TBM Computational analysisComputational Framework

Boolean Lattice Data Structure

The Möbius Transform

Data Fusion Algorithm

Case Studies

The Fast Möbius Transform

Ludovico Pinzari

Computational Framework

Fusion Algorithm Time Space Transform

BRUTE FORCE

Mobius Transform X X X

Fast Mobius Transform

Ω insieme universale

||22 ||2

||2 ||2 ||2 ||2

||2||2

||2 ||2

NB: O ( ) + O( x ) ~ O( x )

||2 ||2 ||2 ||2 ||2

O ( ) + O( ) ~ O( ) ||2 ||2||2

||2

Boolean Lattice Data Structure position Bit array Ω m

[0] 0 0 0 Ø m(Ø)

[1] 0 0 1 a m(a)

[2] 0 1 0 b m(b)

[3] 0 1 1 a,b m(a,b)

[4] 1 0 0 c m(c)

[5] 1 0 1 a,c m(a,c)

[6] 1 1 0 b,c m(b,c)

[7] 1 1 1 a,b,c m(a,b,c)

Ø insieme vuoto

Boolean Lattice Data Structure

Ø (0 0 0)

c (1 0 0)b (0 1 0) a (0 0 1)

abc (1 1 1)

ab (0 1 1)bc (1 1 0) ac (1 0 1)

The Möbius Transform

• Implicability function

]10[2: b

b(A) = bel(A) + m(Ø) =

AXX

AXm,

)(

• Belief function]10[2: bel

bel(A) =

XAX

AXm,

)(

Vincolo: b(Ω) = 1

Ø

w,x 0

w,y 0

0.20x,y

0.05

w

0.05x

0

0

y

z

0.10w,x,y

0.05

w,x,z

0.25

0

w,y,z

x,y,z

w,x,y,z 0

w,z 0.05

x,z 0

0.05y,z

0.20

The Möbius Transform • Implicability function

]10[2: b

b(A) =

Vincolo: b(Ω) =

Ω = w,x,y,z Insieme Universale

XXm )(

m =

AXX

AXm,

)(A = w,y,z

= 1

X |X| = 3 |X| = 2 |X|= 1 |X| = 0

m (w,y,z) 0.25 - - -

m (w,y) - 0 - -

m (w,z) - 0.05 - -

m (y,z) - 0.05 - -

m (w) - - 0.05 -

m (y) - - 0 -

m (z) - - 0 -

m (Ø) - - - 0

∑∑ = 0.40 0.25 0.1 0.05 0

B(A) = 0.40

Ø

w,x 0

w,y 0

0.20x,y

0.05

w

0.05x

0

0

y

z

0.10w,x,y

0.05

w,x,z

0.25

0

w,y,z

x,y,z

w,x,y,z 0

w,z 0.05

x,z 0

0.05y,z

0.20

A0.40

The Möbius Transform

• Implicability function m->b]10[2: b

b(A) = bel(A) + m(Ø) =

AXX

AXm,

)(

• Inverse Transform

m(A) =

Vincolo: b(Ω) = 1

b -> m ?]10[2: m

Am =

||

0

||

)1(U

i

iA

AX

AXb )(.

The Möbius Transform

• Proof: b->mb(A) =

AX

Xb )(Am =

=

m (A) + m (w,y) + m (w,z) + m (y,z) + m (w) + m (y) + m (z) + m(Ø)

m(A) = b (A) –[ m (w,y) + m (w,z) + m (y,z) + m (w) + m (y) + m (z) + m(Ø)]

=

b(A) -

|2|

m (w,y) = b (w,y) – [ m (w) + m (y) + m(Ø) ] m (w,z) = b (w,z) – [ m (w) + m (z) + m(Ø) ] m (y,z) = b (y,z) – [ m (y) + m (z) + m(Ø) ] |1|

AX

Xm )(

m (y) = b (y) – [ m(Ø)] m (z) = b (z) – [ m(Ø)]

|0|

m (Ø) = b (Ø)

A = w,y,z

The Möbius Transform

• Proof: b->mm(A) = b (A)

|2|

– [ b (y,z) + b (w,z) + b (w,y) ] |1|

m (A) = total A value of all subsets of size |A|

|0|

A = w,y,z

+ [ b (w) + b (y) + b (z) ]

– [ b (Ø)]

– total A value of all subsets of size |A| - 1 + total A value of all subsets of size |A| - 2 ...

... – [ b (Ø)]

Ø

w,x

w,y

x,y

w

x

y

z

w,x,y

w,x,z

0.25

w,y,z

x,y,z

w,x,y,z

w,z

x,z

y,z

A0.40

0

0.05

0

0

0.05

0.05

0.05

The Möbius Transform

• Commonality function m->q

q(A) =

• Inverse Transform q->m

m(A) =

q(Ø) = 1

]10[2: q

XAAX

AXm,

)(

]10[2: m

XA

AXq )(.

||

||

||

)1(U

Ai

iA

Ø

w,x 0

w,y

0.20x,y

w

0.05x

y

z

0.10w,x,y

0.05

w,x,z

0

w,y,z

x,y,z

w,x,y,z

w,z

x,z 0

y,z

0.20

A0.60

Ø

w,x

w,y

x,y

w

0.05

x

y

z

w,x,y

w,x,z

w,y,z

x,y,z

w,x,y,z

w,z

x,z

y,z

A0.60

0.35

0.50

0.25

0.30

0.25

0.20

0.20

The Möbius Transform Complexity • Möbius Transform

||2

|Ω|

• Fast Möbius Transform

|Ω| - 1

|Ω| - 2..Ø

||2

||2

||2

||2

.

. = ||2 x ||2

||2

0

||

.

.

.

||

||

1||

||

2||

||

2 ||||

0k k||2 x

|Ω|

|Ω| - 1

|Ω| - 2..

Ø

.

.

||

||

1||

||

2||

||

0

||

2 ||||

0k k= ||2

NB: Ɵ( ) ||22 Ɵ( ) ||2

Best Case = Medium Case = Worst Case Focal elements Power Set indipendent

The Möbius Transform Implementation

• Implicability function m->b]10[2: b

b(A) = bel(A) + m(Ø) =

AXX

AXm,

)(

b =

Vincolo: b(Ω) = 1]10[2: m

Am =

.

BfrM

• Matrix transform m->bm

m: bba vettore b: implicability vettore

BfrM: matrice

||2x

1||2x

1||2x

||2

BfrM:

BfrM(A,B) = 1 iff

AB

AB

0 otherwise

The Möbius Transform • Ω=a,b BfrM m->b

b = .

BfrM

• Inverse Transform MfrB b->m

m

AB

1111

0101

0011

0001

,

ba

b

a

, baba

A

B

row Aa,b

a b

Ø

=

m(Ø)

m(a)

m(b)m(a,b)

b(Ø)

b(a)

b(b)b(a,b)

m b

1111

0101

0011

0001

,

ba

b

a

, baba

A =

m(Ø)

m(a)

m(b)m(a,b)

b(Ø)

b(a)

b(b)b(a,b)

m b

BAB

|A| |A|-1 |A|-2

+ 1 - 1

+ 1

0 - 1

Aa,b

b a

Ø

• Implicability

The Möbius Transform • Ω=a,b QfrM m->q

q = .

QfrM

• Inverse Transform MfrB q->m

m

AB

1000

1100

1010

1111

,

ba

b

a

, baba

A

B

row Aa,b

a b

Ø

=

m(Ø)

m(a)

m(b)m(a,b)

q(Ø)

q(a)

q(b)q(a,b)

m q

1000

1100

1010

1111

,

ba

b

a

, baba

A =

m(Ø)

m(a)

m(b)m(a,b)

b(Ø)

b(a)

b(b)b(a,b)

m b

B

|A| |A|-1 |A|-2

+ 1 - 1

+ 1

0 - 1

Aa,b

b a

Ø

• commonality

AB

The Möbius Transform Implementation

• OSS Det(BfrM)≠ 0

Det(QfrM)≠ 0

BfrM1

QfrM1

BfrMQfrMT

BJBJBfrMT 1

001

010

100

001

010

100

987

654

321

321

654

987

789

456

123

987

654

321

=

=

001

1

11

.

Bijective Functional

The Möbius Transform Implementation

• m

JBJBBfrQ 1

b B• m q JBJQfrM

BMfrB 1

JBJMfrQ 1

• b q BJBJQfrB 1

m->b (+) (X)

|Ω|=2 = 4

|Ω|= 3 = 8

|Ω|= 4

||2 ||22

||2

||2= 16

= 16||22 = 64

||22 = 65536

mbill-conditioning problemExpensive computationFor matrix multiplicationAnd inverse.

The Fast Möbius Transform

m b

Implicability function

v0 v1 v2

+

Ø

a

b

ab

Ø

Ø a

b

b + ab

Ø

Ø + a

Ø + b

Ø + a b + ab+

m(Ø)

m(a)

m(b)

m(a.b)

Data Fusion

• Dempster’s Rule of Combination

m12

CB

ACB

CmBm

CmBm

)()(

)()(

21

21

1==mm 21

K conflict

Can we solve in linear time ?

Data Fusion: Case Study

m2a,b

b

Ø

a 0.5 0.5

a,b

Ø

a 0.7

0.3

b

m1

ma=0.5 mb=0.5 Ϝ1

Ϝ2

ma

,b=

0.3

ma

=0.

7

0.15

0.35 0.35

0.15

Ω x Ω

Data Fusion:Case Study

A B C = A m (B) m (C) m (B) . m (C)a,b a,b a,b 0.3 0 0

TOTAL ∑ = 0a a a 0.7 0.5 0.35

a a,b 0.7 0 0a,b a 0.3 0.5 0.15

TOTAL ∑ = 0.50b b b 0 0.5 0

b a,b 0 0 0a,b b 0.3 0.5 0.15

TOTAL ∑ = 0.15

Conjunctive Combination Rule: Brute Force Approach

U

1 2

U

1 2

U

U

U

U

U

U

Data Fusion:Case Study

Conflict B C = Ø m (B) m (C) m (B) . m (C)Ø Ø Ø 0 0 0

Ø a 0 0.5 0Ø b 0 0.5 0Ø a,b 0 0 0a Ø 0.7 0 0b Ø 0 0 0a,b Ø 0.3 0 0a b 0.7 0.5 0.35b a 0 0.5 0

TOTAL ∑ = 0.35

Conjunctive Combination Rule: Brute Force Approach

U

1 2

U

1 2

U

U

U

U

U

U

U

U

Data Fusion:Case Study

)()(1 21. CmBmk

CB

= 1 – 0.35 = 0.65

Normalization constant

k1 m12(a)

m12(b)

m12(a,b)

0

=

0

0.77

0.23

0

m12a,b

b

Ø

0.77 0.23

a

||22

Bit-array: worst case |Ϝ1| |Ϝ2|Ω x Ω =||2

=

Computational cost =

Data Fusion:FMT Conjunctive Combination Rule

1) Compute Commonality functions using FMT

Ϝ

Ϝ

||2

m1 m2( , ) ( , )q1q

2

qi1 . q i2 i = 1, ..,m1 m2

-1

2) Compute the product in the new domain

3) Compute the orthogonal sum using the inverse FMT Computational cost: ||2

Data Fusion: FMT

0

0.7

00.3

1

1

0.30.3

m1

Ϝ

q1

1

0.5

0.50

q2

0

0.5

0.50

m2

Ϝ

x

xxx

3.0000

03.000

0010

0001 1

0.5

0.50

q2

x

Diag(q1)

=

q12

1

0.5

0.150

Ϝ-1 0.35

0.5

0.150

m12

DATA FUSION DESIGNSequencing

Combination Rule

Dempster’s Rule is an associative operator.Thus is order independent. However is conflict sensitive!

A solution is to reduce the system entropy.Filter the conflict between the agents.

Another way is to use a clustering algorithm andUse the most suitable comb rule related to the bba’s.

DATA FUSION DESIGNHow can we compare 2 body of evidence ?

Observing the conflict magnitude related to the orthogonal sum.

Apply an Euclidean metric between bba’s. (mass vectors)

A new metric based on the probability confidence interval.

DATA FUSION DESIGNComputational and design issues

Conflict magnitude

)()( 21. CmBmk

CB

• Computational problem related to the orthogonal sum.

• Hard to identify the specific body of evidence framework.

• Hard to design a clustering algorithm

DATA FUSION DESIGNComputational and design issues

Well known and tested metric is the Josuellem distance.

TmmSmmssd )21()21(21)2,1(

),( BAS BAif1

2||

,|BA|

|BA| BA

Computational complexity:

O ( ) + O( x ) ~ O( x )

||2 ||2 ||2 ||2 ||2

• Ω = a,b 8 sums and 20 multiplication

Ludovico’s metric (probability confidence interval)

Based on the Taxicab (Manhattan) distance.

Bel(A)

Bel(B).

Bel(Z)

A

B

...Z

Pl(A)

Pl(B).

Pl(Z).

Unc(A)

Unc(B).

Unc(Z).

Bel(X) Pl(X) Unc(X)X\

Z

AX

B |Bel(x)Bel(x)| 21

Z

AX

P |Pl(x)Pl(x)| 21

Z

AX

U |Unc(x)Unc(x)| 21

Z

AXBel 11

Bel(x)

Z

AXPl 11

Pl(x)

Z

AXUnc 11

Unc(x)

Z

AXBel 22

Bel(x)

Z

AXPl 22

Pl(x)

Z

AXUnc 22

Unc(x)

Ludovico’s metric Depends on the configuration and on the Jaccard

dissimilarity between Sets

• Jaccard dissimilarity|YX|

|YX|1),(

YXd

Metric’s Properties:

0),( YXd• Non-negative:• reflexive: YXiffYXd 0),(

• symmetric: ),(),( XYdYXd

• Triangle inequality: ),(),(),( YZdZXdYXd

• NB: YXiffYXd 1),(

Depends on the configuration

• Bayesiana + Bayesiana

• Superset + Superset

• Bayesiana + Superset

4),(

PBYXd

UPB

PYXd

2),(

PPlPl

PYXd

21

2),(

• Superset + Pseudo-BayesianaPB

UUncUncP

YXd

2

2),(

21

• Pseudo-Bayesiana + Pseudo-Bayesiana

o a) total belief-overlapping 2)(2

)(),(

PB

UPBBPYXd

o b) partial belief-overlapping2121

),(PlPlBelBel

PBYXd

Computational Complexity Time and Space Complexity

O (|Ω |) |Ω |=2

6 sums and a division

How to filter the conflict D: Distance matrix between agents

• S -Similar Matrixj))Max(d(i,

j)d(i,1),( jiSim

• Support Degree ),()(,

jiSimiSupZ

jiAi

• Credibility agentsniCrd #1-n

Sup(i))(

How to filter the conflict Discounting procedure

• Discounting factor )()( iCrdi

• Filter the Noise

\2)()( || xxmxm iii

))(1(1)( iii mm