Skip to main content
SpringerLink
Log in

Testing the Hypothesis of the Independence of Two-Dimensional Random Variables Using a Nonparametric Algorithm for Pattern Recognition

  • Published:
Optoelectronics, Instrumentation and Data Processing Aims and scope

Abstract

A new method for testing the hypothesis of the independence of two-dimensional random variables is proposed. The method under consideration is based on the use of a nonparametric algorithm for pattern recognition that meets the maximum likelihood criterion. In contrast to the traditional problem statement, there is no training sample a priori. The initial information is represented by statistical data that make up the values of two-dimensional random variables. The laws of distribution of random variables in classes are estimated from the initial statistical data for the conditions of their dependence and independence. When choosing the optimal blur coefficients for nonparametric estimates of probability densities, the maximum of the likelihood functions is used as a criterion. Under these conditions, estimates of the probability of pattern recognition errors in classes are calculated. Based on the minimum value of the estimate of the probability of an error in pattern recognition, a decision is made on the independence or dependence of random variables. The effectiveness of the developed method is confirmed by the results of computational experiments when testing the hypothesis of the independence or linear dependence of two-dimensional random variables.

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.

Fig. 1

Similar content being viewed by others

REFERENCES

  1. A. V. Lapko and V. A. Lapko, ‘‘Properties of nonparametric estimates of multidimensional probability density of independent random variables,’’ Inf. Sci. Control Syst. 31 (1), 166–174 (2012).

    Google Scholar 

  2. A. V. Lapko and V. A. Lapko, ‘‘Nonparametric estimation of probability density of independent random variables,’’ Inf. Sci. Control Syst. 29 (3), 118–124 (2011).

    Google Scholar 

  3. A. V. Lapko and V. A. Lapko, ‘‘Effect of a priori information about independence multidimensional random variables on the properties of their nonparametric density probability estimates,’’ Sist. Upr. Inf. Tekhnol. 48 (2.1), 164–167 (2012).

  4. A. V. Lapko and V. A. Lapko, ‘‘Properties of the nonparametric decision function with a priori information on independence of attributes of classified objects,’’ Optoelectron., Instrum. Data Process. 48, 416–422 (2012). https://doi.org/10.3103/S8756699012040139

    Article  Google Scholar 

  5. V. S. Pugachev, Theory of Probability and Mathematical Statistics (Fizmatlit, Moscow, 2002).

  6. A. V. Lapko and V. A. Lapko, ‘‘Nonparametric algorithms of pattern recognition in the problem of testing a statistical hypothesis on identity of two distribution laws of random variables,’’ Optoelectron., Instrum. Data Process. 46, 545–550 (2010). https://doi.org/10.3103/S8756699011060069

    Article  Google Scholar 

  7. A. V. Lapko and V. A. Lapko, ‘‘Comparison of empirical and theoretical distribution functions of a random variable on the basis of a nonparametric classifier,’’ Optoelectron., Instrum. Data Process. 48, 37–41 (2012). https://doi.org/10.3103/S8756699012010050

    Article  Google Scholar 

  8. A. V. Lapko and V. A. Lapko, ‘‘A technique for testing hypotheses for distributions of multidimensional spectral data using a nonparametric pattern recognition algorithm,’’ Comput. Optics 43, 238–244 (2019). https://doi.org/10.18287/2412-6179-2019-43-2-238-244

    Article  ADS  Google Scholar 

  9. E. Parzen, ‘‘On estimation of a probability density function and mode,’’ Ann. Math. Stat. 33, 1065–1076 (1962). https://doi.org/10.1214/aoms/1177704472

    Article  MathSciNet  MATH  Google Scholar 

  10. V. A. Epanechnikov, ‘‘Non-parametric estimation fo a multivariate probability density,’’ Theory Probab. Its Appl. 14, 153–158 (1969). https://doi.org/10.1137/1114019

    Article  MathSciNet  Google Scholar 

  11. R. P. W. Duin, ‘‘On the choice of smoothing parameters for parzen estimators of probability density functions,’’ IEEE Trans. Comput. C-25, 1175–1179 (1976). https://doi.org/10.1109/TC.1976.1674577

    Article  MATH  Google Scholar 

  12. Z. I. Botev and D. P. Kroese, ‘‘Non-asymptotic bandwidth selection for density estimation of discrete data,’’ Methodol. Comput. Appl. Probab. 10, 435 (2008). https://doi.org/10.1007/s11009-007-9057-z

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. A. W. Bowman, ‘‘A comparative study of some kernel-based non-parametric density estimators,’’ J. Stat. Comput. Simul. 21, 313–327 (1982). https://doi.org/10.1080/00949658508810822

    Article  MATH  Google Scholar 

  15. P. Hall, ‘‘Large-sample optimality of least squares cross-validation in density estimation,’’ Ann. Statist. 11, 1156–1174 (1983).

    MathSciNet  MATH  Google Scholar 

  16. M. Jiang and S. B. Provost, ‘‘A hybrid bandwidth selection methodology for kernel density estimation,’’ J. Stat. Comput. Simul. 84, 614–627 (2014). https://doi.org/10.1080/00949655.2012.721366

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Dutta, ‘‘Cross-validation revisited,’’ Commun. Stat. Simul. Comput. 45, 472–490 (2016). https://doi.org/10.1080/03610918.2013.862275

    Article  MathSciNet  MATH  Google Scholar 

  18. N.-B. Heidenreich, A. Schindler, and S. Sperlich, ‘‘Bandwidth selection for kernel density estimation: a review of fully automatic selectors,’’ AStA Adv. Stat. Anal. 97, 403–433 (2013). https://doi.org/10.1007/s10182-013-0216-y

    Article  MathSciNet  MATH  Google Scholar 

  19. Q. Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice (Princeton Univ. Press, Princeton, 2007).

    MATH  Google Scholar 

  20. A. V. Lapko and V. A. Lapko, ‘‘Method of fast bandwidth selection in a nonparametric classifier corresponding to the a posteriori probability maximum criterion,’’ Optoelectron., Instrum. Data Process. 55, 597–605 (2019). https://doi.org/10.3103/S8756699019060104

    Article  ADS  Google Scholar 

  21. D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, New Jersey, 2015). https://doi.org/10.1002/9780470316849

  22. S. J. Sheather, ‘‘Density estimation,’’ Stat. Sci. 19, 588–597 (2004). https://doi.org/10.1214/088342304000000297

    Article  MATH  Google Scholar 

  23. B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman and Hall, London, 1986).

    Book  Google Scholar 

  24. A. S. Sharakshane, I. G. Zheleznov, and V. A. Ivnitskii, Complex Systems (Vysshaya Shkola, Moscow, 1977).

Download references

Funding

The research was carried out with the financial support of the Russian Foundation for Basic Research, the Government of the Krasnoyarsk krai, and the Krasnoyarsk Regional Science Foundation (project no. 20-41-240001).

Author information

Authors and Affiliations

  1. Institute of Computational Modelling, Siberian Branch, Russian Academy of Sciences, 660036, Krasnoyarsk, Russia

    A. V. Lapko & V. A. Lapko

  2. Reshetnev Siberian State University of Science and Technology, 660037, Krasnoyarsk, Russia

    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.

Additional information

Translated by T. N. Sokolova

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lapko, A.V., Lapko, V.A. Testing the Hypothesis of the Independence of Two-Dimensional Random Variables Using a Nonparametric Algorithm for Pattern Recognition. Optoelectron.Instrument.Proc. 57, 149–155 (2021). https://doi.org/10.3103/S8756699021020114

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S8756699021020114

Keywords:

Search

Navigation