Delay-Embedding Approach to Multi-Agent System Construction and Calibration
Campbell, Alexander B., Pham, Binh L., & Tian, Yu-Chu (2005) Delay-Embedding Approach to Multi-Agent System Construction and Calibration. In Zerger, A. & Argent, R.M. (Eds.) MODSIM 2005 International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, 12-15 December 2005, University of Melbourne, Australia.
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.
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