Journal of Korean Society of Industrial and Systems Engineering (산업경영시스템학회지)

Society of Korea Industrial and System Engineering (한국산업경영시스템학회)
권호 표지
DOI QR코드

DOI QR Code

Analysis of Multivariate Process Capability Using Box-Cox Transformation

Box-Cox변환을 이용한 다변량 공정능력 분석

  • Moon, Hye-Jin (Department of Industrial and Management Engineering, Incheon National University) ;
  • Chung, Young-Bae (Department of Industrial and Management Engineering, Incheon National University)
  • Received : 2019.03.22
  • Accepted : 2019.06.14
  • Published : 2019.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

The process control methods based on the statistical analysis apply the analysis method or mathematical model under the assumption that the process characteristic is normally distributed. However, the distribution of data collected by the automatic measurement system in real time is often not followed by normal distribution. As the statistical analysis tools, the process capability index (PCI) has been used a lot as a measure of process capability analysis in the production site. However, PCI has been usually used without checking the normality test for the process data. Even though the normality assumption is violated, if the analysis method under the assumption of the normal distribution is performed, this will be an incorrect result and take a wrong action. When the normality assumption is violated, we can transform the non-normal data into the normal data by using an appropriate normal transformation method. There are various methods of the normal transformation. In this paper, we consider the Box-Cox transformation among them. Hence, the purpose of the study is to expand the analysis method for the multivariate process capability index using Box-Cox transformation. This study proposes the multivariate process capability index to be able to use according to both methodologies whether data is normally distributed or not. Through the computational examples, we compare and discuss the multivariate process capability index between before and after Box-Cox transformation when the process data is not normally distributed.

Keywords

References

  1. Anderson, T.W. and Darling, D.A., A test of goodness of fit, Journal of the American Statistical Association, 1954, Vol. 49, No. 268, pp. 765-769. https://doi.org/10.1080/01621459.1954.10501232
  2. Box, G.E.P. and Cox, D.R., An Analysis of Transformations, Journal of the Royal Statistical Society, 1964, Series B, Vol. 26, No. 2, pp. 211-252.
  3. Chang, Y.S., Heuristic Process Capability Indices Using Distribution-decomposition Methods, Journal of the Korean Society for Quality Management, 2013, Vol. 41, No. 2, pp. 233-248 https://doi.org/10.7469/JKSQM.2013.41.2.233
  4. Hosseinifard, S.Z., Abbasi, B., Ahmad, S., and Abdollahian, M., A transformation technique to estimate the process capability index for non-normal processes, International Journal of Advanced Manufacturing Technology, 2009, Vol. 40, pp. 512-517 https://doi.org/10.1007/s00170-008-1376-x
  5. Looney, S.W., How to Use Tests for Univariate Normality to Assess Multivariate Normality, The American Statistician, 1995, Vol. 49, No. 1, pp. 64-70. https://doi.org/10.2307/2684816
  6. Moon, H.J. and Chung, Y.B., Multivariate Process Capability Index Using Inverted Normal Loss Function, Journal of Society of Korea Industrial and Systems Engineering, 2018, Vol. 41, No. 2, pp. 174-183. https://doi.org/10.11627/jkise.2018.41.2.174
  7. Razali, N.M. and Wah, Y.B., Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests, Journal of Statistical Modeling and Analytics, 2011, Vol. 2, No. 1, pp. 21-33.
  8. Royston, J.P., Some Techniques for Assessing Multivarate Normality Based on the Shapiro-Wilk W, Journal of the Royal Statistical Society. Series C, 1983, Vol. 32, No. 2, pp. 121-133.
  9. Small, N.J.H., Marginal Skewness and Kurtosis in Testing Multivariate Normality, Journal of the Royal Statistical Society. Series C, 1980, Vol. 29, No. 1, pp. 85-87.
  10. Taam, W., Subbaiah, P., and Liddy, J.W., A Note on Multivariate Capability Indices, Journal of Applied Statistics, 1993, Vol. 20, pp. 339-351. https://doi.org/10.1080/02664769300000035
  11. Wu, C.H., Lin, S.J., Yang, D.L. and Pearn, W.L., Box-Cox Transformation Approach for Evaluating Non-Normal Processes Capability Based on the Cpk Index, Journal of Testing and Evaluation, 2014, Vol. 42, No. 4, pp. 949-961
  12. Yap, B. W. and Sim, C. H., Comparisons of various types of normality tests, Journal of Statistical Computation and Simulation, 2011, Vol. 81, No. 12, pp. 2141-2155. https://doi.org/10.1080/00949655.2010.520163
  13. Zhang, T. and Yang, B., Box-Cox Transformation in Big Data, Technometrics, 2017, Vol. 59, No. 2, pp. 189-201 https://doi.org/10.1080/00401706.2016.1156025