﻿ 冲突证据决策新方法及应用<sup>*</sup>
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A new conflict evidence decision method and its application
ZHAO Jing, GUAN Xin, LIU Haiqiao
School of Aviation Operational Support, Navy Aviation University, Yantai 264001, China
Received: 2019-01-02; Accepted: 2019-03-22; Published online: 2019-05-08 13:38
Foundation item: National Natural Science Foundation of China (91538201, 61671463, 61571454); National Defense Science and Technology Excellence Youth Talent Fund (2017-JCJQ-2Q-003); Taishan Scholar Engineering Special Fund (ts201712072)
Corresponding author. GUAN Xin, E-mail: 597268914@qq.com
Abstract: Research on conflict evidence decision methods is an important research topic of evidence theory. In view of the existing problems in the evidence theory improvement process, such as large computational complexity, unreasonable normalization process and unsatisfactory evidence combination, this paper proposes a method based on quadratic combination for conflict evidence decision-making. Firstly, the paper proposes a new flowchart of conflict evidence decision method based on quadratic combination. Secondly, a new multiplicative normalization rule is proposed, and a new multiplicative normalization rule is analyzed by example to verify its rationality. Finally, the shortcomings of the existing conflict measurement function are analyzed, a new conflict measurement function is proposed, and the rationality of the conflict measurement function is analyzed. Through the analysis of examples and the comparison with the existing evidence combination rules, it is shown that the proposed method not only improves the calculation amount, but also improves the combination results.
Keywords: conflict evidence combination     conflict metric function     multiplicative normalization rule     evidence theory     quadratic combination

1) 基于信息熵，通过信息的无序性，表征证据的不确定性。Höhle[28]在似然函数基础上提出Confusion度量；Yager[29]在似真度函数基础上提出Dissonance度量；Klir等[30-31]基于命题的模，给出Discord度量和Strife度量；Harmanec和Klir[32]在概率转换的基础上，提出Aggregated不确定性度量；Jousselme等[33]基于pignistic概率转换基础上，提出Ambiguity度量；Deng[34]在Shannon熵的基础上提出Deng熵，用来衡量证据的不确定性。上述给出的多种基于信息熵的不确定性度量方法，通过算例分析，都能一定程度上表征信息的不确定性。在实际应用中，当传感器获取信源的信息时，信息在产生、传输过程中，不存在不确定性，而在信息的接收方面，随着传感器的可靠性，不同传感器获取的信源信息存在着不确定性。这时，便可以利用上述提出的多种基于证据本身的不确定性，来表征证据的可靠性，从而对证据进行修正，达到有效的组合结果。本文对这一类的冲突证据组合方法不做深入的研究，根据实际应用的需求，重点分析冲突证据不确定衡量的第2类方法，即如何衡量证据之间的冲突度或者说相似度。

2) 基于距离衡量证据的不确定性[35-37]。Jousselme等[35]提出的Jousselme距离是应用最为广泛的证据距离；Yu等[9]提出支持概率(supporting probability)距离，通过计算证据被支持程度，确定证据的折扣权重；Zhou等[38]对现有的证据距离进行运算，得到新的权重系数；Liu等[15]在概率转换DSmP的基础上，提出一种新的证据距离；Yang和Han[39]利用Tran&Duckstein区间距离衡量证据的不确定性，并得到很好的融合效果。

1 理论基础

X为识别对象，UX可能取值的集合，并且U中的所有元素互不相容，则称U为一识别框架。

 (1)

1) Dempster组合规则

 (2)

2) PCR6组合规则

 (3)
2 基于二次组合的冲突证据决策 2.1 基于冲突度量的折扣证据不确定推理模型

Dempster组合规则在处理冲突证据时，会产生悖论现象，对于这一问题，相关学者给出多种改进方法，主要分为两类：第1类是对原有的DS证据理论的组合公式进行改进；第2类是对原始证据进行修正。目前大多数学者都接受第2类改进方式，认为证据的冲突根本上是来源于获取数据的冲突、不精确。

 图 1 折扣证据组合模型 Fig. 1 Discount evidence combination model

1) 确定冲突度量函数。这是关键的一步，目前的相关修正数据源的改进算法，都是基于对不同的冲突度量函数的改进。冲突度量函数不仅需要满足物理意义，即与证据之间的距离成正相关；还需要满足数学条件，即满足范数的几条基本条件。

① 对称性。

m1(·), m2(·), CM(m1, m2)=CM(m2, m1)。

② 正定性。

m1(·), m2(·)∈U，当m1m2时，CM(m1, m2)≥0；当m1=m2时，CM(m1, m2)=0。

③ 齐次性。

CM(r1m1, r2m2)=r1r2CM(m1, m2), rR

④ 三角不等式。

CM(m1, m2)≤CM(m1, m3)+CM(m2, m3)。

2) 计算两两证据之间的距离，得到距离矩阵D，元素dij表示第i条矩阵与第j条证据之间的距离。距离矩阵是一个对称方阵，并且对角线的元素为0。

3) 对距离矩阵的每一列进行归一化处理，得到的矩阵第i列的一条向量，表示第i条矩阵对每一条证据的支持程度，从而便于后续的运算。

4) 将归一化后矩阵的每一行相加，得到一条向量，向量的第s个元素表示第s条证据被其他证据支持的总程度。

5) 对步骤4)得到的向量进行归一化处理，得到归一化可信度权重。

6) 利用可信度权重对证据进行折扣计算。常用的折扣公式为

 (4)

7) 对折扣后的证据进行组合，得到最后的决策证据。

2.2 基于二次组合的冲突证据决策模型

 (5)

 图 2 基于二次组合的冲突证据决策模型 Fig. 2 Conflict evidence decision model based on quadratic combination

1) 利用PCR6组合规则对原有证据进行组合，得到初步的组合结果。PCR6组合被广泛地应用在冲突证据的组合规则上，并得到较好的组合效果。组合后得到的证据mi能够一定程度上反映证据的总体水平。

2) 确定冲突度量函数，与2.1节描述的一样，度量函数需要满足范数的基本条件，这里不再赘述。

3) 计算每一条证据到组合后证据之间的距离，直接计算得到可信度向量。

4) 对证据的可信度向量进行归一化。在常规的修正证据源的冲突证据中，证据的归一化是每一项元素除以元素的和。但是从式(2)和式(3)可以看出，证据的组合规则中，证据之间的元素是进行乘法运算，即m1(Ai)m2(Aj)…mn(Ak)，所以求和形式的归一化不满足计算的要求。归一化应该满足：

 (6)

 (7)

5) 利用可信度对向量进行加权处理，采用折扣的方法对证据进行折扣，由于归一化公式的改变，折扣公式需要进行修正，得

 (8)

6) 利用PCR6证据组合规则对证据进行组合。

2.3 乘性组合规则的有效性分析

2.2节中提出基于乘性法则的归一化公式，如式(6)和式(7)所示，下面通过算例，验证乘性归一化法则在冲突证据理论决策中的合理性。

 组合规则 mi(A) mi(B) mi(AB) mi(C) mi(AC) mi(BC) mi(ABC) PCR6 0.5436 0.0939 0.0403 0.2544 0.0224 0.0454 0 PCR6+ 0.5386 0.0938 0.0314 0.0588 0.0095 0.0205 0.2475 PCR6× 0.6379 0.0751 0.0622 0.0523 0.0641 0.1083 0

2.4 预处理合理性分析

2.2节中给出基于二次组合的冲突证据决策模型，首先给出PCR6组合结果为基准参考，但是后文又需要对PCR6组合规则的效果进行讨论，从逻辑上就存在了矛盾。

 图 3 预处理误差 Fig. 3 Preprocessing error

3 冲突度量函数的确定 3.1 经典的Jousselme距离冲突度量函数

 (9)

Jousselme距离能一定程度上反映BPAs之间的距离，在数学上是严谨的，并得到广泛的应用。

Jousselme距离已经在多个领域得到了广泛的支持，但是考虑到，Jousselme距离需要求解矩阵D，运算量较大(计算量问题，本文第5节会进行进一步深入分析)。鉴于此，本文将引入向量角的概念来衡量证据之间的距离，向量之间的夹角余弦可以表示为

 (10)

3.2 基于pignistic向量角的冲突证据度量函数

 (11)

 (12)

2) 正定性

m1=m2时，, 所以

3) 齐次性

4) 三角不等式

4 基于二次组合的冲突证据决策的计算量分析

 (13)

 (14)

 (15)

 (16)

 图 4 基于Jousselme距离和二次组合的冲突证据决策计算量图 Fig. 4 Graph of calculated amount of conflict evidence decision based on Jousselme distance and quadratic combination
 图 5 基于Jousselme距离和二次组合的冲突证据决策计算量的对数图 Fig. 5 Graph of logarithm of calculated amount of conflict evidence based on Jousselme distance and quadratic combination

5 基于二次组合的冲突证据决策算例分析

 mi(·) BPA A B AB C AC BC ABC m1 0.7 0.1 0.1 0.1 0 0 0 m2 0.6 0.2 0 0.1 0 0 0.1 m3 0.6 0.05 0 0.05 0 0 0.3 m4 0.4 0.3 0 0.2 0.1 0 0 m5 0.1 0.7 0 0.1 0 0 0.1 m6 0.9 0.05 0 0.05 0 0 0

 mi′(·) BPA A B AB C AC BC ABC m′1 0.74 0.11 0.04 0.11 0 0 0 m′2 0.64 0.21 0 0.11 0 0 0.05 m′3 0.72 0.06 0 0.06 0 0 0.16 m′4 0.40 0.30 0 0.20 0.10 0 0 m′5 0.03 0.21 0 0.03 0 0 0.73 m′6 0.90 0.05 0 0.05 0 0 0

 组合规则 mi(A) mi(B) mi(AB) mi(C) mi(AC) mi(BC) mi(ABC) PCR1 0.57 0.22 0.02 0.10 0.02 0 0.08 PCR2 0.57 0.22 0.02 0.10 0.02 0 0.08 PCR3 0.67 0.24 0 0.06 0 0 0.04 PCR5 0.46 0.35 0.01 0.07 0.01 0 0.10 PCR6 0.72 0.21 0 0.03 0 0 0.04 本文方法 0.77 0.07 0 0.02 0 0 0.14

6 结论

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#### 文章信息

ZHAO Jing, GUAN Xin, LIU Haiqiao

A new conflict evidence decision method and its application

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(9): 1838-1847
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0787