, 2012 Botho Collage Conference 2012 10
nicaA iii ,,2,1,
1.0
c ca
n
n
n
xxxcccaaa
Y
,,,,,,,,,
,,
21
21
21
xca
xcaxxcaAx
Junzo WatadaFuzzy Regression Model
Oct. 19, 2012 Botho Collage Conference 2012 11
Ax nn xAxAxAY 2211
where nicaA iii ,,2,1,
n
n
n
xxxcccaaa
Y
,,,,,,,,,
,,
21
21
21
xca
xcaxxcaAx
Junzo Watada1. Fuzzy Regression Analysis
Oct. 19, 2012 Botho Collage Conference 2012 12
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Upper Bound
Lower Bound
Possibility
Junzo Watada
Fuzzy Regression Model
Oct. 19, 2012 Botho Collage Conference 2012 13
LP Problem
),,2,1(0
tosubject
minimize1
mj
xxy
xxy
x
jjj
jjj
m
jj
c
ca
ca
c Vagueness
upperbound
lowerbound
coefficient
13
Junzo Watada
Introduction
• Hybrid uncertainty is a pivotal factor in building models.
• That is, fuzziness, possibility and probability
• Several models are illustrated here to understand fuzzy random variables.
Oct. 19, 2012 Botho Colledge Conference 2012 14
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 15
Junzo Watada
Evaluation
Oct. 19, 2012 Botho Colledge Conference 2012 16
Measure some dimensions of the target apparatus
Measure some features of the target subject
Measure some dimensions of the target car
Thes
e nu
mer
ical
mea
sure
men
ts d
o no
t exp
ress
th
eir b
eaut
ifuln
ess.
Usu
ally
the
beau
tiful
ness
sh
ould
be
expr
esse
d us
ing
hum
an li
ngui
stic
ex
pres
sion
s in
stea
d of
the
aggr
egat
ion
of
num
eric
al d
imen
sion
s an
d fe
atur
es.
Oil Palm Fruit Evaluation
Oct. 19, 2012 CJS2011 at Hejnice, Czech Republic 1717
Oil Palm Fruit Anatomy
DECISION MAKING
QUALITY INSPECTION
Graded Fruit
Condition
FruitletsColor
Surface
(MPOB, 2003)
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 18
Junzo WatadaLinguistic Evaluation
Oct. 19, 2012 19
Junzo Watada
2. Summery of Fuzzy Regression Models
Oct. 19, 2012 Botho Colledge Conference 2012 20
Junzo WatadaFuzzy Linear Function
Oct. 19, 2012 Botho Colledge Conference 2012 21
numberfuzzyare,where, 10110 AAxAAY
n
n
n
xxxcccaaa
Y
,,,,,,,,,
,,
21
21
21
xca
xcaxxcaAx
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90
regression
Possibility110 xAAY
0A
1x
y
Junzo WatadaFuzzy Linear Function
Oct. 19, 2012 Botho Colledge Conference 2012 22
numberfuzzyare,where, 10110 AAxAAY
n
n
n
xxxxccccaaaa
Y
,,,,,,,,,,,,,,
210
210
210
xca
xcaxxcaAx
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90
regression
Possibility
110 xAAY
0A
1x
x cxa
x cxa
y
y
y
Junzo WatadaFuzzy Regression Analysis of Crisp Data
Oct. 19, 2012 Botho Colledge Conference 2012 23
numberfuzzyare,where, 10110 AAxAAY
n
n
n
xxxcccaaa
Y
,,,,,,,,,
,,
21
21
21
xca
xcaxxcaAx
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90
Possibility
110 xAAY
1x
y
x cxa
x cxa
y
y
ii y,x
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012
LP Problem
Vagueness
upper bound
lower boundcoefficient
24
),,2,1(0
tosubject
minimize1
mj
y
y
jjj
jjj
m
jj
c
x cxa
x cxa
xc
[001] H. Tanaka, S. Uejima and K. Asai, “Linear Regression Analysis with Fuzzy Model,“ IEEE Transactions on Systems, Man and Cybernetics, 12(6), 903-907, 1982.
[042] H. Tanaka, Junzo Watada, Possibilistic linear systems and their application, Int. J. Fuzzy Sets and Systems, Vol. 27, No. 3, pp. 275--289, 1988.
[044] H. Tanaka, I. Hayashi and Junzo Watada, Possibilistic linear regression for fuzzy data, European J. of Operational Research, Vol. 40, No. 3, pp. 389--396, 1989.
Fuzzy Regression Analysis of Crisp Data
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 25
2.1. Crisp Data
x y
1 x1 y1
2 x2 y2
n xn yn
Junzo WatadaFuzzy Regression Analysis of Crisp Data
Oct. 19, 2012 Botho Colledge Conference 2012 26
numberfuzzyare,where, 10110 AAxAAY
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90
Possibility
110 xAAY
1x
y
),,2,1( mj
y
y
jjj
jjj
x cxa
x cxa
x cxa
x cxa
y
y
ii y,x
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012
LP Problem
Vagueness
upper bound
lower bound
coefficient
27
),,2,1(0
tosubject
minimize1
mj
y
y
jjj
jjj
m
jj
c
x cxa
x cxa
xc
Fuzzy Regression Analysis of Crisp Data
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 28
2.2. Y values are fuzzy
1 x1
2 x2
n xn
x ),( YYY
),( 111 YYY
),( 222 YYY
),( nnn YYY
Junzo WatadaFuzzy Regression Analysis of Fuzzy Data (Y)
Oct. 19, 2012 Botho Colledge Conference 2012 29
numberfuzzyare,where, 10110 AAxAAY
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Possibility
110 xAAY
1x
y
),,2,1( mj
Y
Y
jjj
jjj
x cxa
x cxa
x cxa
x cxa
y
y
ii Y,x
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012
LP Problem
Vagueness
upper bound
lower boundcoefficient
30
),,2,1(0
tosubject
minimize1
mj
Y
Y
jjj
jjj
m
jj
c
x cxa
x cxa
xc
Fuzzy Regression Analysis of Fuzzy Data (Y)
Junzo WatadaFuzzy Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 31
[067] Y. Toyoura, M. Kambara, M. Uemura, T. Miyake, K. Kawasaki and Junzo Watada, Evaluation of Fuzzy Regression Analysis and Its Application to Oral Age Model, Official Journal of Biomedical Fuzzy Systems Association, Vol. 6, No. 1, pp. 41--49, 2000.
[191] Junzo Watada, Fuzzy Regression Analysis of Software Bug Structure, 3rd Czech-Japan Seminar on Data Analysis and Decision Making under uncertainty, at Osaka university, 2000.10
[231] Junzo Watada, Recent Development of Fuzzy Regression Model, Proceedings, the 6th Czech-Japan Seminar, Valtice, Czech Republic., pp. -, 2003.9
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 32
3 X and Y are Fuzzy Data
1
2
n
),( YYY
),( 111 YYY
),( 222 YYY
),( nnn YYY
),( XXX
),( 111 XXX
),( 222 XXX
),( nnn XXX
(-5,-4)X(-3,-2)=(8,15)
(-5,-4)X(2,3)=(-15,-8)
(4,5)X(2,3)=(8,15)
Junzo WatadaFuzzy Regression Analysis of Fuzzy-Fuzzy Data
Oct. 19, 2012 Botho Colledge Conference 2012 33
numberfuzzyare,where, 10110 AAxAAY
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Possibility
110 xAAY
1x
y
X cXa
X cXa
Y
Y
ii Y,X
?
How to solve this fuzzy regression model
?(-5,-4)X(-3,-2)=(8,15)
(-5,-4)X(2,3)=(-15,-8)
(4,5)X(2,3)=(8,15)
Junzo Watada
[J053] Junzo Watada, Hideo Tanaka, Kiyoji Asai, "Fuzz Quantification Model Type I, Japan Society of Behavioral Metrics, Vol. 11, No. 1, pp. 66--73, 1984 in Japanese.
[B023] Junzo Watada, Applied Fuzzy System, Ed. by Toshiro Terano, Kiyoji Asai, Michio Sugeno, Chapter 5 Applications in Business, 5. 5 Multiattribute Decision-Making, AP Professional, pp. 244--252, 1994.5
[J159] Junzo Watada, Shuming Wang and Witold Pedrycz, Building Confidence-Interval-Based Fuzzy Random Regression Models, IEEE Trans. Fuzzy Systems, vol. 17, issue 6, pp. 1273-1283, Dec. 2009.
[J073] Yabuuchi, Y., Watada, J., Fuzzy robust regression analysis based on a hyper elliptic function, Journal of the Operations Research Society of Japan 39 (4), pp. 512-524, 1996
[J170] Azizul Azhar Ramli, Junzo Watada, and Witold Pedrycz, Performance Measurement in Manufacturing Enterprises: New Paradigm on Intelligent Data Analysis (IDA) Implementation, JCSES, Accepted 2010
[J182] Azizul Azhar Ramli, Junzo Watada, Witold Pedrycz: Real-Time Fuzzy Regression Analysis: A Convex Hull Approach. European Journal of Operational Research, vol. 210, issue 3, pp. 606-617 (2011)
[185] Yoshiyuki Yabuuchi and Junzo Watada, Fuzzy Robust Regression Model by Possibility Maximization, JACIII, Accepted, 2011
Oct. 19, 2012 Botho Colledge Conference 2012 34
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 35
4. Fuzzy-Fuzzy Random Data
1
2
n
),( YYY
),( 111 YYY
),( 222 YYY
),( nnn YYY
),( XXX
),( 111 XXX
),( 222 XXX
),( nnn XXX
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 36
Junzo WatadaFuzzy Regression Analysis of Fuzzy Random Data
Oct. 19, 2012 Botho Colledge Conference 2012 37
numberfuzzyare,where, 10110 AAxAAY
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Possibility
110 xAAY
1x
y
X cXa
X cXa
Y
Y
ii Y,X?How to solve this fuzzy regression model
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 38
[Cxxx] Junzo Watada, Shuming Wang and Witold Pedrycz, Building a fuzzy random regression model with confidence interval, Proc. of the 11th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty, Japan, pp. 39-44, 2008.
[J159] Junzo Watada, Shuming Wang and Witold Pedrycz, Building Confidence-Interval-Based Fuzzy Random Regression Models, IEEE Trans. Fuzzy Systems, vol. 17, issue 6, pp. 1273-1283, Dec. 2009.
[B038] Junzo Watada, Shuming Wang, and Witold Pedrycz, "Formulation of Fuzzy Random Regression Model," In: (Eds) by Antonio E.B. Ruano and Annamaria R. Varkonyi-Koczy (editors), New Advances in Intelligent Signal Processing on occasion of the ten years existence of the IEEE WISP (IEEE International Symposium on Intelligent Signal Processing) series, in Studies in Computational Intelligence, Springer
[B035] Junzo Watada, Shuming Wang, Regression model based on fuzzy random variables, Chapter 26, Seising Rodulf (Ed.), Views on Fuzzy Sets and Systems from Different Perspectives - Philosophy and Logic, Criticisms and Applications -, Studies in Fuzziness and Soft Computing, Volume 243, Spring-Verlag, Berlin, pp. 533-546, 2009.
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 39
5. Linguistic Data
1
2
n
),( YYY
),( 111 YYY
),( 222 YYY
),( nnn YYY
),( XXX
),( 111 XXX
),( 222 XXX
),( nnn XXX ?
How to solve this Linguistic regression model
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 40
TotalTraining Corrosion St. Cracking St. Loading and Env. State Assessment
Sample X1 X2 X3 Y
1 extremely good extremely bad below normal not safe
2 bad very bad below normal not safe
3 very bad very bad below normal not safe 4 very bad very good below normal questionable5 good extremely good below normal safe 6 good bad normal not safe7 bad very bad normal not safe
8 very bad good normal not safe
9 bad very good normal questionable10 very bad very good normal questionable11 extremely good bad severe not safe12 very good bad severe not safe
13 good very bad severe not safe
14 very bad good severe not safe
15 very good good severe not safe
16 very good very good severe questionable
Attributes
Junzo WatadaFuzzy Regression Analysis of Linguistic Data
Oct. 19, 2012 Botho Colledge Conference 2012 41
numberfuzzyare,where, 10110 AAxAAY
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Possibility
110 xAAY
1x
y
X cXa
X cXa
Y
Y
ii Y,X?
How to solve this linguistic regression model
?
Junzo Watada
[C204] Watada, J., Fu, K.S., Yao, J.T.P., DAMAGE ASSESSMENT USING FUZZY MULTIVARIANT ANALYSIS., Purdue University, School of Civil Engineering, Structural Engineering (Technical Report) CE-STR, 1984
[C209] A. Hinkle, Junzo Watada and Junzo T. P. Yao, Linguistic Assessment of Fatigue Damage, Proceeding, North American Fuzzy Information Processing Society, at New Orleans, 1986.7
[B031] Junzo Watada and Witold Pedrycz, A Fuzzy Regression Approach to Acquisition of Linguistic Rules, Chapter 32, Witold Pedrycz, Andrzej Skowron and Vladik Kreinovich (Eds.), Handbook of Granular Computing, John Wiley & Sons, Chichester, Chapter 32, pp. 719-740, July 2008.
Oct. 19, 2012 Botho Colledge Conference 2012 42
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 43
5. Linguistic Random Data
1
2
n
),( YYY
),( 111 YYY
),( 222 YYY
),( nnn YYY
),( XXX
),( 111 XXX
),( 222 XXX
),( nnn XXX ?
How to solve this Linguistic Random regression model
Junzo WatadaLinguistic Evaluation
Oct. 19, 2012 44
Junzo WatadaFuzzy Regression Analysis of Linguistic Random Data
Oct. 19, 2012 Botho Colledge Conference 2012 45
numberfuzzyare,where, 10110 AAxAAY
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Possibility
110 xAAY
1x
y
X cXa
X cXa
Y
Y
ii Y,X?
How to solve this linguistic random regression model ?
Junzo Watada3. Fuzzy Regression Analysis
Oct. 19, 2012 Botho Colledge Conference 2012 46
Upper Bound
Lower Bound0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Possibility
[001] H. Tanaka, S. Uejima and K. Asai, “Linear Regression Analysis with Fuzzy Model,“ IEEE Transactions on Systems, Man and Cybernetics, 12(6), 903-907, 1982.
[042] H. Tanaka, Junzo Watada, Possibilistic linear systems and their application, Int. J. Fuzzy Sets and Systems, Vol. 27, No. 3, pp. 275--289, 1988.
[044] H. Tanaka, I. Hayashi and Junzo Watada, Possibilistic linear regression for fuzzy data, European J. of Operational Research, Vol. 40, No. 3, pp. 389--396, 1989.
Junzo WatadaFuzzy Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 47
Ax nn xAxAxAY 2211
where nicaA iii ,,2,1,
n
n
n
xxxcccaaa
Y
,,,,,,,,,
,,
21
21
21
xca
xcaxxcaAx
[067] Y. Toyoura, M. Kambara, M. Uemura, T. Miyake, K. Kawasaki and Junzo Watada, Evaluation of Fuzzy Regression Analysis and Its Application to Oral Age Model, Official Journal of Biomedical Fuzzy Systems Association, Vol. 6, No. 1, pp. 41--49, 2000.
[191] Junzo Watada, Fuzzy Regression Analysis of Software Bug Structure, 3rd Czech-Japan Seminar on Data Analysis and Decision Making under uncertainty, at Osaka university, 2000.10[
231] Junzo Watada, Recent Development of Fuzzy Regression Model, Proceedings, the 6th Czech-Japan Seminar, Valtice, Czech Republic., pp. -, 2003.9
1.0
c ca
Junzo Watada
Fuzzy Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012
LP Problem
Vagueness
upper bound
lower boundcoefficient
48
[074] Yoshihiro Toyoura, Junzo Watada, Yoshiyuki Yabuuchi, Hajime Ikegame, Seiichi Sato, Katsuhiko Watanabe, Masahiro Tohyama, Fuzzy Regression Analysis of Software Bug Structure, Central European Journal of Operations Research(CEOR), Vol. 12, No. 1, pp. 13-23, 2004.2.
[253] Yoshiyuki Yabuuchi & Junzo Watada, Possibilistic Forecasting Model and Its Application to Analyze the Economy in Japan, Knowledge-Based Intelligent Information Engineering Systems, Springer, Vol. LNCS 3215No., pp. 151-158, 2004.9
[293] Yoshiyuki Yabuuchi, Junzo Watada, Model Building Based on Central Position for a Fuzzy Regression Model, Proceedings, CzJp2006, Czech Japan Seminar 2006, pp. 114-119, Aug 18 (Fri)- 22 (Tue) 2006., pp. 114-229, 2006.
),,2,1(0
tosubject
minimize1
mj
xxy
xxy
x
jjj
jjj
m
jj
c
ca
ca
c
Junzo WatadaRegression Analysis based on Convex Hull
[193] Y. Toyoura, Junzo Watada, Efficient Fuzzy Regression Model based on Convex Hull, 3rd Czech-Japan Seminar on Data Analysis and Decision Making under uncertainty, at Osaka university, 2000.10
[073] Y. Toyoura, Junzo Watada, Efficient Fuzzy Regression Model based on Convex Hull, Central European Journal of Operations Research(CEOR), 2002.
Oct. 19, 2012 Botho Colledge Conference 2012 49
Robust Regression Analysis [058] Yoshiyuki Yabuuchi, Junzo Watada, “Fuzzy Robust Regression Analysis Based on
Ecrisp Function,” Japan Society Journal of Operations Research, Vol. 39, No. 4, pp. 512--524, 1996.12
[111] Kunio Shibata, Junzo Watada, Yoshiyuki Yabuuchi, “A Fuzzy Robust Regression Analysis Approach to Evaluation of Electronic Apparatus Industry In Japan ,” Japan Society of Management Engineering, 2007
Junzo WatadaSwitching Regression Analysis
[169] Junzo Watada and Hirohito Mizunuma, Fuzzy Switching Regression Model based on Genetic Algorithm, Invited Proceeding, The 7th International Fuzzy Systems Association World Congress(IFSA'97) in Prague, Czech Republic, 1997.6
[071] Junzo Watada, Y. Toyoura, Formulation of Fuzzy Switching Auto-Regression Model, International Journal of Chaos Theory and Applications,, Vol. 7, No. 1,2, pp. 67--76, 2002.7
[086] 薮内賢之, 和多田淳三, ファジィスイッチング回帰モデルの構成 (解説), 日本知能情報ファジィ学会誌, Vol. 16, No. 1, pp. 53-59, 2004.12
Oct. 19, 2012 Botho Colledge Conference 2012 50
Auto Regression Analysis[076] Junzo Watada, Y. Toyoura, Formulation of Fuzzy Switching Auto-Regression Model,
International Journal of Chaos Theory and Applications, Vol. 7, No. 1, 2, pp. 67-76, 2002.7
[215] Yoshiyuki Yabuuchi, Yoshihiro Toyoura & Junzo Watada, Fuzzy AR Model of Stock Price, Proceedings, 5th Czech-Japan Seminar (CZJP2002), at Koyasan, ., pp. 127-132, 2002.9
Junzo WatadaTime-series Analsysis
[034] 和多田淳三、田中英夫、横山宏、浅居喜代治, ファジィ時系列モデルと予測問題への応用, 日本経営工学会, Vol. 34, No. 3, pp. 180-186, 1983.8
[153] Junzo Watada, H. Tanaka and K. Asai, Analysis of Time-Series Data by Possibilistic Model, Processing, Int.Workshop on Fuzzy System Applications, at Fukuoka, pp. 228--229, 1988.8
[015] J. Watada, Fuzzy Regression Analysis, J. Kacprzyk & M. Fedrizzi eds., Fuzzy time-series analysis and forecasting of sales , Vol.ume, Omnitech Press, Warsaw, Poland, pp. 211--217, 1992.5
[021] Junzo Watada, Fuzzy Information Engineering, ed. by D. Duboir and M. M. Yager, Possibilistic Time-series Analysis and its analysis of Consumption, John Wiley & Sons, Inc., pp. 187--200, 1996.4
Oct. 19, 2012 Botho Colledge Conference 2012 51
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 52
4. Fuzzy-Fuzzy Random Data
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 53
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 54
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 55
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 56
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 57
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 58
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 59
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 60
Junzo Watada1. Fuzzy Regression Analysis
Oct. 19, 2012 Botho Colledge Conference 2012 61
0
50
100
150
200
250
300
0 10 20 30 40 50 60 70 80 90 100
Upper Bound
Lower Bound
Possibility
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 62
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 63
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 64
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 65
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 66
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 67
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 68
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 69
TotalTraining Corrosion St. Cracking St. Loading and Env. State Assessment
Sample X1 X2 X3 Y
1 extremely good extremely bad below normal not safe
2 bad very bad below normal not safe
3 very bad very bad below normal not safe 4 very bad very good below normal questionable5 good extremely good below normal safe 6 good bad normal not safe7 bad very bad normal not safe
8 very bad good normal not safe
9 bad very good normal questionable10 very bad very good normal questionable11 extremely good bad severe not safe12 very good bad severe not safe
13 good very bad severe not safe
14 very bad good severe not safe
15 very good good severe not safe
16 very good very good severe questionable
Attributes
5. Linguistic Data
Junzo Watada
[C204] Watada, J., Fu, K.S., Yao, J.T.P., DAMAGE ASSESSMENT USING FUZZY MULTIVARIANT ANALYSIS., Purdue University, School of Civil Engineering, Structural Engineering (Technical Report) CE-STR, 1984
[C209] A. Hinkle, Junzo Watada and Junzo T. P. Yao, Linguistic Assessment of Fatigue Damage, Proceeding, North American Fuzzy Information Processing Society, at New Orleans, 1986.7
[B031] Junzo Watada and Witold Pedrycz, A Fuzzy Regression Approach to Acquisition of Linguistic Rules, Chapter 32, Witold Pedrycz, Andrzej Skowron and Vladik Kreinovich (Eds.), Handbook of Granular Computing, John Wiley & Sons, Chichester, Chapter 32, pp. 719-740, July 2008.
Oct. 19, 2012 Botho Colledge Conference 2012 70
Junzo WatadaLinguistic Evaluation
Oct. 19, 2012 71
5. Linguistic Random Data
Junzo Watada
3.Linguistic Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 72
Variable
Lnumber(visiting customers) = “bad”
Lvolume(production) = “extremely bad”
πnumber(visiting customers) ≡ πstate(number)≡U(bad)
Possibility
Fuzzy number
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 73
TotalTraining Corrosion St. Cracking St. Loading and Env. State Assessment
Sample X1 X2 X3 Y
1 extremely good extremely bad below normal not safe
2 bad very bad below normal not safe
3 very bad very bad below normal not safe 4 very bad very good below normal questionable5 good extremely good below normal safe 6 good bad normal not safe7 bad very bad normal not safe
8 very bad good normal not safe
9 bad very good normal questionable10 very bad very good normal questionable11 extremely good bad severe not safe12 very good bad severe not safe
13 good very bad severe not safe
14 very bad good severe not safe
15 very good good severe not safe
16 very good very good severe questionable
Attributes
Junzo Watada
Linguistic Variable
Oct. 19, 2012 Botho Colledge Conference 2012 74
(1) Translation
1.0
0.00.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
Deg
ree
Fuzzy Number
)15.0,15.0,00.1(
)15.0,15.0,80.0(
)15.0,15.0,60.0(
)15.0,15.0,40.0(
)15.0,15.0,20.0(
)15.0,00.0,00.0(
)badextremely (
)badvery (
)bad(
)good(
)goodvery (
)goodextremely (
UUUUUU
Dictionary
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 75
TotalTraining Corrosion St. Cracking St. Loading and Env. State Assessment
Sample X1 X2 X3 Y
1 (0.00,0.00,0.15) (1.00,0.15,0.15) (0.00,0.00,0.40) (1.00,0.40,0.00)2 (0.60,0.15,0.15) (0.80,0.15,0.15) (0.00,0.00,0.40) (1.00,0.40,0.00)3 (0.80,0.15,0.15) (0.80,0.15,0.15) (0.00,0.00,0.40) (1.00,0.40,0.00)4 (0.80,0.15,0.15) (0.20,0.15,0.15) (0.00,0.00,0.40) (0.50,0.20,0.20)5 (0.40,0.15,0.15) (0.00,0.00,0.15) (0.00,0.00,0.40) (0.00,0.00,0.40)6 (0.40,0.15,0.15) (0.60,0.15,0.15) (0.50,0.20,0.20) (1.00,0.40,0.00)7 (0.60,0.15,0.15) (0.80,0.15,0.15) (0.50,0.20,0.20) (1.00,0.40,0.00)8 (0.80,0.15,0.15) (0.40,0.15,0.15) (0.50,0.20,0.20) (1.00,0.40,0.00)9 (0.60,0.15,0.15) (0.20,0.15,0.15) (0.50,0.20,0.20) (0.50,0.20,0.20)10 (0.80,0.15,0.15) (0.20,0.15,0.15) (0.50,0.20,0.20) (0.50,0.20,0.20)11 (0.00,0.00,0.15) (0.60,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)12 (0.20,0.15,0.15) (0.60,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)13 (0.40,0.15,0.15) (0.80,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)14 (0.80,0.15,0.15) (0.40,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)15 (0.20,0.15,0.15) (0.40,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)16 (0.20,0.15,0.15) (0.20,0.15,0.15) (1.00,0.40,0.00) (0.50,0.20,0.20)
Attributes
Junzo WatadaProcess of Linguistic Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 76
UUU k,,,21
LLL k,, ,21
Uw
Translation EstimationVocabularyMatching
AttributesFuzzy Number
Total AssessmentFuzzy Number
Total AssessmentLinguistic Words
AttributesLinguistic Words
Dictionary
Lw
),,,(21
UUUV LLLfL kw
[146] A. Hinkle, Junzo Watada and Junzo T. P. Yao, Linguistic Assessment of Fatigue Damage, Proceeding, North American Fuzzy Information Processing Society, at New Orleans, 1986.7
[077] Junzo Watada, T. Watanabe, Formulation of Linguistic Regression Model Based on Natural Words, Journal of Intelligent and Fuzzy Systems., pp. 1-15, 2002.
[211] Junzo Watada, T. Watanabe, Formulation of Linguistic Regression Model Base on Natural Words, Journal of Intelligent and Fuzzy Systems, pp. 1--15, 2002.
[084] Y. Toyoura, J. Watada, M. Khalid, & R. Yusof, Formulation of linguistic regression model based on natural words, Soft Computing Journal, Vol. 8, No. 10, pp. 681-688, 2004.11
Junzo Watada
Process of Linguistic Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 77
UUU k,,,21
LLL k,, ,21
Uw
Translation EstimationVocabularyMatching
AttributesFuzzy Number Total Assessment
Fuzzy Number
Total AssessmentLinguistic WordsAttributes
Linguistic Words
Dictionary
Lw
)L,,L,Lf(L UUUVkw
21
Junzo Watada
Linguistic Regression
Oct. 19, 2012 Botho Colledge Conference 2012 78
(2) K
f LLL U,,U,UV21
Junzo Watada
Process of Linguistic Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 79
UUU k,,,21
LLL k,, ,21
Uw
Translation EstimationVocabularyMatching
AttributesFuzzy Number Total Assessment
Fuzzy Number
Total AssessmentLinguistic WordsAttributes
Linguistic Words
Dictionary
Lw
),,,(21
UUUV LLLfL kw
Junzo WatadaEvaluation of Linguistic Variable
Oct. 19, 2012 Botho Colledge Conference 2012 80
(3) Linguistic MatchingAssign the most appropriate word to the fuzzy number in the dictionary given
)()(max maxLiWVtDW0 tt
i
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
度合
ファジィ数
0L
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
度合
ファジィ数
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
度合
ファジィ数
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
度合
ファジィ数
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
1.0
0.0
0.0 0.2 0.4 0.6 0.8 1.0
extremelygood
verygood
good bad verybad
extremelybad
度合
ファジィ数
0L
Junzo WatadaLinguistic Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 81
K
ii i
Kf
1)(L
)(L)(L)(L)(L
UA
U,,U,UV210
It is most important to determine an fuzzy valuation function pursued in an expert(s).
Where ω is an training sample to build a model
Two evaluations of the model are employed to determinethe optimal fuzzy coefficients Ai.
)()()( )(L)(L 0yyh
iRy
Fitness:
Vagueness of System:
K
iii aa
1
LUS
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 82
Modeling the damage assessment of buildings by experts
Linguistic Var. Fuzzy Number extremely good (0.00,0.00,0.15)
very good (0.20,0.15,0.15)
good (0.40,0.15,0.15)
bad (0.60,0.15,0.15) very bad (0.80,0.15,0.15) extremely bad (1.00,0.15,0.15) extremely good (0.00,0.00,0.15) very good (0.20,0.15,0.15)
good (0.40,0.15,0.15)
bad (0.60,0.15,0.15) very bad (0.80,0.15,0.15) extremely bad (1.00,0.15,0.15)
below normal (0.00,0.00,0.40)
X3 normal (0.50,0.20,0.20)
severe (1.00,0.40,0.00)
safe (0.00,0.00,0.40)
Y questionable (0.50,0.20,0.20) not safe (1.00,0.40,0.00)
X2
X1Corrosion State
Cracking State
Loading andEnvironment
State
TotalAssssment
Dictionary
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 83
TotalTraining Corrosion St. Cracking St. Loading and Env. State Assessment
Sample X1 X2 X3 Y
1 extremely good extremely bad below normal not safe
2 bad very bad below normal not safe
3 very bad very bad below normal not safe 4 very bad very good below normal questionable5 good extremely good below normal safe 6 good bad normal not safe7 bad very bad normal not safe
8 very bad good normal not safe
9 bad very good normal questionable10 very bad very good normal questionable11 extremely good bad severe not safe12 very good bad severe not safe
13 good very bad severe not safe
14 very bad good severe not safe
15 very good good severe not safe
16 very good very good severe questionable
Attributes
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 84
TotalTraining Corrosion St. Cracking St. Loading and Env. State Assessment
Sample X1 X2 X3 Y
1 (0.00,0.00,0.15) (1.00,0.15,0.15) (0.00,0.00,0.40) (1.00,0.40,0.00)2 (0.60,0.15,0.15) (0.80,0.15,0.15) (0.00,0.00,0.40) (1.00,0.40,0.00)3 (0.80,0.15,0.15) (0.80,0.15,0.15) (0.00,0.00,0.40) (1.00,0.40,0.00)4 (0.80,0.15,0.15) (0.20,0.15,0.15) (0.00,0.00,0.40) (0.50,0.20,0.20)5 (0.40,0.15,0.15) (0.00,0.00,0.15) (0.00,0.00,0.40) (0.00,0.00,0.40)6 (0.40,0.15,0.15) (0.60,0.15,0.15) (0.50,0.20,0.20) (1.00,0.40,0.00)7 (0.60,0.15,0.15) (0.80,0.15,0.15) (0.50,0.20,0.20) (1.00,0.40,0.00)8 (0.80,0.15,0.15) (0.40,0.15,0.15) (0.50,0.20,0.20) (1.00,0.40,0.00)9 (0.60,0.15,0.15) (0.20,0.15,0.15) (0.50,0.20,0.20) (0.50,0.20,0.20)10 (0.80,0.15,0.15) (0.20,0.15,0.15) (0.50,0.20,0.20) (0.50,0.20,0.20)11 (0.00,0.00,0.15) (0.60,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)12 (0.20,0.15,0.15) (0.60,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)13 (0.40,0.15,0.15) (0.80,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)14 (0.80,0.15,0.15) (0.40,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)15 (0.20,0.15,0.15) (0.40,0.15,0.15) (1.00,0.40,0.00) (1.00,0.40,0.00)16 (0.20,0.15,0.15) (0.20,0.15,0.15) (1.00,0.40,0.00) (0.50,0.20,0.20)
Attributes
Junzo Watada
Numerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 85
)loading(
)cracking(
)corrosion(
LLL
)065.0,065.0,373.0(
)000.0,000.0,813.0(
)000.0,000.0,322.0(
U,U,UV321
f
Fuzzy Evaluation Function
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 86
TrainingSample Assessed Fuzzy Num.rAssessed Word Givn Word by Experts
1 (0.81,0.12,0.29) not safe not safe
2 (0.84,0.17,0.56) not safe not safe3 (0.91,0.17,0.40) not safe not safe4 (0.42,0.17,0.41) questionable questionable5 (0.13,0.05,0.41) safe safe6 (0.80,0.31,0.40) not safe not safe7 (1.03,0.31,0.40) not safe not safe8 (0.77,0.31,0.35) not safe not safe9 (0.54,0.22,0.40) questionable questionable10 (0.61,0.31,0.39) questionable questionable11 (0.86,0.40,0.39) not safe not safe12 (0.93,0.45,0.38) not safe not safe13 (1.15,0.45,0.39) not safe not safe14 (0.96,0.45,0.39) not safe not safe15 (0.76,0.45,0.40) not safe not safe16 (0.60,0.45,0.39) questionable questionable
Predicted Valu by the Model
Junzo WatadaNumerical Example
Oct. 19, 2012 Botho Colledge Conference 2012 87
Evaluation using testing samples
New Corrosion St. Cracking St. Loading and Env. State Linguistic Value Linguistic Value
Samples X1 X2 X3 by the Model by Experts
21 good extremely bad below normal not safe not safe 22 very good very good normal questionable questionable 23 bad extremely good below normal safe safe 24 very good extremely good below normal safe safe 25 bad very bad normal not safe not safe
Attribute Total Assessment Y
Oil Palm Fruit Evaluation
Oct. 19, 2012 Botho Collage Conference 2012 8888
Oil Palm Fruit Anatomy
DECISION MAKING
QUALITY INSPECTION
Graded Fruit
Condition
FruitletsColor
Surface
(MPOB, 2003)
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 89
Junzo WatadaLinguistic Evaluation
Oct. 19, 2012 90
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 91
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 92
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 93
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 94
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 95
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 96
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 97
Junzo Watada
Oct. 19, 2012 Botho Colledge Conference 2012 98
Junzo Watada
99
Thank You !Thank You !
Junzo Watada
Oct. 19, 2012 Botho Collage Conference 2012 100
Junzo Watada
Fuzzy Regression Model
Oct. 19, 2012 Botho Colledge Conference 2012 101
LP Problem
),,2,1(0
tosubject
minimize1
mj
xxy
xxy
x
jjj
jjj
m
jj
c
ca
ca
c Vagueness
upper bound
lower bound
coefficient
101
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