Skip to main content
SpringerLink
Log in

Analysis of optimization methods for nonparametric estimation of probability density in large samples

  • Published:
Measurement Techniques Aims and scope

Abstract

The article proposes a procedure for selecting the blur factor of kernel functions for nonparametric density estimation of a one-dimensional random variable given large amounts of statistical data, for example, obtained via the remote sensing of natural objects. The proposed procedure uses regression density estimation. A procedure is presented for synthesizing a regression density estimate. The estimate synthesis is based on the original sample compression by decomposing the range of a random variable. To this end, the Heinhold-Gaede rule and a formula for choosing the optimal number of sampling intervals are applied. The article considers two approaches to selecting the blur factor of regression density estimation using the conventional method and that proposed by the authors to optimize nonparametric density estimation. The conventional method for optimizing nonparametric density estimation is based on minimizing its standard deviation. In the proposed method, the choice of the blur factors of kernel functions relies on the conditions for the minimum approximation error of regression density estimation. The article analyzes the approximation properties of regression density estimation using two optimization methods. The conditions of their competence in estimating the probability densities of random variables following lognormal distribution are established. The results obtained for a one-dimensional random variable can be used to optimize the regression density estimation of a multi-dimensional random variable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Lapko, A.V., Lapko, V.A.: Yadernye Ocenki Plotnosti Veroyatnosti i ih Primenenie [Kernel Probability Density Estimates and their Application; in Russian]. Reshetnev University Publ, Krasnoyarsk (2021)

    Google Scholar 

  2. Lapko, A.V., Lapko, V.A.: Optoelectron. Instrum. Data Process., 50. No 2, 148–153 (2014). https://doi.org/10.3103/S875669901402006X

    Article  Google Scholar 

  3. Rudemo, M.: Empirical choice of histogram and kernel density estimators. Scand. J. Stat. (9), 65–78 (1982)

  4. A. W. Bowman, J. Stat. Comput. Sim., 21, No. 3–4 (1985). https://doi.org/10.1080/00949658508810822

  5. Hall, P.: Ann. Stat. 11(4), 1156–1174 (1983). https://doi.org/10.1214/aos/1176346329

    Article  Google Scholar 

  6. Jiang, M., Provost, S.B.: J Stat Comput Simul 84(3), 614–627 (2014). https://doi.org/10.1080/00949655.2012.721366

    Article  MathSciNet  Google Scholar 

  7. Dutta, S.: Commun. Stat. Simulat. 45(2), 472–490 (2016). https://doi.org/10.1080/03610918.2013.862275

    Article  Google Scholar 

  8. Sturges, H.A.: J Am Stat Assoc 21, 65–66 (1926). https://doi.org/10.1080/01621459.1926.10502161

    Article  Google Scholar 

  9. Storm, R.: Teoriya Veroyatnostej. Matematicheskaya Statistika. Statisticheskij Kontrol’ Kachestva [Probability Theory. Mathematical Statistics. Statistical Quality Control; in Russian]. Mir Publ, Moscow (1970)

    Google Scholar 

  10. Heinhold, I., Gaede Ingeniur-Statistic, K.W.: in German], Springler Verlag. München, Wien (1964)

    Google Scholar 

  11. Lapko, A.V., Lapko, V.A.: Meas Tech 56(7), 763–767 (2013). https://doi.org/10.1007/s11018-013-0279-x

    Article  Google Scholar 

  12. Lapko, A.V., Lapko, V.A.: Meas. Tech. 63(7), 534–542 (2020). https://doi.org/10.1007/s11018-020-01820-1

    Article  Google Scholar 

  13. Parzen, E.: Ann Math Stat 33(3), 1065–1076 (1962). https://doi.org/10.1214/aoms/1177704472

    Article  Google Scholar 

  14. Epanechnikov, V.A.: Theor Probab Appl 14(1), 156–161 (1969). https://doi.org/10.1137/1114019

    Article  MathSciNet  Google Scholar 

  15. Gradov, V.M., Ovechkin, G.V., Ovechkin, P.V., Rudakov Komp’yuternoe Modelirovanie, I.V.: [Computer Modeling; in Russian], Kurs: INFRA‑M Publ. Moscow (2019)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Krasnoyarsk, Russian Federation

    A. V. Lapko & V. A. Lapko

  2. Reshetnev Siberian State University of Science and Technology, Krasnoyarsk, Russian Federation

    A. V. Lapko & V. A. Lapko

Authors
  1. A. V. Lapko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. A. Lapko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Lapko.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Translated from Izmeritel’naya Tekhnika, No. 11, pp. 26–32, November 2023. Russian DOI: https://doi.org/10.32446/0368-1025it.2023-11-26-32

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Original article submitted 06/20/2023. Accepted 09/07/2023.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lapko, A.V., Lapko, V.A. Analysis of optimization methods for nonparametric estimation of probability density in large samples. Meas Tech (2024). https://doi.org/10.1007/s11018-024-02298-x

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11018-024-02298-x

Keywords

UDC

Search

Navigation