Spectral Decomposition of Transition Matrix

Sutawanir Darwis ( Statistics Research Division, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Bandung)
Kuslan Kuslan ( Risk Management Group, LIPPO Bank, Indonesia)

Jurnal Matematika & Sains
Vol 13, No 3 (2008)

The transition probabilities of a two-state Markov process can be determined explicitly. The modeling of forcetransition of two state Markov process using double decrement approach is well developed in the literature. However,the approaches are mainly analytic or illustrative and are based on small data set. The study based on large data set arerarely published. For higher number of states, the computation of transition probabilities is laborious, and analternative method is needed. This work aims to propose a spectral approach of forces of transition that attempts toaddress the issues. The method is based on results that are available when a Markov process with constant forces oftransition is assumed. In this case, transition probabilities are obtained regardless of the number of states. A differentialequation is used to express the relationship between forces of decrements and transition probabilities, and by assumingconstant force, the explicit solution is reduced to spectral decomposition of force of decrements. The results are thevisualization of transition probabilities, and a contribution for the development of double decrement table. The maincontributions of this work are a spectral representation of transition probabilities and a multistate approach to doubledecrement modeling.