DOI: https://doi.org/10.15588/1607-3274-2020-2-6

RISKS ESTIMATION METHOD BY CLUSTERED EXTREME DATA OF PROCESS COVARIATES

I. V. Tereshchenko, A. I. Tereshchenko, S. V. Shtangey

Abstract


Context. This paper presents a method for solving the problem of detecting and taking into account the influence of various (external and/or internal) factors on extreme and risky values of the multivariate observed parameters (covariates) of technological and/or diagnostic processes. Taking into account external and internal influence factors on covariates, by analogy with critical process parameters, is a significant addition to the extreme values statistics and the estimations the influence of the variability of process’s covariates on the expected losses, i.e. value at risk. Risk-oriented analysis is an actual tool for the data behavior investigation of the multivariate observations of process’s parameters. 

Objective. To disclose a method for detecting and taking into account the factors influence on the distribution functions parameters of the observed extreme values of process’s covariates and determine the influence of these distribution functions parameters on estimates of risks values. 

Method. The method consistently uses: the procedures of multivariate statistical cluster analysis, transformation the matrix of observed extreme values of process’s covariates into data frame with factor variables, estimation the extremal index and distribution functions parameters of nonclustered and clustered the observed extreme data of covariates and estimation the risk values on the calculated values of distribution functions parameters. The proposed sequence of actions is aimed at implementing the information technology of statistical causal analysis of the influence of factors on the variability of process’s covariates and their risk values due to the application of the clustering procedure for observed multivariate extreme values of covariates. The method is implementing the R-language packages software. 

Results. Clustering of the multivariate observed extreme values of process’s covariates allows to identifying the influence of environmental (manufacturing) factors and estimates the covariates’ risky values taking into account of this influence.

Conclusions. The method is an information technology of statistical causal analysis of factors influence on the variability of process’s covariates and theirs risk values due to the application of the clustering procedure of covariates’ multivariate values. The prospect of further research is to improve the methods of causal multivariate statistical analysis of the various factors influence on the exogenous and endogenous parameters of manufacturing and other processes in order to reduce the variability of these parameters and, as a result, minimize the risks. 


Keywords


Extreme value theory, generalized extreme value distribution, generalized Pareto distributions, value at risk, extreme value index, cluster analysis, process approach.

Full Text:

PDF

References


ISO/TC 176/SC 2/N 544R3. ISO 9000 Introduction and Support Package: Guidance on the Concept and Use of the Process Approach for management systems [Electronic resource]. Access mode: https://www.iso.org/files/live/sites/isoorg/files/archive/pdf/e n/04_concept_and_use_of_the_process_approach_for_mana gement_systems.pdf

ISO 9001: 2015. Quality management systems [Electronic resource]. Access mode: https://www.iso.org/standard/62085.html

ISO 31000 Risk management [Electronic resource]. Access mode: https://www.iso.org/iso-31000-risk-management.html

Beirlant J., Goegebeur Y., Segers J. et al. Statistics of Extremes: Theory and Applications, Wiley Series in Probability and Statistics, Chichester, John Wiley & Sons, 2004, 522 p.

Castillo E., Hadi A. S., Balakrishnan N. et al. Extreme Value and Related Models with Applications in Engineering and Science. New York, Wiley, 2004, 362 p.

Kreinovich V., Nguyen H. T., Sriboonchitta S., Kosheleva O. Modeling Extremal Events Is Not Easy: Why the Extreme Value Theorem Cannot Be as General as the Central Limit Theorem [Electronic resource]. Access mode: http://digitalcommons.utep.edu/cs_techrep/923

Repository CRAN [Electronic resource]. Access mode: https://cran.rproject.org/web/packages/fExtremes/fExtremes.pdf

Coles S. An Introduction to Statistical Modeling of Extreme Values. London, Springer, 2001, 209 p.

Repository CRAN [Electronic resource]. Access mode: https://cran.r-project.org/web/packages/texmex/index.html.

Novak S. Y. Extreme Value Methods with Applications to Finance. Florida, CRC Press, 2011, 399 p.

Yan J. Extreme Value Modeling and Risk Analysis: Methods and Applications / J. Yan, D. K. Dey. – Florida: CRC Press, 2016. – 540 p.

Tiemann T. K. Introductory Business Statistics with Interactive Spreadsheets [Electronic resource]. Access mode: http://people.wcsu.edu/lightwoods/mat120%20s19/IBStats% 20CanEd.pdf

Zivot E., Wang J. Modeling Financial Time Series with SPLUS: Second Edition. New York, Springer-Verlag, 2006, 705 p.

Bensalah Y. Steps in Applying Extreme Value Theory to Finance: A Review [Electronic resource]. Access mode: https://www.banqueducanada.ca/wpcontent/uploads/2010/01/wp00-20.pdf

Ferro C. A. T., Segers J. Inference for clusters of extreme values, Journal of the Royal Statistical Society. Series B: Statistical Methodology, 2003, Vol. 65, No. 2, pp. 545–556.

Heffernan J. E., Tawn J. A. A conditional approach for multivariate extreme values, Journal of the Royal Statistical Society. Series B: Statistical Methodology, 2004, Vol. 66, No. 3, pp. 497–546.

An overview of non-parametric clustering and computer vision [Electronic resource]. Access mode: https://web.archive.org/web/20080113181815/http://www.n erd-cam.com/cluster-results/

Tutorial with introduction of Clustering Algorithms [Electronic resource]. Access mode: http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/

Coghlan A. A Little Book of R For Multivariate Analysis, Release 0.1 [Electronic resource]. Access mode: http://alittle-book-of-r-for-time-series.readthedocs.org/

Havrylko Ye., Kurchenko O., Tereshchenko I. et al. The method of multivariate statistical analysis of the time multivariate critical quality attributes of manufacture process with the data factorization, Radio Electronics, Computer Science, Control, 2019, № 1, pp. 167–177.

Branch Information Technologies of Quality Management / [V. Nakonechnyi, S. Toliupa, I. Tereshchenko et al.] // Problems of Infocommunications. Science and Technology (PIC S&T): International Scientific-Practical Conference, Kharkov, 9–12 Oct. 2018: proceedings. – Kharkov : IEEE, 2018. – P. 783–788.

Gomes M. I., Guillou A. Extreme Value Theory and Statistics of Univariate Extremes: A Review, International Statistical Review, 2015, Vol. 83, No. 2, pp. 263–292.

Gomes M. I. Generalized Means in Statistical EVT [Electronic resource]. Access mode: https://www.researchgate.net/publication/337635760_Gener alized_Means_in_Statistical_EVT

Bezak N., Brilly M., Šraj M. Comparison between the peaks-over-threshold method and the annual maximum method for flood frequency analysis, Hydrological Sciences Journal, 2014,Vol. 59, No. 5, pp. 959–977.

Gomes M. I., Caeiro F., Figueiredo F. et al. Corrected-Hill versus partially reduced-bias value-at-risk estimation, Journal Communications in Statistics – Simulation and Computation, 2020, Vol. 49, No. 4, pp. 867–885.

Caeiro F., Henriques-Rodrigues L., Gomes P. D. A simple class of reduced bias kernel estimators of extreme value parameters, Computational and Mathematical Methods, 2019, Vol. 1, No. 3, pp. 1–12.

[Ragno E., Kouchak A. A., Cheng L. et al. A generalized framework for process-informed nonstationary extreme value analysis, Advances in Water Resources, 2019-08, Vol. 130, pp. 270–282.

Chai W. A., Leira B. J., Naess A. Probabilistic methods for estimation of the extreme value statistics of ship ice loads, Cold Regions Science and Technology, 2018-02, Vol. 146, pp. 87–97.

Majumdar S. N., Pal A., Schehr G. Extreme value statistics of correlated random variables: A pedagogical review, Physics Reports, 2020-01-22, Vol. 840, pp. 1–32.

Tsay R. S. Testing serial correlations in high-dimensional time series via extreme value theory, Journal of Econometrics, 2020, Vol. 216, No. 1, pp. 106–117.

Carreau J., Toulemonde G. Extra-parametrized extreme value copula: Extension to a spatial framework [Electronic resource]. Access mode: https://hal.inria.fr/hal02419118/document

Quinn N., Bates P., Neal J. et al. The spatial dependence of flood hazard and risk in the United States, Water Resources Research, 2019, Vol. 55, No. 3, pp. 1890–1911.

Abad P., Benito S., López C. A comprehensive review of Value at Risk methodologies, The Spanish Review of Financial Economics, 2014-01, Vol. 12, No. 1, pp. 15–32.

Zrazhevska N. G., Zrazhevsky А. G. Classification methods for risk measures VaR and CVaR calculation and estimation, System Research & Information Technologies, 2016, No. 3, pp. 126–141.

Stephenson A., Gilleland E. Software for the analysis of extreme events: The current state and future directions, Extremes, 2005-01, Vol. 8, No. 3, pp. 87–109.

Gilleland E., Ribatet M., Stephenson A. G. A software review for extreme value analysis, Extremes, 2013-03-01, Vol. 16, No. 1, pp. 103–119.

Natarajan D. ISO 9001 Quality Management Systems (Management and Industrial Engineering). Cham, Springer, 2017-03-31, 181 p.

Greenwood J. A., Landwehr J. M., Matalas N. C. et al. Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form, Water Resources Research, 1979, Vol. 15, No. 5, pp. 1049–1054.


GOST Style Citations


1. ISO/TC 176/SC 2/N 544R3. ISO 9000 Introduction and Support Package: Guidance on the Concept and Use of the Process Approach for management systems [Electronic resource]. –
Access mode: https://www.iso.org/files/live/sites/isoorg/files/archive/pdf/en/0 4_concept_and_use_of_the_process_approach_for_managemen t_systems.pdf

2. ISO 9001: 2015. Quality management systems [Electronic resource]. – Access mode: https://www.iso.org/standard/62085.html

3. ISO 31000 Risk management [Electronic resource]. – Access mode: https://www.iso.org/iso-31000-risk-management.html

4. Statistics of Extremes: Theory and Applications / [J. Beirlant, Y. Goegebeur, J. Segers et al.]. – Wiley Series in Probability and Statistics. – Chichester : John Wiley & Sons, 2004. – 522 p.

5. Extreme Value and Related Models with Applications in Engineering and Science / [E. Castillo, A. S. Hadi, N. Balakrishnan et al.]. – New York: Wiley, 2004. – 362 p.

6. Kreinovich V. Modeling Extremal Events Is Not Easy: Why the Extreme Value Theorem Cannot Be as General as the Central Limit Theorem [Electronic resource] / V. Kreinovich, H. T. Nguyen, S. Sriboonchitta, O. Kosheleva. – Access mode: http://digitalcommons.utep.edu/cs_techrep/923

7. Repository CRAN [Electronic resource]. – Access mode: https://cran.rproject.org/web/packages/fExtremes/fExtremes.pdf

8. Coles S. An Introduction to Statistical Modeling of Extreme Values / S. Coles. – London: Springer, 2001. – 209 p.

9. Repository CRAN [Electronic resource]. – Access mode: https://cran.r-project.org/web/packages/texmex/index.html

10. Novak S. Y. Extreme Value Methods with Applications to Finance / S. Y. Novak. – Florida: CRC Press, 2011. – 399 p.

11. Yan J. Extreme Value Modeling and Risk Analysis: Methods and Applications / J. Yan, D. K. Dey. – Florida : CRC Press, 2016. – 540 p.

12. Tiemann T. K. Introductory Business Statistics with Interactive Spreadsheets [Electronic resource] / T. K. Tiemann. – Access mode: http://people.wcsu.edu/lightwoods/mat120%20s19/IBStats%20 CanEd.pdf

13. Zivot E. Modeling Financial Time Series with S-PLUS: Second Edition / E. Zivot, J. Wang. – New York: Springer-Verlag, 2006. – 705 p.

14. Bensalah Y. Steps in Applying Extreme Value Theory to Finance: A Review [Electronic resource] / Y. Bensalah. – Access mode: https://www.banqueducanada.ca/wpcontent/uploads/2010/01/wp00-20.pdf

15. Ferro C. A. T. Inference for clusters of extreme values / C. A. T. Ferro, J. Segers // Journal of the Royal Statistical Society. Series B: Statistical Methodology. – 2003. – Vol. 65, № 2. – P. 545–556.

16. Heffernan J. E. A conditional approach for multivariate extreme values / J. E. Heffernan, J. A. Tawn // Journal of the Royal Statistical Society. Series B: Statistical Methodology. – 2004. – Vol. 66, № 3. – P. 497–546.

17. An overview of non-parametric clustering and computer vision [Electronic resource]. – Access mode: https://web.archive.org/web/20080113181815/http://www.nerdcam.com/cluster-results/

18. Tutorial with introduction of Clustering Algorithms [Electronic resource]. – Access mode: http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/

19. Coghlan A. A Little Book of R For Multivariate Analysis, Release 0.1 [Electronic resource] / A. Coghlan. – Access mode: http://a-little-book-of-r-for-timeseries.readthedocs.org/

20. The method of multivariate statistical analysis of the time multivariate critical quality attributes of manufacture process with the data factorization / [Ye. Havrylko, O. Kurchenko, I. Tereshchenko et al.] // Radio Electronics, Computer Science, Control. – 2019. – № 1. – P. 167–177.

21. Branch Information Technologies of Quality Management / [V. Nakonechnyi, S. Toliupa, I. Tereshchenko et al.] // Problems of Infocommunications. Science and Technology (PIC S&T):
International Scientific-Practical Conference, Kharkov, 9–12 Oct. 2018: proceedings. – Kharkov: IEEE, 2018. – P. 783–788.

22. Gomes M. I. Extreme Value Theory and Statistics of Univariate Extremes: A Review / M. I. Gomes, A. Guillou // International Statistical Review. – 2015. – Vol. 83, № 2. – P. 263–292.

23. Gomes M. I. Generalized Means in Statistical EVT [Electronic resource] / M. I. Gomes. – Access mode: https://www.researchgate.net/publication/337635760_Generaliz ed_Means_in_Statistical_EVT

24. Bezak N. Comparison between the peaks-over-threshold method and the annual maximum method for flood frequency analysis / N. Bezak, M. Brilly, M. Šraj // Hydrological Sciences Journal. – 2014. – Vol.59, № 5. – P. 959–977.

25. Corrected-Hill versus partially reduced-bias value-at-risk estimation / [M. I. Gomes, F. Caeiro, F. Figueiredo et al.] // Journal Communications in Statistics – Simulation and Computation. – 2020. – Vol. 49, № 4. – P. 867–885.

26. Caeiro F., Henriques-Rodrigues L., Gomes P. D. A simple class of reduced bias kernel estimators of extreme value parameters / F. Caeiro, L. Henriques-Rodrigues, D. P. Gomes // Computational and Mathematical Methods. – 2019. – Vol. 1, № 3. – P. 1–12.

27. A generalized framework for process-informed nonstationary extreme value analysis / [E. Ragno, A. A. Kouchak, L. Cheng et al.] // Advances in Water Resources. – 2019-08. – Vol. 130 – P. 270–282.

28. Chai W. A. Probabilistic methods for estimation of the extreme value statistics of ship ice loads / W. Chai, B. J. Leira, A. Naess // Cold Regions Science and Technology. – 2018-02. – Vol. 146. – P. 87–97.

29. Majumdar S. N. Extreme value statistics of correlated random variables: A pedagogical review / S. N. Majumdar, A. Pal, G. Schehr // Physics Reports. – 2020-01-22. – Vol. 840. – P. 1-32.

30. Tsay R. S. Testing serial correlations in high-dimensional time series via extreme value theory / R. S. Tsay // Journal of Econometrics. – 2020. – Vol. 216, № 1. – P. 106–117.

31. Carreau J. Extra-parametrized extreme value copula: Extension to a spatial framework [Electronic resource] / J. Carreau, G. Toulemonde. – Access mode: https://hal.inria.fr/hal-02419118/document

32. The spatial dependence of flood hazard and risk in the United States / [N. Quinn, P. Bates, J. Neal et al.] // Water Resources Research. – 2019. – Vol.55, № 3. – P. 1890-1911.

33. Abad P. A comprehensive review of Value at Risk methodologies / P. Abad, S. Benito, C. López // The Spanish Review of Financial Economics. – 2014-01. – Vol. 12, № 1. – P. 15-32.

34. Zrazhevska N. G. Classification methods for risk measures VaR and CVaR calculation and estimation / N. G. Zrazhevska, А. G. Zrazhevsky // System Research & Information Technologies. – 2016. – № 3. – P. 126-141.

35. Stephenson A. Software for the analysis of extreme events: The current state and future directions / A. Stephenson, E. Gilleland // Extremes. – 2005-01. – Vol. 8, № 3. – P. 87–109.

36. Gilleland E. A software review for extreme value analysis / E. Gilleland, M. Ribatet, A. G. Stephenson // Extremes. – 2013-0301. – Vol. 16, № 1. – P. 103–119.

37. Natarajan D. ISO 9001 Quality Management Systems (Management and Industrial Engineering) / D. Natarajan. – Cham : Springer, 2017-03-31. – 181 p.

38. Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form / [J. A. Greenwood, J. M. Landwehr, N. C. Matalas et al.] // Water Resources Research. – 1979. – Vol. 15, № 5. – P. 1049– 1054.







Copyright (c) 2020 I. V. Tereshchenko, A. I. Tereshchenko, S. V. Shtangey

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Address of the journal editorial office:
Editorial office of the journal «Radio Electronics, Computer Science, Control»,
National University "Zaporizhzhia Polytechnic", 
Zhukovskogo street, 64, Zaporizhzhia, 69063, Ukraine. 
Telephone: +38-061-769-82-96 – the Editing and Publishing Department.
E-mail: rvv@zntu.edu.ua

The reference to the journal is obligatory in the cases of complete or partial use of its materials.