Statistical analysis of CARMA models: an advanced tutorial

CARMA (Collective Adaptive Resource-sharing Markovian Agents) is a process-algebra-based quantitative language developed for the modeling of collective adaptive systems. A CARMA model consists of an environment in which a collective of components with attribute stores interact via unicast and broadcast communication, providing a rich modeling formalism. The semantics of a CARMA model are given by a continuous-time Markov chain which can be simulated using the CARMA Eclipse Plug-in. Furthermore, statistical model checking can be applied to the trajectories generated through simulation using the MultiVeStA tool. This advanced tutorial will introduce some of the theory behind CARMA and MultiVeStA as well as demonstrate its application to collective adaptive system modeling

Constitutive hybrid processes

Introduction When modeling a physical system, it is common practice to describe the components that constitute the system, using so-called constitutive relations on the physical variables that play a role in the system. The intersection of all these relations then forms a model of the system as a whole. The behavior of physical systems is usually assumed to be continuous and, therefore, the constitutive relations are often stated as differential algebraic equations. When part of the continuous behavior occurs very fast, however, as is for example the case when studying impact phenomena, it may be convenient to describe this behavior as being discontinuous. The constitutive relations that are used to describe the system, should in that case not only contain algebraic differential equations (for the large time-scale behavior), but using also equations that describe the discontinuous behavior (for the behavior during impact). In this report, we describe the constitutive relations of many more-or-less standard components in physical modeling, using the hybrid process algebra HyPA [4]. This algebra allows us to describe combinations of continuous and discontinuous behavior as one, hybrid, process (hence, the title of this report). As a vehicle for our thoughts, we use a graphical language named bond graphs [11] to formalize our physical models, before engaging in the construction of constitutive relations for them. Bond graphs generalize all domains of physics, such as electronics, hydraulics, and mechanics, in one framework. Recently, they have been extended with elements that are suitable for describing discontinuous behavior [10, 9, 1, 12]. This report, can therefore also be considered an attempt to give a formal semantics to hybrid bond graphs. Our expectation is, that after we have explained how to derive hybrid constitutive processes using hybrid bond graphs, it will also be easier to derive these processes directly, without using bond graphs as an intermediate step. Nevertheless, the construction of a bond graph sometimes gives additional insight in the workings of a system, and can facilitate analysis in many ways (see for example [8, 14, 3, 2]). In general, different model representations have strengths in different kinds of analysis. In the next section, we give a short discussion on the modeling of physical systems through constitutive relations, using an example from mechanical engineering. Then, we briefly explain the traditional bond graph modeling method and discuss the need for abstraction from small timescale behavior. In section 3 we briefly discuss the syntax and semantics of hybrid process algebra [4]. In section 4, we turn back to the bond graph modeling formalism, to see how the constitutive relations of the bond graph elements can be extended to include discontinuous behavior. In the last section, we give modeling examples that show how hybrid bond graph models can be made of several physical systems, and how these bond graph models can be turned into constitutive hybrid processes describing the systems algebraically

Constitutive hybrid processes

Introduction When modeling a physical system, it is common practice to describe the components that constitute the system, using so-called constitutive relations on the physical variables that play a role in the system. The intersection of all these relations then forms a model of the system as a whole. The behavior of physical systems is usually assumed to be continuous and, therefore, the constitutive relations are often stated as differential algebraic equations. When part of the continuous behavior occurs very fast, however, as is for example the case when studying impact phenomena, it may be convenient to describe this behavior as being discontinuous. The constitutive relations that are used to describe the system, should in that case not only contain algebraic differential equations (for the large time-scale behavior), but using also equations that describe the discontinuous behavior (for the behavior during impact). In this report, we describe the constitutive relations of many more-or-less standard components in physical modeling, using the hybrid process algebra HyPA [4]. This algebra allows us to describe combinations of continuous and discontinuous behavior as one, hybrid, process (hence, the title of this report). As a vehicle for our thoughts, we use a graphical language named bond graphs [11] to formalize our physical models, before engaging in the construction of constitutive relations for them. Bond graphs generalize all domains of physics, such as electronics, hydraulics, and mechanics, in one framework. Recently, they have been extended with elements that are suitable for describing discontinuous behavior [10, 9, 1, 12]. This report, can therefore also be considered an attempt to give a formal semantics to hybrid bond graphs. Our expectation is, that after we have explained how to derive hybrid constitutive processes using hybrid bond graphs, it will also be easier to derive these processes directly, without using bond graphs as an intermediate step. Nevertheless, the construction of a bond graph sometimes gives additional insight in the workings of a system, and can facilitate analysis in many ways (see for example [8, 14, 3, 2]). In general, different model representations have strengths in different kinds of analysis. In the next section, we give a short discussion on the modeling of physical systems through constitutive relations, using an example from mechanical engineering. Then, we briefly explain the traditional bond graph modeling method and discuss the need for abstraction from small timescale behavior. In section 3 we briefly discuss the syntax and semantics of hybrid process algebra [4]. In section 4, we turn back to the bond graph modeling formalism, to see how the constitutive relations of the bond graph elements can be extended to include discontinuous behavior. In the last section, we give modeling examples that show how hybrid bond graph models can be made of several physical systems, and how these bond graph models can be turned into constitutive hybrid processes describing the systems algebraically

About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page

Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples

A Computational Algebra Approach to the Reverse Engineering of Gene Regulatory Networks

This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. The complexity of the algorithm is quadratic in the number of variables and cubic in the number of time points. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster.Comment: 28 pages, 5 EPS figures, uses elsart.cl