Skip to main content
SpringerLink
Log in

Statistical Measures

  • Chapter
  • First Online:
Control Performance Assessment: Theoretical Analyses and Industrial Practice

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 245))

Abstract

Statistical measures constitute the second major group of the CPA indexes. Control engineer is especially interested in two aspects that might be measured with statistics. It is the variable average value and its variability. Statistical analysis often starts with the evaluation of simple statistics like maximum, minimum, mean, median or standard deviation. On the other hand, the variable histogram gives graphical representation of the of its statical properties. The chapters describes the indexes derived from the normal Gaussian distribution. Nonetheless much attention is put to the non-Gaussian approaches. The rationale for that part comes from the analysis of the real industrial data as they are rarely of normal properties. Fat-tailed stable probabilistic density functions are discussed in details as they deliver good and robust scale factors. The chapter finishes with the short presentation of the robust statistics data are very useful in the CPA task.

Sanity is not statistical.

–George Orwell

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ailliot, P., Thompson, C., Thomson, P.: Mixed methods for fitting the GEV distribution. Water Resour. Res. 47(5), w05551 (2011)

    Article  Google Scholar 

  2. Anderson, T.W., Darling, D.A.: A test of goodness-of-fit. J. Am. Stat. Assoc. 49, 765–769 (1954)

    Article  Google Scholar 

  3. Axensten, P.: Cauchy CDF, PDF, inverse CDF, parameter fit and random generator. http://www.mathworks.com/matlabcentral/fileexchange/11749-cauchy/ (2006)

  4. Borak, S., Misiorek, A., Weron, R.: Models for heavy-tailed asset returns. In: Cizek, P., Härdle, K.W., Weron, R. (eds.) Statistical Tools for Finance and Insurance, 2nd edn, pp. 21–56. Springer, Berlin (2011)

    Chapter  Google Scholar 

  5. Choudhury, M.A.A.S., Shah, S.L., Thornhill, N.F.: Diagnosis of poor control-loop performance using higher-order statistics. Automatica 40(10), 1719–1728 (2004)

    Article  MathSciNet  Google Scholar 

  6. Cramer, H.: On the composition of elementary errors. Scand. Actuar. J. 1, 13–74 (1928)

    Article  Google Scholar 

  7. Croux, C., Dehon, C.: Robust estimation of location and scale. Wiley StatsRef: Statistics Reference Online (2014)

    Google Scholar 

  8. Daszykowski, M., Kaczmarek, K., Heyden, Y.V., Walczak, B.: Robust statistics in data analysis–a review: basic concepts. Chemom. Intell. Lab. Syst. 85(2), 203–219 (2007)

    Article  Google Scholar 

  9. Domański, P.D.: Non-gaussian properties of the real industrial control error in SISO loops. In: Proceedings of the 19th International Conference on System Theory, Control and Computing, pp. 877–882 (2015)

    Google Scholar 

  10. Domański, P.D.: Non-Gaussian statistical measures of control performance. Control Cybern. 46(3), 259–290 (2017)

    MATH  Google Scholar 

  11. Domański, P.D.: Statistical measures for proportional-integral-derivative control quality: simulations and industrial data. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 232(4), 428–441 (2018)

    Article  Google Scholar 

  12. Domański, P.D., Gintrowski, M.: Alternative approaches to the prediction of electricity prices. Int. J. Energy Sect. Manag. 11(1), 3–27 (2017)

    Article  Google Scholar 

  13. Domański, P.D., Ławryńczuk, M.: Assessment of the GPC control quality using non-Gaussian statistical measures. Int. J. Appl. Math. Comput. Sci. 27(2), 291–307 (2017)

    Article  MathSciNet  Google Scholar 

  14. Engle, R.: Risk and volatility: econometric models and financial practice. Am. Econ. Rev. 94(3), 405–420 (2004)

    Article  Google Scholar 

  15. Ferguson, T.S.: Maximum likelihood estimates of the parameters of the cauchy distribution for samples of size 3 and 4. J. Am. Stat. Assoc. 73(361), 211–213 (1978)

    Article  Google Scholar 

  16. Gremm, M.: The origin of fat tails. https://ssrn.com/abstract=2337830 (2013). SSRN

  17. Hawkins, D.M.: Identification of Outliers. Chapman and Hall, London, New York (1980)

    Book  Google Scholar 

  18. Huber, P.J., Ronchetti, E.M.: Robust Statistics, 2nd edn. Wiley (2009)

    Google Scholar 

  19. Johnson, N., Kotz, S., Balakrishnan, N.: Continuous univariate distributions. No. t. 2 in Wiley series. In: Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley (1995)

    Google Scholar 

  20. Koutrouvelis, I.A.: Regression-type estimation of the parameters of stable laws. J. Am. Stat. Assoc. 75(372), 918–928 (1980)

    Article  MathSciNet  Google Scholar 

  21. Kuruoglu, E.E.: Density parameter estimation of skewed alpha-stable distributions. IEEE Trans. Signal Process. 49(10), 2192–2201 (2001)

    Article  MathSciNet  Google Scholar 

  22. Li, L.: Bayes estimation of parameter of laplace distribution under a new LINEX-based loss function. Int. J. Data Sci. Anal. 3(6), 85–89 (2017)

    Article  Google Scholar 

  23. Lilliefors, H.: On the Kolmogorov-Smirnov test for normality with mean and variance unknown. J. Am. Stat. Assoc. 62, 399–402 (1967)

    Article  Google Scholar 

  24. Mantegna, R.N., Stanley, H.E.: Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys. Rev. Lett. 73, 2946–2949 (1994)

    Article  MathSciNet  Google Scholar 

  25. McCulloch, J.H.: Simple consistent estimators of stable distribution parameters. Commun. Stat. Simul. Comput. 15(4), 1109–1136 (1986)

    Article  MathSciNet  Google Scholar 

  26. Pasquini, M., Serva, M.: Clustering of volatility as a multiscale phenomenon. Eur. Phys. J. B Condens. Matter Complex Syst. 16(1), 195–201 (2000)

    Article  Google Scholar 

  27. Royuela-del Val, J., Simmross-Wattenberg, F., Alberola-Lopez, C.: Libstable: fast, parallel, and high-precision computation of \(\alpha \)-stable distributions in R, C/C++, and MATLAB. J. Stat. Softw. Art. 78(1), 1–25 (2017)

    Google Scholar 

  28. Scarf, P.A.: Estimation for a four parameter generalized extreme value distribution. Commun. Stat. Theory Methods 21(8), 2185–2201 (1992)

    Article  MathSciNet  Google Scholar 

  29. Schuster, S.: Parameter estimation for the Cauchy distribution. In: 2012 19th International Conference on Systems, Signals and Image Processing (IWSSIP), pp. 350–353 (2012)

    Google Scholar 

  30. Shapiro, S.S., Wilk, M.B.: An analysis of variance test for normality (complete samples). Biometrika 52(3–4), 591–611 (1965)

    Article  MathSciNet  Google Scholar 

  31. Shen, X., Xu, Zhang Y.H., Meng, S.: Observation of alpha-stable noise in the laser gyroscope data. IEEE Sens. J. 16(7), 1998–2003 (2016)

    Article  Google Scholar 

  32. Taleb, N.N.: Real-world Statistical Consequences of Fat Tails: Papers and Commentary. STEM Academic Press, Technical Incerto Collection (2018)

    Google Scholar 

  33. Verboven, S., Hubert, M.: LIBRA: a Matlab library for robust analysis. Chemom. Intell. Lab. Syst. 75, 127–136 (2005)

    Article  Google Scholar 

  34. Viswanathan, G.M., Fulco, U.L., Lyra, M.L., Serva, M.: The origin of fat tailed distributions in financial time series, Cornell University Library (2002). arXiv:cond-mat/0112484v4

  35. Wong, T.S.T., Li, W.K.: A note on the estimation of extreme value distributions using maximum product of spacings. Lect. Notes Monogr. Ser. 52, 272–283 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Control and Computation Engineering, Warsaw University of Technology, Warsaw, Poland

    Paweł D. Domański

Authors
  1. Paweł D. Domański
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paweł D. Domański .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Domański, P.D. (2020). Statistical Measures . In: Control Performance Assessment: Theoretical Analyses and Industrial Practice. Studies in Systems, Decision and Control, vol 245. Springer, Cham. https://doi.org/10.1007/978-3-030-23593-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation