Outline Outline Monit Monitoring Location and ring Location and - PowerPoint PPT Presentation

7/25/2011 A A Dis Distribut bution-free Contro Control Char l Chart f t for r Outline Outline Monit Monitoring Location and ring Location and Scale Scale  Introduction Introduction  Normal Distribution  Normal Distribution Normal Distribution Normal Distribution  Joint Monitoring of Mean and Variance Joint Monitoring of Mean and Variance  Connections to statistical testing problems Connections to statistical testing problems S. Chakraborti  Classes of charts Classes of charts Department of Information Systems, Statistics and Management Science  Brief literature review Brief literature review University of Alabama  Nonparametric Nonparametric U.S.A.  Joint monitoring of location and scale Joint monitoring of location and scale and  Proposed chart Proposed chart A. Mukherjee Department of Mathematics  Implementation Implementation Alto University  Example Example Finland  Performance Performance  On On- -going/Future Work going/Future Work ISSPC 2011 Conference PUC, Rio de Janeiro, Brazil July, 2011 Joint Monitoring Joint Monitoring -- -- Normal Normal Joint Monitoring Joint Monitoring  Variables Data  Two Chart Schemes  Process output normally distributed, X ~ N(µ, σ 2 )  Shewhart-type  Monitor both mean µ and variance σ 2  EWMA-type  Approaches  Others  Two Chart Schemes:  One Chart Schemes  Use a mean chart and a variance chart  Shewhart-type  Driven by the independence of sample mean  Hybrid—EWMA and variance under normality  Others  Several charts are available  See Cheng and Thaga (2006) for an overview and  One Chart Schemes: references up to 2005  Use a plotting statistic which is a function of  See McCracken et al. (2011) (in-progress) for a more two statistics, one for the mean and one for recent overview the variance (also driven by normality)  Several charts are available 1

7/25/2011 Some Recent Literature Some Recent Literature Some More Literature Some More Literature o Zhang, J., Zou, C., & Wang, Z. (2011). “A new chart for detecting the process mean and o Zhou, Q., Luo, Y., & Wang, Z. (2010). “A control chart based on likelihood ratio test for variability,” Communications in Statistics - Simulation and Computation, 40 (5), 728 -743. detecting patterned mean and variance shifts,” Computational Statistics & Data Analysis, o Maboudou-Tchao, E. & Hawkins, D. (2011). “Self-Starting Multivariate Control Charts for , , ( ) g 54 (6) 1634-1645 54 (6), 1634-1645. Location and Scale,” Journal of Quality Technology, 43 (2), 113-126. o Hawkins, D.M., & Deng, Q. (2009). “Combined Charts for Mean and Variance o Huang, C.C., & Chen F.L. (2010). “Economic Design of Max Charts,” Communications in Information,” Journal of Quality Technology, 41 (4), 415-425. Statistics - Theory and Methods,39 (16), 2961-2976. o Chao, M.T. & Cheng, S.W. (2008). “On 2-D Control Charts,” Quality Technology & o Khoo, M.B.C. et al. (2010a). “Using one EWMA chart to jointly monitor the process mean Quantitative Management, 5 (3), 243-261. and variance,” Computational Statistics, 25 , 299–316. o Wu, Z., Zhang, S., & Wang, P. (2007). “A CUSUM scheme with variable sample sizes and o Khoo, M.B.C. et al. (2010b). “Monitoring process mean and variability with one double sampling intervals for monitoring the process mean and variance,” Quality and Reliability EWMA Chart”, Communications in Statistics - Theory and Methods, 39 (20), 3678 -3694. Engineering International, 23 (2), 157-170. o Li, Z., Zhang, J., & Wang, Z. (2010) “Self-starting control chart for simultaneously o Reynolds, M. R. & Stoumbos, Z. G. (2006). “Comparisons of some exponentially weighted monitoring process mean and variance,” International Journal of Production Research, moving average control charts for monitoring the process mean and variance,” Technometrics, 48 (15), 4537-4553. 48 (15) 4537 4553 48 (4), 550-567 o Zhang, J., Zou, C., & Wang, Z. (2010). “A Control Chart Based on Likelihood Ratio Test o Yeh, A.B., Lin, D.K.J., & Venkataramani, C. (2004). “Unified CUSUM charts for for Monitoring Process Mean and Variability,” Quality and Reliability Engineering monitoring process mean and variability,” Quality Technology & Quantitative Management, International, 26 , 63-73. 1 (1), 65-86. o Zhang J. , Li Z., & Wang Z. (2010). “A multivariate control chart for simultaneously o Costa, A.B.F., & Rahim, M.A. (2004). “Monitoring process mean and variability with one monitoring process mean and variability,” Computational Statistics & Data Analysis, 54 non-central chi-square chart.” Journal of Applied Statistics, 31 (10), 1171-1183. (10), 2244-2252. o Chen, G., Cheng, S.W., & Xie H. (2001). “Monitoring process mean and variability with one EWMA chart,” Journal of Quality Technology, 33 (2), 223-233. o ……. Distribution Distribution- -free/Nonparametric free/Nonparametric Parametric Control Charts: Parametric Control Charts: Key Key issues issues Nonparametric statistical inference is a collective term given to inferences that are valid under less restrictive assumptions than with classical Distribution-free/Nonparametric: Di t ib ti f /N t i ( (parametric) statistical inference. The assumptions that can be relaxed t i ) t ti ti l i f Th ti th t b l d  Form of the distribution is assumed known include specifying the probability distribution of the population from which Statistical methods that require  e.g. normal the sample was drawn and the level of measurement required of the sample data. For example, we may have to assume that the population is  Is this ever really true? minimal assumptions about the symmetric, which is much less restrictive than assuming the population is the normal distribution. The data may be ranks, i.e., measurements on an  Chart properties (of normal theory charts) are ordinal scale, instead of precise measurements on an interval or ratio scale. form of the distribution to make an not always robust Or the data may be counts. In nonparametric inference, the null distribution of the statistic on which the inference is based does not depend on the inference based on a test statistic inference based on a test statistic,  False alarm rate  False alarm rate probability distribution of the population from which the sample was drawn. probability distribution of the population from which the sample was drawn In other words, the statistic has the same sampling distribution under the  ARL, SDRL, … all are affected a p-value, a control chart or a null hypoth-esis, irrespective of the form of the parent population. This statistic is therefore called distribution-free, and, in fact, the field of  Charts may lose value for practice! confidence interval! nonparametric statistics is some-times called distribution-free statistics. Nonparametric methods are often based on ranks, scores, or counts. This  Not applicable with all types of data such as allows us to make less restrictive assumptions and still make an inference -- Gibbons and Chakraborti (2010) ranks such as calculate a P-value or find a confidence interval. Strictly speaking, the term nonparametric implies an inference that may or may 2

7/25/2011 Distribution Distribution- -free/Nonparametric free/Nonparametric Nonparametric Control Charts Nonparametric Control Charts A test of hypothesis is nonparametric (NP) or Advantages: distribution-free (DF) if the Type I error probability is  Often a natural thing to do  Often a natural thing to do – intuitive intuitive the same for all continuous distributions  Distributional assumption not required -- Gibbons and Chakraborti (2010)  In-control properties are known (stable) Nonparametric Statistical Inference, 5 th ed., CRC Press --- similar definition for a NP or DF confidence interval in terms of  Robust the coverage!  Comparable detection power A control chart is nonparametric (NP) or distribution- free (DF) if the in control run length distribution is the free (DF) if the in-control run length distribution is the same for all continuous distributions Challenges: -- Chakraborti, van der Laan and Bakir (2001)  Not as widely known So all in-control performance characteristics remain  Not always easy to construct the same and known for all continuous distributions!!!  Not available for all problems Example: the sign test and the sign control chart  Erase doubts about efficiency Nonparametric Control Charts Nonparametric Control Charts Nonparametric Control Charts Nonparametric Control Charts Types of charts: Some Literature: Univariate: Univariate:  Shewhart charts  Shewhart charts  EWMA charts  Overview Papers:  CUSUM charts  Chakraborti et al. (2010, 2007, 2001)  Other charts  Other recent papers  Balakrishnan et al. (2009, 2010), Li et al. (2010), Based on Hawkins and Deng (2010), Zou et al. (2010),  Signs Human et al. (2010), Memar and Niaki (2010),  Signed-ranks Chatterjee and Qiu (2009), Zhou et al. (2009), …  Sample Quantiles (Order statistics) Multivariate:  Ranks  Boone (2010) and references therein  ….  Other papers 3