Time Series Invariants under Boundedness: Existence

Introduction

One of the most exciting properties of the complex systems is that even though each of them responds in a highly specific manner to the tiniest changes of the environment, most of them remain remarkably stable in the following sense: a stable complex system exhibits specific steady pattern of behavior whose properties are insensitive to the time window of any observation. A remarkable general property of that stability is that though the response is highly specific, all stable complex systems share certain universal properties. This well established empirical fact prompts us to put forward the idea that the response of stable complex systems is decomposable to a specific steady part and universal noise one. For reasons that would become clearer later we suggest to call the specific part of the response homeostasis and the non-specific one to call fluctuations. An exclusive property of stable complex systems, proven in Chapter 2, is that their response, represented by corresponding power spectrum, resolves explicit additive separation to a specific pattern and a universal noise band. Taking for granted the proof of this claim until Chapter 2, now we focus our attention on establishing those exclusive properties of a noise band which commence from stability of the corresponding complex system alone.

The derivation of the time-series invariants follows Chapter 1 of our previous work (Koleva, 2005).

The first step in this detailed presentation is to establish a general frame for description of the enormous diversity of irregular variations which characterizes the response of the stable complex systems. These irregularities constitute a significant part of our daily life: traffic noise, heartbeat, public opinion, currency exchange rate, electrical current, chemical reactions - they all permanently and irregularly vary in space and in time. The remarkable property of this behavior is that, though wildly varying, each of these systems “keeps” its “individuality” intact - at every moment we can distinguish without any moment of hesitation between a heartbeat in a mammal and a daily record of currency exchange rate. All these diverse phenomena can be conceptualized by a precise definition of a fluctuation that can be expressed as: a fluctuation is any deviation from the steady pattern which serves as ”identity card” of the corresponding system. The advantage of this definition is that it renders the notion of a fluctuation independent of the enormous diversity of the “driving “forces”: human emotions, varying interests in economic systems, and physical interactions. The existence of such definition opens the door to a powerful general study whose outcomes are relevant to each and every system regardless to its origin and particularities. In turn, this makes the subject an indispensable part of the fundamental science. Indeed, for a long time an abstract definition of the fluctuations does exist - irregular deviations from the average that leaves the system stable; the average stays steady while the amplitude and the frequency of the deviations are not precisely predictable. The application of this definition in the analysis of the behavior of a fluctuating system leaves only one possible approach: the systems` response can only be matched through the properties of an irregular time series. This general understanding involves a lot of efforts and ingenuity in order to study characteristics such as the frequency and size of the fluctuations, first exit time etc. The overall thread that unites the enormous number of studies dealing with this type of analyses is that they are specific to the statistics of the fluctuations. That is why the mainstream research has been set on various statistical approaches and the associated classifications according to appropriate parameters.