Quantitative Emergence -- A Refined Approach Based on Divergence Measures

The article addresses the phenomenon of emergence from a technical viewpoint. A technical system exhibits emergence when it has certain kinds of properties or qualities that are irreducible in the sense that they are not traceable to the constituent parts of the system. In particular, we show how emergence in technical systems can be detected and measured gradually using techniques from the field of probability theory and information theory. To detect or measure emergence we observe the system and extract characteristic attributes from those observations. As an extension of earlier work in the field, we propose emergence measures that are well-suited for continuous attributes (or hybrid attribute sets) using either non-parametric or model-based probability density estimation techniques. We also replace the known entropy-based emergence measures by divergence measures for probability densities (e.g., the Kullback-Leibler divergence or the Hellinger distance). We discuss advantages and drawbacks of these measures by means of some simulation experiments using artificial data sets and a real-world data set from the field of intrusion detection.