### 1. Introduction

### 2. Related Work

### 3. Jøsang’s Fusion Rule and Its Issues

*i*to entity

*X*

*;*

_{j}*X*

*separately;*

_{j}*X*= {

*x*

*|*

_{i}*i*= 1, …,

*k*} be a frame and let

*ω*

*be an opinion on*

_{X}*X*with belief

*b*and uncertainty mass

*u*. Let

*a*be a base rate vector on

*X*. The function

*E*(

*X*) from

*X*to [0, 1]

*k*is expressed as:

*X*.

The two sensors for the observing the process during disjointed time periods. In this case the observations are independent, and it is natural to simply add the observations from these two sensors. The resulting fusion is called cumulative fusion.

The two sensors for the observing the process during the same time period. In this case, the observations are non independent, and it is natural to take the average of the observations by using the two sensors. The resulting fusion is called averaging fusion.

### Cumulative Fusion Rule

*ω*

*and*

^{A}*ω*

*be opinions that are respectively held by agents*

^{B}*A*and

*B*over the same frame

*X*= {

*x*

*|*

_{i}*i*= 1, … ,

*l*}. Let

*ω*

^{A}^{⋄}

*be the opinion such that:*

^{B}*ω*

^{A}^{⋄}

*is called the cumulatively fused*

^{B}*bba*of

*ω*

*and*

^{A}*ω*

^{B}*,*which represents the combination of independent opinions of

*A*and

*B*. By using the symbol ‘⊕’ to designate this belief operator, we define

*ω*

^{A}^{⋄}

^{B}*≡ ω*

*⊕*

^{A}*ω*

^{B}*.*

### Averaging Fusion Rule

*ω*

*and*

^{A}*ω*

*be opinions that are respectively held by agents*

^{B}*A*and

*B*over the same frame

*X*= {

*x*

*| i = 1, … ,*

_{i}*l*}. Let

*ω*

^{A}^{⋄}

*be the opinion such that:*

^{B}*ω*

^{A}^{⋄}

*is called the averaged opinion of*

^{B}*ω*

*and*

^{A}*ω*

^{B}*,*which represents the combination of the dependent opinions of

*A*and

*B*. By using the symbol ‘⊕’ to designate this belief operator, we define

*ω*

^{A}^{⋄}

^{B}*≡ ω*

*⊕*

^{A}*ω*

^{B}*.*

*A*,

*B*,

*C*) fusion opinion by averaging fusion rule,

*b*

^{A}^{⋄}

^{B}^{⋄}

*(*

^{C}*x*

*) ≠*

_{i}*b*

^{A}^{⋄}

^{B}^{⋄}

*(*

^{C}*x*

*). That is to say, the three agents’ fusion opinion result is not uniqueness by Eq. (5). This is because the fusion sequence is different, which is obviously unreasonable.*

_{i}### 4. The ISLM-ED Model

Only one sensor observes the process during disjoint time periods: in this case, the observations are independent. We need to consider the time factor, and the resulting fusion is called cumulative fusion.

The different sensors observe the process during the same time period: in this case, the observations are independent. We need to consider the weight of the observation sensors, and the resulting fusion is called the averaging fusion.

The different sensors observe the process during disjoint time periods: in this case, we need to consider both the time factor and the weight of the sensors, and the resulting fusion is called the multi-agent unified fusion operator.

### 4.1 Cumulative Fusion Operator

*X*and the same agent’s opinions about

*x*for cumulative fusion in different observation periods, because the cumulative fusion opinion is influenced by the observation period, the result may be different. Fig. 1 shows the distance diagram according to observation period.

*ω*

*by the same agent in N periods, (*

_{i}*i*= 1, 2,…, N), the influence on the fusion opinion can be studied according to the 3 cases listed below and as shown in Fig. 2:

Line 1 can be understood as the nearer time to the current, the greater the influence.

Line 2 can be understood as the further time to the current, the greater the influence. For example, a person’s first impressions are the strong.

Line 3 can be understood as the same influence on the fusion opinion among different periods. For example, the observation is a random event, and as such there is no effect on the fusion opinion among periods.

*i*) denotes the influence coefficient of the opinion in

*i*

_{th}period to the cumulative fusion.

*i*

_{th}agents’ opinion are:

### 4.2 Averaging Fusion Operator

*X*, the different agents’ opinions about

*x*for averaging fusion in the same observation period. Because the different influences of the fusion opinion come from different agents, we need to consider the weight of the agents only.

*W*

*and*

_{A}*W*

*are weights of the agents*

_{B}*A*and

*B*, respectively. Both

*W*

*and*

_{A}*W*

*, then*

_{B}*A*,

*B*] opinion about

*x*, as if

*A*and

*B*. By using the symbol ‘ ⊕ ’ to designate this operator, we define

*W*

_{A}_{,}

*=*

_{B}*W*

*+*

_{A}*W*

*.*

_{B}*j*

_{th}observer is

*ω*, and express it as

### 4.3 Multi-Agent Unified Fusion Operator

*X*, the different agents’ opinions about

*x*for averaging fusion in the different periods. In this case, we need to integrate the former two fusion operators.

*i*

*observer in*

_{th}*j*

*periods is*

_{th}*j*∈[2,M], the opinions are fused by the multi-agent unified fusion operator. As such, let

### 4.4 Fusion Operator Character Proving

#### Commutativity proving

#### Associativity law proving

*ω*

*= {*

^{A}*b⃗*

^{A}*,u*

*,*

^{A}*W*

*}, where*

^{A}*ω*

*= {*

^{B}*b⃗*

^{B}*, u*

^{B}*,W*

*} and*

^{B}*ω*

*= {*

^{C}*b⃗*

^{C}

_{,}*u*

^{C}

_{,}*W*

*} are the opinion of the agents of*

^{C}*A*,

*B*,

*C*on the

*X*. Proving the associativity law only needs to prove that the following equation hold. Take

*b*as an example of fusion.

*u*satisfies associativity.

### 4.5 Dynamic Function of the Base Rate with Evidence Driven

*r*

*denotes the positive evidence number of the*

_{i}*i*

_{th}period and

*s*

*denotes the negative evidence number of the*

_{i}*i*

_{th}period.

*a*

*denotes the base rate of the*

_{i}*i*

_{th}period. First, it needs to analyze the evidence in terms of how to drive the change of the base rate through the first and second observation. Then, it needs to design the dynamic function of the base rate.

*r*

_{1}=

*r*

_{2}, it means that the positive evidence number of the second observation is equal to the first. Because the sensor observes M times in each period, we have

*s*

_{1}=

*s*

_{2}. That is to say, the result of the second observation is the same as the first observation. Therefore, the base rate should not have any changes occur.

*r*

_{2}>

*r*

_{1}, the positive evidence number of the second observation is more than the first. So, the base rate should increase, that is, mean

*a*

_{2}>

*a*

_{1}, and vice versa.

**|**

*r*

_{2}-

*r*

_{1}

**|/**M)

^{λ}and λ be the adjustment coefficient, (λ>=1). The larger the λ is, the less sensitive the base rate is to the change of the evidence. If we do not consider the time effectiveness of the evidence, we have:

*r*

_{1}>

*r*

_{2}, a

_{2}= a

_{1}+(1-a

_{1})η;

*r*

_{1}<

*r*

_{2}, a

_{2}=(a

_{1}-η)/(1-η).

_{i+1}by a

_{i},

*r*

_{i}_{+1}-

*r*

*|/ M)*

_{i}^{λ}, which gives us:

_{i+1}already includes the effect of the former

*i*period of evidence.

**|**

*r*

_{i}_{+1}-

*kr*

_{i}**|/**

*s*

*)*

_{i}^{λ}, 0<

*k*<1, and

*k*is the discount coefficient.

### 4.6 Non-Informative Prior Weight with Evidence Driven

*C*before the observation period. This uncertainty that reflects on the evidence is expressed as the changes of the positive evidence numbers in different periods. There is a close relationship between

*C*and the observation evidence. We give the definition of the non-informative prior weight

*C*based on the obtained evidence in different observation periods.

*r*

*denotes the positive evidence number of the*

_{i}*i*

_{th}period. MAX

*and MIN*

_{i}*denotes the maximum and minimum value of the positive evidence number in the former*

_{i}*i*periods.

*C*’s design idea by analysis and comparison.

*C*.

### 5. Simulation Experiment

^{th}–42,000

^{th}records from the corrected data as experiment data. Assume that there are 600 records in each observation period and that there are 10 observation periods. Six observers gave each of the 600 records separately. That is to say, each observer provided 100 records. In order to use the new consensus operator, assume that the six observers’ weights were {0.5, 0.9, 0.7, 0.6, 0.8, and 0.8}. The statistical normal event number as positive evidence, the description see Table 3.

_{0}=0.8. The discount coefficient was k=1, which means without consider the time effectiveness of evidence.