Regression differentiation and regression integration of finite series

Abstract

Data analysis is a complex and multi-dimensional concept. It is explained by both the objective complexity of the data itself and the subjective nature of the expert analyzing them. Therefore, adequate formalization of this requires a completely new apparatus, on the one hand, capable of overcoming the objective complexity of data (irregularity and inaccuracy), and, on the other hand, the vague nature of expert judgment. The development of Discrete Mathematical Analysis (DMA) is an important step in this direction. DMA is highly expert oriented and occupies an intermediate position in data analysis between strict mathematical methods (statistical analysis, TFA, etc.) and soft combinatorial methods (simulation modeling, neural networks, etc.). This paper proposes new mathematical constructions of regression derivatives and regression integrals for discrete time series defined in general on an irregular grid. An important role in their study is played by the recently created by the authors projection method for solving systems of linear algebraic equations described at the end of this paper. The obtained constructions of regression derivatives and regression integrals have a hierarchical character in the spirit of wavelet and fractal analysis. The results of this work define another direction in further studies, namely, the propagation of regression differentiation and regression integration in finite mathematics under scenarios of the classical one.