Direct and reverse problems of reconstruction of evolution operators in the analysis of the dynamics of multidimensional processes

Physical processes of scattering of electromagnetic waves and the processes of generation of anomalies in the transmission of information in communication channels are considered from the unified positions of the transformation of multidimensional data streams by the corresponding evolution operators acting as projection operators. The work was based on the results of one of the authors studies devoted to the problems of synthesis of three-dimensional holographic filters for the processing of coherent optical signals, location of objects moving in heterogeneous environment, as well as the results of the authors' joint work in the field of protection of information transmitted over computer networks. In this paper, a certain generalization of the results of these studies is made. Thus, the direct problem of reconstructing a stationary evolution operator describing the elastic scattering of a coherent electromagnetic field by spatial inhomogeneities of a stationary medium is posed and solved. It is shown that, up to small noise, determined by the dispersion of medium inhomogeneities, the evolution operator is represented as a projection operator or an active filter of the angular (spatial) field frequencies. On the basis of the obtained results, the inelastic scattering of a non-stationary electromagnetic field by a nonstationary medium is regarded as its transformation by a nonstationary evolution operator, also having the form of a projection operator or an active filter, but transforming the temporal frequencies of the field. By analogy with scattering problems, the evolution operator, in the form of a projection operator reconstructed from observable network information flows of multidimensional telemetric data, is applied to the description of anomaly generation processes in the transmission of information in network communication channels. The evolution operator is used to form special statistics of a multidimensional process that can be used for its sequential statistical analysis and classification. Examples are given of the practical use of these statistics for the detection and classification of network traffic anomalies. In particular, a computational experiment was conducted to generate statistical distributions of the values of informative characteristics of real network traffic and to classify various types of network attacks. As a result of the experiments, the efficiency of the reconstruction method based on evolution operator application has been confirmed.