2,309 research outputs found
Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks
Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations
Stochastic hybrid system : modelling and verification
Hybrid systems now form a classical computational paradigm unifying discrete and continuous system aspects. The modelling, analysis and verification of these systems are very difficult.
One way to reduce the complexity of hybrid system models is to consider randomization. The need for stochastic models has actually multiple motivations. Usually, when building models complete information is not available and we have to consider stochastic versions. Moreover, non-determinism and uncertainty are inherent to complex systems. The stochastic approach can be thought of as a way of quantifying non-determinism (by assigning a probability to each
possible execution branch) and managing uncertainty. This is built upon to the - now classical - approach in algorithmics that provides polynomial complexity algorithms via randomization.
In this thesis we investigate the stochastic hybrid systems, focused on modelling and analysis.
We propose a powerful unifying paradigm that combines analytical and formal methods. Its
applications vary from air traffic control to communication networks and healthcare systems.
The stochastic hybrid system paradigm has an explosive development. This is because of its
very powerful expressivity and the great variety of possible applications. Each hybrid system model can be randomized in different ways, giving rise to many classes of stochastic hybrid systems.
Moreover, randomization can change profoundly the mathematical properties of discrete and continuous aspects and also can influence their interaction. Beyond the profound foundational and semantics issues, there is the possibility to combine and cross-fertilize techniques from analytic mathematics (like optimization, control, adaptivity, stability, existence and uniqueness of trajectories, sensitivity analysis) and formal methods (like bisimulation, specification, reachability
analysis, model checking). These constitute the major motivations of our research. We
investigate new models of stochastic hybrid systems and their associated problems. The main difference from the existing approaches is that we do not follow one way (based only on continuous or discrete mathematics), but their cross-fertilization. For stochastic hybrid systems we introduce concepts that have been defined only for discrete transition systems. Then, techniques
that have been used in discrete automata now come in a new analytical fashion. This is partly explained by the fact that popular verification methods (like theorem proving) can hardly work even on probabilistic extensions of discrete systems. When the continuous dimension is added, the idea to use continuous mathematics methods for verification purposes comes in a natural
way.
The concrete contribution of this thesis has four major milestones:
1. A new and a very general model for stochastic hybrid systems;
2. Stochastic reachability for stochastic hybrid systems is introduced together with an approximating method to compute reach set probabilities;
3. Bisimulation for stochastic hybrid systems is introduced and relationship with reachability analysis is investigated.
4. Considering the communication issue, we extend the modelling paradigm
Unbiased and Consistent Nested Sampling via Sequential Monte Carlo
We introduce a new class of sequential Monte Carlo methods called Nested
Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested
Sampling method of Skilling (2006) in terms of sequential Monte Carlo
techniques. This new framework allows convergence results to be obtained in the
setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An
additional benefit is that marginal likelihood estimates are unbiased. In
contrast to NS, the analysis of NS-SMC does not require the (unrealistic)
assumption that the simulated samples be independent. As the original NS
algorithm is a special case of NS-SMC, this provides insights as to why NS
seems to produce accurate estimates despite a typical violation of its
assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels
in an automated manner via a preliminary pilot run, and present a new method
for appropriately choosing the number of MCMC repeats at each iteration.
Finally, a numerical study is conducted where the performance of NS-SMC and
temperature-annealed SMC is compared on several challenging and realistic
problems. MATLAB code for our experiments is made available at
https://github.com/LeahPrice/SMC-NS .Comment: 45 pages, some minor typographical errors fixed since last versio
Reduction of Markov Chains using a Value-of-Information-Based Approach
In this paper, we propose an approach to obtain reduced-order models of
Markov chains. Our approach is composed of two information-theoretic processes.
The first is a means of comparing pairs of stationary chains on different state
spaces, which is done via the negative Kullback-Leibler divergence defined on a
model joint space. Model reduction is achieved by solving a
value-of-information criterion with respect to this divergence. Optimizing the
criterion leads to a probabilistic partitioning of the states in the high-order
Markov chain. A single free parameter that emerges through the optimization
process dictates both the partition uncertainty and the number of state groups.
We provide a data-driven means of choosing the `optimal' value of this free
parameter, which sidesteps needing to a priori know the number of state groups
in an arbitrary chain.Comment: Submitted to Entrop
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