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Geometric fluid approximation for general continuous-time Markov chains.

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This study introduces a novel spectral analysis method for continuous-time Markov chains (CTMCs), bypassing population structure requirements. This approach enables accurate fluid approximations for complex Markov systems using diffusion maps and Gaussian process regression.

Keywords:
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Area of Science:

  • Computational Mathematics
  • Applied Probability
  • Data Science

Background:

  • Fluid approximations are effective for Markov systems but require specific population structures.
  • Existing methods for continuous-time Markov chains (CTMCs) are limited by assumptions on state-space dynamics.

Purpose of the Study:

  • To develop a general method for fluid approximation of CTMCs.
  • To overcome limitations of existing methods by removing the need for population structure.
  • To provide a robust framework for analyzing complex discrete-state systems.

Main Methods:

  • Spectral analysis of the CTMC transition matrix.
  • Application of diffusion maps for manifold learning.
  • Gaussian process regression for inferring drift vector fields.
  • Construction of an ordinary differential equation for the fluid limit.

Main Results:

  • A general method for fluid approximation of CTMCs is established.
  • The method successfully embeds discrete states into a continuous space.
  • A drift vector field is inferred, enabling the approximation of CTMC mean evolution.
  • The fluid limit is accurately approximated without prior population structure assumptions.

Conclusions:

  • The proposed spectral analysis method offers a more general approach to fluid approximation for CTMCs.
  • This technique enhances the analysis of complex systems by leveraging hidden continuous dynamics.
  • The integration of diffusion maps and Gaussian processes provides a powerful tool for fluid limit computation.