Neural Computing and Applications

, Volume 31, Issue 11, pp 7415–7428 | Cite as

Retrospective analysis for phase I statistical process control and process capability study using revised sample entropy

  • Shing I. ChangEmail author
  • Zheng Zhang
  • Siim Koppel
  • Behnam Malmir
  • Xianguang Kong
  • Tzong-Ru Tsai
  • Donghai Wang
Original Article


This study explored a new nonparametric analytical method for identifying heterogeneous segments in time-series data for data-abundant processes. A sample entropy (SampEn) algorithm often used in signal processing and information theory can also be used in a time series or a signal stream, but the original SampEn is only capable of quantifying process variation changes. The proposed algorithm, the adjusted sample entropy (AdSEn), is capable of identifying process mean shifts, variance changes, or mixture of both. A simulation study showed that the proposed method is capable of identifying heterogeneous segments in a time series. Once segments of change points are identified, any existing change-point algorithms can be used to precisely identify exact locations of potential change points. The proposed method is especially applicable for long time series with many change points. Properties of the proposed AdSEn are provided to demonstrate the algorithm’s multi-scale capability. A table of critical values is also provided to help users accurately interpret entropy results.


Sample entropy Change points Process capability analysis Statistical process control 



Sample entropy


Adjusted sample entropy


Independent and identically distributed


Historical data series


Cumulative sum


Exponential moving average


Approximate entropy



This work was based on Mr. Zhang’s thesis as partial fulfillment of his MS degree in the Department of Industrial and Manufacturing Systems Engineering at Kansas State University, Manhattan, KS. Mr. Malmir’s support was based on the Biomass Research and Development Initiative Program with Grant No. 2012-10006-20230 from the US Department of Agriculture National Institute of Food and Agriculture. Dr. Kong was supported by the Fundamental Research Funds for the Central Universities projects under Grant Nos. 7214487602 and 72134876 and co-supported by Shaanxi Provincial scientific and technological research projects under Grant No. DF0102130401. He was a research visiting scholar to the IMSE department at Kansas State University.

Compliance with ethical standards

Conflict of interest

The authors have no conflict of interest to declare.


  1. 1.
    Kotz S, Johnson NL, Hubele NF, Spiring F, Cheng S, Yeung A, Leung B, Rodriguez RN, Bothe D, Rlu M-W (2002) Process capability indices: a review, 1992–2000. Discussions. J Qual Technol 34(1):2–53CrossRefGoogle Scholar
  2. 2.
    Montgomery DC (2009) Statistical quality control, vol 7. Wiley, New YorkzbMATHGoogle Scholar
  3. 3.
    Sullivan JH (2002) Detection of multiple change points from clustering individual observations. J Qual Technol 34(4):371CrossRefGoogle Scholar
  4. 4.
    Dupuis D, Sun Y, Wang HJ (2015) Detecting change-points in extremes. Stat Interface 8(1):19MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jarušková D (2015) Detecting non-simultaneous changes in means of vectors. Test 24(4):681–700MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brodsky B, Darkhovsky B (1993) Nonparametric methods in change point problems. Kluwer, DordrechtCrossRefGoogle Scholar
  7. 7.
    Page E (1955) A test for a change in a parameter occurring at an unknown point. Biometrika 42(3/4):523–527MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hinkley DV (1971) Inference about the change-point from cumulative sum tests. Biometrika 58(3):509–523MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bhattacharya P, Frierson D Jr (1981) A nonparametric control chart for detecting small disorders. Ann Stat 9(3):544–554MathSciNetCrossRefGoogle Scholar
  10. 10.
    Darkhovsky B, Brodsky B (1987) Nonparametric method of the quickiest detection of a change in mean of a random sequence. Theory Probab Appl 32(4):703–711Google Scholar
  11. 11.
    Pettitt A (1980) A simple cumulative sum type statistic for the change-point problem with zero-one observations. Biometrika 67:79–84MathSciNetCrossRefGoogle Scholar
  12. 12.
    Deshayes J, Picard D (1985) Off-line statistical analysis of change-point models using non parametric and likelihood methods. In: Basseville M (ed) Detection of abrupt changes in signals and dynamical systems. Springer, Berlin, pp 103–168CrossRefGoogle Scholar
  13. 13.
    Yin Y (1988) Detection of the number, locations and magnitudes of jumps. Commun Stat Stoch Models 4(3):445–455MathSciNetCrossRefGoogle Scholar
  14. 14.
    Xie Y, Huang J, Willett R (2013) Change-point detection for high-dimensional time series with missing data. IEEE J Sel Top Signal Process 7(1):12–27CrossRefGoogle Scholar
  15. 15.
    Liu S, Yamada M, Collier N, Sugiyama M (2013) Change-point detection in time-series data by relative density-ratio estimation. Neural Netw 43:72–83CrossRefGoogle Scholar
  16. 16.
    Shannon C (1948) A mathematical theory of communication. Bell Syst Tech J 27: 379–423 and 623–656. (MathSciNet): MR10, 133eGoogle Scholar
  17. 17.
    Kolmogorov AN (1998) On tables of random numbers. Theor Comput Sci 207(2):387–395MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pincus SM (1991) Approximate entropy as a measure of system complexity. Proc Natl Acad Sci 88(6):2297–2301MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pincus SM, Goldberger AL (1994) Physiological time-series analysis: what does regularity quantify? Am J Physiol Heart Circ Physiol 266(4):H1643–H1656CrossRefGoogle Scholar
  20. 20.
    Shardt YA, Huang B (2013) Statistical properties of signal entropy for use in detecting changes in time series data. J Chemom 27(11):394–405CrossRefGoogle Scholar
  21. 21.
    Liu H, Han M (2014) A fault diagnosis method based on local mean decomposition and multi-scale entropy for roller bearings. Mech Mach Theory 75:67–78CrossRefGoogle Scholar
  22. 22.
    Zheng Y, Sun C, Li J, Yang Q, Chen W (2011) Entropy-based bagging for fault prediction of transformers using oil-dissolved gas data. Energies 4(8):1138–1147CrossRefGoogle Scholar
  23. 23.
    Jha PK, Jha R, Datt R, Guha SK (2011) Entropy in good manufacturing system: tool for quality assurance. Eur Eur J Oper Res 211(3):658–665CrossRefGoogle Scholar
  24. 24.
    Zhang Z (2012) Manufacturing complexity and its measurement based on entropy models. Int J Adv Manuf Technol 62(9–12):867–873CrossRefGoogle Scholar
  25. 25.
    Nair SS, Joseph KP (2014) Chaotic analysis of the electroretinographic signal for diagnosis. BioMed Res Int. CrossRefGoogle Scholar
  26. 26.
    Richman JS, Moorman JR (2000) Physiological time-series analysis using approximate entropy and sample entropy. Am J Physiol Heart Circ Physiol 278(6):H2039–H2049CrossRefGoogle Scholar
  27. 27.
    Lake DE, Richman JS, Griffin MP, Moorman JR (2002) Sample entropy analysis of neonatal heart rate variability. Am J Physiol Regul Integr Comp Physiol 283(3):R789–R797CrossRefGoogle Scholar
  28. 28.
    Grassberger P, Procaccia I (1983) Estimation of the Kolmogorov entropy from a chaotic signal. Phys Lett A 28(4):2591Google Scholar
  29. 29.
    Grassberger P (1988) Finite sample corrections to entropy and dimension estimates. Phys Lett A 128(6–7):369–373MathSciNetCrossRefGoogle Scholar
  30. 30.
    Borchers HW (2014) Package “pracma”. (1.7.3). Obtained from: Accessed 6 June 2018
  31. 31.
    Lake DE, Moorman JR (2010) Accurate estimation of entropy in very short physiological time series: the problem of atrial fibrillation detection in implanted ventricular devices. Am J Physiol Heart Circ Physiol Ajpheart 00561:2010Google Scholar
  32. 32.
    Tano I, Vännman K (2013) A multivariate process capability index based on the first principal component only. Qual Reliab Eng Int 29(7):987–1003CrossRefGoogle Scholar
  33. 33.
    Xie H-B, Guo J-Y, Zheng Y-P (2010) Using the modified sample entropy to detect determinism. Phys Lett A 374(38):3926–3931CrossRefGoogle Scholar
  34. 34.
    Kong D-R, Xie H-B (2011) Use of modified sample entropy measurement to classify ventricular tachycardia and fibrillation. Measurement 44(4):653–662CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2018

Authors and Affiliations

  • Shing I. Chang
    • 1
    Email author
  • Zheng Zhang
    • 1
  • Siim Koppel
    • 1
  • Behnam Malmir
    • 1
  • Xianguang Kong
    • 3
  • Tzong-Ru Tsai
    • 4
  • Donghai Wang
    • 2
  1. 1.Department of Industrial and Manufacturing Systems EngineeringKansas State UniversityManhattanUSA
  2. 2.Department of Biological and Agricultural EngineeringKansas State UniversityManhattanUSA
  3. 3.School of Electromechanical EngineeringXidian UniversityXianChina
  4. 4.Department of StatisticsTamkang UniversityTamsui District, New Taipei CityTaiwan

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