A Method for Inferring Hierarchical Dynamics in Stochastic Processes

Abstract

Complex systems may often be characterized by their hierarchical dynamics. In this paper do we present a method and an operational algorithm that automatically infer this property in a broad range of systems; discrete stochastic processes. The main idea is to systematically explore the set of projections from the state space of a process to smaller state spaces, and to determine which of the projections that impose Markovian dynamics on the coarser level. These projections, which we call Markov projections, then constitute the hierarchical dynamics of the system. The algorithm operates on time series or other statistics, so a priori knowledge of the intrinsic workings of a system is not required in order to determine its hierarchical dynamics. We illustrate the method by applying it to two simple processes; a finite state automaton and an iterated map. a Strictly speaking, Lie's work is not limited to classical mechanics and can be applied to any differential equation.

Markov Chain Aggregation for Agent-Based Models

“…From the lumpability point of view, but also from the point of view of observation, a macro process obtained by aggregation over the agent attributes neglects important information about the microscopic details. In other words, for heterogeneous networks in general, the micro-level process and the macro process do not commute, neither in an observational sense nor in the strict mathematical sense (Görnerup and Jacobi, 2008;Pfante et al, 2013). The only case where full aggregation provides us with a exact macro description is the homogeneous mixing case for which we have provided a Markov chain analysis in Sec.…”

“…A useful complementary view on lumpability is provided by looking at it from an information-theoretic perspective, namely, in terms of the time series a model generates. A method to derive a Markovian state space aggregation on the basis of an information-theoretic view on time series data (typically created by some simple models) has been proposed by Görnerup and Jacobi (2008) and is inspired by the framework of computational mechanics (Crutchfield and Young, 1989;Shalizi and Crutchfield, 2001;Shalizi and Moore, 2003).…”

“…Now, in this context, lumpability of the process with respect to the partition X means that the macro sequencex exhibits the Markov property. In that case, Görnerup and Jacobi (2008) refer to the projection Π as Markov projection, a notion that will be adopted throughout this thesis.…”

“…With its focus on time series and computations on it, the devision of a process or sequence into past, present and future, and the intense use of information theory, the approach due to Görnerup and Jacobi (2008) is located within a larger program called computational mechanics (Crutchfield and Young, 1989;Shalizi and Crutchfield, 2001;James et al, 2011, and references therein). The main idea in computational mechanics is to group histories which give rise to the same conditional probability distribution over futures into equivalence classes -so-called causal states -and to construct in this way a minimal causal model -called ǫ-machines -for the prediction of the process at question.…”

“…Time-series-based aggregation schemes as the one proposed by Görnerup and Jacobi (2008) as well as ǫ-machines (Crutchfield and Young, 1989;Shalizi and Crutchfield, 2001) are appealing from the theoretical point of view, but their application to ABM aggregation is limited by their computational complexity (cf. Görnerup and Jacobi, 2008, 13).…”