证据理论中基于区间概率的不确定测度

Uncertainty measure in evidence theory based on interval probability

在证据理论的应用中,不确定测度作为一种新的评价准则,可以对知识进行量化评价。相关研究者对证据不确定测度进行了研究,但现有不确定测度在应用中存在一定的局限性,尤其是对信任函数的变化不够敏感。针对现有不确定测度对证据变化不够敏感问题,从信度区间的角度提出了一种新的证据不确定性测度。首先基于证据理论和概率论之间的关系,将证据理论中基本概率赋值函数转化为单元素子集上的信度区间,区间的上下界分别为似然函数和信任函数,然后结合证据理论中的不一致性和不精确性,定义了一种新的不确定测度SU,并通过数值算例对SU的计算过程和数学性质进行了分析,最后对SU在证据组合中的应用进行了验证。算例结果表明,在其他不确定测度无法辨别不同证据的不确定性时,SU仍能较好反映不同证据间不确定性的差异,而且对于参数化的基本概率赋值函数,SU对参数变化比较敏感。可见,考虑了区间数中值和区间长度的不确定测度SU可以很好地对证据不确定性进行度量;而且,当所有焦元都为单元素子集时,SU退化为香农熵的形式,证据不确定性的最大值与辨识框架中元素的个数有关,这在实际应用中更具意义。

Uncertainty measure in evidence theory supplies a new criterion to assess the quality and quantity of knowledge conveyed by belief structures. As generalizations of uncertainty measure in the probabilistic framework,several uncertainty measures for belief structures have been developed. However,the inconsistency between evidential and probabilistic frameworks causes limitations to existing measures. They are quite insensitive to the change of belief functions.To solve these problems,the definition of uncertainty measure for belief structures was considered from the perspective of belief intervals. Based on the relation between evidence theory and probability theory,belief structures were transformed to belief intervals on singleton subsets,with the belief function and the plausibility function as lower and upper bounds,respectively. An uncertainty measure SU for belief structures was then defined based on interval probabilities in the framework of evidence theory,without changing the theoretical frameworks. The center and the span of the interval were used to define the total uncertainty degree of the belief structure. Numerical examples were applied to indicate the calculation and performance of the proposed measure. Finally,the proposed uncertainty measure was used in the application of evidence theory to show its effectiveness. It is demonstrated that the proposed uncertainty measure can discriminate the uncertainty degree of different evidence bodies,while other measures fail. Moreover,the proposed uncertainty measure is sensitive to the parameters in evidence bodies. It is proved that SU is identical to Shannon entropy and AM for Bayesian belief structures. Moreover,the proposed uncertainty measure has a wider range determined by the cardinality of discernment frame,which is more practical.