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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Belarusian National Technical University
2018-08-01
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| Series: | Системный анализ и прикладная информатика |
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| Online Access: | https://sapi.bntu.by/jour/article/view/210 |
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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. |
| format | Article |
| id | doaj-art-3f8f1d5334f34cf4a4bf4b82bcd4ff6d |
| institution | Kabale University |
| issn | 2309-4923 2414-0481 |
| language | English |
| publishDate | 2018-08-01 |
| publisher | Belarusian National Technical University |
| record_format | Article |
| series | Системный анализ и прикладная информатика |
| spelling | doaj-art-3f8f1d5334f34cf4a4bf4b82bcd4ff6d2025-02-03T11:37:41ZengBelarusian National Technical UniversityСистемный анализ и прикладная информатика2309-49232414-04812018-08-0102232810.21122/2309-4923-2018-2-23-28162APPROXIMATELY SINGULAR WAVELETV. M. Romanchak0Belarusian National Technical UniversityThe 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/210wavelet transformthe parzen–rosenblatt window methodnonparametric estimatornadaraya-watson kernel regression |
| spellingShingle | V. M. Romanchak APPROXIMATELY SINGULAR WAVELET Системный анализ и прикладная информатика wavelet transform the parzen–rosenblatt window method nonparametric estimator nadaraya-watson kernel regression |
| 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 | wavelet transform the parzen–rosenblatt window method nonparametric estimator nadaraya-watson kernel regression |
| url | https://sapi.bntu.by/jour/article/view/210 |
| work_keys_str_mv | AT vmromanchak approximatelysingularwavelet |