Delay-Embedding Approach to Multi-Agent System Construction and Calibration

Abstract

Agent-based modelling offers a way to break from the crude
assumptions of mean-field type models, which ignore space
correlations between elements of the system and replace local
interactions with uniform long-range ones. Multi-Agent Systems
(MAS) explicitly model spatially distributed individuals; however
the richness of such a model can also be a liability due to the
sensitive dependence of such high-dimensional systems. This has
implications for choice of MAS architecture, programming of rules,
confidence in predictions, and calibration of model parameters.

Delay-embedding, also known as geometry from a time
series, provides a deep theoretical foundation for the analysis
of time series generated by nonlinear deterministic dynamical
systems. The profound insight of embedding is that an accessible
variable can explicitly retrieve unseen internal degrees of
freedom.

In the domain of complex systems modelling, however, there
typically exist an abundance of observables, in which case
reconstructing hidden degrees of freedom may be problematic or
even nonsensical. Also, many observables often implies high
dimensionality, which generally precludes a dynamical systems
approach in the first instance. Un-cautious use of
delay-embedding, from which it is easy to get a result
regardless of physical justification, has in the past led to a
degree of negative press for this idea.

However, the recent extensions of Takens' delay-embedding theorem to
deterministically and stochastically forced systems
provide a rigorous framework in which to reconstruct using multiple observables. This
holds great significance for pattern discovery in complex data
series, which we define to be more than one series - spatial,
temporal or a mixture - of an underlying complex system. In
particular, the concept of a bundle embedding highlights a
way to usefully employ the 'surplus' observables in the embedding
process. More generally, forced embeddings provide a methodology
to breakdown complex system data sets in a modular fashion, while
still retaining nonlinear relationships.

Cluster-Weighted Modelling is a sophisticated approach to density
estimation that, when applied to the output of a delay-embedding
process, is able to obtain a statistical representation of the
dynamics. These two concepts - forced embeddings and density
estimation - provide a promising theory and a practical
probabilistic interpretation respectively to the 'inverse problem'
of system identification.

Expert-knowledge based MAS construction and density-estimation of
delay-embedded data can therefore be thought of as two
complementary approaches to the goal of bottom-up, complex systems
modelling. The original contribution of this paper is to present the latter
as a highly data-driven approach to MAS construction in its own right,
and, perhaps more importantly, as an aid to constructing and calibrating
the more expert-knowledge rule-driven approach. The emphasis is on a
solid theoretical and conceptual foundation.

To illustrate the feasibility of our approach, preliminary implementation results for an
ecological modelling scenario are presented and discussed.