New approach in correlation analysis

摘要

In this paper, a new method for restoration of the desired correlations is proposed. It allows to evaluate the “content” of the external factors (l = 1,2,...,L) setting in the form of data arrays y(l) m (x)(m = 1,2,...,M) inside the given Ym (x) function that is supposed to be subjected by the in?uence of these factors. As contrasted to the conventional correlation analysis, the proposed method allows ?nding the “in?uence” functions bl (x)(l = 1,2,...,L) and evaluating the “remnant” array Gm (x) that is remained as a “quasi-independent” part from the in?uence of the factors y(l) m (x). The general expression works as a speci?c “balance” and reproduces the wellknows cases, when bl (x) = Cl (it is reduced to the linear least square method with Gm (x) ～ = 0) and coincides with the remnant function Ym (x) ～ = Gm (x), when the in?uence functions becomes negligible (bl (x) ～ = 0). The available data show that the proposed method allows to extract a small signal from the “pattern” background and it conserves its stability/robustness in the presence of a random ?uctuationsoise. The method is rather ?exible and allows to consider the cases of strong correlations, when the external factors act successively, forming the causeand-e?ect chains. It can be generalized for expressions containing the bonds expressed in the form of memory functions. The proposed method adds new quantitative ties expressed in the form of the desired functions to the conventional correlation relationships expressed in the form of the correlation coe?cients forming, in turn, the correlation matrices. New relationships allow to understand deeper the existing correlations and make them more informative, especially in detection of the desired deterministic and stable bonds/laws that can be hidden inside.