The report discusses a new empirical approach to reconstruction of unknown high-dimensional, spatially distributed systems possessing by a broad spectrum of temporal scales.
The main goal of the reconstruction is prognosis of the system evolution, and natural (environment and life) systems are main application objects of proposed approach. The goal necessitates correct reconstruction of deterministic properties of the system. The target applications make it impossible to reconstruct correctly even so crucial deterministic attribute as embedding dimension: observed time series of natural variables are nearly always too noisy and short.
To overcome this critical contradiction the suggested approach combines several advanced methods of time series analysis that provide a step-by-step procedure:
i. Decomposition of the underlying system into weakly coupled space-time patterns (“nonlinear modes”) that make the basic contribution to the observed time series;
ii. Construction of the separate modes models in the form of low dimensional random dynamical systems;
iii. Optimization of the mode models basing on bayesian (probabilistic) approach, aiming to find optimal models that have optimal complexity;
iv. Employment of the optimal models of extracted modes as building blocks for construction of the simplest possible adequate model of the system as a whole.
The apparent merit of the proposed approach is its objectiveness, in the first place. The intrinsic feature of the constructed model is description of the properties of the system that determine its evolution registered in experiments. Second, complexity (i.e., dimension and number of structural parameters) of such models are optimal in terms of their adequacy to available data, in particular, adequacy to the volume of information contained in the data about the system.
The methods of realization of the first three steps are described in the report. The abilities of these methods are illustrated by dint of both really measured and generated numerically vector time series of climatic observables.