Differentiation in Fuzzy Calculus and its Applications

1
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Differentiation in Fuzzy Calculus and its
Applications to Decision Making and Financial
Engineering

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• (???????????)

2
1 ????????Y-CG????????????????
(IFSA99,IJFS99)
1)???????Y-CG??
2)????????????
????? ????ROI ????
?????? (ROI???)
9?? ??????
??????? ???????? (10,000 ?)
?????? ?Y-CG??
?????
(David Hertz??????)
3
1 ????????Y-CG????????????????
(IFSA99,IJFS99)
?????? ???????? (10,000 ?)
9??????? ??
3????? (????/???????/ ???????)
????????????????? ????????? ??????? ?????-??
?? ROI
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1.?? 2.?? 3.?????? 4.???????? 5.???
6.???? 7.???? 8.??? 9.???
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(David Hertz??????)
ROI? ???????
???????? ???????? ROI??????
?? ??
???????? Y-CG??
??????? (ROI??)
?????
4
1 ????????Y-CG????????????????
9???????????
?????/????/?????
? ????????()
3
17
12
?3.1 ????????????????
5
1 ????????Y-CG????????????????
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??????()
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0
3
6
545
435
10
370
34
25
575
510
385
????? (years)
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? ????(million )
375
250
300
350
500
450
15
10
5
700
1050
950
?3.1 9?????????
6
1 ????????Y-CG????????????????
David Hertz????
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???? ????????????? ??? ?????? ???? ??? ??
???-???? ROI
??? ???
ROI 25.2
??
ROI??? 17.0
?? ??
????? ???
?? ???
ROI???????
7
1 ????????Y-CG????????????????
(IFSA99,IJFS99)
ROI ()
f(x0)
F(x0) (m10)
??(/ton)
?3.4 ??x0(/ton) ?ROI f(x0) ()???
(Page 33)
8
1 ????????Y-CG????????????????
(IFSA99,IJFS99)
? (48) of page 30
?ROI ???
0.3
????
????
0.39
??(/ton)
????
?3.5 ?????? ? ???
(Page 35)
9
3.Choquet????????????????????
(FUZZ-IEEE99)
3.1 ??????????????????????? ?
(????)
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µ
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f
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Choquet??? ?? ????? ????????? ???? ?????
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(C)?f dµ
X
????
10
3.Choquet????????????????????
(FUZZ-IEEE99)
3.1 ??????????????????????? ?
?? ????????????????????????????????? 1.
?????????µ????? ? µ(??????) 0.5
? µ(????) 0.4 (37) ?
µ(????) 0.3 ? µ(???) 0.2 2.
V(??????????????????????)? ?????? ?
V(????) 0.35 ? V(??????) 0.32 (38)
? V(???) 0.2 ? V(????) 0.13
11
3.Choquet????????????????????
(JACI2000)
3.2 ??????????? ?
?????????????????
?????????????
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??????
??????
MA
MBO
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?????????
?????????????
12
3.Choquet????????????????????
(JACI2000)
3.2 ??????????? ?
? ????????????????????????
µ????
??????????
Choquet ????? ?????
????
Moodys? ?????
(C)?f(x) dµ
(?????)
X
(????????)
????
(?????????)
???????? ?????????µ???
??? ???µ
?????? ?????? ????????
Choquet?????
????
(?????)
??????? ??????
????
(?????????)
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13
3.Choquet????????????????????
(JACI2000)
3.2 ??????????? ?
????????????????????????
µ????
Choquet ???? ??????? yj
??????????
Moodys? ????? Yj
? ? ?
?17??? ????? ????? ??????
f
(C)? f(x) dµ
X
???????? ?????????µ???
(????/????) x1???? x2???????? x3????????????
?? x4???? x5???? x6??????? x7???????
x8ROE x9ROA x10????? (????/?????????) x1
1??? x12??? x13?? x14??
L(µ(xi))?(Yj-yj)2
µ???????????? ??????(Aaa),?????(Aa1), ??????(Aa2
),???????(Aa3), ???(A1),?????(A2),????(A3), ???(Ba
a1),????(Baa2), ???(Baa3),?????????(Ba1), ???(Ba2)
,?????????(Ba3), ????(B1),?????(B2), ????(B3),????
(Caa1)
5000? ??
(TOYOKEIZAI)
??? ???µ
µ(????)0.39,µ(????????)0.41,µ(?????????????)0.4
0, µ(????)0.38,µ(????)0.40,µ(???????)0.40,
µ(???????)0.47,µ(ROE)0.40,µ(ROA)0.40,
µ(??)0.37 µ(?????)0.41,µ(???)0.49,µ(???)0.39
,µ(??)0.40
14
3.Choquet????????????????????
3.2 ??????????? ?
Aa
A
Ba
B
C
X?Choquet??????? Y????
?3.1 ????????????????????????(81?)
?Choquet???????????
15
3.Choquet????????????????????
3.2 ??????????? ?
??????????/???????????
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???? Moodys,SP,??DB,?
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16
3.Choquet????????????????????
3.3 ????????????? ?
???????????????????????????
???(???)????????(???)?????
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17
3.Choquet????????????????????
3.3 ????????????? ?
??????????????(???????)
(99?4?30?)
5?
6?
7?
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840 440 200 80 35
-140 -130 -90 -45 -10
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190 270 420 600 845
6275 7324 4464 1595 1292
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???? HV 15.4 ???? IV 24.0
??? 12621 ???? 11361 ??? 68612 ???? 97107
???? ???
18
3.Choquet????????????????????
(IEEE SMC99)
3.3 ????????????? ?
1999?2?19?5?20??????(60???) ???????????????
µ
µ
FP
?????? ?????? ?????? ?????
S 1r
????????16199.99
?5.9 ??????????????????
19
3.Choquet????????????????????
(FUZZ-IEEE2000)
3.4 ?????????????????(????) ?
3200????? ?????? ??????
1???? ???
1???? ???? (ROI)
?
1???? ????F
Choquet??? x??????
????µ (????? ????????)
???? ??
????? ????? ???? ??F
X???? ???? Choquet??
????? ????? ?? f

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?5.7 ??????????1????????????
20
3.Choquet????????????????????
(FUZZ-IEEE2000)
3.4 ????????????????? ?
? ?????????????
F(x,µ) ?????
(page98)
x ( 0 ?x?2000) f(x)
0.9(x-2000)2000 (2000?x?4000)
0.8(x-4000)3800 (4000?x?5000) .
3,228
(196)
2,244
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0
(0?xlt1000)
1,422
x2 6000
500 3
-
(1000?xlt2000)
500
2x2 10000
7 90
1300 9
-
x-
(197)
F(x,µ)
(2000?xlt3000)
2x2 10000
52 135
7000 9
-
x
(3000?xlt4000)
0
1000
2000
3000
4000
5000
52400 27
??
x2 7500
221 540
-
(4000?xlt5000)
x
?5.12 ?????????????
21
3.Choquet????????????????????
(FUZZ-IEEE2000)
3.4 ????????????????? ?
???????
F(x,µ)
1?????????? ????????
?????
??3,200??????
1????????
????? ???????
??
?????
??3,200 ??????
5000
2000
3000
1000
4000
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?5.13 ?????1????????? ???????????1?
??????????
22
????
1.?????? 1 T.Kaino, K.Hirota Y-CG Derivative
of Fuzzy Relations and its Application to
Sensitivity Analysis,
The International Journal of Fuzzy
Systems,Vol.1, No.2,pp.129-132 (Dec.1999) 2
T.Kaino, K.HirotaDifferentiation of the Choquet
Integral and Its Application to Long-term
Debt Ratings,
Journal of Advanced Computational
Intelligence.(to appear) 2.???? 3 T.Kaino,
K.Hirota "Derivative of Fuzzy Relations and Its
Application to Capital Investment
Decision Making Problem",
Proc. of IFSA'99, Taiwan, pp. 995-998, 1999.
4 T.Kaino, K.Hirota "Differentiation of the
Choquet Integral of a Nonnegative Measurable
Function",
Proc. of FUZZ-IEEE'99, Seoul, Vol.?,
pp.1322-1327, 1999. 5 T.Kaino, K.Hirota
Differentiation of Nonnegative Measurable
Function Choquet Integral over
Real Fuzzy Measure Space and
its Application to Financial Option Trading
Model, Proc. of
IEEE SMC99, Tokyo,?, pp.73-78, 1999. 6
T.Kaino, K.Hirota Differentiation of Choquet
Integral for Nonnegative Measurable Function
and its Application
to Capital Investment Decision Making Problem,
Proc. of
FUZZ-IEEE2000, Texas, pp.89-93, 2000.
3.????????? 7 ??, ?? Choquet????????????????
??, ?15?????? ????
?????????????, ??, pp. 805-806, 1999. 8
??,??Choquet??????????/????????,
?10?????????? ?????????????, ??,
pp.92-95, 2000.