Generalized Self-Similarity Theory and Models

In order to understand various anisotropic scale invariance systems, the generalized scale invariance (GSI) concept was brought forward to present a formalism stating the most general conditions under which large and small scales can be related.Two different anisotropic scale invariance quantification models were developed: the scale invariant generator (SIG) model quantifies anisotropies by estimating the GSI generator in frequency domain, a form of scale transformation defined in GSI representing how the scaling field is stratified and how it rotates, and the family of balls that best describes the scaling field; the spectrum-area (S-A) model quantifies anisotropies by estimating the anisotropic scaling exponent defined in GSI through a power-law function representing the relationship between area of the set with spectral energy density above P on the 2D frequency domain and P. S-A is not only an anisotropic scale invariance quantification technique but also a mixing data decomposition technique, which can decompose mixing data into multiple components based on anisotropic scaling properties in frequency domain.This paper introduces the GSI concept, the SIG model and S-A model systematically and proposes an idea to combine the SIG model and S-A model so that the new combined model can not only decompose mixing data into multiple components but also quantify the decomposed components' anisotropic scale invariance as well.