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