Consider a Markov chain x(t), t = 0, 1, 2, ..., with a finite state space, N = {1, 2, ..., n}, transition probability matrix P = (p(,ij)) i, j (epsilon) N, and an initial probability vector V = (v(,i)) i (epsilon) N. For m (LESSTHEQ) n let A = {A(,1), A(,2), ..., A(,m)} be a partition on the set N. Define the process / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / The new process y(t), called a function of Markov chain, need not be Markov. If y(t) is again Markov, whatever the initial probability vector of x(t), x(t) is said to be lumped to y(t) with respect to the partition A. If y(t) is again Markov for only certain initial probability vectors of x(t), x(t) is said to be weakly lumped to y(t) with respect to the partition A. / Conditions under which x(t) can be lumped or weakly lumped to y(t) with respect to A, are introduced. Relationships between the two processes x(t) and y(t) and the properties of the new process y(t) are discussed. / Criteria are developed to determine whether a given Markov chain can be weakly lumped with respect to a given partition in terms of an analysis of systems of linear equations. Necessary and sufficient conditions on the transition probability matrix of a Markov chain, a partition, A, on N and a subset S of probability vectors for weak lumpability to occur are given in terms of the solution classes to these systems of linear equations. Finally, given that weak lumping occurs, the class S of all initial probability vectors which allow weak lumping is determined as is the transition probability matrix of the lumped process, y(t). / Lumpability and weak lumpability are also studied for Markov chains which are not irreducible. This involves a study of the interplay between two partitions of the state space N, the partition C, induced by the closed sets of states of the Markov chain and the partition A, with respect to which lumpability is to be considered. Under the assumptions that lumpability occurs the relationships which must exist between sets of the two partitions A and C are obtained in detail. It is found, for example that if neither partition is a refinement of the other and (A,C) form an irreducible pair of partitions over N then for each A (epsilon) A and C (epsilon) C, A (INTERSECT) C (NOT=) (phi). Further conditions which the transition probability matrix P must satisfy if lumpability is to hold are obtained as are relationships which must exist between P and P*. / Suppose a process y(t) is known to arise as a result of a weak lumping or lumping from some unknown Markov chain x(t). Let (chi)(t) be the class of all Markov chains x(t) with n states which yield this weak lumping or lumping. The problem of characterizing this class and a class S of initial probability vectors which allow this lumping is considered. A complete solution is given when n = 3 and m = 2. / The importance of lumpability in application is discussed. / Source: Dissertation Abstracts International, Volume: 41-11, Section: B, page: 4172. / Thesis (Ph.D.)--The Florida State University, 1980.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_74361 |
| Contributors | ABDEL-MONEIM, ATEF MOHAMED., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 100 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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