Statistical Identification Of Q Matrix Of Markov Chain On Trees

The theoretical study of continuous-time homogeneous Markov chains is usually based on a natural assumption of a known transition rate matrix(TRM).However,the TRM of a Markov chain in realistic systems always be unknown,but which can be identified by the hitting time and sojourn time distributions(mixed exponential distributions)of some observed states.This paper mainly studies the Markov chain on trees,as for specific no-loop Markov chain,which can be seen as trees by changing the number of its children.Combined the implicit constraint relationship between the derivatives of the hitting time and sojourn time distributions of a certain state set and the Q matrix,the inver-sion approach is used to study the statistical identification of continuous-time Markov Chain on trees with finite state space.At first,Markov chain on binary tree is studied,and which can be identified by the hitting time and sojourn time's probability density functions(PDFs)of all leaves.Secondly,the statisti-cal identification of Markov chain on several special trees,if the leaf states is fewer,which can be identified by theirs hitting time and sojourn time's PDFs;if the leaf states is much more than non-leaf states,which can be identified by non-leaf states' PDFs of hitting time and sojourn time.At last,the statistical identification of Markov chain on a general tree,which can be identified by the hitting time and sojourn time's PDFs of all leaf states.