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.