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2017 2nd International Conference on Information Technology and Management Engineering (ITME 2017) ISBN: 978-1-60595-415-8

An Analysis of the Relationship between Weapon Equipment System of

Systems Capability Based on Conditional Belief Function

Dong PEI

1

, Da-guo QIN

1

and Guang-zhi BU

2 1

Department of Space Command, Equipment Academy, Beijing 101416, China 2

Complex System Simulation Lab, Beijing Institute of System Engineering, Beijing 100101, China

Keywords: Conditional belief function, System of systems capability, Weight coefficient, Causality strength.

Abstract. The relation analysis of weapon equipment system of systems capability is the prerequisite for the capability evaluation and contribution evaluation of weapon equipment system of systems. In order to study the system of systems capability evaluation based on evidential network, we first classify the relations between the capabilities, and defined different conditional belief functions for different relations of the capabilities, and study the Conversion relationship between weight coefficient, causal strength for system of systems capability and conditional belief functions. Finally, the feasibility of the method is verified by a concrete example.

Introduction

There are many factors influencing weapon equipment system of systems capability, which include the qualitative information and the quantitative information, but also the uncertainty, ambiguity and incomplete information under a large number of uncertain conditions.

In order to deal with the uncertainty, a large number of literatures have appeared in academic circles in recent years, mostly using the Bayesian network method. But the Bayesian network not only needs the accurate probability judgment, but also needs the fuzzy estimate. Evidential network is an extension of Bayesian network, which can not only deal with random uncertainties, but also deal with cognitive uncertainty and become a new approach to deal with uncertain knowledge.

The evidential network is based on the conditional Belief function theory. Based on the capability evaluation of the weapon system of systems based on the evidential network, this paper makes a deep analysis on the relationship between weapon equipment system of systems capabilities, and obtains the conditional Belief function between the capabilities, and studies the conversion of weight coefficient, causal Strength and conditional Belief function.

The Basic Theory of Conditional Belief Function

In the belief function theory, Θ is a mutually exclusive and complete set of finite elements, and it is

called recognition framework. 2Θis its power set. The belief allocation that supports proposition A is called the Basic Belief Assignment (BBA), which is a function that maps from 2Θ to [0,1] as follows:

( ) 1 A

m A

∈Θ

=

(1)

If the basic Belief assignment of any subset A satisfies the formula m

( )

A >0, then A is called a

focal element of Θ.

(2)

( ), ( | )

0, X A

m B X B A

m B A

otherwise

⊆ Θ

 = 

(2)

Analysis of the Capability Relationship of Weapon Equipment System of Systems Based on Conditional Belief Function

In the system of systems capability relationship analysis, the literature [6] analyzes and defines the relationship between the capabilities of the weapon equipment system of systems from different angles. In [7], a multi-view model building method based on ontology is proposed, and a meta-model of six capability views is established.

To facilitate the definition of the conditional belief function, this paper divides the system of systems capability relation into Aggregation, generalization and dependence, in which dependence is divided into value-dependent and semantic-dependent. In the following, the corresponding conditional belief function structure is defined for different types of system of systems capability relations.

(1) Aggregation

The aggregate relationship is also called a constituent relation or an inclusive relation.

The set of the system of systems capability that contains the aggregation relation is a hierarchical structure, and the aggregation relation is an AND relationship. The conditional belief function or conditional basic belief structure for the aggregation relation is defined as follows:

(

)

(

)

(

) (

)

(

)

(

) (

)

(

)

1 2

1 2 1 2 1 2

1 2 1 2 1 2

1 2

1 1, 1 1

1 1, 0 1 0, 1 ; if the weight of is greater than

0 0, 1 0 1, 0 ; if the weight of is greater than

0 0, | | 0 1 | | | |

m C C C

m C C C m C C C C C

m C C C m C C C C C

m C C C

= = = =

= = = > = = =

= = = > = = =

= = = =

(3)

where C is the parent capability in the aggregation relation and C1 and C2 are the sub-abilities in the aggregation relation. In the demand satisfaction assessment, "1" means "satisfied" or "high", and "0" means "not satisfied" or "low". The weights may be quantified using a weighting factor.

(2) Generalization

The set of the system of systems capability contains the generalization relation is a hierarchical structure. The generalization relationship is the OR relationship. The conditional Belief function or conditional basic Belief for the generalization relation is defined as follows:

(

)

(

)

(

)

(

)

(

)

(

)

(

) (

)

(

)

(

)

1 2

1 2 1 2

1 2 1 2

1 2

1

1 1, 1 1

1 1, 0 0 1, 0 ;

1 0, 1 0 0, 1 ;

1 1 1, 2 0 1 1 0, 2 1 ; if the weight of is greater than

0 1 0, 2 1 0 1 1, 2 0 |

| |

| |

; if the weight of |

| |

|

m C C C

m C C C m C C C

m C C C m C C C

m C C C m C C C C C

m C C C m C C C C

= = = =

= = = > = = =

= = = > = = =

= = = > = = =

= = = > = = =

(

)

(

)

2

1 2

is greater than

0| 0, 0 1

C

m C= C = C = =

(4)

Among them, C is super ability in generalization relation, C1 and C2 are sub-abilities in generalization relation. The weights may be quantified using a weighting factor.

(3) Value dependent

(3)

(

)

(

)

(

) (

)

(

)

(

) (

)

(

)

1 2

1 2 1 2 1 2

1 2 1 2 1 2

1 2

1 1, 1 1

1 1, 0 1 0, 1 ; if is more dependent on than

0 0, 1 0 1, 0 ; if is more dependent on than

0 0, 0

|

1 |

|

| |

|

m C C C

m C C C m C C C C C C

m C C C m C C C C C C

m C C C

= = = =

= = = > = = =

= = = > = = =

= = = =

(5)

where C is the parent capability in the value dependency relation, and C1 and C2 are the sub-abilities in the value dependency relation. The degree of dependence can be characterized by a causal coefficient.

(4) Semantic dependence

Semantic dependence is also called semantic association. The set of the system of systems capability that contains the semantic dependencies may form a network structure. The semantic dependency relationship is an AND relationship. The conditional belief function or conditional basic Belief for a semantic dependency relationship is defined as follows:

(

)

(

)

(

)

(

)

(

)

(

)

1 2

1 2 1 2

1 2

1 2 1 2

1

1 1, 1 1

1 1, 0 1 0, 1 ;

If the degree of and semantic relevance is greater than

0 0, 1 0 1, 0 ;

If the degree of and semantic relevance is g |

reat

| |

| |

er

m C C C

m C C C m C C C

C C C

m C C C m C C C

C C

= = = =

= = = > = = =

= = = > = = =

(

)

(

)

2

1 2

than

0| 0, 0 1

C

m C= C = C = =

(6)

where C is the parent capability in the semantic dependency relation, and C1 and C2 are the sub-capabilities in the semantic dependency relation. The degree of semantic association can also be characterized by a causal coefficient.

Conversion between Weight Coefficient, Causal Intensity and Conditional Belief function

The Conditional Belief Function is Calculated from the Weighting Factor

The following table shows the weight assignment of sub-abilities C1 and C2 for capability C. The identification framework of parent competence and child ability are {1,0}. This subsection considers the conditional belief function from the weighting factor. The conditional belief function is represented by the conditional basic belief.

Table 1. Weight coefficients.

Parent Capability Sub-capability Weight

C(1,0)

C1(1,0)

θ

1

C2(1,0)

θ

2

Table 2. Conditional belief unction.

m(C=1|(C1=0, C2=0))=0

m(C=1|(C1=1, C2=0))=x1

m(C=1|(C1=0, C2=1))=x3

m(C=1|(C1=1, C2=1))=1

m(C=0|(C1=0, C2=0))=1

m(C=0|(C1=1, C2=0))=x2

m(C=0|(C1=0, C2=1))=x4

m(C=0|(C1=1, C2=1))=0

m(C={1, 0}|(C1=0,C2=0))=0

m(C={1, 0}|(C1=1,C2=0))=x5

m(C={1, 0}|(C1=0,C2=1))=x6

(4)

1 2 1 2 5

3 4 4 5 6

1 1 1 1

3 2 3 2

2 2 2 2

4 1 4 1

complete conditions Incomplete conditions

( 5 0; 6 0) (with cognitive u

1 1

1 1

;

ncertainty)

x x x x x

x x x x x

x x

x x

x x

x

x x

x

θ θ

θ θ

θ θ

θ θ

+ = + = −

 

 

+ = + =

= =

 

     

= =

     

   

 

 

= =

 

 

(7)

Complete condition is that the basic belief is only assigned to two mutually exclusive states, not assigned to the whole set, that is, x5 = 0; x6 = 0. Specific distribution formula as shown in the above

formula.

Incomplete condition is that, because of the existence of cognitive uncertainty, the basic belief is assigned not only to the two mutually exclusive state, but also assigned to the whole set. First determine the probability of cognitive uncertainty, that is, the basic trustworthiness x5, x6, then assign

the basic belief of other states. Specific distribution formula as shown in the above formula.

Calculate the Conditional Belief Function from the Causal Intensity

[image:4.612.99.511.71.166.2]

The following figure is a single-value causal diagram, C is the top layer capability, C1, C2 are the sublevel capabilities, B3, B4 are the underlying indicators, P1, P2, P31, P42 for is causal intensity factor, nodes with no connection are independent of each other. Consider the conditional belief function m C C C( | 1 2) from the causal intensity.

Figure 1. Single-value cause and effect diagram.

Conditional belief function, like posterior probability of Bayesian theory, can be derived from causal reasoning. Proceed as follows.

1) Find the first-order cut-set expression of node events

A cut set (CS) is a group of events (including basic events, node events, logic gate events and connection events) in a logical AND (AND) relationship. A cut set (CS) consisting only of events adjacent to a node event is called a first-order cut set, abbreviated CSs-1[8-10].

The method for finding CSs-1 of node events is: transforming the cause-and-effect diagram into the causal tree expression. According to the definition of the first-order cut set, we can write the logical expression of the root node of every micro-causal tree.

The first-order divisions of Fig. 1 are given by the following equations.

1 31 3; 2 42 4; 1 1 2 2

C =P B C =P B C=PCP C (8) 2) Find the final cut set expression for the node event

The cut set expression denoted by the basic event and the join event is called the final cut set of the node event, abbreviated as CSs-f.

The CSs-1 expression is expanded to eliminate all node events and the final cut set is obtained. If there is a directed ring in the expanding process, the de-ringing rule needs to be used to de-ring.

(5)

1 31 3; 2 42 4; 2 31 3 2 42 4

C =P B C =P B C=P P BP P B (9)

3) Find the disjoint cut sets (DCSs-f) of node events

The CSs-f expression for a node event X can be denoted as

1

m i i

X C

=

=

, where

1

i

n i ij

j

C V

=

=

, Ci is a

CS, then X =C1+C C2 1+C C C3 1 2++C C Cm 1 2Cm1.

The above formula is the DCSs-f expression of X, and the '+' is the exclusive-OR operator. The DCSs-f of Fig. 1 is as follows. In the process of disjointing, we need to use Di - Morgan law for equal power (A A⋅ =A), complementary (A A⋅ =0) and absorption (AAB=A) logic operations.

1

1 31 3; 2 42 4; 1 31 3 2 42 4 2 42 4 1 31 2 42 4 1 31 3

C =P B C =P B C=P P B +P P B P +P P B P P +P P B P P B (10)

4) The conditional belief function of the event of interest H m H E( | ) is calculated under the condition of evidence E given

Given evidence E is observed some node events or basic events, E is the evidence set

1 2 k

E=EE ∩∩E , k is the number of evidence received. H is the event to be investigated.

1 2

, , , , k

H E E E are basic events or intermediate events. According to Bayesian formula,

1 2

1 2

Pr( )

Pr( ) Pr( ) ( | )

Pr( ) Pr( ) Pr( )

k k

H E E E

HE H E

m H E

E E E E E

∩ ∩ ∩ ∩

= = =

∩ ∩ ∩

(11)

The conditional belief function of Figure 1. is shown in the following formula. The logical AND operation need to use the absorption operation (A+AB= A).

1

1 2 31 3 42 4; 1 2 31 3 42 4( 1 2 ); ( | 1 2) 1 (1 1)(1 2)

C C =P B P B CC C =P B P B P+P P m C C C = − −PP (12)

The relationship between the nodes in the causal graph is OR by default, and the conditional belief function of the AND relationship can be deduced from the above method as shown in the following formula.

1 2 1 2

( | )

m C C C =P P (13)

As can be seen from the above results, the conditional belief function m C C C( | 1 2) depends only on

causal intensity P1 and P2, but not on basic events B3 and B4.

Example

Maritime regional air defense system-of-systems capability

Reconnaissance and early warning

capability

Fire interception capability Command and control capability

Remote early warning capability

Tracking and monitoring capability

Decision-making capability

Communication capability

Missile intercept capability

[image:5.612.203.411.534.724.2]

Aircraft intercept capability

(6)

The above picture shows the capability structure of the air defense system at sea.

The relationship between the maritime regional air defense system of systems capability and the reconnaissance and early warning capability, the command and control capability, the fire interception capability belongs to the combination relationship. The relationship between the reconnaissance and early warning capability and early warning capability, tracking monitoring capability belongs to the dependency relation; The relationship between the command and control capability and decision-making capability, the communication capability belongs to the combination relation; The relationship between the fire interception capability and missile intercept capability, the aircraft intercept capability belongs to the generalization relation.

According to the aforementioned method, the combination relationship and the generalization relation are quantized by the weight coefficient, and the dependency relationship is measured by the causal coefficient. Among them, the weight coefficient is assigned as follows:

The weight of reconnaissance and early warning capability is 0.3, the weight of command and control capability is 0.3, the weight of firepower interception ability is 0.4;

The weight of decision-making capability is 0.4, the weight of communication capability is 0.6; The weight of missile interception capability is 0.6, and the weight of aircraft interception

capability is 0.4.The causalintensity assignment is as follows:

the causal intensity between the early warning capability and the reconnaissance early-warning capability is 0.8, the causal intensity between racking surveillance capability and reconnaissance early warning capability is 0.7.

[image:6.612.88.519.378.686.2]

The conditional belief parameter table obtained as described above is shown below.

Table 3. Conditional belief function.

Node Conditional Belief Function

C2

m(C2=1|C5=1, C6=1)=0.56,m(C2=0|C5=1, C6=1)=0.44

m(C2=1|C5=1, C6=0)=0.24,m(C2=0|C5=1, C6=0)=0.76

m(C2=1|C5=0, C6=1)=0.14,m(C2=0|C5=0, C6=1)=0.86

m(C2=1|C5=0, C6=0)=0.06,m(C2=0|C5=0, C6=0)=0.94

C3

m(C3=1|C7=1, C8=1)=1

m(C3=1|C7=1, C8=0)=0.4,m(C3=0|C7=1, C8=0)=0.6

m(C3=1|C7=0, C8=1)=0.6,m(C3=0|C7=0, C8=1)=0.4

m(C3=0|C7=0, C8=0)=1

C4

m(C4=1|C9=1, C10=1)=1

m(C4=1|C9=1, C10=0)=0.6 ,m(C4=0|C9=1, C10=0)=0.4

m(C4=1|C9=0, C10=1)=0.4 ,m(C4=0|C9=0, C10=1)=0.6

m(C4=0|C9=0, C10=0)=1

C1

m(C1=1|C2=1, C3=1, C4=1)=1

m(C1=1|C2=1, C3=1, C4=0)=0.6,m(C1=0|C2=1,C3=1, C4=0)=0.4

m(C1=1|C2=1,C3=0, C4=1)=0.7,m(C1=0|C2=1,C3=0, C4=1)=0.3

m(C1=1|C2=0, C3=1, C4=1)=0.7,m(C1=0|C2=0,C3=1, C4=1)=0.3

m(C1=1|C2=1, C3=0, C4=0)=0.3,m(C1=0|C2=1,C3=0, C4=0)=0.7

m(C1=1|C2=0, C3=1, C4=0)=0.3,m(C1=0|C2=0,C3=1, C4=0)=0.7

m(C1=1|C2=0, C3=0, C4=1)=0.4,m(C1=0|C2=0,C3=0, C4=1)=0.6

m(C1=0|C2=0, C3=0, C4=1)=1

(7)

(

)

(

)

(

)

(

)

(

)

(

)

2 5 6 1 2

2 5 6

2 5 6 2

2 5 6

2 5 6 1 2

2 5 6

1 1, 1 * 0.8* 0.7 0.56

0 1, 1 1 0.56 0.44

1 1, 0 1* 1 0.8* 1 0.7 0.24

0 1, 0 1 0.24 0.76

1 0, 1 1 * 1 0.8 *0.7 0.14

0 0, 1 1 0.14 |

|

|

|

|

|

m C C C P P

m C C C

m C C C P P

m C C C

m C C C P P

m C C C

= = = = = =

= = = = − =

= = = = − = − =

= = = = − =

= = = = − = − =

= = = = − =

( ) ( )

( ) ( )

(

)

(

)

2 5 6 1 2

2 5 6

0.86

1 0, 0 1 * 1 1 0.8 * 1 0.7 0.06

0 0, 0 1 0.0 |

0.94

| 6

m C C C P P

m C C C

= = = = − − = − − =

= = = = − =

( )( )( )( )

(14)

The conditional belief functions of C3 and C4 nodes are transformed from weight coefficients. To C3, for example, Substituting θ1=0.4 and θ2 =0.6 into equation () yields: x1 = 0.4, x2 = 0.6, x3 = 0.6, x4 = 0.4.

Then, there are:

(

)

(

)

(

)

(

)

(

)

(

)

3 7 8

3 7 8 1 3 7 8 2

3 7 8 3 3 7 8 4

3 7 8

1 1, 1 1

1 1, 0 0.4, 0 1, 0 0.6

1 0, 1 0.6, 0 0, 1 0

|

| |

.4

0 0

| |

, 0 1 |

m C C C

m C C C x m C C C x

m C C C x m C C C x

m C C C

= = = =

= = = = = = = = = =

= = = = = = = = = =

= = = =

(15)

The C1 node conditional belief function is transformed from the weight coefficient. Unlike C3, it has three child nodes.

Conclusion

In order to study the capability evaluation of weapon equipment system of systems based on evidential network, this paper studies the acquisition of the conditional belief function from the two aspects of the weight coefficient and the causality coefficient respectively according to the relationship between the weapon equipment system of systems capability. The final example shows that the method is feasible, which can quantify the uncertainties in the capability evaluation of weapon equipment system of systems and improve the credibility of capability evaluation of weapon equipment system of systems to a certain extent.

The next step is to study the capability evaluation of weapon equipment system of systems based on conditional evidential network, and further study the uncertainty in the capability evaluation of weapon equipment system of systems on the basis of conditional belief function.

References

[1]W. Laamari, B. Ben Yaghlane, Reasoning in Singly-Connected Directed Evidential Networks with Conditional Beliefs[J]. A. Likas, K. Blekas, and D. Kalles (Eds.): SETN 2014, LNAI 8445, 2014:221–236.

[2]W. Laamari, N. Ben Hariz, and B. Ben Yaghlane, Approximate Inference in Directed Evidential Networks with Conditional Belief Functions Using the Monte Carlo Algorithm[J]. A. Laurent et al. (Eds.): IPMU 2014, Part III, CCIS 444, 2014:486–497.

[3]N. Ben Hariz and B. Ben Yaghlane, Learning Parameters in Directed Evidential Networks with Conditional Belief Functions[J]. F. Cuzzolin (Ed.): BELIEF 2014, LNAI 8764, 2014:294–303.

(8)

[5]W. Laamari and B. Ben Yaghlane, Propagation of Belief Functions in Singly-Connected Hybrid Directed Evidential Networks[J]. Beierle and A. Dekhtyar (Eds.): SUM 2015, LNAI 9310, 2015:234–248.

[6]Qing-song Zhao, Research on Description of Weapon Equipment System of Systems Capability Spaces [J]. Journal of National University of Defense Technology, 2009, 31 (1): 135-140.

[7]Ben Chen, Capability Views Model for Weapon System of Systems[J]. Journal of National University of Defense Technology, 2011, 33 (6): 163-168.

[8]Xin-yuan Liang, The Research on Theory and Algorithm of Causality Diagram of Complex System [D]. Chongqing, Chongqing University, 2005: 9-17.

[9]Hong-chun Wang, The Research on Theory and Algorithm of Uncertainty Reasoning Based on Causality Diagram [D]. Chongqing, Chongqing University, 2005: 21-38.

Figure

Figure 1. Single-value cause and effect diagram.
Figure 2. Maritime regional air defense capability.
Table 3. Conditional belief function.

References

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