Belief entropy rate: a method to measure the uncertainty of interval-valued stochastic processes

Entropy rate, as an effective tool in information theory, can measure the uncertainty of stochastic processes modeled by probability mass function. However, when the stochastic process to be measured cannot be accurately modeled, i.e., it is more sense to describe the phenomenon with an interval, the stochastic process needs a more general method to represent. In this paper, the interval-valued stochastic process is modeled with the basic belief assignment in an ordered frame of discernment and the corresponding belief entropy rate is proposed to measure its uncertainty. Two common stochastic processes are discussed. The first is the case of independent identically distributed stochastic processes, where the belief entropy rate is formally the same as the Shannon entropy rate. The second is Markov processes. We construct the evidential Markov chain and calculate its belief entropy rate. Compared with the Shannon entropy rate, the belief entropy rate is easier to implement the Markov chains. By validating in real dataset, the proposed method can better deal with interval information with stronger practicability. When encountering tiny disturbances, the variance of the Shannon entropy rate is more than 50 times the variance of the belief entropy rate, which reflects the stronger robustness of the belief entropy rate.