title: From empirical data to time-inhomogeneous continuous Markov processes.
abstract: We present an approach for testing for the existence of continuous generators of discrete stochastic transition matrices. Typically, existing methods to ascertain the existence of continuous Markov processes are based on the assumption that only time-homogeneous generators exist. Here a systematic extension to time inhomogeneity is presented, based on new mathematical propositions incorporating necessary and sufficient conditions, which are then implemented computationally and applied to numerical data. A discussion concerning the bridging between rigorous mathematical results on the existence of generators to its computational implementation is presented. Our detection algorithm shows to be effective in more than 60% of tested matrices, typically 80% to 90%, and for those an estimate of the (nonhomogeneous) generator matrix follows. We also solve the embedding problem analytically for the particular case of three-dimensional circulant matrices. Finally, a discussion of possible applications of our framework to problems in different fields is briefly addressed.
authors.affiliation: Mathematical Department, FCUL, University of Lisbon, 1749-016 Lisbon, Portugal.
authors.forename: Pedro
authors.initials: P
authors.lastname: Lencastre
authors.affiliation: Instituto Dom Luiz, University of Lisbon, 1749-016 Lisbon, Portugal.
authors.forename: Frank
authors.initials: F
authors.lastname: Raischel
authors.affiliation: Centre for Networks and Collective Behaviour, Department of Mathematical Sciences, University of Bath, Claverton Down, BA2 7AY, Bath, United Kingdom.
authors.forename: Tim
authors.initials: T
authors.lastname: Rogers
authors.affiliation: ForWind-Center for Wind Energy Research, Institute of Physics, Carl-von-Ossietzky University of Oldenburg, DE-26111 Oldenburg, Germany.|Institut für Physik, Universität Osnabrück, Barbarastrasse 7, 49076 Osnabrück, Germany.
authors.forename: Pedro G
authors.initials: PG
authors.lastname: Lind