Ordinal belief entropy

Entropies are widely applied in measuring the degree of uncertainties existing in frame of discernment. However, all of these entropies regard the frame as a whole that has already been determined, which does not conform to actual situations. In real life, everything comes in a sequence. So, how to measure uncertainties of the dynamic process of determining sequence of propositions contained in a frame of discernment is still an open issue, and no related research has been proceeded. Therefore, a novel ordinal entropy to measure uncertainty of frame of discernment considering the order of propositions is proposed in this paper. Compared with other traditional entropies, it manifests effects on degree of uncertainty brought by orders of propositions. For example, assume there exist three propositions, for ordinal belief entropy, the potential categories of situations are C-3(1) C-2(1) C-1(1), which illustrates that the proposed entropy is able to measure more complex environment and more matches actual circumstances. But for other entropies, they have only one certain value for descriptions of actual situations and the ability of measuring environment is limited. In a general rule, if number of propositions is n, then there are Pi(n)(k)=1 C-k(1) categories of description of situations with respect to ordinal belief entropy. Besides, some numerical examples are provided to verify the correctness and validity of the proposed entropy in this paper.