Informatics and Applications
2026, Volume 20, Issue 3, pp 47-57
ASYMPTOTIC ANALYSIS OF THRESHOLD PROCESSING METHODS IN SPARSE MODELS WITH A POISSON NUMBER OF OBSERVATIONS
- E. I. Melezhnikov
- O. V. Shestakov
Abstract
The problem of threshold processing of a sparse signal with noisy observations is considered. The number of observations is assumed to be random and generated by a Poisson process with a given intensity. The behavior of the mean-square risk and its estimate based on Stein's unbiased risk estimate (SURE) is investigated. Special attention is paid to the effect of randomness in the sample size on signal recovery accuracy and on the properties of risk estimation. An upper bound for the risk under optimal threshold selection is obtained and it is shown that its asymptotic order coincides with that in the deterministic case. In addition, a central limit theorem and a strong law of large numbers for the SURE risk estimate are proved. Thus, the stability of the asymptotic properties of thresholding procedures is established when passing to a model defined by a Poisson process. The obtained results are applicable to problems of streaming data analysis, where observations arrive at random time moments and the sample size is not fixed in advance. This extends the applicability of thresholding methods to statistical models with a random number of observations.
[+] References (15)
- Donoho, D., and I. M. Johnstone. 1994. Ideal spatial adaptation via wavelet shrinkage. Biometrika 81(3):425- 455. doi: 10.1093/biomet/81.3.425.
- Donoho, D., and I. M. Johnstone. 1995. Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90(432):1200-1224. doi: 10.1080/ 01621459.1995.10476626.
- Donoho, D., and J. Jin. 2004. Higher criticism for detecting sparse heterogeneous mixtures. Ann. Stat. 32(3):962- 994. doi: 10.1214/009053604000000265.
- Lin, Y., and J. Cai. 2010. A new threshold function for signal denoising based on wavelet transform. Conference (International) on Measuring Technology andMechatronics Automation Proceedings. Piscataway, NJ: IEEE. 200-203. doi: 10.1109/ICMTMA.2010.347.
- Huang, H.-C., and T. C. M. Lee. 2010. Stabilized thresholding with generalized sure for image denoising. 17th Conference (International) on Image Processing Proceedings. Piscataway, NJ: IEEE. 1881-1884. doi: 10.1109/ICIP. 2010.5652353.
- He, C., J. Xing, J. Li, Q. Yang, and R. Wang. 2015. A new wavelet thresholding function based on hyperbolic tangent function. Math. Probl. Eng. 2015:528656. 10 p. doi: 10.1155/2015/528656.
- Shestakov, O. V. 2022. Nesmeshchennaya otsenka riska porogovoy obrabotki s dvumya porogovymi znacheniyami [Unbiased thresholding risk estimate with two threshold values]. Informatika i ee Primeneniya - Inform. Appl. 16(4):14-19.doi: 10.14357/19922264220403. EDN: DZBVLC.
- Kudryavtsev, A.A., and O. V. Shestakov. 2024. Properties of the SURE estimates when using continuous thresholding functions for wavelet shrinkage. Mathematics 12(23):3646.14 p. doi: 10.3390/math12233646. EDN: AETFBD.
- Vorontsov, M. O., and O. V. Shestakov. 2024. Asimptoticheskaya normal'nost' i sil'naya sostoyatel'nost' otsenki riska pri ispol'zovanii FDR-poroga v usloviyakh slaboy zavisimosti [Asymptotic normality and strong consistency of risk estimate when using the FDR threshold under weak dependence condition]. Informatika i ee Primeneniya - Inform. Appl. 18(3):69-79. doi: 10.14357/ 19922264240309. EDN: ZOQVTO.
- Shestakov, O. V. 2024. Statisticheskie svoystva otsenki srednekvadratichnogo riska metoda blochnoy porogovoy obrabotki v zadachakh neparametricheskoy regressii so sluchaynoy setkoy [Statistical properties of the mean- square risk estimate for the block threshold processing method in nonparametric regression problems with a random grid]. Informatika i ee Primeneniya - Inform. Appl. 18(4):26-33.doi: 10.14357/19922264240404. EDN: WHZQPX.
- Shestakov, O. V. 2025. Reshenie obratnykh statisticheskikh zadach s pomoshch'yu metodov porogovoy obrabotki, dopuskayushchikh postroenie nesmeshchennoy otsenki srednekvadratichnogo riska [Solving inverse statistical problems using threshold processing methods that allow the construction of an unbiased estimate of the mean-square risk]. Informatika i ee Primeneniya - Inform. Appl. 19(1):74-81. doi: 10.14357/19922264250110. EDN: ZSIBDL.
- Melezhnikov, E. I., O. V. Shestakov, and E. S. Stepanov. 2026. Asymptotic analysis of a thresholding method for sparse models with application to network delay detection. Mathematics 14(1):148.13 p. doi: 10.3390/math14010148. EDN: RRBORZ.
- Stein, C. 1981. Estimation of the mean of a multivariate normal distribution. Ann. Stat. 9(6):1135-1151. doi: 10.1214/aos/1176345632.
- Mallat, S. 1999. A wavelet tour of signal processing. New York, NY: Academic Press. 857 p.
- Kingman, J. F C. 1992. Poisson processes. Oxford: Clarendon Press. 112 p.
[+] About this article
Title
ASYMPTOTIC ANALYSIS OF THRESHOLD PROCESSING METHODS IN SPARSE MODELS WITH A POISSON NUMBER OF OBSERVATIONS
Journal
Informatics and Applications
2026, Volume 20, Issue 3, pp 47-57
Cover Date
2026-30-09
DOI
10.14357/19922264260304
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
thresholding; mean square risk; risk estimate; sparse model; Poisson process; central limit theorem; strong law of large numbers
Authors
E. I. Melezhnikov  ,  and O. V. Shestakov , , 
Author Affiliations
Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M. V Lomonosov Moscow State University, 1-52 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
Moscow Center for Fundamental and Applied Mathematics, M.V. Lomonosov Moscow State University, 1 Leninskie Gory, GSP-1, Moscow 119991, Russian Federation
Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
|