Granular Analysis of Time Sequence Based on Quotient Space

This paper aims to carry out granular analysis of time sequence based on quotient space. Granular methods have long before been adopted to analyze time sequence, but the granularity was based on time, for example, day mean, month mean, year mean and so on in finance forecast. In this paper, the granularity is based on space and some significant results are obtained: we can, in certain circumstances, get characteristics of time sequence in an original space when carrying out granular analysis of it in its coarser-grain space; granular analysis of a Markov chain is equivalent to an hidden Markov model (HMM), contrarily, any HMM is equivalent to granular analysis of a Markov chain. These results deepened our understanding of HMM from the perspective of granular analysis. We can not only use the methods of HMM to study time sequence, but also use the methods of granular analysis based on quotient space theory to solve the problems of HMM.