Time Series Invariants under Boundedness: Information Symbols and Early Warning Signs

Introduction

The empirical analysis of complex systems behavior makes it clear that even though they are very different in nature and each and every of them respond in a highly specific manner to the tiniest variations of the environment, their behavior shares the common property of persistent coexistence of specific and universal properties. This persistency poses the fundamental question whether behind it there is a common constraint embedded in the self-organization of the complex systems. What makes the task enormously difficult is that the ubiquity of that separation is shared by an enormous diversity of the environmental variations and the corresponding response. This makes a lot of scientists to preclude any general rule able to govern the observed ubiquitous coexistence between specific and universal properties. To remind, this behavior has been observed for systems as diverse as quasar pulsation, heartbeat in mammals, currency exchange rates, DNA sequences etc. Contrary to this wide-spread understanding we put forward the idea that, though there is no general rule to govern the complex systems behavior, there do exist a general operational protocol that governs their behavior. Our reason for adopting this point of view is the clear indication that only stable complex systems retain common behavior. In turn, the constraint of stability naturally draws the limitations of our approach: it is available for stable complex systems only.

Our reason for highlighting the stability of complex systems as a key property for explanation of their common behavior is grounded on the assumption that the response of a complex system is a collective phenomenon whose self-organization explicitly involves another persistent property of complex systems: that of their multi-level hierarchical semantic-like self-organization. As it will be demonstrated throughout the book, the strategic aim of that hierarchy is to tighten the overall response of the corresponding system. This is implemented by means of diversifying the environmental impact onto different hierarchical levels so that the impact of one level on its neighbors to be kept permanently bounded. This explains how and why the “environment” is kept bounded for such diversity of scales and systems, i.e. it ranges from atomic to cosmic scales, from quantum systems to quasars. In the next Chapter we will demonstrate how the boundedness, set on quantum level, restrains local fluctuations of e.g. concentrations on the next level of hierarchy, to be bounded. Later, in Chapter 7, we will demonstrate that the boundedness of the impact of one hierarchical level onto another is a generic property of the multi-level hierarchical semantic-like self-organization.

The major goal of semantic-like hierarchy is sustaining boundedness of each and every hierarchical level. In turn, as it will be proven in the section Power Spectrum as a Letter under the mild constraint of boundedness alone, the steady pattern of the response of a stable complex system, presented through a BIS, maintains its characteristics with constant accuracy which is insensitive to the statistics of the corresponding environmental impact. Consequently, this result gives not only a credible general explanation for the persistent coexistence of specific and universal properties, but substantiates the use of steady patterns as “letters”, i.e. as building blocks of semantics. Later, in Chapter 4 and Chapter 7, we will demonstrate that the executed semantics is organized in a non-extensive way similar to that of a human language. A great value of this result is that it opens the door for an explicit formulation of “early warning signs” for development of any changes in the steady pattern far before any of them reaches full size. Moreover, the early warning signs explicitly discriminate between events with hazardous effects on a given system from “non-hazardous” ones.