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W8INSE6220

# W8INSE6220 - 1 3 The Multivariate Quality Control Problem...

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1 INSE 6220 -- Week 8 Advanced Statistical Approaches to Quality Multivariate Control Charts Principal Component Analysis Dr. A. Ben Hamza Concordia University 2 Many situations require the simultaneous monitoring or control of two or more related quality characteristics • For example, consider a bearing with both an inner diameter ( x 1 ) and an outer diameter ( x 2 ) that together determine the usefulness of a part. Monitoring the characteristics independently can be very misleading. Use of multiple independent charts distorts the simultaneous monitoring of the averages • Type I error and probability of a point correctly plotting in control are not equal to advertised levels for the individual control charts • Distortion in process-monitoring procedures increases as the number of quality characteristics increases to Pr(all p means plot in control) = (1   ) p where p = total number of variables monitored ( independent quality characteristics) and = Probability [type I error] x The Multivariate Quality Control Problem 3 The Multivariate Quality Control Problem 4 The Multivariate Normal Distribution 1 2 ( , ,. .., )' p   p p p 2 1 1 1 2 ... ... ... ... ... >> help mvnpdf Therefore, the multivariate normal probability density function is given by 1 /2 1/2 1 1 ( ) exp ( ) ( ) (2 ) | | 2 p f x x x

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5 Description of Multivariate Data The Sample mean Vector and Covariance Matrix Suppose that we have a random sample from a multivariate normal distribution – say,           1 2 1 1 2 2 1 1 , ,. .., 1 1 1 1 1 1 1 n n i i n i i i n j ij j i n jk ij j ik k i x x x x x n S x x x x n s x x n s x x x x n 6 The Hotelling Control Chart Subgroup Data         2 2 2 2 2 0 2 1 1 1 2 2 12 1 1 2 2 2 2 2 1 2 12 2 n x x x x   7 8 In the case where the two quality characteristics are dependent, then , and the corresponding control ellipse is shown in the figure below: 12 0
9 10 Chi-Square Control Chart This set of quality characteristic means is represented by the p x1 vector 1 2 p x x x x The upper limit on the control chart is 2 , p UCL The test statistic plotted on the chi-square control chart for each samples is where is the vector of in-control means for each quality characteristic and is the covariance matrix     2 1 0 n x x 1 2 ' [ , ,. .., ] p   11 Estimating the mean and covariance 12

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13 The Hotelling Control Chart , , 1 ( 1)( 1) 1 0 p mn m p p m n UCL F mn m p LCL    The
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W8INSE6220 - 1 3 The Multivariate Quality Control Problem...

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