Complex Principal Oscillation Pattern Analysis

Abstract

Complex principal oscillation pattern (CPOP) analysis is introduced as an extension of conventional POP analysis. Both are intended to resolve regular evolving patterns from processes with many degrees of freedom. While POP analysis, like many other techniques, deals with the concept of the system state as a real vector, it is argued that this notion be extended into the complex domain. The approach used here results from a critical review of the theory of linear systems of first order. It turns out that these systems cannot appropriately model standing oscillations. The notion of the traveling rate of a mode is defined, and it is demonstrated that the mode's frequency and traveling rate are directly coupled via the system matrix. One consequence is that clean standing oscillations cannot be modeled by linear systems of first order. CPOP analysis introduces a new vector of state. By defining the complex state ?state + i · momentum,? both the conventional state itself and its momentum are simultaneously described. The method is capable of resolving oscillatory patterns of any given traveling rate from a stationary process. First experiments show that the CP0Ps evolve more regularly and with less noise than corresponding P0Ps. A prediction scheme that is appropriate for CP0Ps is defined by introducing a transformation technique that can be considered as a causal Hilbert transform. With this scheme prediction skills that are significantly stronger than those of the POP model are gained.