DOI QR코드

DOI QR Code

Markov 과정의 최초통과시간을 이용한 지수가중 이동평균 관리도의 평균런길이의 계산

Average run length calculation of the EWMA control chart using the first passage time of the Markov process

  • 박창순 (중앙대학교 응용통계학과)
  • 투고 : 2016.10.12
  • 심사 : 2016.11.03
  • 발행 : 2017.02.28

초록

많은 확률과정이 Markov 특성을 만족하거나 근사적으로 만족하는 것으로 가정된다. Markov 과정에서 특히 관심을 끄는 것은 최초통과시간이다. 최초통과시간에 대한 연구는 Wald의 축차분석에서 시작하여 근사적 특성에 대한 많은 연구가 되어왔고 컴퓨터의 발달로 통계계산적 방법이 사용되면서 근사적 결과가 참값에 가까운 값을 계산할 수 있게 되었다. 이 논문은 Markov 과정의 예로서 지수가중 이동평균 관리도를 사용할 때 평균런길이를 계산하는 과정과 계산상의 주의점, 문제점 등을 연구하였다. 이 결과는 다른 모든 Markov 과정에 적용될 수 있으며 특히 Markov 연쇄로의 근사는 확률과정의 특성의 연구에 유용하고 계산적 접근을 용이하게 한다.

Many stochastic processes satisfy the Markov property exactly or at least approximately. An interested property in the Markov process is the first passage time. Since the sequential analysis by Wald, the approximation of the first passage time has been studied extensively. The Statistical computing technique due to the development of high-speed computers made it possible to calculate the values of the properties close to the true ones. This article introduces an exponentially weighted moving average (EWMA) control chart as an example of the Markov process, and studied how to calculate the average run length with problematic issues that should be cautioned for correct calculation. The results derived for approximation of the first passage time in this research can be applied to any of the Markov processes. Especially the approximation of the continuous time Markov process to the discrete time Markov chain is useful for the studies of the properties of the stochastic process and makes computational approaches easy.

키워드

참고문헌

  1. Champ, C. W. and Rigdon, S. E. (1991). A comparison of the Markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts, Communications in Statistics-Simulation and Computation, 20, 191-204. https://doi.org/10.1080/03610919108812948
  2. Chan, Y., Han, B., and Pascual, F. (2015). Monitoring the Weibull shape parameter with type II censored data, Quality and Reliability Engineering International, 31, 741-760. https://doi.org/10.1002/qre.1631
  3. Chang, T. C. and Gan, F. F. (1994). Optimal designs of one-sided EWMA charts for monitoring a process variance, Journal of statistical Computation and Simulation, 49, 33-48. https://doi.org/10.1080/00949659408811559
  4. Crowder, S. V. (1987a). A simple method for studying run-length distributions of exponentially weighted moving average charts, Technometrics, 29, 401-407.
  5. Crowder, S. V. (1987b). Average run length of exponentially weighted moving average charts, Journal of Quality Technology, 19, 161-164. https://doi.org/10.1080/00224065.1987.11979055
  6. Crowder, S. V. and Hamilton, M. D. (1992). An EWMA for monitoring a process standard deviation, Journal of Quality Technology, 24, 12-21. https://doi.org/10.1080/00224065.1992.11979369
  7. Gan, F. F. (1993). Exponentially weighted moving average control charts with reflecting boundaries, Journal of statistical Computation and Simulation, 46, 45-67. https://doi.org/10.1080/00949659308811492
  8. Gan, F. F. (1995). Joint monitoring of process mean and variance using exponentially weighted moving average control charts, Technometrics, 37, 446-453. https://doi.org/10.1080/00401706.1995.10484377
  9. Gan, F. F. (1998). Designs of one- and two-sided exponential EWMA charts, Journal of Quality Technology, 30, 55-69. https://doi.org/10.1080/00224065.1998.11979819
  10. Gan, F. F. and Chang, T. C. (2000). Computing average run lengths of exponential EWMA charts, Journal of Quality Technology, 32, 183-187. https://doi.org/10.1080/00224065.2000.11979989
  11. Gianino, A. B., Champ, C. W., and Rigdon, S. E. (1990). Solving integral equations by the collocation method. In ASA Proceedings of the Statistical Computing Section (pp. 101-102), American Statistical Association, Washington.
  12. Hunter, J. S. (1986). The exponentially weighted moving average, Journal of Quality Technology, 18, 203-210. https://doi.org/10.1080/00224065.1986.11979014
  13. Knoth, S. (2003). EWMA schemes with non-homogeneous transition kernels, Sequential Analysis, 22, 241-255. https://doi.org/10.1081/SQA-120025169
  14. Knoth, S. (2004). Fast initial response features for EWMA control charts, Statistical Papers, 46, 47-64.
  15. Knoth, S. (2005). Accurate ARL computation for EWMA-S2 control charts, Statistics and Computing, 15, 341-352. https://doi.org/10.1007/s11222-005-3393-z
  16. Knoth, S. (2007). Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance, Sequential Analysis, 26, 251-264. https://doi.org/10.1080/07474940701404823
  17. Lucas, J. M. and Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: properties and enhancements, Technometrics, 32, 1-12. https://doi.org/10.1080/00401706.1990.10484583
  18. MacGregor, J. F. and Harris, T. J. (1993). The exponentially weighted moving variance, Journal of Quality Technology, 25, 106-118. https://doi.org/10.1080/00224065.1993.11979433
  19. Nichols, M. D. and Padgett, W. J. (2005). A bootstrap control chart for Weibull percentiles, Quality and Reliability Engineering International, 22, 141-151.
  20. Park, C. (2007). An algorithm for the properties of the integrated process control with bounded adjustments and EWMA monitoring, International Journal of Production Research, 45, 5571-5587. https://doi.org/10.1080/00207540701325397
  21. Park, C., Lee, J., and Kim, Y. (2004). Economic design of a variable sampling rate EWMA chart, IIE Transactions, 36, 387-399. https://doi.org/10.1080/07408170490426116
  22. Park, C. and Reynolds, M. R. (1999). Economic design of a variable sampling rates X chart, Journal of Quality Technology, 31, 363-443. https://doi.org/10.1080/00224065.1999.11979943
  23. Park, C. and Reynolds, M. R. (2008). Economic design of an integrated process control procedure with repeated adjustments and EWMA monitoring, Journal of the Korean Statistical Society, 37, 155-174. https://doi.org/10.1016/j.jkss.2007.10.005
  24. Park, C. S. and Won, T. Y. (1996). Selection of the economically optimal parameters in the EWMA control chart, Korean Journal of Applied Statistics, 9, 91-109.
  25. Pascual, F. (2010). EWMA charts for the Weibull shape parameter, Journal of Quality Technology, 42, 400-416. https://doi.org/10.1080/00224065.2010.11917836
  26. Ramalhoto, M. F. and Morais, M. (1999). Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals, Journal of Applied Statistics, 26, 129-160. https://doi.org/10.1080/02664769922700
  27. Waldmann, K. H. (1986). Bounds for the distribution of the run length of geometric moving average charts, Applied Statistics, 35, 151-158. https://doi.org/10.2307/2347265