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TRANSCRIPT
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November 14, 2007
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What is Rough Set?
A�ÎÒ:
U k��, U ={x1, x2, . . . , xn
}.
R �d'X(÷vg�!é¡ÚD45).
[x]R �da, [x]R ={y ∈ U
∣∣ (x, y) ∈ R}.
U/R �d'X R y©Ø� U , ¤��da�8Ü.
¯K:
Question
�½ X ⊆ U , XÛ^�da
[xi1 ]R , [xi2 ]R , · · · , [xik ]R
£ãL� X?
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What is Rough Set?
A�ÎÒ:
U k��, U ={x1, x2, . . . , xn
}.
R �d'X(÷vg�!é¡ÚD45).
[x]R �da, [x]R ={y ∈ U
∣∣ (x, y) ∈ R}.
U/R �d'X R y©Ø� U , ¤��da�8Ü.
¯K:
Question
�½ X ⊆ U , XÛ^�da
[xi1 ]R , [xi2 ]R , · · · , [xik ]R
£ãL� X?
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What is Rough Set?
�½Ø� U ;^���d'Xò U ?1y©;�½8I8Ü X;X �eCq RX =
{x ∈ U
∣∣ [x]R ⊆ X}.
X �>.�.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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What is Rough Set?
�½Ø� U ;^���d'Xò U ?1y©;�½8I8Ü X;X �eCq RX =
{x ∈ U
∣∣ [x]R ⊆ X}.
X �>.�.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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What is Rough Set?
�½Ø� U ;^���d'Xò U ?1y©;�½8I8Ü X;X �eCq RX =
{x ∈ U
∣∣ [x]R ⊆ X}.
X �>.�.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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What is Rough Set?
�½Ø� U ;^���d'Xò U ?1y©;�½8I8Ü X;X �eCq RX =
{x ∈ U
∣∣ [x]R ⊆ X}.
X �>.�.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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What is Rough Set?
�½Ø� U ;^���d'Xò U ?1y©;�½8I8Ü X;X �eCq RX =
{x ∈ U
∣∣ [x]R ⊆ X}.
X �>.�.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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�½ X ⊆ U , �^ U/R ¥���5£ã!L� X, Ø�½U°(/?1©�~~�±^'u X ��éeCq!þCq5.½ X, ù��o÷8Vg��)©
½Â (Pawlak(1982)[2])
� R ´Ø� U þ��d'X, é8Ü X ⊆ U , óé
(RX, RX
)¡� X 3Cq�m (U,R) þ���o÷C
q, Ù¥
RX ={x ∈ U
∣∣ [x]R ⊆ X}
,
RX ={x ∈ U
∣∣ [x]R ∩X 6= ∅}
.(1)
RX!RX ©O¡� X � R eCqÚ R þCq©
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�½ X ⊆ U , �^ U/R ¥���5£ã!L� X, Ø�½U°(/?1©�~~�±^'u X ��éeCq!þCq5.½ X, ù��o÷8Vg��)©
½Â (Pawlak(1982)[2])
� R ´Ø� U þ��d'X, é8Ü X ⊆ U , óé
(RX, RX
)¡� X 3Cq�m (U,R) þ���o÷C
q, Ù¥
RX ={x ∈ U
∣∣ [x]R ⊆ X}
,
RX ={x ∈ U
∣∣ [x]R ∩X 6= ∅}
.(1)
RX!RX ©O¡� X � R eCqÚ R þCq©
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��ûüL�~f
(a) ��&EûüL
Ø� ^ � á 5 ûüá5
¾< ÞÛ *SÛ N§ 6a
e1 ´ ´ �~ Ä
e2 ´ ´ p ´
e3 ´ ´ ép ´
e4 Ä ´ �~ Ä
e5 Ä Ä p Ä
e6 Ä ´ ép ´
e7 Ä Ä p ´
e8 Ä ´ ép Ä
(b) êizL��ûüL
U C D
a b c d
1 1 1 1 0
2 1 1 2 1
3 1 1 3 1
4 0 1 1 0
5 0 0 2 0
6 0 1 3 1
7 0 0 2 1
8 0 1 3 0
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(a) ��&EûüL
Ø� ^ � á 5 ûüá5
¾< ÞÛ *SÛ N§ 6a
e1 ´ ´ �~ Ä
e2 ´ ´ p ´
e3 ´ ´ ép ´
e4 Ä ´ �~ Ä
e5 Ä Ä p Ä
e6 Ä ´ ép ´
e7 Ä Ä p ´
e8 Ä ´ ép Ä
(b) êizL��ûüL
U C D
a b c d
1 1 1 1 0
2 1 1 2 1
3 1 1 3 1
4 0 1 1 0
5 0 0 2 0
6 0 1 3 1
7 0 0 2 1
8 0 1 3 0
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ûüL^�á5�«©Ý
ûüL�«©ÝXeL¤«(dué¡5��ÑÙen�Ü©).
1 2 3 4 5 6 7 812 c3 c c4 a a, c a5 a, b, c a, b a, b, c b, c6 a, c a, c a, c c b, c7 a, b, c a, b a, b, c b, c b, c8 a, c a, c a, c c b, c b, c
N´��^�á5�{� {a, c}.
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^�á5��{
ÏLá5�{, ûüL{z�Xe�/ª:
LLL: �{�ûüL
U C Da c d
1 1 1 02 1 2 13 1 3 14 0 1 05 0 2 06 0 3 17 0 2 18 0 3 0
dL�, D/{d} ={{1, 4, 5, 8}, {2, 3, 6, 7}
};
U/{a, c} ={{1}, {2}, {3}, {4}, {5, 7}, {6, 8}
}.
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ûü5K
P D0 = {1, 4, 5, 8}, D1 = {2, 3, 6, 7}, K RD0 = {1, 4},RD1 = {2, 3}. ? ��(½�ûü5K:
r1 : (a, 1) ∧ (c, 1) 7−→ (d, 0); (2)r2 : (a, 0) ∧ (c, 1) 7−→ (d, 0); (3)r3 : (a, 1) ∧ (c, 3) 7−→ (d, 1); (4)r4 : (a, 1) ∧ (c, 2) 7−→ (d, 1). (5)
ù�ÒlÃS,�&E¥���<�Jøë��ûü5K:
(ÞÛ, ´)�(N§, �~) 7−→ (6a, Ä); (6)(ÞÛ, Ä)�(N§, �~) 7−→ (6a, Ä); (7)(ÞÛ, ´)�(N§, ép) 7−→ (6a, ´); (8)
(ÞÛ, ´)�(N§, p) 7−→ (6a, ´). (9)
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ûü5K
P D0 = {1, 4, 5, 8}, D1 = {2, 3, 6, 7}, K RD0 = {1, 4},RD1 = {2, 3}. ? ��(½�ûü5K:
r1 : (a, 1) ∧ (c, 1) 7−→ (d, 0); (2)r2 : (a, 0) ∧ (c, 1) 7−→ (d, 0); (3)r3 : (a, 1) ∧ (c, 3) 7−→ (d, 1); (4)r4 : (a, 1) ∧ (c, 2) 7−→ (d, 1). (5)
ù�ÒlÃS,�&E¥���<�Jøë��ûü5K:
(ÞÛ, ´)�(N§, �~) 7−→ (6a, Ä); (6)(ÞÛ, Ä)�(N§, �~) 7−→ (6a, Ä); (7)(ÞÛ, ´)�(N§, ép) 7−→ (6a, ´); (8)
(ÞÛ, ´)�(N§, p) 7−→ (6a, ´). (9)
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A^^�
o÷8^�:ROSE2(Rough Sets Data Explorer: http://idss.cs.put.poznan.pl);
RSES2(Rough Set Exploration System: http://logic.mimuw.edu.pl);
RS-SYSTEMS(http://www.rs-systems.com/).
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C°Ý�{{0
ù´²;o÷8ã«;ã¥�Ú¬�¹k�þ�k^&E, C°Ý�g�Ò´N�8Ü��¹§Ý, l ¦TÚ¬�B\�eCq8;b�©a�(Ç β = 0.8, Kã¥�Ú¬ÑB\�eCq8.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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C°Ý�{{0
ù´²;o÷8ã«;ã¥�Ú¬�¹k�þ�k^&E, C°Ý�g�Ò´N�8Ü��¹§Ý, l ¦TÚ¬�B\�eCq8;b�©a�(Ç β = 0.8, Kã¥�Ú¬ÑB\�eCq8.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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C°Ý�{{0
ù´²;o÷8ã«;ã¥�Ú¬�¹k�þ�k^&E, C°Ý�g�Ò´N�8Ü��¹§Ý, l ¦TÚ¬�B\�eCq8;b�©a�(Ç β = 0.8, Kã¥�Ú¬ÑB\�eCq8.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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C°Ýo÷8½Â
½Â (Ziarko (1993)[3])
� X ´k�Ø� U ���f8. é β ∈ (0.5, 1], ½Â X� β–eCq!β–þCq©O�
Rβ(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
β⊆X
}, (10)
Rβ(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
1−β⊂ X
}. (11)
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β–þCq8ã«
β–þCq: Rβ(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
1−β⊂ X
}.
b½ 1− β = 0.2, Kã¥�Ú¬Ø2áuþCq8.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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β–þCq8ã«
β–þCq: Rβ(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
1−β⊂ X
}.
b½ 1− β = 0.2, Kã¥�Ú¬Ø2áuþCq8.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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β–þCq8ã«
β–þCq: Rβ(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
1−β⊂ X
}.
b½ 1− β = 0.2, Kã¥�Ú¬Ø2áuþCq8.
8Ü X �>.
X �eCq(��)
X �þeCq��(>.�)
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C°Ýo÷8ã«
b�©a�(Ç β = 0.8.
8Ü X �>.
X � β–eCq
X � β–>.�
�±w�C°Ý�{¦�eCq8O�!þCq8~�, l ¦8Ü£ã�°ÝO�.
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C°Ýo÷8�ÿнÂ
½Â (Katzberg & Ziarko (1996)[4])
?�8Ü X ∈ U Úá58 R, é 0 6 l < u 6 1, 8Ü X� l–eCq!u–þCq©O½Â�
Ru(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
u⊆X
}, (12)
Rl(X) =⋃ {
Xi ∈ U/R∣∣∣ Xi
l⊂X
}. (13)
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Ü©?, Ç��, ù3�, o��.
o÷8nØ��{.
�ÆÑ��, �®, 2001.
Z. Pawlak.
Rough sets.
International Journal of Computer Information Science, 5:341–356,1982.
W. Ziarko.
Variable precision rough set model.
Journal of Computer and System Sciences, 46:39–59, 1993.
J. D. Katzberg and W. Ziarko.
Variable precision extension of rough sets.
Fundamenta Informaticae, 27:155–168, 1996.
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Thank you!
Author: HUANG Zheng-huaAddress: School of Mathematics & Statistics
Wuhan UniversityWuhan, 430072, China
Email: [email protected]