The polynomial forecasts improvement based on the algorithm of optimal polynomial degree selecting

Abstract

The object of research in the paper is extrapolation problems based on interpolation polynomials. Polynomial-based prediction methods are well known. However, the problem is that such methods often give very large errors in practice. The permissible error of extrapolation even by one grid step is not ensured by the high accuracy of interpolation using polynomials. The paper proposes an algorithm that allows to significantly improve polynomial forecasts by optimizing the procedure for choosing the power of the polynomial, on the basis of which the forecast is built. The algorithm is based on the procedure for building all polynomial forecasts according to known data and analysis of these forecasts. In particular, the presence of monotonicity and a tendency to convergence allows determining the optimal degree of the polynomial. In the absence of monotonicity, provided that certain ratios are met, the forecast can be constructed as the arithmetic average of all polynomial forecasts. An important result is the estimation of the error of the forecasting method by averaging polynomial forecasts. The development of the algorithm became possible due to the use of a special method of constructing a one-step polynomial forecast. The method differs in that it allows to build a forecast without using the cumbersome procedure of calculating the unknown coefficients of the polynomial. The numerical results presented in the work demonstrate the effectiveness of the forecasting technique based on the average of polynomial forecasts. In particular, for the test functions, the relative error was about 2–5 %, while polynomials of different degrees in the worst case yielded more than 50 %. The obtained results can be useful for building short-term forecasts of series of economic dynamics, forecasting the behavior of arbitrary processes with a dominant deterministic component