TY - JOUR
T1 - Geometric fluid approximation for general continuous-time Markov chains
AU - Michaelides, Michalis
AU - Hillston, Jane
AU - Sanguinetti, Guido
PY - 2019/9
Y1 - 2019/9
N2 - Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous statespace endowed with a dynamics for the approximating process.We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ODE whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).
AB - Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous statespace endowed with a dynamics for the approximating process.We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ODE whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).
KW - Continuous-time Markov chains
KW - Markov jump processes
KW - Fluid approximation
KW - Diffusion maps
KW - Gaussian processes
U2 - 10.1098/rspa.2019.0100
DO - 10.1098/rspa.2019.0100
M3 - Article
VL - 475
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
T2 - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
SN - 1364-5021
IS - 2229
ER -