Transformation method of decision-making probability based on correlation degree

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11

2015 11 Systems Engineering — Theory & Practice : 1000-6788(2015)11-2932-07 : TP391 ! " # $ % & ')()* + 35

Vol.35, No.11 Nov., 2015 : A

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Transformation method of decision-making probability based on correlation degree ZHAO Yu-xin, JIA Ren-feng, LIU Chang, SHEN Zhi-feng (College of Automation, Harbin Engineering University, Harbin 150001, China)

Abstract To slove the problem in the transformation of basic probability assignment to decision-making probability, this paper proposed a novel transformation method based on correlation degree. The correlation degree between basic probability assignment of singleton proposition and decision-making probability was used to evaluate the transformation method, and the decision-making probability of each proposition was achieved by linear combination, which was the transformation method of decision-making probability based on proportional belief and proportional plausibility. The proposed method was compared to the other usual methods with an example. The experimental result shows that the proposed method is more reasonable and effective. Keywords basic probability assignment; correlation degree; belief; plausibility; decision-making probability

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: 2014-02-28 : (51379049); (LC2013C21); : (1980–), , , hrbeu.edu.cn; (1987–), , ,

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sJtJu , : JY J J J J J J J J¡J¢J£J J¤ 2933 ¥ $ (plausibility function transform, PFT), ¦ § ¨ © E F G # $ ª ¯ « ¬ , à A % § ¨ ªM¼ F G ¯ « ¬ , ¯ ­ ®=<]½ ¯ ° . Dezert ÀM´M¯ DSmP ÞMß=<]½MÏMÐ , äMå ±=²]â , µ ³ G ¯ ´ µM¬MÆ ¶ · . ÀM´ © <º¹MªM¼ # $ ÞMß @=<º¹ E ¼ #"$ ÞMß , <º¹MªM¼ # $ ÞMß # $ M¯ »M¼=<]½ ¼ · , Daniel ¸ 4 5 ½ ¾¾M ¢M£M%Mì íÞM;ßM<º¯ ¹ Ø E ¿ ¼ # $ ÞMß # $ ¯= M<º¯ ¹ »M# ¼=$ <]ÞM½ ßM¯ ¯M° ¬ ,Á ¬ 9 ÀM¸ ´ ¢M£ . Ø ©MþMî ÿ Ç"# ¸ $ + ÞMÂ ß ¼M¯ ¯MÈ ¢M£ # $ À Daniel , # $ Ã Ä Å ÞMß . 2 ÞMß û MÍM¯ ,BPA M ¯ M ¢ £ ¯ " ¼ Æ <ʹ ª ¼ #"$ Þ ß"@Ë<ʹ E" ¼ #"$ Þ ß Ú Û"Ì Ü"Í Â , ´" +"Í Â ¯ ¢ £ ý " . Î Ã , Ï ª"Ð ¹"! Ñ , ×MÄ"å © É "¾ ÞMßM¯ Â Ü . 1 Daniel ÒÔÓÔÕÔÖÔ×ÙØÙÚÙÛÝÜ 1.1 Daniel Þ ß à á â ã ä <º¹MªM¼ # $ ÞMßM¯ # $ å æ : 11

[15]

[16]

P ropBelP (A) =

m(A) P m(X) m(B)

X

(1)

MÞ ß 9 å MÍM M¡ Í ¤ ¯ ªM¼Ë<ºç #"$ ¢ £ , # $ M¯ »M¼=<]½ ¼ · , 4 5 ½ ¾M¢ 2 M£ ì í . <º¹ E ¼ # $ ÞMßM¯ # $ å æ : A∈X⊆2Θ B∈X

P ropP lP (A) =

P l(A) P m(X) P l(B)

X

(2)

MÞ ß 9 å MÍM M¡ Í ¤ ¯ E ¼=<Êç # $ ¢M£ , # $ M¯ »M¼ è=<]½ ¯ ° , ¬ 9 ¸ 2 ¢M£ . é ê ë 1.2 Æ ¬MºM»MÜM¼Mñ —— î ¯ ï ð ñ , ¾ ò [17] ó ô © õ ö ¥ ñM ¯ ÷ î @ + Â î , ø É ÜMð × » ù © + ý ÂMì í G . ú û 1 ü õ ö ¥ ñ X Y ¯ ýMý A∈X⊆2Θ B∈X

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(

a1 , a2 , · · · , ak p1 , p2 , · · · , pk

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Y

)

,Y =

¯ ÷ î » ù ý

(

ú û

)

,

:

HY (X) = −

ú û

b1 , b2 , · · · , bk q1 , q2 , · · · , qk

H(X) = −

n X

qk log pk

(3)

pk log pk

(4)

k=1 n X

õ ö ¥ ñ X Y Ä Å ¯ + Â î » ù ýMÎ ( ¯ ÷ î Ä @ , H(X; Y ) = H (X) + H (Y ) ¥ M ñ ¯ ÷ õ ö + ÂM GM + ÂM G ¸ » ù ý : k=1

2

Y

3

X

H(Y ) HY (X) H(X) γX (Y ) = HX (Y )

γY (X) =

% M²MÜMð + M Â G

γ(X; Y ) =

H(X ⊗ Y ) H(X) + H(Y ) = H(X; Y ) HY (X) + HX (Y )

(5) (6) (7) (8)

:

1 ∀X, Y

? ?

X, Y ý ý

0 ≤ γX (Y ), γY (X), γ(X; Y ) ≤ 1

(9)

γX (Y ) = γY (X) = γ(X; Y ) = 1

(10)

,

Ù Ù

& ÜMð % ¥ + ñ ÂM G γ¯ õ C ö ¥ ñ XÐ Y ¯´ ý Ä Mþ ÜÔÓ þ ¯M °MÜM± ¯ M¼M¾ ñ ò (0) = −∞ , ý & ¯ ïµ , õ, ? ö ¥ ñ Xý +M0¸ õ ö ¥ , ñ Y ¯ ÷ D î » ù log ý : 2934

HY (X) = H(X) =

n X

35

. [18]

ÀM´ ©MþMÿ î

qk e−5pk

(11)

pk e−5pk

(12)

k=1 n X

¯ å æ ¿MÆM½MÇMÈMÉM¢ M¾M¯MÌMÍ , ¯ Ä © Ð ¯ ! ºMÜ . 2 ÔØÔÚ ! " #ÙÒÙÓÝÕÙÖ ÇMÈ % &=<º¹MªM¼ # $ ÞMß"@=<º¹ ¼ # $ ÞMß ¯ ' ( ¯M¬ Á ¾ 9 ¯ MÜý ,© ÀM$ ´ ©MþMÿ Ø ¸ +  ¼M¯M¢M£ # $ E ÞM ß . 2 ÞMß <º¹MªM,¼ ) # *M$ Î ÞM( ß @=<º¹ , E ¼ # å $+ ÞM ßM+ ÚG Û ÌMÜ Í Â , , # $ M¯ »M¼ -M¬ ¼ · ¬ ¯ ° , M´  . / 0 1 2 3 . 4 5 6 7 8 9 : 0 / ; < = > ? @BA C /BD E , A C / F G 0 E H I J K L / M N , O I J P Q > R S TVU W N I J 6 7 / P Q X Y Z , [ > R S TVU \ ] N I J 6 7 / P Q X Y Z . ^V_ X ` a b < > c d ` a b < e f / Y g h i , 0 1 2 3 I J > j k g h i l m n o p . q Q d ` a b < / BPA r F , s t c u / 0 1 2 3 v u w x F , y z { z | } ~ / k g V , D E ? @ A C . j d ` a b < / BPA I J / 0 1 2 3 / , } _ N o I J / 0 1 2 3 BPA / N v R I J 6 7 /B . q Q TVU W N I J 6 7 /B BN r F , w 6 7 / I J P Q r , BPA / W , w I J 6 7 r , I J K / M N = u w Y Z ; q Q TVU \ ] N I J 6 7 / N r F , w 6 7 r TVU W N I J 6 7 , I J K / M N = u w Y Z . ¡ y ¢ , 4 5 6 7 / I J £ ¤ q ¥ : k=1

¦ §

m(A) P m(X) + α2 m(B)

X

N ewP (A) = α1

A∈X⊆2Θ B∈X

,

P l(A) P m(X) P l(B)

(14)

γP l = 1 − α1 γBel + γP l

(15)

γ γ ¨ © ª TVU W N I J 6 7 T«U \B] N IBJ 6B7¬§VdB` aBbB< / 3 / £ ¤ (13) v { . z ­ ® Bel

¦ § ¯ ª

Pl

N ewP (A) = α

X

m(A) P m(X) + (1 − α) m(B)

A∈X⊆2Θ B∈X

,

α=

A

(13)

A∈X⊆2Θ B∈X

γBel γBel + γP l

α1 = α2 =

X

X

BPA

I J / 0 1 2

P l(A) P m(X) P l(B)

(16)

A∈X⊆2Θ B∈X

γBel γBel + γP l

(17)

> d ` a b < , y z I J £ ¤ ° { ­ ®  N ewP (A) = m(A) +

X

 m(A) P l(A) α P  m(X) + (1 − α) P m(B) P l(B)

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B∈X

B∈X

[19]

Pm(A) m(B)

B∈X

(18)

.

PP l(A) P l(B)

B∈X

N ewP (A) = m(A) +

[αf (A) + (1 − α)g(A)]m(X)

A∈X⊆2Θ ,|X|>1

(19)

Â

11

Ã

ÄÆÅÆÇ , È : ÆÉ ÊÆËÆÌÆÍÆÎÆÏÆÐÆÑÆÒÆÓÆÔÆÕÆÖ

à á

1

× 1 ÆØ ÙÆÚÆÛÆÜÆÝÆÞÆß Y ç ½ ¡ P T (·) â ã ä ¥ å æ ,O

è éëê

Bel(A) ≤ P T (A) ≤ P l(A) X

P1 (A) = m(A) +

Á ¯ ª

(20)

f (A)m(X), P2 (A) = m(A) +

A∈X⊆2Θ ,|X|>1

° ¯ ª

2935

X

g(A)m(X),

A∈X⊆2Θ ,|X|>1

N ewP (A) = αP1 (A) + (1 − α)P2 (A), X

P1 (A) = m(A) +

f (A)m(X) ≥ m(A) = Bel(A),

A∈X⊆2Θ ,|X|>1

P1 (A) = m(A) +

y z ì . { í ° ¯ ª N ewP (A) >

X

f (A)m(X) ≤ m(A) +

A∈X⊆2Θ ,|X|>1

A∈X⊆2Θ ,|X|>1

Bel(A) ≤ P1 (A) ≤ P l(A),

P1 (A)

/ î ½ ? @ , y z

Bel(A) ≤ P2 (A) ≤ P l(A), P2 (A)

X

Bel(A) ≤ N ewP (A) ≤ P l(A).

m(X) = P l(A),

 35 ø ï ð ñ ò ó ô õ ö ÷ ù ú 1 û c ü ý þ BPA, P T (·) â ã ä 2 3 Y ç ½ , O P T (A) = m(A) (21) V ^ £ ¤ è éë¯ ª m(A) > ü ý þ BPA, y z m(X) (19) { ÿ N ewP (A) = m(A). ù ú 2 c _ { A, P T (·) â ã ä = 0, 0 (22) ^ èBé ¯ ª A > ª { , yBz Bel(A)P T=(A)0, P=l(A) = 0, Bel(A) ≤ N ewP (A) ≤ P l(A) {Bÿ N ewP (A) = 0. à á 2 c _ © Θ ¡ ¸ R(·), R(Θ) → Θ, P T (·) â ã ä 2936

¸ R(·) ª

è éë¯ ª

P T ∗ (R(A)) = P T (R(A))

, R(Θ) → Θ,

(23)

y z

R(A) = A, m∗ (R(A)) = m(A),

y z

X

N ewP ∗ (R(A)) = m∗ (R(A)) +

[αf (A) + (1 − α)g(A)]m(X),

R(A)∈X⊆2Θ ,|X|>1

y z

X

N ewP ∗ (R(A)) = m(A) +

[αf (A) + (1 − α)g(A)]m(X) = N ewP (A).

¡ y ¢ , 4 5 6 7 ã ä Daniel l · 0 1 2 3 I J ¸ E ¹ . ² U c 4 5 6 7 ± ² ¼ ´ ¾ ¿ , z c ¦ @ ½ ¾ ¿ í . ¥ º » ¼ 1 À i © Θ = {θ , θ , θ , θ }, w ¥ 4 2 3 q h 1 y ¶ 1 ÆÚÆÛ A∈X⊆2Θ ,|X|>1

3

1

θ1 0.30

θ2 0.15

θ3 0.01

2

θ4 0.09

3

.

4

θ1 ∪ θ 2 ∪ θ 3 0.25

θ1 ∪ θ 2 ∪ θ 4 0.05

θ2 ∪ θ 3 ∪ θ 4 0.05

θ1 ∪ θ 2 ∪ θ 3 ∪ θ 4 0.10

4 5 6 7 ¾ ¿ q ¥ ± ² : T«UBW NBIBJB0B1B2B3 6B7 , ¨B© ±B²B· θ , θ , θ , θ 0B1B2B3 P (θ ), P (θ ), P (θ ), 1) h 2 y ¶ . P (θ ) q T«UB\B] NBIBJB0B1B2B3 6B7 , ¨B© ±B²B· θ , θ , θ , θ 0B1B2B3 P (θ ), P (θ ), P (θ ), h 3 y ¶ . P (θ ) q j d ` a b < BPA ! ® > b < 2 3 ¨ " X, j TVU W N I J 6 7 TVU \ ] N I J 6 7 ÿ # 2) 0 1 2 3 ¨ © ! ® > 2 3 ¨ " Y $ Y q h 4 y ¶ . 4 % & 'ÆÚÆÛ ( ) 2 ØÆÙÆÚÆÛ P (θ ) 3 ØÆÙÆÚÆÛ P (θ ) θ θ θ θ m(·)

1

2

1

2

3

4

1

1

1

2

1

3

1

2

3

4

2

1

2

2

2

3

4

4

1

1

2

2

i

i

P1 (θ1 ) P1 (θ2 ) P1 (θ3 ) P1 (θ4 )

P2 (θ1 ) P2 (θ2 ) P2 (θ3 ) P2 (θ4 )

0.5453 0.3027 0.0193 0.1327

0.4740 0.3021 0.0918 0.1321

± ² T«UBW NBIBJB6B7 T«UB\B] NBIBJB6B7¬§VdB`Ba bB< B ¨ B © γ , γ . * ± ² { ÿ : 3)

Bel

1

2

BPA

Pl

HY1 (X) = 0.3676, HX (Y1 ) = 0.1081, HY2 (X) = 0.4200, HX (Y2 ) = 0.1140,

γBel =

4

IBJB 0B1B2B3 B

H(X) = 0.2047, H(Y1) = 0.1882, H(Y2) = 0.2372,

Á

3

X 0.3000 0.1500 0.0100 0.0900 Y1 0.5453 0.3027 0.0193 0.1327 Y2 0.4740 0.3021 0.0918 0.1321

H(X) + H(Y1 ) H(X) + H(Y2 ) = 0.8259, γP l = = 0.8277. HY1 (X) + HX (Y1 ) HY2 (X) + HX (Y2 )

Â

11

Ã

4)

± ² ? @ A C

ÄÆÅÆÇ , È : ÆÉ ÊÆËÆÌÆÍÆÎÆÏÆÐÆÑÆÒÆÓÆÔÆÕÆÖ 5 ØÆÙÆÚÆÛ = 0.4995,

α. γBel α= γBel + γP l 1 − α = 0.5005. 5) P1 (θ1 ) P2 (θ1 ) P1 (θ2 ) P2 (θ2 ) P1 (θ3 ) P2 (θ3 ) P1 (θ4 ) P2 (θ4 ) , θ1 , θ2 , θ3 , θ4 P (θ1 ), P (θ2 ), P (θ3 ), P (θ4 ) 5 . 6 , DSmP ε , ε = 0.5. 6 , θ1 . BetP CuzzP , ,

Á

¨B© c

P (θi )

P (θ1 )

P (θ2 )

P (θ3 )

P (θ4 )

0.5096

0.3024

0.0556

0.1324

6 = > ? @ A B C D > ? E F $ ¾ ¿ î ½ ? @ $ ÿ $ 0 1 2 3 · θ θ θ θ γ + , h BetP(·) 0.4250 0.2917 0.1350 0.1483 0.8057 0.3500 0.3000 0.2050 0.1450 0.7441 q h § jy 4¶ 5 6 7 ¦.-./ } 6 7 ¾ ¿ n c T PFT(·) CuzzP(·) 0.4241 0.2897 0.1341 0.1521 0.8066 ^V_ 6 7 §10 2 3 4 D E k 5 3 § DSmP (·) 0.4530 0.2929 0.1096 0.1445 0.8219 6Bh PBQ 0.5453 0.3027 0.0193 0.1327 0.8259 · w U § 7 gB6B7 ÿ # PropBelP(·) B { z ! PropPlP(·) 0.4740 0.3021 0.0918 0.1321 0.8277 6B7 G H ÕÆÖ (·) 0.5096 0.3024 0.0556 0.1324 0.8359 PBQ 8B>BbB< 0B1B2B3 + F 6 7 8 > j X ` a b < W 9 : l ; < d ` a b < k g ¨ I 6 ¤ J K ¨ | }ML1N ¨ W O 4 P ® W Q R , | _ 0 1 . PFT 6 7 ÿ # b < θ 0 1 2 3 P (θ ) = 0.35, b < θ 0 1 2 3 P (θ ) = 0.3 S T > x F , 4 _ 0 1 U V · @ 0 1 . DSmP 6 7 W ] X _ Y 7 g 6 7 , Z ÿ # P Q v > x [ . PropBelP 6 7 | } n L1N W , ÿ # PBQ \ \ >BxB , { >BwB6B7 IBJ ¼ ´ ¼ _B B , O 4 ]BF ^B1 _ ` . PropPlP 6B7 W ] v |B} n LaN W , Z ÿ # P Q T r , | _ V · @ ^ 1 . 4 5 6 7 ÿ # N γ T ¦ - / } 6 7 8 9 F , w 6 7 ÿ # P Q r BPA W , b c ÿ # ^ 1 2 3 Yed n ] $ | _ ^ 1 .

2937

1

2

1

2

3

4

ε=0.5

1

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f g h i _ D-S í j u } § , k k ^ 1 U ª n l X 6 m § 2 · Y < + X 6 m , n o p q } + F r U n Q R s + G , â 9 c ; < 6 m ¾ ¿ t , 6 b V · @ ^ 1 . Z > x X u v ¥ , y w x W > x y º , ^ 1 U x z V · ^ 1 , b ^ 1 2 3 I J r : ^ k < ; < , ª ^ 1 U { | } D $ { ~ P Q . 4 Y < ^ 1 2 3 I J 6 º u } , v z ^ 1 2 3 I J 9 ½ . 2 L1N n , c 6 4 < 6 x, y, z, w P ® nM VF , c L1w x9 W ¾ ¿ n t , ÿ # d kY 4 6 < 6 { # 4 2 3 i q 2 h 6 7m y ¶ h . ª n ° k 4 < Q 6 , â Y ¾ ¿ , Z , â 9 · Y E s 3 , z q Q , y ,z ì 6 ° ¡ P q ® ì8 y ´ ¶ N . Q N R , q s ¸ q h 9 y ¶ . t { ¢ N£ 4 5 6 7 ¾ ¿ ± ² , ÿ # k 4 < 6 { # 2 3 q h 10 y ¶ . 7 ¤ ¥ ¦ § ¨ © ' ÆÚÆÛ 8 ª « > ¬ x y z w x∪w x∪y∪z∪w ® ¯ ° ± {x,Õ ­ y, z}1 {x,Õ ­y, w}2 {x,Õ ­ z, w}3 {y,Õ ­z, w}4 m(·) 0.15 0.20 0.10 0.10 0.20 0.25 9 ² ³ ´ µ 10 ¤ ¥ ¦ § ¨ © 'ÆÚÆÛ x y z w ® ¯ −10 x y z w −10 −10 −10 ¶ ® ¯ −20 −30 −60 −100 p(·) 0.3364 0.2744 0.1452 0.2440 s t # · ¸ Y 7 < c 6 ½ m g ¹B6 > m+ X Q 6 R m s, ¾Bn ¿ ±B6 ² R Eº # < + 6 X m > , n X Q 6 R m » # + G ¼ ? { k | 5 @ , ¾ ¸ , ª + { z ^ » 1 ¼ e o f .p 6 m 1 Q R s : 0.3364 × (−10) + 0.2744 × (−10) + 0.1452 × (−10) + 0.2440 × (−100) = −31.96. 6 m 2 Q R s : 0.3364 × (−10) + 0.2744 × (−10) + 0.1452 × (−60) + 0.2440 × (−10) = −17.26. 6 m 3 Q R s : 0.3364 × (−10) + 0.2744 × (−30) + 0.1452 × (−10) + 0.2440 × (−10) = −15.49. 6 m 4 Q R s : 0.3364 × (−20) + 0.2744 × (−10) + 0.1452 × (−10) + 0.2440 × (−10) = −13.36. 6 ¡ º ±B²BPBQ z ! · , 6 m 4 Q R s + G , yBzB 6BuBw 2 ¿ 6 m 4 ¾B¿ , O c < 6 ¾ ¿ . y, z, w 3 4

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