1  

Fusion of Masses Defined on Infinite Countable Frames of Discernment

Florentin Smarandache University of New Mexico, Gallup, USA

Arnaud Martin

ENSIETA, Brest, France Abstract. In this paper we introduce for the first time the fusion of information on infinite discrete frames of discernment and we give general results of the fusion of two such masses using the Dempster’s rule and the PCR5 rule for Bayesian and non-Bayesian cases. Introduction. Let { }1 2, ,..., ,...ix x x xθ ∞= be an infinite countable frame of discernment, with

i jx x∩ =Φ for i j≠ , and 1 2( ), ( )m m⋅ ⋅ two masses, defined as follows:

( ) [ ]1 0,1i im x a= ∈ and ( ) [ ]2 0,1i im x b= ∈ for all { }1, 2,..., ,...i i∈ ∞ , such that

( )11

1ii

m x∞

=

=∑ and ( )21

1ii

m x∞

=

=∑ ,

therefore 1( )m ⋅ and 2 ( )m ⋅ are normalized. Bayesian masses.

1. Let’s fusion 1( )m ⋅ and 2 ( )m ⋅ , two Bayesian masses:

1 2

1 1 2

2 1 2

12

... ... ... | ( )

... ... ..........

... ... ..........

i j

i j

i j

x x x x x conflicting mass

m a a a a

m b b b b

m a

∞ Φ

1 1 2 21

... ... .......... 1- i i j j i ii

b a b a b a b a b∞

=∑

where 12 ( )m ⋅ represents the conjunctive rule fusion of 1( )m ⋅ and 2 ( )m ⋅ .

a) If we use Dempster’s rule to normalize 12 ( )m ⋅ , we need to divide each 12 ( )im x by the sum of masses of all non-null elements, and we get:

12

1

( ) i iDS i

i ii

a bm xa b

∞

=

=

∑,

2  

for all i .

b) Using 5PCR the redistribution of the conflicting mass i j i ja b b a+ between ix and jx (for all j i≠ ) is done in the following way:

j i ji

i j i j

a ba b a b

αα= =

+, whence

2i j

ii j

a ba b

α =+

and

j i ji

i j i j

b ab a b a

ββ= =

+, whence

2i j

ii j

b ab a

β =+

.

Therefore

5

2 2

121

( ) i j j iPCR i i i

j i j j ij i

a b a bm x a b

a b a b

∞

=≠

⎛ ⎞= + +⎜ ⎟⎜ ⎟+ +⎝ ⎠

∑ ,

for all i . Non-Bayesian masses.

2. Let’s consider two non-Bayesian masses ( )3m ⋅ and ( )4m ⋅ :

1 2

3 1 2

4 1 2

... ... ... ( )

... ... ..........

... ... .........

i j

i j

i j

x x x x x conflicting mass

m c c c c C

m d d d d

θ∞ Φ

( )341

.

.................. + ............... 1- i i i i i i i ii

D

m c d c D Cd CD CD c d c D Cd∞

=

+ − + +∑

where ( ) [ ]3 0,1i im x c= ∈ for all i , and ( ) [ ]3 0,1m Cθ = ∈ ,

and ( ) [ ]4 0,1i im x d= ∈ for all i , and ( ) [ ]4 0,1m Dθ = ∈ ,

such that ( )3m ⋅ and ( )4m ⋅ are normalized:

1

1ii

C c∞

=

+ =∑ and 1

1ii

D d∞

=

+ =∑ .

34 ( ) + i i i i im x c d c D Cd= + for all { }1, 2,....,i∈ ∞ , and ( )34m C Dθ = ⋅ , where ( )34m ⋅ represents the conjunctive combination rule.

a) If we use the Dempster’s rule to normalize, we get:

( )34

1

+( ) i i i iDS i

i i i ii

c d c D Cdm xCD c d c D Cd

∞

=

+=

+ + +∑

for all i , and

3  

( )

34

1

( )DS

i i i ii

CDmCD c d c D Cd

θ ∞

=

=+ + +∑

.

b) If we use 5PCR , we similarly transfer the conflicting mass as in the previous 1.b) case, and we get:

5

2 2

341

( ) i j j iPCR i i i i ii

j i j j ij i

c d c dm x c d c D Cd

c d c d

∞

=≠

⎛ ⎞= + + + +⎜ ⎟⎜ ⎟+ +⎝ ⎠

∑

for all i , and

534 ( )PCRm C Dθ = ⋅

Numerical Examples. We consider infinite positive geometrical series whose ratio 0 1r< < as masses for the sets 1 2, ,...,x x x∞ , so the series are congruent: If 1 2, ,..., ...nP P P is an infinite positive geometrical series whose ratio 0 1r< < , then

1

1 1ii

PPr

∞

=

=−∑

Example 1 (Bayesian).

Let ( )112i im x = for all { }1, 2,....,i∈ ∞ .

( )11 1

11 2 112 1

2

i ii i

m x∞ ∞

= =

= = =−

∑ ∑

since the ratio of this infinite positive geometric series is 12

.

And ( )223i im x = for all { }1, 2,....,i∈ ∞

( )21 1

22 3 113 1

3

i ii i

m x∞ ∞

= =

= = =−

∑ ∑

since the ratio of this infinite positive geometric series is 13

.

4  

1 2

1 2

2 2

12 2

... ... ...

1 1 1 1 ... ... .......... 2 2 2 22 2 2 2 ... ... .......... 3 3 3 3

2 2 6 6

i j

i j

i j

x x x x x

m

m

m

∞ Φ

1

22 2 2 36 ... ... ......... 1 1 = 16 6 6 51

6

i j ii

∞

=

− = −−

∑

( )12m ⋅ is the conjunctive rule.

a) Normalizing with the Dempster’s we get:

12

1

2 22 5 56 6( ) 22 6 2 6

66116

i i

DS i i i

ii

m x ∞

=

= = = ⋅ =

−

∑

for all i . b) Normalizing with 5PCR we get:

5

2 2

121

1 2 1 42 2 3 2 6( ) 1 2 1 26

2 3 2 3

i j j i

PCR i ijj i i j j i

m x∞

=≠

⎛ ⎞⋅ ⋅⎜ ⎟= + +⎜ ⎟

⎜ ⎟+ +⎝ ⎠

∑

Example 2 (non-Bayesian).

Let 31( ) 3i im x = for all { }1, 2,....,i∈ ∞ , and 3

1( )2

m θ = .

( )3 31 1

11 1 1 3( ) 112 3 2 1

3

i ii i

m m xθ∞ ∞

= =

+ = + = + =−

∑ ∑ ,

so 3( )m ⋅ is normalized.

And ( )414i im x = for all i , and ( )4

23

m θ = .

( )4 41 1

12 1 2 4( ) 113 4 3 1

4

i ii i

m m xθ∞ ∞

= =

+ = + = + =−

∑ ∑ ,

so 4 ( )m ⋅ is normalized.

5  

1 2

3 2

4 2

... ... ...

1 1 1 1 1 ... ... .......... 3 3 3 3 21 1 1 ... ... 4 4 4

i j

i j

i

x x x x x

m

m

θ∞ Φ

34 1

1 1 2 1 1 + +116 12 3 2 4

1 2 .......... 4 3

1 2 1 1 .............. + + ......................... 12 3 2 4 6

j

i i i i i iim +

∞∑− − =+= ⋅

⎛ ⎞⎜ ⎟⋅ ⎝ ⎠

1 2 125 12 3 2 4 = - - =1 16 1 1 11 3 4

12

⋅−− −−

5 1 1 1 8 = - -

11 3 6 336conflicting mass− =

a) Normalizing with Dempster’s rule we get:

3433 1 2 1( ) + +125 12 2 43

DS im x i ii⎛ ⎞

= ⎜ ⎟+ ⋅⎝ ⎠

for all i , and

( )3433 1 3325 6 150DSm θ = ⋅ = .

b) Normalizing with 5PCR we get

5

2 2

341

1 1 1 11 2 1 3 4 3 4( ) + +1 1 1 1 112 2 43

3 4 3 4

i j j i

PCR ijj i i j j i

m x i ii∞

=≠

⎛ ⎞⋅ ⋅⎜ ⎟= + +⎜ ⎟+ ⋅ ⎜ ⎟+ +

⎝ ⎠

∑

for all i , and

534

1( )6PCRm θ = .

References: 1-3. F. Smarandache, J. Dezert, Advances and Applications of DSmT for Information

Fusion, Vols. 1-3, AR Press, 2004, 2006, and respectively 2009.