How to implement the belief functions

Mar 7, 2011 - DST. Framework bbas. Adding a constraint: if Θ = {[1 2 3 5],[1 2 4 6],[1 3 4 7]} and ..... rule with parameter optimization. IEEE Transactions on.
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How to implement the belief functions Arnaud Martin [email protected] Universit´ e de Rennes 1 - IRISA, Lannion, France Autrans, April, 5 2011

1/45 How to implement the belief functions?, A. Martin

07/03/11

Order DST Framework bbas

Plan

I

Natural order

I

Smets codes

I

General framework How to obtain bbas?

I

I I I

Random bbas Distance based model probabilistic based model

2/45 How to implement the belief functions?, A. Martin

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Order DST Framework bbas

Plan

I

Natural order

I

Smets codes

I

General framework How to obtain bbas?

I

I I I

Random bbas Distance based model probabilistic based model

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Natural order or Binary order

(1/5) Order DST Framework bbas

Discernment frame: Θ = {θ1 , θ2 , . . . , θn } Power set: all the disjunctions of Θ: 2Θ = {∅, {θ1 }, {θ2 }, {θ1 ∪ θ2 }, . . . , Θ} Natural order: 2Θ = {∅, {θ1 }, {θ2 }, {θ1 ∪ θ2 }, {θ3 }, {θ1 ∪ θ3 }, {θ2 ∪ θ3 }, {θ1 ∪ θ2 ∪ θ3 }, {θ4 }, . . . , Θ}

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(2/5) Order DST Framework bbas

Natural order or Binary order

Natural order: ∅ 0 θ3 4 = 23−1 + 1 θ4 8 = 24−1 + 1 θi i−1 2 +1

θ1 1 θ 1 ∪ θ3 5 ...

θ2 2 θ2 ∪ θ3 6 ...

θ1 ∪ θ2 3 = 22 − 1 θ1 ∪ θ2 ∪ θ3 7 = 23 − 1 ...

...

...

Θ 2n

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Natural order or Binary order

(3/5) Order DST Framework bbas

Bba in Matlab: Example: m1 (θ1 ) = 0.5, m1 (θ3 ) = 0.4, m1 (θ1 ∪ θ2 ∪ θ3 ) = 0.1 m2 (θ3 ) = 0.4, m2 (θ1 ∪ θ3 ) = 0.4 F1=[1 4 7]’; F2=[4 5]’; M1=[0.5 0.4 0.1]’; M2=[0.4 0.6]’;

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(4/5) Order DST Framework bbas

Natural order or Binary order

Combination

mConj (X) =

X

m1 (Y1 )m2 (Y2 )

(1)

Y1 ∩Y2 =X

θ1 ∩ (θ1 ∪ θ3 ): 1 ∩ 5 In binary on base 3: 1=100 and 5=101=100 |001 100&101 = 100

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Natural order or Binary order

(5/5) Order DST Framework bbas

In Matlab: sizeDS=3; F1=[1 4 7]’; F2=[4 5]’; M1=[0.5 0.4 0.1]’; M2=[0.4 0.6]’; Fres=[]; Mres=[]; for i=1:size(F1) for j=1:size(F2) Fres=[Fres bi2de(de2bi(F1(i),sizeDS)&de2bi(F2(j),sizeDS))]; Mres=[Mres M1(i)*M2(j)]; end end 7/45 How to implement the belief functions?, A. Martin

07/03/11

Order DST Framework bbas

Plan

I

Natural order

I

Smets codes

I

General framework How to obtain bbas?

I

I I I

Random bbas Distance based model probabilistic based model

8/45 How to implement the belief functions?, A. Martin

07/03/11

Order DST Framework bbas

Plan

I

Natural order

I

Smets codes

I

General framework How to obtain bbas?

I

I I I

Random bbas Distance based model probabilistic based model

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Smets codes for DST

Order (1/9) DST Framework bbas

Smets gaves the codes of the Mobius transform (see Only Mobius Transf) for conversions: I

bba and belief: mtobel, beltom

I

bba and plausibility: mtopl, pltom

I

bba and communality: mtoq, qtom

I

bba and implicability: mtob, btom

I

bba to pignistic probability: mtobetp

I

etc...

e.g. in Matlab: m1=[0 0.4 0.1 0.2 0.2 0 0 0.1]’; mtobel(m1) gives: 0 0.4000 0.1000 0.7000 0.2000 0.6000 0.3000 1.0000 9/45 How to implement the belief functions?, A. Martin

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Order (2/9) DST Framework bbas

Smets code Conjunctive combination X

mConj (X) =

m Y

mj (Yj )

Y1 ∩...∩Ym =X j=1

The practical way: q(X) =

m Y

qj (X)

j=1

Disjunctive combination X

mDis (X) =

m Y

mj (Yj )

Y1 ∪...∪Ym =X j=1

The practical way: b(X) =

m Y

bj (X)

j=1 10/45 How to implement the belief functions?, A. Martin

07/03/11

DST code

Order (3/9) DST Framework bbas

In Matlab For the conjunctive rule of combination: m1=[0 0.4 0.1 0.2 0.2 0 0 0.1]’; m2=[0 0.2 0.3 0.1 0.1 0 0.2 0.1]’; q1=mtoq(m1); q2=mtoq(m2); qConj=q1.*q2; mConj=qtom(qConj) mConj = 0.4100 0.2200 0.2000 0.0500 0.0900 0 0.0200 0.0100

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DST code

Order (4/9) DST Framework bbas

In Matlab For the disjunctive rule of combination: m1=[0 0.4 0.1 0.2 0.2 0 0 0.1]’; m2=[0 0.2 0.3 0.1 0.1 0 0.2 0.1]’; b1=mtob(m1); b2=mtob(m2); bConj=b1.*b2; bDis=b1.*b2; mDis=btom(bDis) mDis = 0 0.0800 0.0300 0.3100 0.0200 0.0800 0.1300 0.3500

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DST code

Order (5/9) DST Framework bbas

Once bbas are combined, to decide just use the functions mtobel, mtopl or mtobetp, etc. In Matlab mtopl(mConj) 0 0.2800 0.2800 0.5000 0.1200 0.3900 0.3700 0.5900 mtobetp(mConj) 0.4209 0.4040 0.1751 mtopl(mDis) 0 0.8200 0.8200 0.9800 0.5800 0.9700 0.9200 1.0000 mtobetp(mDis) 0.3917 0.3667 0.2417

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DST code

Order (6/9) DST Framework bbas

DST code for the combination: I criteria=1 Smets criteria I criteria=2 Dempster-Shafer criteria (normalized) I criteria=3 Yager criteria I criteria=4 disjunctive combination criteria I criteria=5 Dubois criteria (normalized and disjunctive combination) I criteria=6 Dubois and Prade criteria (mixt combination) I criteria=7 Florea criteria I criteria=8 PCR6 I criteria=9 Cautious Denoeux Min for non-dogmatics functions I criteria=10 Cautious Denoeux Max for separable functions I criteria=11 Hard Denoeux for functions sous-normales I criteria=12 Mean of the bbas 14/45 How to implement the belief functions?, A. Martin

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Order (7/9) DST Framework bbas

PCR6 (Martin et Osswald, 2006, 2007)

Transfers the partial conflict on focal elements given this conflict proportionnaly to the masses. mPCR5 (X) = mConj (X) +

X Y ∈2Θ ,



m1 (X)2 m2 (Y ) m2 (X)2 m1 (Y ) + m1 (X) + m2 (Y ) m2 (X) + m1 (Y )



X∩Y =∅

mPCR6 (X)

=

mConj (X)  s −1 Y mσ (j 0 ) (Yσ (j 0 ) )   j j     j 0 =1     s − 1   X  m (X)+ m (Y ) j σ (j 0 ) σ (j 0 ) 

+

s X

2

mi (X)

j=1

X

s−1 ∩ Yσ (j 0 ) ∩X=∅ j j 0 =1 (Yσ (1) ,...,Yσ (s−1) )∈(2Θ )s−1 j j

j 0 =1

j

j

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DST code

Order (8/9) DST Framework bbas

decisionDST code for the decision: I

criteria=1 maximum of the plausibility

I

criteria=2 maximum of the credibility

I

criteria=3 maximum of the credibility with rejection

I

criteria=4 maximum of the pignistic probaiblity

I

criteria=5 Appriou criteria

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DST code

Order (9/9) DST Framework bbas

test.m: m1=[0 0.4 0.1 0.2 0.2 0 0 0.1]’; m2=[0 0.2 0.3 0.1 0.1 0 0.2 0.1]’; m3=[0.1 0.2 0 0.4 0.1 0.1 0 0.1]’; m3d=discounting(m3,0.95); M comb Smets=DST([m1 m2 m3d],1); M comb PCR6=DST([m1 m2],8); class fusion=decisionDST(M comb Smets’,1) class fusion=decisionDST(M comb PCR6’,1) class fusion=decisionDST(M comb Smets’,5,0.5) 17/45 How to implement the belief functions?, A. Martin

07/03/11

Order DST Framework bbas

Plan

I

Natural order

I

Smets codes

I

General framework How to obtain bbas?

I

I I I

Random bbas Distance based model probabilistic based model

18/45 How to implement the belief functions?, A. Martin

07/03/11

Order DST Framework bbas

Plan

I

Natural order

I

Smets codes

I

General framework How to obtain bbas?

I

I I I

Random bbas Distance based model probabilistic based model

18/45 How to implement the belief functions?, A. Martin

07/03/11

General framework

Order DST (1/12) Framework bbas

I

Main problem of the DST code: all element must be coded (not only the focal elements)

I

Only usable for belief functions defined on power set (2Θ )

I

General belief functions framework works for power set and hyper power set (DΘ )

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Order DST (2/12) Framework bbas

DSmT DSmT introduced by Dezert, 2002. I

DΘ closed set by union and intersection operators

I

DΘ is not closed by complementary, A ∈ DΘ ; A ∈ DΘ

I

if |Θ| = n: 2Θ