Information-Preserving Markov Aggregation

We present a sufficient condition for a non-injective function of a Markov chain to be a second-order Markov chain with the same entropy rate as the original chain. This permits an information-preserving state space reduction by merging states or, equivalently, lossless compression of a Markov source on a sample-by-sample basis. The cardinality of the reduced state space is bounded from below by the node degrees of the transition graph associated with the original Markov chain. We also present an algorithm listing all possible information-preserving state space reductions, for a given transition graph. We illustrate our results by applying the algorithm to a bi-gram letter model of an English text.Comment: 7 pages, 3 figures, 2 table

Compositional Performance Modelling with the TIPPtool

Stochastic process algebras have been proposed as compositional specification formalisms for performance models. In this paper, we describe a tool which aims at realising all beneficial aspects of compositional performance modelling, the TIPPtool. It incorporates methods for compositional specification as well as solution, based on state-of-the-art techniques, and wrapped in a user-friendly graphical front end. Apart from highlighting the general benefits of the tool, we also discuss some lessons learned during development and application of the TIPPtool. A non-trivial model of a real life communication system serves as a case study to illustrate benefits and limitations

Language-based Abstractions for Dynamical Systems

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of effectively performing analyses. This has motivated a large body of research, across many disciplines, into abstraction techniques that provide smaller ODE systems while preserving the original dynamics in some appropriate sense. In this paper we give an overview of a recently proposed computer-science perspective to this problem, where ODE reduction is recast to finding an appropriate equivalence relation over ODE variables, akin to classical models of computation based on labelled transition systems.Comment: In Proceedings QAPL 2017, arXiv:1707.0366

Weak Markovian Bisimulation Congruences and Exact CTMC-Level Aggregations for Concurrent Processes

We have recently defined a weak Markovian bisimulation equivalence in an integrated-time setting, which reduces sequences of exponentially timed internal actions to individual exponentially timed internal actions having the same average duration and execution probability as the corresponding sequences. This weak Markovian bisimulation equivalence is a congruence for sequential processes with abstraction and turns out to induce an exact CTMC-level aggregation at steady state for all the considered processes. However, it is not a congruence with respect to parallel composition. In this paper, we show how to generalize the equivalence in a way that a reasonable tradeoff among abstraction, compositionality, and exactness is achieved for concurrent processes. We will see that, by enhancing the abstraction capability in the presence of concurrent computations, it is possible to retrieve the congruence property with respect to parallel composition, with the resulting CTMC-level aggregation being exact at steady state only for a certain subset of the considered processes.Comment: In Proceedings QAPL 2012, arXiv:1207.055

Relevant states and memory in Markov chain bootstrapping and simulation

Markov chain theory is proving to be a powerful approach to bootstrap and simulate highly nonlinear time series. In this work, we provide a method to estimate the memory of a Markov chain (i.e. its order) and to identify its relevant states. In particular, the choice of memory lags and the aggregation of irrelevant states are obtained by looking for regularities in the transition probabilities. Our approach is based on an optimization model. More specifically, we consider two competing objectives that a researcher will in general pursue when dealing with bootstrapping and simulation: preserving the “structural” similarity between the original and the resampled series, and assuring a controlled diversification of the latter. A discussion based on information theory is developed to define the desirable properties for such optimal criteria. Two numerical tests are developed to verify the effectiveness of the proposed method