APPROXIMATELY SINGULAR WAVELET
The problem of approximation is relevant for most engineering applications. In this connection, the universal methods of approximation are of interest. The method of nonparametric approximation is developing in the paper – the method of singular wavelets. The method includes an effective numerical a...
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| Format: | Article |
| Language: | English |
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Belarusian National Technical University
2018-08-01
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| Series: | Sistemnyj Analiz i Prikladnaâ Informatika |
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| Online Access: | https://sapi.bntu.by/jour/article/view/210 |
| _version_ | 1797873709783973888 |
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| author | V. M. Romanchak |
| author_facet | V. M. Romanchak |
| author_sort | V. M. Romanchak |
| collection | DOAJ |
| description | The problem of approximation is relevant for most engineering applications. In this connection, the universal methods of approximation are of interest. The method of nonparametric approximation is developing in the paper – the method of singular wavelets. The method includes an effective numerical algorithm based on the summation of a recursive sequence of functions. The universal algorithm of approximation makes it possible to apply it to approximate one-dimensional and multidimensional functions, in decision support systems, in the processing of stochastic information, pattern recognition, and solution of boundary-value problems.The introduction explain the idea of the method of singular wavelets – to combine the theory of wavelets with the Nadaraya-Watson kernel regression estimator. Usually, Nadaraya-Watson kernel regression are considered as an example of non- parametric estimation. However, one parameter, the smoothing parameter, is still present in the traditional kernel regression algorithm. The choice of the optimal value of this parameter is a complex mathematical problem, and numerous studies have been devoted to this question. In the approximation by the method of singular wavelets, summation of Nadaraya-Watson kernel regression estimates with the smoothing parameter takes place, which solves the problem of the optimal choice of this parameter.In the main part of the paper theorems are formulated that determine the properties of the regularized wavelet transform. Sufficient conditions for uniform convergence of the wavelet series are obtained for the first time. To illustrate the effectiveness of the numerical approximation algorithm, we consider an example of the quasi-interpolation of the Runge function by wavelets with a uniform distribution of interpolation nodes. |
| first_indexed | 2024-04-10T01:20:27Z |
| format | Article |
| id | doaj.art-3f8f1d5334f34cf4a4bf4b82bcd4ff6d |
| institution | Directory Open Access Journal |
| issn | 2309-4923 2414-0481 |
| language | English |
| last_indexed | 2024-04-10T01:20:27Z |
| publishDate | 2018-08-01 |
| publisher | Belarusian National Technical University |
| record_format | Article |
| series | Sistemnyj Analiz i Prikladnaâ Informatika |
| spelling | doaj.art-3f8f1d5334f34cf4a4bf4b82bcd4ff6d2023-03-13T09:47:41ZengBelarusian National Technical UniversitySistemnyj Analiz i Prikladnaâ Informatika2309-49232414-04812018-08-0102232810.21122/2309-4923-2018-2-23-28162APPROXIMATELY SINGULAR WAVELETV. M. Romanchak0Белорусский национальный технический университетThe problem of approximation is relevant for most engineering applications. In this connection, the universal methods of approximation are of interest. The method of nonparametric approximation is developing in the paper – the method of singular wavelets. The method includes an effective numerical algorithm based on the summation of a recursive sequence of functions. The universal algorithm of approximation makes it possible to apply it to approximate one-dimensional and multidimensional functions, in decision support systems, in the processing of stochastic information, pattern recognition, and solution of boundary-value problems.The introduction explain the idea of the method of singular wavelets – to combine the theory of wavelets with the Nadaraya-Watson kernel regression estimator. Usually, Nadaraya-Watson kernel regression are considered as an example of non- parametric estimation. However, one parameter, the smoothing parameter, is still present in the traditional kernel regression algorithm. The choice of the optimal value of this parameter is a complex mathematical problem, and numerous studies have been devoted to this question. In the approximation by the method of singular wavelets, summation of Nadaraya-Watson kernel regression estimates with the smoothing parameter takes place, which solves the problem of the optimal choice of this parameter.In the main part of the paper theorems are formulated that determine the properties of the regularized wavelet transform. Sufficient conditions for uniform convergence of the wavelet series are obtained for the first time. To illustrate the effectiveness of the numerical approximation algorithm, we consider an example of the quasi-interpolation of the Runge function by wavelets with a uniform distribution of interpolation nodes.https://sapi.bntu.by/jour/article/view/210вейвлет преобразованиеокно парзена–розенблаттанепараметрическая аппроксимацияядерная оценка надарая–ватсона |
| spellingShingle | V. M. Romanchak APPROXIMATELY SINGULAR WAVELET Sistemnyj Analiz i Prikladnaâ Informatika вейвлет преобразование окно парзена–розенблатта непараметрическая аппроксимация ядерная оценка надарая–ватсона |
| title | APPROXIMATELY SINGULAR WAVELET |
| title_full | APPROXIMATELY SINGULAR WAVELET |
| title_fullStr | APPROXIMATELY SINGULAR WAVELET |
| title_full_unstemmed | APPROXIMATELY SINGULAR WAVELET |
| title_short | APPROXIMATELY SINGULAR WAVELET |
| title_sort | approximately singular wavelet |
| topic | вейвлет преобразование окно парзена–розенблатта непараметрическая аппроксимация ядерная оценка надарая–ватсона |
| url | https://sapi.bntu.by/jour/article/view/210 |
| work_keys_str_mv | AT vmromanchak approximatelysingularwavelet |