Decomposing Hotelling s T 2 Statistic Using Four Variables

Transcription

1 ISSN: Engineering and echnology (An ISO 397: 007 Certified Organization) Vol. 3, Issue 4, April 014 Decomposing Hotelling s Statistic Using Four Variables Nathan S. Agog 1, Hussaini G. Dikko, Osebekwin E. Asiribo 3 P.G. Student, Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria 1 Lecturer, Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria Professor, Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria 3 Abstract: he Mason, Young and racy (MY) decomposition approach has gained much acceptance due to its effectiveness in detecting variables that contributes to an out-of-control condition. he decomposition approach enables one to assess the value of each variable independently as while as their oint contribution. One of the setbacks of this approach is its computational complexity. In this work, the Hotelling s decomposition is extended from three variables to four variables using the MY decomposition technique. wenty four (4) models were obtained from the decomposition, where each model gives the same value proving the invariance property of the Hotelling s statistic. Keywords: Invariance Property, Matrix Permutation, Multivariate Statistical Process Control (MSPC), MY decomposition. I. INRODUCION he decomposition of the Hotelling s statistic into orthogonal components is considered to be one of the most effective tools for detecting variables that significantly contributes to out-of-control signal. his concept reflects the contribution of each individual variable as well as the oin contribution between two or more variables in a multivariate quality control chart. Multivariate control charts are widely used in practice to monitor the simultaneous performance of several related quality characteristics. he origin of multivariate control chart can be attributed to Hotelling [1]. A multivariate control scheme has a better sensitivity than the univariate control charts in monitoring multivariate quality process (Lu et al., []) Woodall and Montgomery [1] in one of their discussion paper stated that multivariate process control is one of the most rapidly developing sections of statistical process control. he demand to implement MSPC in a production process for quality improvement increases daily. According to Woodall [13], statistical methods play a very important role in quality improvement in manufacturing industries. he most commonly used multivariate statistical technique is the statistical Hotelling s control chart. here are a lot of literatures focusing on multivariate control charting methods based on Hotelling s statistic in detecting mean shift such as Sullivan and Woodall [9], Mason and Young [5], ong et al., [10] and many others. Much application of the Hotelling s decomposition technique has been on two and three process variables as seen in the case of Mason et al.,[4], Ulen and Demir [11], Sani and Abubakar [8] and many others. In this article, our interest is to extend this approach by deriving the decomposition model using four variables. his decomposition will not only be helpful to quality control engineers but also to researchers in the field of quality control. Copyright to IJIRSE

2 ISSN: Engineering and echnology (An ISO 397: 007 Certified Organization) Vol. 3, Issue 4, April 014 II. DECOMPOSIION MEHOD Mason et al., [3], proposed the use of decomposition in detecting out-of-control variable. he decomposition is considered to be the most valuable technique in detecting influential variable(s). he main idea of this method is to decompose the statistic into independent parts, each of which reflects the contribution of an individual variable. he characteristic that is significantly contributing to the signal is more readily identified by decomposing the Mason et al., [4], presented a practical application of the decomposition on a bivariate data (two process variables, p=). Where there were two possible decompositions; Sani and Abubakar [8], presented the decomposition of three process variables (p=3) by Mason and Young [6] to demonstrate the invariance property of the Hotelling s statistic. he decomposition of the Hotelling s statistic for three process variables is as follows; 1..1,.1.1, , , , ,3 his work extends the decomposition of Hotelling s statistic from three variable (p=3) to four variables (p=4). he number of decompositions increased from 6 to 4 possible ways (p! =3! = (6 decompositions) to p! = 4! = (4 decompositions)). hese decompositions give the same overall statistic as stated by Mason et al.,[3] III. MASON, YOUNG AND RACY (MY) DECOMPOSIION ECHNIQUE he Hotelling s control chart is the most commonly used multivariate control charts when all the quality characteristics are normally distributed. Let x1, x,..., xp be p quality characteristics. Assuming they are normally distributed with multivariate mean, and covariance matrix,. he statistic for a p dimensional observation vector ' x1 x x p (,,..., ) ' 1 S can be represented as, ( ) ( ) (1) his approach is to decompose the statistic into independent components, each of which reflects the contribution of individual variables. he statistic in (1) can be partitioned into two independent parts (Rencher [7]). hese components are given by, () p1 p.1,,..., p1 Since the first term in () is a statistic, it can also be separated into two orthogonal parts; (3) p1 p p1.1,,..., p1 Continuing in the same fashion, we can generate one of the many possible MY decompositions of the statistic. 1..1,... p.1,,3..., p1 (4) Copyright to IJIRSE

3 ISSN: Engineering and echnology (An ISO 397: 007 Certified Organization) Vol. 3, Issue 4, April 014 III.1. DECOMPOSIION MODEL USING FOUR VARIABLES he process is initiated by selecting any one of the p variables, followed by any of the (p-1) remaining variables to condition on the first selected variable. Next, select any of the remaining (p-1) variables to be condition on the first two selected variables. And finally, select any of the (p-3) variables to condition on the first three selected variables. Iterating in the same procedure will generate all the decomposition equations which compose the same over all statistic. For the first p variables, assuming we select p=1(remaining nd, 3 rd and 4 th variables). he decomposition is as follows; 1..1, 4.1,, ,3.1,3, ,4 3.1,,4 he next step is to keep the selected p variable and the (p-1) remaining variables conditioned on the p selected variable constant (that is, the first and the second decomposition terms). hen select any of the remaining (p-) variables not used in the first decomposition to be condition on the first two kept variables. And finally, select any of the remaining (p-3) variables not used during the first decomposition to condition on the first three selected variables. hese yield the following results; , 3.1,,4.1.1,3 4.1,, ,4.1,3,4 Iterating in the same procedure will generate all the possible decomposition equations which compose the same overall statistic. his condition satisfies the invariance property which states that the value of the cannot change by permuting the observation vector (Mason and Young [6]). Hence, the complete decomposition using four variables (p = 4) is as follows; 1..1, 4.1,, ,3.1,3, ,4 3.1,, , 3.1,,4.1.1,3 4.1,, ,4.1,3, , 4.1,, ,3 1.,3, ,3 3.1,, , 3.1,, ,3 4.1,, ,4 1.,3, ,3 4.1,, ,3 1.,3,4 Copyright to IJIRSE

4 ISSN: Engineering and echnology ,4.1,3, ,3.1,3, ,3 4.1,, ,4 1.,3, ,4 3.1,, ,4 1.,3, ,4.1,3, ,4.1,3, ,4 3.1,, ,4 1.,3,4 (An ISO 397: 007 Certified Organization) Vol. 3, Issue 4, April 014 IV. COMPUING HE DECOMPOSIION ERMS One of the decomposition for the observation vector ( x1, x, x3, x 4) is given as (5) ( x, x, x, x ).1.1,3 4.1,, he computation starts by first determining the value of the conditional term, 4.1,,3 ( x, x, x, x ) ( x, x, x ) ,,3. From the above equation, (6) ( x, x, x, x ) is given as; he variance-covariance matrix and the mean vector for the observation vector S S S S S S1 S S3 S4 44 S31 S3 S3 S34 S41 S4 S43 S4 hus the computation of and ( x, x, x, x ) (4) x1 x x3 x 4 is as follows; ( ) S ( ) (8) (4) (4) ' 1 (4) (4) ( x, x, x, x ) 44 o obtain ( x, x, x ) 1 3, the original estimates of the mean vector and covariance structure is partitioned to obtain the (3) mean vector and covariance matrix of the sub vector ( x, x, x ). he corresponding partition is given as; S1 S1 S 13 x1 S33 S1 S S3 and x S31 S3 S 3 x 3 hus the computation is as follows; ( x, x, x ) 1 4 (3) (3) ' 1 (3) (3) ( x1, x, x3 ) ( ) S ( ) (10) Also the decomposition of ( x, x, x ) 1 3 is given by (7) (9) Copyright to IJIRSE

5 ISSN: Engineering and echnology (An ISO 397: 007 Certified Organization) Vol. 3, Issue 4, April 014 (11) ( x, x, x ).1.1,3 1 3 he equation above can be obtained by first computing the conditional term.1,3 ( x, x, x ) ( x, x ) 1 3 ( x1, x3).1,3 as follows; (1) o obtain the term, the original estimates of the mean vector and the covariance structure is partitioned to obtain the mean vector and covariance matrix of the sub vector as; S S S 1 13 S31 S3 and () x1 x3 Hence, the computation of the term ( x, x ) () () ' 1 () () ( x, x ) is as follows; () ( x, x ). he corresponding partition is given ( ) S ( ) (14) Also, the decomposition for ( x, x ).1 ( x, x ) is given by (15) he term is computed by ( x x ) s1 3.1 hus, is obtained by computing the value of the sub vector (1) 1 (13) ( x ). Hence, the unconditional term (16) is computed as; (17) 3.1 ( x, x ) ( x ) 1 ( x x ) s (18) = 1,,,p. is the square of a univariate t statistic for the observed value of the th variable of vector. V. CONCLUSION he MY decomposition has a beautiful way of presenting the contribution of independent variables and also the relationship between two or more variables. his work has presented a simplified way decomposing the statistic by using four process variables where 4 decompositions were obtained. Each of these decompositions can be used to identify variable(s) that significantly contributes to an out-of-control signal. REFERENCES 1. Hotelling, H. Multivariate quality control. In C. Eisenhart, M.W. Hastay and W.A. Wallis, eds., echniques of Statistical Analysis. New York: McGraw Hill, pp , Lu,.S., M. ie,.n. Goh and C.D. Lai, Control chart for multivariate attribute Processes, International Journal Production Research, Vol. 36, pp , Mason R.L., racy N.D. and Young J.C.,. Decomposition of for multivariate control chart interpretation Journal of Quality echnology Vol. 7, pp , Mason, R. L.,racy N.D., and Young J.C. A practical application for interpreting multivariate control chart signal. Journal of Quality echnology, Vol. 9, No.4, pp , Mason, R.L. and Young, J.C.. Improving the sensitivity of the statistic in multivariate process control. Journal of Quality echnology, Vol. 31, No., pp , Copyright to IJIRSE

6 ISSN: Engineering and echnology (An ISO 397: 007 Certified Organization) Vol. 3, Issue 4, April Mason, R.L. and Young, J.C., Multivariate Statistical Process Control with Industrial Applications, Philadelphia: ASA-SIAM. pp , Rencher, A.C., he contribution of individual variables to Hotelling s, Wilks Ʌ and R, Biometrics, Vol. 49, pp , Sani, S.A., and Abubakar, Y., Demonstrating the Invariance Property of the Hotelling Statistic, International Journal of Innovative Research in Science, Engineering and echnology, Vol., No. 7, pp , Sullivan, J.H. and Woodall, W.H., A comparison of multivariate control charts for individual observations, Journal of Quality echnology, Vol. 8, No. 4, pp , ong, L.I., Wang, C.H. and Huang, C.L., Monitoring defects in IC fabrication using a Hotelling control chart, IEE rans. Semiconductor Manufacturing, Vol. 18, No. 1, pp , Ulen, M. and Demir, I., Application of multivariate statistical quality control in pharmaceutical industry, Balkan Journal of Mathematics, Vol. 1, pp , Woodall, W.H and Montgomery, D.C., Research issues and ideas in statistical process control. Journal of Quality echnology, Vol. 31, No. 4, pp , Woodall, W.H., Controversies and contradictions in statistical process control, Journal of Quality echnology, vol. 3, No.4, pp , 000. Copyright to IJIRSE