Meeting 03/11/2014

Summary

In this kick-off meeting we reviewed various classes of Markov models we encounter in our research, and general problems encountered in their study.

Classes of Markov models

Finite-state Markov processes operate in discrete or continuous time.

  • Discrete-time models are determined by a transition matrix A satisfying Aij>0 and ∑i Aij =1
  • Continuous-time models are determined by a rate matrix A satisfying Aij>0 for i≠j and ∑i Aij =0.
  • Extensions to Markov processes studied at DKE include
  • Time-dependent Markov processes in which the rate matrix depends on time: A=A(t). These include:
    • periodic time-dependent models A(T+t)=A(t)
    • state-dependent models in ion-channel gating A=A(V(t)) where V is the action potential, which depends on the Markov state dV/dt = f(V,x,…).
    • Cox processes: A(t) = k(t) A where k(t) is a stochastic process
  • Markov games in which the transition matrix depends on choices qk made by multiple players: A=A(q1,…,qn). Each player tries to maximise rewards rk(x,q1,…,qn). For one player, these are Markov decision processes.Dynamics of Markov models
  • The fundamental problem for Markov processes is to compute the probability distribution at a given time in the future from a given initial distribution. The equations are
  • Discrete time: xn = An x0
  • Continuous time: x(t) = eAt x0
  • Another important problem is to compute the stationary distributions, which satisfy
  • Discrete time: A x = x
  • Continuous time: A x = 0Special structural properties
  • Markov models may have structural properties which may make their evolution easier or harder to compute.
  • For Markov games, the fundamental problem is to characterise and compute equilibrium strategies.
  • Multiple time-scales: The process has states which change very quickly compared to other states. Typically, the rate matrix has the block form  A=[ 1/ε  Aff, Afs; Asf, Ass]  where xf and xs  are fast and slow variables. Systems with multiple time scales are especially challenging to compute the evolution of.
  • Modular systems: The states of the process partition into subsets, with transitions between states in different sets occurring very slowly. The transition/rate matrix has the block form [ A11, ε A12; ε A21,  A22 ]. Modularity should help compute the evolution, especially of large systems.
  • Regressive/delay models: The evolution depends on the current state and states up to k steps in the past. The evolution has the form xn+1 = A [xn ; xn-1 ; … ; xn-k]. Can be reformulated as a standard Markov model.Research ProblemsEvolutionExisting techniques include Taylor series, domain reduction exp(At)=exp(At/2)2, similarity transformations and eigenvalue decompositions exp(At)P = P exp(Dt) P-1, and Padé approximants. The Baker-Campbell-Hausdorff / Zassenhaus expansions may be useful for modular systems, and the Magnus expansion for time-dependent systems.Computing the stationary distribution reduces to solving a system of linear equations, but for large systems is still a challenging and relevant problem (e.g. internet page ranking). Multigrid/reduction methods have been developed by Chun Wen.The basic workflow for Markov modelling of physical systems is:The identification problem is to find the best model fitting the available experimental data, or a set of plausible system models.The experiment design problem is to find a good experimental protocol for identifying the system; in particular, we want to identify parameter combinations which are most important for the system behaviour.Stochastic/DeterministicEquilibrium strategies
  • For Markov decision processes without discounting, and for Markov games, there are no good algorithms for computing optimal/equilibrium strategies. Maybe we could find good algorithms by formulating and solving necessary conditions for equilibria as linear complementarity problems.
  • Markov models are inherently stochastic, but when sufficiently many Markov models are aggregated, they act as a deterministic model with evolution given by the probability density vector. For moderate numbers of processes, stochastic fluctuations may still be non-negligable. When is the continuum limit valid, and how do we take stochastic fluctuations into account when these are important?
  • Ultimately, we are often interested in making predictions, and explicit models are then only a means to this end. An interesting question is whether we can directly make predictions from data without an explicit model, or with only a partial model.
  • The model-order reduction problem is to find a lower-dimensional model with approximately the same external/observable behaviour. This is still unsolved for time/state-dependent models. A subproblem is to simplify the functional form of nonlinear dependencies.
  • Data -> [Identification] ->  (Physical Model) -> [Reduction] -> (Computational Model) -> [Analysis] -> Prediction
  • Prediction/Workflow
  • Stationary distribution
  • Good ways of computing the evolution using the matrix exponential exp(At) are still lacking, especially for systems with multiple time scales. We would also like to find rigorous bounds on the probability distribution vectors.
  • There are a number of unsolved problems related to Markov processes: