Local transformations with a singular wavelet
https://doi.org/10.37661/1816-0301-2020-17-1-39-46
Abstract
The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously, a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets.
The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of the sum of discrete local wavelet transforms is given.
About the Author
V. M. Romanchak
Belarusian National Technical University
Russian Federation
Vasily M. Romanchak, Cand. Sci. (Eng.), Associate Professor of the Department of Engineering Mathematics
Russian Federation
Vasily M. Romanchak, Cand. Sci. (Eng.), Associate Professor of the Department of Engineering Mathematics
References
1. Härdle W. Applied Nonparametric Regression. Cambridge, Cambridge University Press, 1992, 434 p.
2. Parzen E. On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 1962, vol. 33, no. 3, рр. 1065−1076.
3. Watson G. S. Smooth regression analysis. Sankhya: The Indian Journal of Statistics, Ser. A, 1964, vol. 26, pp. 359–372.
4. Nadaraya E. A. Ob ocenke regressii [About a regression assessment]. Teorija verojatnostej i ee primenenie [Probability Theory and Its Application], 1964, vol. 9, no. 1, pp. 157–159 (in Russian).
5. Chui C. An Introduction to Wavelets. San Diego, Academic Press, 1992, 266 р.
6. Daubechies I. Ten Lectures on Wavelets. Philadelphia, Society for Industrial and Applied Mathematics, 1992, 377 р.
7. Serenkov P. S., Rоmanchak V. M., Solomakho V. L. Sistema sbora dannyh o kachestve kak tehnicheskaja osnova funkcionirovanija jeffektivnyh sistem menedzhmenta kachestva [System of collection of data on quality as technical basis of functioning of effective systems of quality management]. Doklady Nacional'noj akademii nauk Belarusi [Doklady of the National Academy of Sciences of Belarus], 2006, vol. 50, no. 4, pp. 100−104 (in Russian).
8. Rоmanchak V. M., Lappo P. M. Approksimacija jekspertnyh ocenok singuljarnymi vejvletami [Approximation of expert estimates by singular wavelets]. Vestnik Grodnenskogo gosudarstvennogo universiteta. Ser. 2. Matematika. Fizika. Informatika, vychislitel'naja tehnika i upravlenie [Bulletin of the Grodno State University. Series 2: Mathematics. Physics. Informatics, Computer Science and Management], 2017, vol. 7, no. 1, pp. 132−139 (in Russian).
9. Rоmanchak V. M. Approksimacija singuljarnymi vejvletami [Approximation by singular wavelets]. Sistemnyj analiz i prikladnaja informatika [Systems Analysis and Applied Informatics], 2018, no. 2, pp. 23–28 (in Russian).
10. Rоmanchak V. M. Singuljarnye vejvlety na konechnom intervale [Singular wavelets on a finite interval]. Informatika [Informatics], 2018, vol. 15, no. 4, pp. 39−49 (in Russian).
Review
For citations:
Romanchak V.M. Local transformations with a singular wavelet. Informatics. 2020;17(1):39-46. (In Russ.) https://doi.org/10.37661/1816-0301-2020-17-1-39-46
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