CN1971549A - Method of fusion processing of multi-source fuzzy information - Google Patents

Abstract

The polygenetic fuzzy message merging handling method is a merging method based on vague proof theory to process the polygenetic heterogeneity fuzzy messages. It belongs to the technical field of detection and information processing. The steps are: choosing the merging handling messages; determining the identification framework of the proof theory; determining the basic degree of credibility distribution function value; determining the similarity of the Vague focus element, modifying the model of proof source, and combining the proof; the contribution factor that the focus element for the trust function of other focus element is determined; and the maximum is the final result according to the valued of focus element trust function. The advantages: the invention expresses the fuzzyness of information combined with the proof theory and vague group, full utilization of the complementary and redundancy between the fuzzy messages, easy to express the fuzzy messages, and the merging result and the trust function is sensitive for the change of the focus element. The method mainly used in the field of message merging, model identification and artificial intelligence.

Description

Method of fusion processing of multi-source fuzzy information

What technical field the present invention relates to is a kind of multi-source foreign peoples Fuzzy Data Fusion disposal route that is used for.Belong to and detect and technical field of information processing.

Background technology is present, according to the advantage of D-S evidence theory in probabilistic expression, measurement and combined aspects, (Fuzzy sets) can handle the characteristics that have fuzzy uncertain information effectively in conjunction with fuzzy set, forms the method for D-S evidence theory to the fuzzy set expansion.But mainly there is the defective of the following aspects in these methods: primitive is under the jurisdiction of the too arbitrary decision of determining of fuzzy burnt first degree; The definition of " degree of comprising " of fuzzy set and " intersecting degree " is not unique; Belief function Bel and likelihood function Pl as upper and lower probability are not made proper explanations; Belief function Bel is insensitive etc. to the marked change of some focus element.

Summary of the invention the objective of the invention is to the defective at above-mentioned existence, has designed a kind of method of fusion processing of multi-source fuzzy information based on the Vague evidence theory.This method is in conjunction with D-S evidence theory and Vague collection, make full use of the Vague collection aspect the expression of fuzzy message advantage and the D-S evidence theory in the advantage of probabilistic expression, measurement and combined aspects, it is had be easy to represent that fuzzy message, conversion fuzzy message are evidence, and fusion results and belief function Bel are to the characteristics such as marked change sensitivity of some focus element.Technical solution of the present invention: concrete implementation step is divided into successively:

(1) information of selection fusion treatment.According to the difference of application, selection need be carried out the information of fusion treatment.

(2) framework of identification in the conclusion evidence theory.According to the needs of recognition objective, in conjunction with the characteristics of Vague evidence theory, the framework of identification in the conclusion evidence theory.Determine corresponding Vague subclass on this basis.

(3) the basic reliability distribution functional value determines.At present, fusion information has polyphyly, and has foreign peoples's characteristics, therefore multi-source information need be converted into the evidence in the evidence theory, makes it have unified representation, is convenient to fusion treatment.Professional or expert system are converted into multi-source information according to experience the basic reliability distribution value of corresponding evidence.

(4) multi-source fuzzy information fusion treatment.The fusion treatment process is divided into two kinds of situations:

(A) fusion of two class evidences

Suppose at same framework of identification X (x ₁, x ₂... x _n) on two evidences are arranged, its basic reliability distribution function is respectively m ₁And m ₂, corresponding Vague is burnt, and unit is respectively { A ₁, A ₂..., A _pAnd { B ₁, B ₂..., B _q, so, make up these two evidences and get new basic reliability distribution function m: P _X→ [0,1], this basic reliability distribution function m is for the burnt unit of all Vague , have:

$m=\left(\tilde{C}\right)=m_1\⊕m_2\left(\tilde{C}\right)=\frac{{\mathrm{\Σ}}_{{\tilde{A}}_i\∩{\tilde{B}}_j=\tilde{C}}S\left(\widetilde{C,}{\tilde{A}}_i\right)m_1\left({\tilde{A}}_i\right)S\left(\widetilde{C,}{\tilde{B}}_j\right)m_2\left({\tilde{B}}_j\right)}{1-{\mathrm{\Σ}}_{{\tilde{A}}_i{\tilde{B}}_j}(1-S({\tilde{A}}_i\∩{\tilde{B}}_j,{\tilde{A}}_i)S({\tilde{A}}_i\∩{\tilde{B}}_j,{\tilde{B}}_j))m_1\left({\tilde{A}}_i\right)m_2\left({\tilde{B}}_j\right)}$

Wherein,

Be the burnt unit of V=ague With

Between similarity, it is defined as

If framework of identification X is (x ₁, x ₂... x _n), Be the burnt unit of the Vague on it, $\tilde{A},\tilde{C}\∈P_x,$ $\tilde{A}=\left(\right[t_{\tilde{A}}\left(x_1\right),1-f_{\tilde{A}}\left(x_1\right)]/x_{1,}[t_{\tilde{A}}\left(x_2\right),1-f_{\tilde{A}}\left(x_2\right)]/x_2,...[t_{\tilde{A}}\left(x_n\right),1-f_{\tilde{A}}\left(x_n\right)]/x_n),$ $\tilde{C}=\left(\right[t_{\tilde{C}}\left(x_1\right),1-f_{\tilde{C}}\left(x_1\right)]/x_1,[t_{\tilde{C}}\left(x_2\right),1-f_{\tilde{C}}\left(x_2\right)]/x_2,...,[t_{\tilde{C}}\left(x_n\right),1-f_{\tilde{C}}\left(x_n\right)]/x_n),$ Then Vague is burnt first With

Between similarity be:

$S(\tilde{C},\tilde{A})=1-\sqrt{\frac{1}{2\left|X\right|}\underset{i=1}{\overset{\left|X\right|}{\mathrm{\Σ}}}\{{(t_{\tilde{A}}\left(x_i\right)-t_{\tilde{C}}\left(x_i\right))}^2+{(f_{\tilde{A}}\left(x_i\right)-f_{\tilde{C}}\left(x_i\right))}^2+{({\mathrm{\π}}_{\tilde{A}}\left(x_i\right)-{\mathrm{\π}}_{\tilde{C}}\left(x_i\right))}^2\}}$

Wherein, | X| be framework of identification X base (| X|=n).

(B) fusion of multiclass evidence

Suppose at same framework of identification X (x ₁, x ₁... x _n) on n evidence arranged, its basic reliability distribution function is respectively m ₁, m ₂..., m _n, corresponding Vague is burnt, and unit is respectively { A _I1, A _I2..., A _IW, so, make up this n evidence and get new basic reliability distribution function m: P _X→ [0,1], combination step is as follows:

Step1：

${\tilde{C}}_k={\tilde{A}}_{i_1}\∩{\tilde{A}}_{i_2}\∩...\∩{\tilde{A}}_{i_n}(i_m=\mathrm{1,2},...,W;m=\mathrm{1,2},...,n)$

Step2：

$m_1\⊕m_2\⊕...\⊕m_n\left({\tilde{C}}_k\right)=\underset{{\tilde{A}}_{i_1}\∩{\tilde{A}}_{i_2}\∩...\∩{\tilde{A}}_{i_n}={\tilde{C}}_k}{\mathrm{\Σ}}{m}_1\left({\tilde{A}}_{i_1}\right)m_2\left({\tilde{A}}_{i_2}\right)...m_n\left({\tilde{A}}_{i_n}\right)=m_{\mathrm{1,2},...,n}\left({\tilde{C}}_k\right)$

Step3：

$N[m_1\⊕m_2\⊕...\⊕m_n]\left({\tilde{C}}_k\right)=\frac{{\mathrm{\Σ}}_{{\tilde{C}}_k}S({\tilde{C}}_k,{\tilde{A}}_{i_1})S({\tilde{C}}_k,{\tilde{A}}_{i_2})...S({\tilde{C}}_k,{\tilde{A}}_{i_n})m_{\mathrm{1,2},...,n}\left({\tilde{C}}_k\right)}{1-{\mathrm{\Σ}}_{{\tilde{A}}_{i_1},{\tilde{A}}_{i_2},...,{\tilde{A}}_{i_n}}(1-S({\tilde{C}}_k,{\tilde{A}}_{i_1})S({\tilde{C}}_k,{\tilde{A}}_{i_2})...S({\tilde{C}}_k,{\tilde{A}}_{i_n}))m_{\mathrm{1,2},...,n}\left({\tilde{C}}_k\right)}$

Wherein, $S({\tilde{C}}_k,{\tilde{A}}_{i_m})(i_m=\mathrm{1,2},...,W;m=\mathrm{1,2},...,n)$ Can adopt the A step to calculate, the k value is by the power set P of framework of identification X _XBase | P _X|=W and evidence number n are determined.

(5) kind judging.Based on the multi-source fuzzy information combined result of Vague evidence theory, determine the belief function value of classification, select the classification of maximum belief function value, as final classification.Concrete steps are as follows:

Step1: based on the multi-source fuzzy information combined result, determine classification belief function contribution because of in.

If framework of identification X is (x ₁, x ₂... x _n), Be the burnt unit of the Vague on it, $\tilde{A},\tilde{B}\∈P_X,$

$\tilde{A}=\left(\right[t_{\tilde{A}}\left(x_1\right),1-f_{\tilde{A}}\left(x_1\right)]/x_1,[t_{\tilde{A}}\left(x_2\right),1-f_{\tilde{A}}\left(x_2\right)]/x_2,...,[t_{\tilde{A}}\left(x_n\right),1-f_{\tilde{A}}\left(x_n\right)]/x_n)$

$\tilde{B}=\left(\right[t_{\tilde{B}}\left(x_1\right),1-f_{\tilde{B}}\left(x_1\right)]/x_1,[t_{\tilde{B}}\left(x_2\right),1-f_{\tilde{B}}\left(x_2\right)]/x_2,...,[t_{\tilde{B}}\left(x_n\right),1-f_{\tilde{B}}\left(x_n\right)]/x_n)$

Then Vague is burnt first Right Contribution factor be

$I_v(\tilde{B};\tilde{A})=\frac{\underset{i}{\mathrm{\Σ}}\min \{t_{\tilde{A}}\left(x_i\right),t_{\tilde{B}}\left(x_i\right)\}}{\underset{i}{\mathrm{\Σ}}(1-f_{\tilde{A}}\left(x_i\right))}$

Step2: foundation $\mathrm{Bel}\left(\tilde{B}\right)=\underset{i}{\mathrm{\Σ}}{I}_v(\tilde{B};{\tilde{A}}_i)m\left({\tilde{A}}_i\right),$ Determine the belief function value of Vague evidence theory.

Step3: calculate Judge final classification.

Embodiment is an object with the water monitoring data of Taihu Lake Xiao Meikou, estimates this regional water quality eutrophication situation, and implementation process of the present invention is described.

(1) information of selection fusion treatment.According to the situation of geographic position, Taihu Lake and surrounding enviroment, choose the bigger factor of water quality eutrophication situation influence: chlorophyll a (Chla), total nitrogen (TN), as evaluation index.Concrete Monitoring Data is that chlorophyll a (Chla) is 0.006mg/L; Total nitrogen (TN) is 1.41mg/L.

(2) framework of identification in the conclusion evidence theory.Based on Taihu Lake area water environment historical situation and Changing Pattern, corresponding standard be divided into poor nutrition, poor-middle nutrition, middle nutrition, in-eutrophy, eutrophy, its value sees Table 1.

Table 1 Taihu Lake eutrophication degree evaluation standard

Nutrient type	Poor nutrition	Poor-middle nutrition	Middle nutrition	In-eutrophy	Eutrophy
						Chla(mg·L -1) TN(mg·L -1)	0.0016 0.079	0.0041 0.160	0.0100 0.310	0.0260 0.650	0.0640 1.200

According to Taihu Lake water quality condition in recent years, determine framework of identification in the D-S evidence theory be Θ=1,2,3}, wherein: the poor nutrition of 1 expression; Nutrition in 2 expressions; 3 expression eutrophy, corresponding Vague subclass is The water quality classification of its concrete numerical value and representative is as follows:

${\tilde{A}}_1=\{\left[\mathrm{1,1}\right]/1,\left[\mathrm{0.6,0.7}\right]/2,\left[\mathrm{0.1,0.3}\right]/3\}$ Poor nutrition

${\tilde{A}}_2=\{\left[\mathrm{0.6,0.8}\right]/1,\left[\mathrm{0.65,0.75}\right]/2,\left[\mathrm{0.2,0.3}\right]/3\}$ Poor-middle nutrition

${\tilde{A}}_3=\{\left[\mathrm{0.55,0.75}\right]/1,\left[\mathrm{1,1}\right]/2,\left[\mathrm{0.55,0.75}\right]/3\}$ Middle nutrition

${\tilde{A}}_4=\{\left[\mathrm{0.3,0.6}\right]/1,\left[\mathrm{0.5,0.8}\right]/2,\left[\mathrm{0.6,0.85}\right]/3\}$ In-eutrophy

${\tilde{A}}_5=\{\left[\mathrm{0.1,0.3}\right]/1,\left[\mathrm{0.55,0.75}\right]/2,\left[\mathrm{1,1}\right]/3\}$ Eutrophy

(3) the basic reliability distribution functional value determines.Rule of thumb Monitoring Data is converted into the basic reliability distribution value of corresponding evidence by monitoring personnel or expert system, sees Table 2.

The basic trust of table 2 evidence distributes (BPA)

(4) Monitoring Data fusion treatment.This example adopts two evidence Chla and TN, belongs to two class evidences and merges.The A one step process merges, and obtains 9 kinds of burnt units of different combinations

Its BPA sees Table 3.

$m\left(\tilde{C}\right)=m_1\⊕m_2\left(\tilde{C}\right)=\frac{{\mathrm{\Σ}}_{{\tilde{A}}_i\∩{\tilde{B}}_j=\tilde{C}}S(\tilde{C},{\tilde{A}}_i)m_1\left({\tilde{A}}_i\right)S(\tilde{C},{\tilde{B}}_j)m_2\left({\tilde{B}}_j\right)}{1-{\mathrm{\Σ}}_{{\tilde{A}}_i{\tilde{B}}_j}(1-S({\tilde{A}}_i\∩{\tilde{B}}_j,{\tilde{A}}_i)S({\tilde{A}}_i\∩{\tilde{B}}_j,{\tilde{B}}_j))m_1\left({\tilde{A}}_i\right)m_2\left({\tilde{B}}_j\right)}$

The BPA value of the fuzzy burnt unit in table 3 evidence combination back

Wherein, ${\tilde{C}}_1={\tilde{A}}_1\∩{\tilde{A}}_3=\left\{\right[\mathrm{0.55,0.75}]/1,[\mathrm{0.6,0.7}]/2,[\mathrm{0.1,0.3}]/3\}$

${\tilde{C}}_2={\tilde{A}}_1\∩{\tilde{A}}_4=\left\{\right[\mathrm{0.3,0.6}]/1,[\mathrm{0.5,0.7}]/2,[\mathrm{0.1,0.3}]/3\}$

${\tilde{C}}_3={\tilde{A}}_1\∩{\tilde{A}}_5=\left\{\right[\mathrm{0.1,0.3}]/1,[\mathrm{0.55,0.7}]/2,[\mathrm{0.1,0.3}]/3\}$

${\tilde{C}}_4={\tilde{A}}_2\∩{\tilde{A}}_3=\left\{\right[\mathrm{0.55,0.75}]/1,[\mathrm{0.65,0.75}]/2,[\mathrm{0.2,0.3}]/3\}$

${\tilde{C}}_5={\tilde{A}}_2\∩{\tilde{A}}_4=\left\{\right[\mathrm{0.3,0.6}]/1,[\mathrm{0.5,0.75}]/2,[\mathrm{0.2,0.3}]/3\}$

${\tilde{C}}_6={\tilde{A}}_2\∩{\tilde{A}}_5=\left\{\right[\mathrm{0.1,0.3}]/1,[\mathrm{0.55,0.75}]/2,[\mathrm{0.2,0.3}]/3\}$

${\tilde{C}}_7={\tilde{A}}_3\∩{\tilde{A}}_3=\left\{\right[\mathrm{0.55,0.75}]/1,[\mathrm{1,1}]/2,[0.55,0.75]/3\}$

${\tilde{C}}_8={\tilde{A}}_3\∩{\tilde{A}}_4=\left\{\right[\mathrm{0.3,0.6}]/1,[\mathrm{0.5,0.8}]/2,[\mathrm{0.55,0.75}]/3\}$

${\tilde{C}}_9={\tilde{A}}_3\∩{\tilde{A}}_5=\left\{\right[\mathrm{0.1,0.3}]/1,[\mathrm{0.55,0.75}]/2,[\mathrm{0.55,0.75}]/3\}$

(5) kind judging.

Step1:, determine classification based on the multi-source fuzzy information combined result The contribution factor of belief function, the results are shown in Table 4.

The fuzzy burnt unit in table 4 evidence combination back

The contribution factor of belief function

Step2: determine the belief function value of Vague evidence theory, see Table 5.

The fuzzy burnt unit in table 5 evidence combination back The belief function value

Step3: calculate

Judge final classification

Promptly the water environment eutrophy situation in this basin is middle nutrition.

Claims (1)

1, method of fusion processing of multi-source fuzzy information.It is characterized in that method step is divided into successively:

(1) information of selection fusion treatment.According to the difference of application, selection need be carried out the information of fusion treatment.

(2) framework of identification in the conclusion evidence theory.According to the needs of recognition objective, in conjunction with the characteristics of Vague evidence theory, the framework of identification in the conclusion evidence theory.Determine corresponding Vague subclass on this basis.

(3) the basic reliability distribution functional value determines.At present, fusion information has polyphyly, and has foreign peoples's characteristics, therefore multi-source information need be converted into the evidence in the evidence theory, makes it have unified representation, is convenient to fusion treatment.Professional or expert system are converted into multi-source information according to experience the basic reliability distribution value of corresponding evidence.

(4) multi-source fuzzy information fusion treatment.The fusion treatment process is divided into two kinds of situations:

(A) fusion of two class evidences

Suppose at same framework of identification X (x ₁, x ₂... x _n) on two evidences are arranged, its basic reliability distribution function is respectively m ₁And m ₂, corresponding Vague is burnt, and unit is respectively { A ₁, A ₂..., A _pAnd { B ₁, B ₂..., B _q, so, make up these two evidences and get new basic reliability distribution function m: P _X→ [0,1], this basic reliability distribution function m is for the burnt unit of all Vague

, have:

$m\left(\tilde{C}\right)=m_1\⊕m_2\left(\tilde{C}\right)=\frac{{\mathrm{\Σ}}_{{\tilde{A}}_i\∩{\tilde{B}}_j=\tilde{C}}S(\tilde{C},{\tilde{A}}_i)m_1\left({\tilde{A}}_i\right)S(\tilde{C},{\tilde{B}}_j)m_2\left({\tilde{B}}_j\right)}{1-{\mathrm{\Σ}}_{{\tilde{A}}_i{\tilde{B}}_j}(1-S({\tilde{A}}_i\∩{\tilde{B}}_j,{\tilde{A}}_i)S({\tilde{A}}_i\∩{\tilde{B}}_j,{\tilde{B}}_j))m_1\left({\tilde{A}}_i\right)m_2\left({\tilde{B}}_j\right)}$

Wherein,

Be the burnt unit of Vague With

Between similarity, it is defined as establishes framework of identification X (x ₁, x ₂... x _n),

Be the burnt unit of the Vague on it, $\tilde{A},\tilde{C}\∈P_X,$

$\tilde{A}=\left(\right[t_{\tilde{A}}\left(x_1\right),1-f_{\tilde{A}}\left(x_1\right)]/x_1,[t_{\tilde{A}}\left(x_2\right),1-f_{\tilde{A}}\left(x_2\right)]/x_2,\·\·\·,[t_{\tilde{A}}\left(x_n\right),1-f_{\tilde{A}}\left(x_n\right)]/x_n)$

$\tilde{C}=\left(\right[t_{\tilde{C}}\left(x_1\right),1-f_{\tilde{C}}\left(x_1\right)]/x_1,[t_{\stackrel{\text{~}}{C}}\left(x_2\right),1-f_{\tilde{C}}\left(x_2\right)]/x_2,\·\·\·,[t_{\tilde{C}}\left(x_n\right),1-f_{\tilde{C}}\left(x_n\right)]/x_n)$

Then Vague is burnt first With Between similarity be:

$S(\tilde{C},\tilde{A})=1-\sqrt{\frac{1}{2\left|X\right|}\underset{i=l}{\overset{\left|X\right|}{\mathrm{\Σ}}}\{{(t_{\tilde{A}}\left(x_i\right)-t_{\tilde{C}}\left(x_i\right))}^2+{(f_{\tilde{A}}\left(x_i\right)-f_{\tilde{C}}\left(x_i\right))}^2+{({\mathrm{\π}}_{\tilde{A}}\left(x_i\right)-{\mathrm{\π}}_{\tilde{C}}\left(x_i\right))}^2\}}$

Wherein, | X| be framework of identification X base (| X|=n).

(B) fusion of multiclass evidence

Suppose at same framework of identification X (x ₁, x ₂... x _n) on n evidence arranged, its basic reliability distribution function is respectively m ₁, m ₂..., m _n, corresponding Vague is burnt, and unit is respectively { A _I1, A _I2..., A _IW, so, make up this n evidence and get new basic reliability distribution function m: P _X→ [0,1], combination step is as follows:

Step1：

${\tilde{C}}_k={\tilde{A}}_{i_1}\∩{\tilde{A}}_{i_2}\∩\·\·\·\∩{\tilde{A}}_{i_n}(i_m=\mathrm{1,2},\·\·\·W;m=\mathrm{1,2},\·\·\·,n)$

Step2：

$m_1\⊕m_2\⊕\·\·\·\⊕m_n\left({\tilde{C}}_k\right)=\underset{{\tilde{A}}_{i_1}\∩{\tilde{A}}_{i_2}\∩\·\·\·\∩{\tilde{A}}_{i_n}={\tilde{C}}_k}{\mathrm{\Σ}}{m}_1\left({\tilde{A}}_{i_1}\right)m_2\left({\tilde{A}}_{i_2}\right)\·\·\·m_n\left({\tilde{A}}_{i_n}\right)=m_{\mathrm{1,2},\·\·\·,n}\left({\tilde{C}}_k\right)$

Step3：

$N[m_1\⊕m_2\⊕\·\·\·\⊕m_n]\left({\tilde{C}}_k\right)=\frac{{\mathrm{\Σ}}_{{\tilde{C}}_k}S({\tilde{C}}_k,{\tilde{A}}_{i_1})S({\tilde{C}}_k,{\tilde{A}}_{i_2})\·\·\·S({\tilde{C}}_k,{\tilde{A}}_{i_n})m_{\mathrm{1,2},\·\·\·,n}\left({\tilde{C}}_k\right)}{1-{\mathrm{\Σ}}_{{\tilde{A}}_{i_1}{\tilde{A}}_{i_2},\·\·\·,{\tilde{A}}_{i_n}}(1-S({\tilde{C}}_k,{\tilde{A}}_{i_2})\·\·\·S({\tilde{C}}_k,{\tilde{A}}_{i_n}))m_{\mathrm{1,2},\·\·\·,n}\left({\tilde{C}}_k\right)}$

Wherein, $S({\tilde{C}}_k,{\tilde{A}}_{i_m})(i_m=\mathrm{1,2},\·\·\·,W;m=\mathrm{1,2},\·\·\·,n)$ Can adopt the A step to calculate, the k value is by the power set P of framework of identification X _XBase | P _X|=W and evidence number n are determined.

(5) kind judging.Based on the multi-source fuzzy information combined result of Vague evidence theory, determine the belief function value of classification, select the classification of maximum belief function value, as final classification.Concrete steps are as follows:

Step1:, determine the contribution factor of the belief function of classification based on the multi-source fuzzy information combined result.

If framework of identification X is (x ₁, x ₂... x _n),

Be the burnt unit of the Vague on it, $\tilde{A},\tilde{B}\∈P_X,$

$\tilde{A}=\left(\right[t_{\tilde{A}}\left(x_1\right),1-f_{\tilde{A}}\left(x_1\right)]/x_1,[t_{\tilde{A}}\left(x_2\right),1-f_{\tilde{A}}\left(x_2\right)]/x_2,\·\·\·,[t_{\tilde{A}}\left(x_n\right),1-f_{\tilde{A}}\left(x_n\right)]/x_n)$

$\tilde{B}=(\[t_{\tilde{B}}\left(x_1\right),1-f_{\tilde{B}}\left(x_1\right)\]/x_1,\[t_{\tilde{B}}\left(x_2\right),1-f_{\tilde{B}}\left(x_2\right)\text{\]/}{x}_2,...,\[t_{\tilde{B}}\left(x_n\right),1-f_{\tilde{B}}\left(x_n\right)\]/x_n)$

Then Vague is burnt first Right

Contribution factor be

$I_v(\tilde{B};\tilde{A})=\frac{\underset{i}{\mathrm{\Σ}}\min \{t_{\tilde{A}}\left(x_i\right),t_{\tilde{B}}\left(x_i\right)\}}{\underset{i}{\mathrm{\Σ}}(1-f_{\tilde{A}}\left(x_i\right))}$

Step2: foundation $\mathrm{Bel}\left(\tilde{B}\right)=\underset{i}{\mathrm{\Σ}}{I}_v(\tilde{B};\widetilde{A_i})m\left(\widetilde{A_i}\right)$ , determine the belief function value of Vague evidence theory.

Step3: calculate $\underset{i}{\max }\left\{\mathrm{Bel}\left({\tilde{B}}_i\right)\right\}$ , judge final classification.