Part 1. A new general fitting method based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth's mean temperature surface data versus time) and for these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.
Part 2. It has been shown that in nature at least two general scenarios of data structuring are possible: (a) self-similar scenario when the measured data form a self-similar (SS) structure; (b) quasi-periodic scenario when the repeated (strongly-correlated) data form random sequences that are almost periodic with respect to each other. In the second case it becomes possible to describe their behavior and express a part of their randomness quantitatively in terms of the deterministic amplitude-frequency response (AFR) belonging to the generalized Prony's spectrum (GPS). This possibility allows to reexamine the conventional concept of measurements and opens a new way for description of a wide set of different data. In particular, it concerns different complex systems when the “best fit” model pretending to description of the data measured is absent but the barest necessity of description of these data in terms of the reduced number of quantitative parameters exists. The possibilities of the proposed approach and detection algorithm of the quasi-periodic (QP) processes were demonstrated on actual data: spectroscopic data recorded for pure water and acoustic data for a test hole. The suggested methodology allows revising the accepted classification of different incommensurable and self-affine spatial structures and finding accurate interpretation of the generalized Prony’s spectroscopy that includes the Fourier spectroscopy as a partial case.
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