Models and algorithms of nonlinear regression analysis of time series

The analysis of data describing certain objects and processes is primarily intended to find dependencies within them and to identify the dynamics of their development. The goals of analysis and forecasting are to prepare materials for making informed decisions. This work examines the stages, methods, and algorithms of conducting analysis aimed at obtaining, primarily, functional dependencies that are suitable not only for description but also for predicting the behavior of the studied objects and processes. The analysis itself is viewed as a multi-stage process that includes data preparation, identifying and removing noise from the data as much as possible, finding a long-term trend, identifying oscillatory components, periodicity of fluctuations, assessing the dynamics of amplitude fluctuations, evaluating the accuracy of possible approximations of the process, and the feasibility of forecasting considering the level of data noise. A number of procedures are proposed to ensure a reasoned verification of hypotheses about the course of processes and to obtain analytical, including differential dependencies based on optimization methods for parameter fitting of nonlinear dependencies. The methods discussed allow for conducting a sufficiently objective data analysis and create conditions for building a reasonable forecast. A numerical analysis was conducted based on multi-year statistics of production dynamics. The scientific novelty of the study lies in the development of a methodology for decomposing the process into trend and oscillatory components. Unlike most existing studies of process dynamics analysis, significant attention is given to accounting for and assessing the noise level by determining the limits of accuracy of the obtained results and, moreover, forecasts, which helps avoid unfounded conclusions and decisions and the construction of "overly" precise results based on insufficiently precise initial data, taking into account the requirements of function smoothness at the existing noise level. The use of limited growth functions and the identification of trend shift points allows for correct qualitative long-term forecasting without unreasonable predictions of a catastrophic course of the studied process. The results obtained allow for an analytical expression of the studied processes, primarily economic ones, which not only enables the approximation of the process behavior but also reveals its physical essence, thus allowing the application of these solutions to a whole class of processes of a similar nature.