Forecasting the Trend Component of Time Series Using a Trigonometric Basis

Abstract

This paper examines the problem of constructing a nonlinear forecast for time series using a trigonometric polynomial framework. Apart from the linear case, time series forecasting remains a complex and insufficiently resolved task. In the development of a technical system, three closely interrelated types of forecasts are typically distinguished:
1. forecasting the operational conditions under which the prediction will be applied;
2. forecasting optimal system parameters;
3. forecasting the functional characteristics of the technical system.
Although this conventional forecasting scheme is conceptually simple, it lacks reliability for medium- and especially long-term predictions. Furthermore, the need for prototypes when forming statistical datasets leads to the rapid obsolescence of information concerning new technical solutions by the time the data sample is constructed.
In this study, we propose a new variational approach to forecasting, in which the regularity criterion serves as the variational functional and trigonometric polynomials are used as the variational basis. The optimal structure of the basis functions is determined through statistical synthesis, followed by the construction of the most effective basis components. We demonstrate that statistical synthesis enables the development of simple and efficient algorithms for generating basis functions. Additionally, we introduce specialized operations on statistical samples that give rise to fundamentally new algorithms for solving multidimensional forecasting problems.
A model experiment confirms the practical convergence and effectiveness of the proposed statistical forecasting method.