Direct evaluation of the desired correlations: Verification on real data

In this paper, a new method for evaluation 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_m ^(l)(x) (m=1, 2,…,M) inside the given Y_m(x) function that is supposed to be subjected by the influence of these factors. As contrasted to the conventional correlation analysis, the proposed method allows finding the “influence” functions b_l(x) (l=1, 2,…,L) and evaluating the “remnant” array G_m(x) that is remained as a “quasi-independent” part from the influence of the factors y_m ^(l)(x). The general expression works as a specific “balance” and reproduces the well-known cases, when b_l(x) =C_l (it is reduced to the linear least square method with G_m(x) ≈ 0) and coincides with the remnant function Y_m(x) ≈G_m(x), when the influence functions becomes negligible (b_l(x) ≈ 0). The available data show that the proposed method allows to extract a small signal S(x) from the “pattern” background and it conserves its stability/robustness in the presence of a random fluctuations/noise. The method is rather flexible and allows to consider the cases of strong correlations, when the external factors act successively, forming the cause-and-effect 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 coefficients 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.