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Abstract
Fluid approximations have seen great success in approximating the macroscale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuoustime 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).
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).
Original language  English 

Article number  20190100 
Number of pages  25 
Journal  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Volume  475 
Issue number  2229 
Early online date  25 Sep 2019 
DOIs  
Publication status  Published  27 Sep 2019 
Keywords
 Continuoustime Markov chains
 Markov jump processes
 Fluid approximation
 Diffusion maps
 Gaussian processes
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 1 Finished
Profiles

Jane Hillston
 School of Informatics  Personal Chair in Quantitative Modelling and Head of School
 Laboratory for Foundations of Computer Science
 Data Science and Artificial Intelligence
Person: Academic: Research Active