The polynomial forecasts improvement based on the algorithm of optimal polynomial degree selecting
DOI:
https://doi.org/10.15587/1729-4061.2023.289292Keywords:
prediction algorithm, problem extrapolation, time series, split differences net, Newton’s polynomials, Pascal’s triangle, convergence of predictions, binomial coefficients, extrapolation errorAbstract
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
References
- Brezinski, С., Redivo-Zaglia, M. (2020). Extrapolation and Rational Approximation: The Works of the Main Contributors. Springer. doi: https://doi.org/10.1007/978-3-030-58418-4
- Israfilov, D. M., Testici, A. (2018). Some Inverse and Simultaneous Approximation Theorems in Weighted Variable Exponent Lebesgue Spaces. Analysis Mathematica, 44 (4), 475–492. doi: https://doi.org/10.1007/s10476-018-0403-x
- Bomba, A. Ya., Turbal, Y. V. (2015). Data Analysis Method and Problems of Identification of Trajectories of Solitary Waves. Journal of Automation and Information Sciences, 47 (10), 13–23. doi: https://doi.org/10.1615/jautomatinfscien.v47.i10.20
- Guliyev, V. S., Ghorbanalizadeh, A., Sawano, Y. (2018). Approximation by trigonometric polynomials in variable exponent Morrey spaces. Analysis and Mathematical Physics, 9 (3), 1265–1285. doi: https://doi.org/10.1007/s13324-018-0231-y
- Hyndman, R. J., Kostenko, A. V. (2007). Minimum sample size requirements for seasonal forecasting model. FORESIGHT, 6. Available at: https://robjhyndman.com/papers/shortseasonal.pdf
- Turbal, Y., Bomba, A., Turbal, M., Alkaleg Hsen Drivi, A. (2021). Some aspects of extrapolation based on interpolation polynomials. Physico-Mathematical Modelling and Informational Technologies, 33, 175–180. doi: https://doi.org/10.15407/fmmit2021.33.175
- Shalaginov, A. V. (2011). Kubicheskaya splayn ekstrapolyatsiya vremennykh ryadov. International conference on System Analysis and Information Technologies SAIT 2011, Institute for Applied System Analysis of National Technical University of Ukraine. Kyiv. Available at: https://cad.kpi.ua/attachments/141_2011_025s.pdf
- Kostinsky, A. S. (2014). On the principles of a spline extrapolation concerning geophysical data. Reports of the National Academy of Sciences of Ukraine, 2, 111–117. doi: https://doi.org/10.15407/dopovidi2014.02.111
- Makridakis, S., Bakas, N. (2016). Forecasting and uncertainty: A survey. Risk and Decision Analysis, 6 (1), 37–64. doi: https://doi.org/10.3233/rda-150114
- Zhan, Z., Fu, Y., Yang, R.-J., Xi, Z., Shi, L. (2012). A Bayesian Inference based Model Interpolation and Extrapolation. SAE International Journal of Materials and Manufacturing, 5 (2), 357–364. doi: https://doi.org/10.4271/2012-01-0223
- Demiris, N., Lunn, D., Sharples, L. D. (2011). Survival extrapolation using the poly-Weibull model. Statistical Methods in Medical Research, 24 (2), 287–301. doi: https://doi.org/10.1177/0962280211419645
- Turbal, Y., Bomba, A., Turbal, M., Sokh, A., Radoveniuk, O. (2019). Pyramidal method of extrapolation for short time series. International Journal of Computing Science and Mathematics, 10 (6), 525. doi: https://doi.org/10.1504/ijcsm.2019.104025
- Monroe, J. I., Hatch, H. W., Mahynski, N. A., Shell, M. S., Shen, V. K. (2020). Extrapolation and interpolation strategies for efficiently estimating structural observables as a function of temperature and density. The Journal of Chemical Physics, 153 (14). doi: https://doi.org/10.1063/5.0014282
- Wang, L.-Y., Lee, W.-C. (2015). One-step extrapolation of the prediction performance of a gene signature derived from a small study. BMJ Open, 5 (4), e007170–e007170. doi: https://doi.org/10.1136/bmjopen-2014-007170
- Bakas, N. P. (2019). Numerical Solution for the Extrapolation Problem of Analytic Functions. Research, 2019. doi: https://doi.org/10.34133/2019/3903187
- Valovyi vnutrishniy produkt (VVP) v Ukraini 2023. Ministerstvo finansiv Ukrainy. Available at: https://index.minfin.com.ua/ua/economy/gdp/
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Yurii Turbal, Ganna Shlikhta, Mariana Turbal, Bogdan Turbal
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.
