Geometric fluid approximation for general continuous-time Markov chains
This provides a novel approximation method for CTMCs in fields like systems biology or queueing theory, but it is incremental as it builds on existing spectral and manifold learning techniques.
The authors tackled the problem of approximating the macro-scale behavior of general continuous-time Markov chains (CTMCs) without relying on a population structure, by constructing a method using spectral analysis and diffusion maps to embed states in a continuous space and infer a drift vector field, resulting in an ODE that approximates the fluid limit.
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 state-space 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).