Map-reduce Implementation of Belief Combination Rules

eric Dambreville

Ensta Bretagne, 2 rue de Verny, Brest, France

DGA MI, Bruz, France

Keywords:

Belief Functions, Combination Rules, Statistic, Map-reduce.

Abstract:

This paper presents a generic and versatile approach for implementing combining rules on preprocessed belief

functions, issuing from a large population of information sources. In this paper, we address two issues, which

are the intrinsic complexity of the rules processing, and the possible large amount of requested combinations.

We present a fully distributed approach, based on a map-reduce (Spark) implementation.

1 INTRODUCTION

Thhis paper addresses the issue of generic computa-

tion of belief combinations in the context of a large-

scale networked community of agents (eg. in a so-

cial network) or sensored sources (eg. surveillance

camera). These agents or sensors produce informa-

tion, which are preprocessed under the form of belief

functions assigned to representative propositions. A

preprocessing of the networked agents/sensors and of

the produced information then infers a large collec-

tion of belief function clusters, which are representa-

tive of agent/sensors viewpoints on a common topic.

The combination of belief functions of a cluster by

means of dedicated combination rules infers a reﬁned

analysis of the relative viewpoints, including agree-

ment and disagreement, on the considered topic.

Emerging works (Zhou et al., 2015a) have been done

on the application of belief function to the analysis

of interaction between agents of a social network on

the basis of shared semantic content. These works

are especially based (Zhou et al., 2015b) on evidential

clustering of agents resulting in a fuzzy identiﬁcation

of communities. These clustering algorithms opti-

mize the evidential similarities/disimilarities between

agents, but do not deeply involve combinations of be-

liefs and the semantic it could extract. Of course, an

issue of belief functions application, especially when

sources are combined, is the efﬁcient computation of

many cases of combination in order to evaluate each

connection between networked agents.

In the domain of surveillance, (Liu et al., 2009)

have underlined eight challenge of a video network.

Among them are the uncertainty of events, inconsis-

tency or conﬂict between multiple sources, the com-

position of elemental events, and the scalability of the

system. (Hong et al., 2014) have applied evidential

networks to the problem of video surveillance in a

controlled application (Smart transport). The struture

implied by the networks makes possible an efﬁcient

reduction of the computational complexity.

When dealing with unstructured network, computa-

tional approaches for computing raw combinations in

numbers is a necessary tool. Now, the issue of the

generated conﬂict has been challenged by many evo-

lutions of the historical conjunctive and Dempster-

Shafer rules (Dubois and Prade, 1986; Lefevre et al.,

2002; Smarandache and Dezert, 2005; Florea et al.,

2006; Martin and Osswald, 2007). The develop-

ment of generic implementation (Dambreville, 2009)

of combination rules is a challenge in itself. Our

work consider both issues by extending a previous

work (Dambreville, 2009) dedicated to the generic

implementation of rules. In the continuation of this

work, this paper considers the problem of parallel

and pooling computation by factoring the combina-

tion process, and a map-reduce (Dean and Ghemawat,

2008) approach is proposed within the framework

Apache SPARK (Zaharia et al., 2010). In order to

factor some combination rules, new algebraic struc-

tures (eg. multisets in the case of Dubois&Prade rule)

are used as a processing space instead of the proposi-

tional framework of belief functions.

Section 2 introduces basic concept on belief func-

tions. Section 3 presents our new contribution for a

parallel and pooling implementation of combination

rules and our previous work (Dambreville, 2009) is

also introduced. Section 4 presents some limited tests.

144

Dambreville, F.

Map-reduce Implementation of Belief Combination Rules.

DOI: 10.5220/0005987001440149

In Proceedings of the 5th International Conference on Data Management Technologies and Applications (DATA 2016), pages 144-149

ISBN: 978-989-758-193-9

2 BELIEF FUNCTIONS

Belief functions are representations of imprecise and

uncertain information over an algebraic framework, a

lattice in its most general form. Most authors consider

belief functions over powersets, as a Boolean algebra,

and this is our hypothesis here. From now is given the

ﬁnite set Ω, the universe or frame of discernment.

2.1 Belief Assignments

The imprecise and uncertain information are char-

acterized by basic belief assignment (bba), m, over

propositions of the framework. Thus, a bba is deﬁned

as the attribution of pieces of belief to subsets of Ω:

m ≥ 0 and

∑

X⊆Ω

m(X) = 1 .

In case of closed world hypothesis, it is assumed that

the belief put on empty set is zeroed, ie. m(

0) = 0 .

This paper does not discuss further about such hy-

pothesis, but its involvement does not imply any dif-

ﬁcult generalization. However, we refer subsequently

to m(

0) as the conﬂict related to m, and consider rules

which are based on a redistribution of the conﬂict.

Given M sources providing information by means of

bba m

1:M

, the fusion of these information are com-

puted by combinations rules. In the case of a closed

world, the classical combination rule of Dempster-

Shafer (Dempster, 1968; Shafer, 1976) is derived

from the conjunctive rule by means of a normaliza-

tion based on the conﬂict. Without lost of gener-

ality (Smets, 1990), Dempster-Shafer rules could be

rewritten as a conjunctive rule without normalization

in the case of open world.

2.2 Combination Rules

Given two sources of information, characterized by

their respective bbas, m

, the conjunctive combi-

nation of m

and m

is deﬁned by:

? m

(X) =

∑

∩Y

) .

The conjunctive rule only works for open world hy-

pothesis, since it is possible to have m

? m

(

0) > 0

while m

(

0) = m

(

0) = 0 .

By interpreting m

? m

(

0) as a measure of conﬂict

and redistributing it, many alternative rules have been

proposed. As example, Dubois & Prade rule (Dubois

and Prade, 1986) and PCR5/PCR6 rule (Smarandache

and Dezert, 2005; Martin and Osswald, 2007) redis-

tribute the conﬂict with different approaches:

• Dubois & Prade Rule. The rule proposed by

Dubois and Prade extends the conjunctive rule

by redistributing disjunctively the conﬂict. Ap-

pendix 5 presents its original deﬁnition,

• PCR Rules. The PCR combination rules, pio-

neered by Dezert and Smarandache (Smarandache

and Dezert, 2005), are based on a local propor-

tional redistribution of the conﬂict.

As an example of implementation, we consider

speciﬁcally Dubois & Prade rule, generalized to many

sources, but the approach is generic and addresses a

potentially large scope of rules.

3 IMPLEMENTATIONS

It is assumed that the bba m

1:M

, to be fused, are taken

amongst a set of bba

1:N

, where 2 ≤ M  N. Typ-

ically, m

j[i]

, where the selection map j ∈ {1 :

{1:M}

is injective in general. The combination of

1:M

is done for selection j. The number of con-

cerned selections could be very large. Our main con-

cern and challenge is to implement the computation

of combination rules for all selections as a distributed

process. A map-reduce approach (Dean and Ghe-

mawat, 2008) is considered for this computation. A

ﬁrst approach is based on a previous work.

Section 3.1 introduces the generic formalism of ref-

eree functions (Dambreville, 2009) for deﬁning com-

bination rules. On this basis, section 3.2 presents a

map-reduce implementation of the combination rules.

Section 3.3 enhances the formalism of referee func-

tions with Markov properties, and improves the deﬁ-

nition of the combination rules, with recursive com-

putational properties. On this basis, section 3.4

presents a map-reduce and recursive implementation.

Notation: Indicator functions are deﬁned by:

I [P] =

(

0 if P is false

1 if P is true

(1)

3.1 Formulation based on Indicators

(Dambreville, 2009) proposed a generic formulation

of combination rules by means of conditional func-

tions (referee functions), denoted F(X|Y

1:M

which have a computational meaning as indicator

functions. In this framework, the combinations of bba

1:M

is expressed under the form:

⊕[m

1:M

F](X) =

∑

1:M

⊆Ω

F(X|Y

1:M

)

∏

i=1

) .

(2)

Map-reduce Implementation of Belief Combination Rules

145

In this formulation, a distinction is made between

the rule processing expressed by the summation, and

the rule deﬁnition which is expressed by the condi-

tional indicator F. It is easy, then, to imply a generic

distributed implementation of this summation, and

we propose an implementation within Spark frame-

work (Zaharia et al., 2010). This generic deﬁnition

by means of indicator function is quite general how-

ever, as shown in (Dambreville, 2009), and typically,

there are referee functions deﬁned for conjunctive or

disjunctive rules, D&P rules, PCR6 rule, and more.

For the concern of this paper, we present only the ref-

eree function related to D&B rules.

3.1.1 Alternative Deﬁnition of D&P Rule

The rule of Dubois and Prade (Dubois and Prade,

1986) is naturally generalized to more than two

sources by redistributing the conﬂict on the disjunc-

tion of the best consensuses:

⊕

··· ⊕

= ⊕[ m

1:M

] , (3)

where:

(X|Y

1:M

) = I

X = arg max

ω∈Ω

∑

i=1

I [ω ∈ Y

]

(4)

Set argmax

ω∈Ω

∑

i=1

I [ω ∈ Y

] is the subset of Ω,

whose elements receive the best vote derived from

their belonging to propositions Y

1:M

3.1.2 Computational Issues

There are actually two aspects to be considered, since

the computation may be computing-intensive as well

as data-intensive. On the one hand, it may be com-

puting intensive, since a belief assignment is a vec-

tor of dimension 2

card(Ω)

; without approximation, the

complexity of any belief computation increases dra-

matically with the size of the frame of discernment,

and this issue is worsened with the number of bba to

be combined. As a perspective of a distributed inten-

sive computation of the rules, is the possibility to han-

dle complex belief representations and their combina-

tions for speciﬁc applicative use or conceptual stud-

ies. On the other hand, the computation may be data

intensive, in the case of multiple combinations among

a large collection of bba, typically issuing from lo-

cal processing related to a collection of sources of

information. In this kind of application, the many

sources of information produce pieces of data, from

which knowledges are extracted by the local process

in the form of bba in the context of a given logical

frame. Then, the extracted bba are combined accord-

ing to a combination plan, caracterized by a selec-

tion function, in order to evaluate the compatibility of

the sources or evaluate a conﬁrmed knowledge. The

combination plan generally implies a large amount of

combination cases. Although our approach may be

applied to both case, our preliminary and limited tests

focus on the second scenario. Now, we do not ad-

dress here the question of the extraction, but only the

question of the combination.

3.2 Map-reduce Implementation

We implemented the generic fusion process (2) by

means of a Map-reduce principle (Dean and Ghe-

mawat, 2008). This implementation has been made

by means of SPARK (Zaharia et al., 2010) Resilient

Distributed Dataset (RDD) with the following steps:

Mapping steps: In these steps, the inner com-

putations are done, that is the joint belief as-

signments,

∏

i=1

), and the deﬁnition maps,

F(X|Y

1:M

). These computations are derived for

all considered selection maps j and all possible non-

zero propositional combinations, Y

1:M

. The amount of

data is potentially exponential with M.

• Deﬁne the set of selection maps j ∈ {1 : N}

{1:M}

to be computed as a RDD of list, that is

J: RDD[List[Int]]. For this purpose, method

flatMap is applied to an iterator describing j,

• From selection map j and the deﬁnition of bba

1:N

, map to the collection of tuples:





j(i)



i=1:M



Only cases, with non zero values for

j(i)

are considered. Methods join and flatMap

are thoroughly used in this process, resulting

in M: RDD[(List[Int], List[(U,U=>Double)])],

where generic type U is used for encoding subsets,

• The referee function is applied through method

flatMap, and yields the collection of tuples:

( j,X),F





j(1:M)



∏

i=1:M

j(i)



j(i)



as the RDD, FM: RDD[((List[Int], U),Double)],

Reducing Step: In this step, all inner computations

are summed up according to summation,

∑

1:M

⊆Ω

• At last RDD FM is reduced by key ( j,X ) with the

addition operator. As a result, the combined bba

are obtained as the collection of tuples:

( j, (X,⊕ [m

1:M

F](X))) .

Method reduceByKey is used with +, yielding

FusedM: RDD[(List[Int], (U,Double))].

DATA 2016 - 5th International Conference on Data Management Technologies and Applications

146

At this time, the caching strategy is not monitored,

and only RDD J, deﬁning the selection, and RDD FM,

deﬁning the ﬁnal result, are persistent.

3.2.1 Computational Issues

While this approach makes possible a fully distributed

computation of the inner elements of the combination

during the mapping steps, the amount of cases kept

in memory increases exponentially with the number

of sources to be combined. Even with a triple com-

bination, the approach consumes a lot of memory,

especially when the set of selected combinations is

densely connected. In order to address this issue, we

propose in next section to bring out and to implement

a Markovian property of the rule deﬁnition.

3.3 Recursive Formulation

As an instrument for a recursive formulation of the

rules, the sets Ψ and Λ are deﬁned for intermediate

computations. The rule is then deﬁned on the basis of

three ﬁnite conditional functions:

(λ

) 7→ σ(λ

) ,

(ψ

,λ

1:n

) 7→ R(ψ

|λ

1:n

) ,

(X, ψ

) 7→ π(X|ψ

) ,

for X,Y

⊆ Ω, ψ

∈ Ψ, λ

∈ Λ and 1 ≤ i,n ≤ M. Func-

tions σ and π are respectively forward and backward

projectors from the space of proposition 2

Ω

to the

spaces of computation, Λ and Ψ. Function R is a ref-

eree function within the spaces of computation. In σ,

parameter m

is the bba related to information source

i, but any other contextual knowledge could be con-

sidered. Based on triplet [σ,R,π], rule ⊕[σ,R,π] is

deﬁned as a composition of conditional inferences:

⊕[ m

1:M

σ,R, π](X) =

∑

∈Ψ

π(X|ψ

)

∑

1:M

⊆Ω

1:M

∈Λ

R(ψ

|λ

1:M

)

∏

i=1

)σ(λ

)) , (5)

for all X ⊆ Ω. Owing to deﬁnition (5), it is noticed

that, although Ψ and Λ may be inﬁnite sets, the sum-

mations are actually ﬁnite: the values are zeroed ex-

cept for a ﬁnite number of them. On such deﬁnition,

the main computation burden comes from the condi-

tional inference R(ψ

|λ

1:n

), while other inferences are

more or less easily factorized. In order to reduce the

computational burden, a Markov hypothesis is made

on R by introducing conditional function ρ:

R(ψ

n+1

|λ

1:n+1

) =

∑

∈Ψ

ρ(ψ

n+1

|ψ

,λ

n+1

)R(ψ

|λ

1:n

) ,

(6)

for ψ

1:M

∈ Ψ, λ

1:M

∈ Λ and 1 ≤ n < M. Under this

hypothesis, ⊕[ m

1:M

σ,R, π] is computed recursively:

1. For λ

1:M

∈ Λ and i = 1 : M , compute projection:

(λ

) =

∑

⊆Ω

)σ(λ

) , (7)

2. Compute ⊕ [m

1:M

R] recursively within space Ψ:

⊕ [ m

R](ψ

) =

∑

∈Λ

(λ

)R(ψ

|λ

) , (8)

⊕ [ m

1:n+1

R](ψ

n+1

) =

∑

n+1

∈Λ

n+1

(λ

n+1

)

∑

∈Ψ

ρ(ψ

n+1

|ψ

,λ

n+1

) ⊕ [m

1:n

R](ψ

) , (9)

for ψ

1:M

∈ Ψ, λ

1:M

∈ Λ and 1 ≤ n < M,

3. Compute backward projection for all X ⊆ Ω :

⊕[ m

1:M

σ,R, π](X) =

∑

∈Ψ

π(X|ψ

) ⊕ [m

1:M

σ,R] (ψ

) . (10)

Combined with map-reduce, the recursion improves

the efﬁciency of the distributed implementation.

3.3.1 Recursive Deﬁnition of D&P Rule

Unprojected deﬁnition (4) is not directly compatible

with a recursive decomposition. In order to take into

account the sources consensuses in a Markov decom-

position, the computation set is chosen as the set of

multisets of underlying set Ω.

⊕

·· · ⊕

= ⊕[ m

1:M

,π

] ,

where σ

is canonical map from sets to multisets,

maps backward from multisets to top sets, and

evaluates the vote by adding on the multisets:

• Λ

= Ψ

= N

Ω

• σ

(λ

) = I



= [I [ω ∈ Y

]]

ω∈Ω



• π

(X|ψ

) = I



X = arg max

ω∈Ω



• R

(ψ

|λ

1:n

) = I

∑

i=1

Markov decomposition of R

comes easily:

(ψ

n+1

|ψ

,λ

n+1

) = I [ψ

n+1

= ψ

+ λ

n+1

] .

3.4 Recursive Implementation

Recursive map-reduce implementations of the rules

are similar to the non-recursive approach described in

section 3.2. But in this case, each recursive step is

computed by means of a map-reduce sequence :

Map-reduce Implementation of Belief Combination Rules

147

Projection Steps: Each bba of (

)

k=1:N

is com-

puted and mapped into its projection (

)

k=1:N

. This

projection is done by a map step and a reduce step:

• All bba are computed as a collection of tuples:

(k, (Y

))) ,

as Bba: RDD[(Int, (U,Double))]. Generic type

U is used for encoding subsets. For this purpose,

flatMap is applied to an iterator of the bba,

• From Bba, the projected weights are then com-

puted as a collection of tuples:

(k, (λ,σ(λ|Y

;

)

))) .

RDD, MapBba: RDD[(Int, (L,Double))], is ob-

tained by applying flatMap to Bba. Generic type

L is used for encoding Λ-parameters,

• MapBba is reduced by key (k, λ) with the addition

operator. As a result, the projected bba,

1:N

, are

obtained as the collection of tuples:

(k, (λ,

(λ))) .

Method reduceByKey is used with +, yielding

ProjBba: RDD[(Int, (L,Double))],

Recursive Steps: First stage (8) is computed:

• For all preﬁxes, j(1), of a selection map j,

ProjBba is mapped to the collection of tuples:



j(1),



ψ,

j(1)

(λ)R(ψ|λ)



FusMapBba: RDD[(List[Int], (P,Double))], is

obtained by applying flatMap to ProjBba.

Generic type P is used for encoding Ψ-parameters,

• FusMapBba is reduced by key ( j(1),ψ) with the

addition operator. As a result, values ⊕



j(1)





are obtained as the collection of tuples:



j(1),



ψ,⊕



j(1)





(ψ)



Method reduceByKey is used with +, yielding

FusProjBba: RDD[(List[Int], (P,Double))],

• Set n ← 1,

and subsequent stages (9) are computed until n = M:

• Set n ← n + 1,

• For all preﬁxes, j(1 : n), of a selection map j,

FusProjBba is mapped to the collection of tuples:



j(1 : n),



j(n)

(λ)ρ(ψ

|ψ,λ) ⊕



j(1:n−1)





(ψ)



FusMapBba: RDD[(List[Int], (P,Double))], is

obtained by applying flatMap to ProjBba.

• FusMapBba is reduced by key ( j(1 : n),ψ) with

the addition operator. Values ⊕



j(1:n)





, are

obtained as the collection of tuples:



j(1 : n),



ψ,⊕



j(1:n)





(ψ)



Method reduceByKey is used with +, yielding

FusProjBba: RDD[(List[Int], (P,Double))],

Backward Projection Steps: At last, the combined

bba are obtained from FusProjBba:

• For all selection maps j, FusProjBba is mapped

to the collection of tuples:



X, π(X|ψ) ⊕



j(1:M)



σ,R



(ψ)



Then, FM: RDD[(List[Int], (U,Double))], is ob-

tained by applying flatMap to FusProjBba,

• FM is reduced by key ( j,X) with the addition oper-

ator. Combined bba are obtained as the collection

of tuples:

( j, (X,⊕ [m

1:M

σ,R, π](X))) .

Method reduceByKey is used with +, yielding

FusedM: RDD[(List[Int], (U,Double))].

4 SOME TESTING CASES

In this preliminary work, some limited tests are made

only for the rule of Dubois & Prade. The tests have

been done with limited computation power, that is a 6-

thread virtual machine with 23 Gio of memory, oper-

ated on a i7-4770 GPU with 32 Gio of memory. These

tests are a ﬁrst glimpse of the improvements by our

approach. But many optimization works are still in

progress and extensions to small clusters of comput-

ers are currently investigated.

A collection of bba is ﬁrst generated randomly on

set Ω = {a, b,c, d}. Then, any triplet combination of

these bba are intended for the computation of Dubois

& Prade rule. The following table evaluates the com-

putation time (in seconds) for the entire triplet set

with different sizes, NN = N(N − 1)(N − 2)/6, and

different computation approaches: 1 thread and non-

recursive (1-nr); n threads and non-recursive (n-nr); n

threads and recursive (n-r).

NN 560 4960 41664 341376

1-nr 1.7 12 104 –

6-nr 1.23 5.25 37.9 3060

6-r 2.96 7.02 37.3 545

These results conﬁrm the efﬁciency of the recur-

sive approach for large concomitant combination se-

quences. The recursive approach is however a burden

for small sequences. On this preliminary work, the

code has not been optimized. For this reason, the ta-

ble is not signiﬁcant at this time in comparison with

other existing optimized libraries.

Moreover, these tests only considered the perfor-

mance of simultaneous computation of large set of

combinations, and especially, a full set of triple com-

binations. This implies important intermediate re-

sults caching. This case of use is then favourable to

DATA 2016 - 5th International Conference on Data Management Technologies and Applications

148

our third, recursive, algorithm, since this approach re-

duces the caching by deﬁnition. But many other as-

pects of this implementation have to be investigated,

in term of performance. Typically, the structure of the

set of combinations should be considered for optimiz-

ing the strategies of the computation ﬂow. Moreover,

an incremental computation of the combinations, may

be also investigated through computation ﬂows more

complex than map-reduce. From this viewpoint, the

reactivity of this parallel computation on possibly

complex single combinations is also a piece of per-

formance to be evaluated precisely or optimized in the

future, in regards to non-parallel approaches.

5 CONCLUSIONS

In this paper, we proposed a generic distributed

processing approach for computing belief combi-

nation rules. The approach is based on a map-

reduce paradigm, and has been implemented in

scala/SPARK. It is derived from the concept of referee

function, introduced in a previous work with the aim

of separating the deﬁnition of the combination rule

from its actual implementation. This work has been

completed by the proposal of a new recursive formal-

ism for the deﬁnition of the rules, and an improved

map-reduce generic implementation of the rules pro-

cessing. Some tests have been made for the rule of

Dubois & Prade, which illustrated this computation

improvement. More tests will be investigated in the

future. Moreover, our intention is to extend this work

to general data ﬂow paradigms for computation.

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APPENDIX

A Dubois & Prade Rule

Dubois and Prade reﬁned the conjunctive rule by re-

distributing disjunctively the conﬂict:

⊕

(X) =

∑

∩Y

)

∑

∩Y

∪Y

) .

Map-reduce Implementation of Belief Combination Rules

149