Hotelling’s <i>T<sup>2</sup></i> Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control

Adepoju Ajibola Akeem; Abubakar Yahaya; Osebekwin Asiribo

doi:doi:10.11648/j.ajtas.20150406.13

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Hotelling’s T² Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control

Adepoju Ajibola Akeem, Abubakar Yahaya, Osebekwin Asiribo

Published in American Journal of Theoretical and Applied Statistics (Volume 4, Issue 6)

Received: 27 August 2015 Accepted: 13 September 2015 Published: 28 September 2015

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Abstract

Multivariate statistical process control (MSPC) is the most acceptable monitoring tool for several variables, and it is advantageous when compare to the simultaneous use of univariate scheme. However, there are some disadvantages in this scheme which include identification of influential variable(s). The Mason, Young and Tracy (MYT) decomposition diagnosis is one of the approaches commonly use to identify the influential variables. This approach aid the breaking down, the overall T square value and show the individual variable contribution, while their joint contributions is also revealed. The challenges of this approach include rigorous derivation of model, computation and complexity more especially when the size of the process characteristics is large. In this research paper we extend the decomposition derivation to five variables. One hundred and twenty (120) models (decomposition partitions) are obtained from the decomposition, revealing the invariance property of the Hotelling’s T square statistic, and eighty (80) unique terms.

Published in	American Journal of Theoretical and Applied Statistics (Volume 4, Issue 6)
DOI	10.11648/j.ajtas.20150406.13
Page(s)	432-437
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Decomposition Chart, Hotelling’s T square, Invariance Property, Matrix Permutation, Multivariate Statistical Process Control (MSPC), MYT Decomposition

References

[1]	Hotelling, H. Multivariate Quality Control. In C. Eisenhart, M. W. Hastay and W. A. Wallis, eds. Techniques of Statistical Analysis. New York: McGraw Hill, 111-184(1947).
[2]	Sullivan, J. H. and Woodall, W. H., A Comparison of Multivariate Control Charts for Individual Observations, Journal of Quality Technology, 28 (4) 398-408 (1996).
[3]	Mason, R. L. and Young, J. C. Improving the sensitivity of the T² statistic in multivariate process control. Journal of Quality Technology, 31(2), 155-165(1999).
[4]	Tong, L. I., Wang, C. H. and Huang, C. L., “Monitoring defects in IC fabrication using a Hotelling T² control chart”, IEE Trans. Semiconductor Manufacturing, 18(1), 140-148(2005).
[5]	Mason R. L., Tracy N. D. and Young J. C., “Decomposition of T² for multivariate control chart interpretation” Journal of Quality Technology 27, 99-108(1995).
[6]	Lowry, C. A. and Montgomery, D. C., A Review of Multivariate Control Chart, IIE Transactions, (27) 800 – 810.
[7]	Mason, R. L., Tracy N. D., and Young J. C. A practical application for interpreting multivariate T² control chart signal. Journal of Quality Technology, 29(4), 396-501(1997).
[8]	Ulen, M. and Demir, I.,” Application of multivariate statistical quality control in pharmaceutical industry”, Balkan Journal of Mathematics, 1, 93-105. (2013).
[9]	Masoud, Y. and Javad E “Identification of the Out of Control Variables in Multivariate Statistical Process Control” Journal of Statistical Modeling and Analytics 1(1), 95-112(2010).
[10]	Sani, S. A., and Abubakar, Y.,” Demonstrating the Invariance Property of the Hotelling T² Statistic”, International Journal of Innovative Research in Science, Engineering and Technology, 2(7), 2515-2519(2013).
[11]	Nathan S. A., Hussaini G. D. and Osebekwin E. A. Decomposing Hotelling’s T² Statistic Using Four Variables International Journal of Innovative Research in Science, Engineering and Technology3(4), 2014.
[12]	Blazek, L. W., Novic B. and Scott M. D. Displaying Multivariate Data Using Polyplots. Journal of Quality Technology, 19(2), 69-74(1987).
[13]	Subramanyan, N. and Houshmand, A. A. Simultaneous representation of multivariate and corresponding univariate charts using line graph. Quality Engineering, 4, 681-682(1995).
[14]	Fuchs, C. and Benjamin, Y. Multivariate profile charts for statistical process control. Technometrics, 36(2), 182-195(1994).
[15]	Iglewicz, B. and Hoaglin, D. C. Use of boxplots for process evaluation. Journal of Quality Technology, 19(4), 180-190(1987).
[16]	Atienza, O. O., Ching L. T. and Wah, B. A. Simultaneous monitoring of univariante and multivariate SPC information using boxplots. International Journal of Quality Science, 3(2), 1988.
[17]	Woodall, W. H. and Montgomery, D. C. Research Issues and Ideas in Statistical Process Control. Journal of Quality Technology 31.376-386(1999).

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Adepoju Ajibola Akeem, Abubakar Yahaya, Osebekwin Asiribo. (2015). Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control. American Journal of Theoretical and Applied Statistics, 4(6), 432-437. https://doi.org/10.11648/j.ajtas.20150406.13

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ACS Style

Adepoju Ajibola Akeem; Abubakar Yahaya; Osebekwin Asiribo. Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control. Am. J. Theor. Appl. Stat. 2015, 4(6), 432-437. doi: 10.11648/j.ajtas.20150406.13

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AMA Style

Adepoju Ajibola Akeem, Abubakar Yahaya, Osebekwin Asiribo. Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control. Am J Theor Appl Stat. 2015;4(6):432-437. doi: 10.11648/j.ajtas.20150406.13

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@article{10.11648/j.ajtas.20150406.13,
  author = {Adepoju Ajibola Akeem and Abubakar Yahaya and Osebekwin Asiribo},
  title = {Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {4},
  number = {6},
  pages = {432-437},
  doi = {10.11648/j.ajtas.20150406.13},
  url = {https://doi.org/10.11648/j.ajtas.20150406.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150406.13},
  abstract = {Multivariate statistical process control (MSPC) is the most acceptable monitoring tool for several variables, and it is advantageous when compare to the simultaneous use of univariate scheme. However, there are some disadvantages in this scheme which include identification of influential variable(s). The Mason, Young and Tracy (MYT) decomposition diagnosis is one of the approaches commonly use to identify the influential variables. This approach aid the breaking down, the overall T square value and show the individual variable contribution, while their joint contributions is also revealed. The challenges of this approach include rigorous derivation of model, computation and complexity more especially when the size of the process characteristics is large. In this research paper we extend the decomposition derivation to five variables. One hundred and twenty (120) models (decomposition partitions) are obtained from the decomposition, revealing the invariance property of the Hotelling’s T square statistic, and eighty (80) unique terms.},
 year = {2015}
}

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TY  - JOUR
T1  - Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control
AU  - Adepoju Ajibola Akeem
AU  - Abubakar Yahaya
AU  - Osebekwin Asiribo
Y1  - 2015/09/28
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N1  - https://doi.org/10.11648/j.ajtas.20150406.13
DO  - 10.11648/j.ajtas.20150406.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
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EP  - 437
PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.ajtas.20150406.13
AB  - Multivariate statistical process control (MSPC) is the most acceptable monitoring tool for several variables, and it is advantageous when compare to the simultaneous use of univariate scheme. However, there are some disadvantages in this scheme which include identification of influential variable(s). The Mason, Young and Tracy (MYT) decomposition diagnosis is one of the approaches commonly use to identify the influential variables. This approach aid the breaking down, the overall T square value and show the individual variable contribution, while their joint contributions is also revealed. The challenges of this approach include rigorous derivation of model, computation and complexity more especially when the size of the process characteristics is large. In this research paper we extend the decomposition derivation to five variables. One hundred and twenty (120) models (decomposition partitions) are obtained from the decomposition, revealing the invariance property of the Hotelling’s T square statistic, and eighty (80) unique terms.
VL  - 4
IS  - 6
ER  -

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Author Information

Adepoju Ajibola Akeem

Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria
Abubakar Yahaya

Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria
Osebekwin Asiribo

Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria

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APA Style

Adepoju Ajibola Akeem, Abubakar Yahaya, Osebekwin Asiribo. (2015). Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control. American Journal of Theoretical and Applied Statistics, 4(6), 432-437. https://doi.org/10.11648/j.ajtas.20150406.13

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ACS Style

Adepoju Ajibola Akeem; Abubakar Yahaya; Osebekwin Asiribo. Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control. Am. J. Theor. Appl. Stat. 2015, 4(6), 432-437. doi: 10.11648/j.ajtas.20150406.13

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AMA Style

Adepoju Ajibola Akeem, Abubakar Yahaya, Osebekwin Asiribo. Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control. Am J Theor Appl Stat. 2015;4(6):432-437. doi: 10.11648/j.ajtas.20150406.13

Copy | Download

@article{10.11648/j.ajtas.20150406.13,
  author = {Adepoju Ajibola Akeem and Abubakar Yahaya and Osebekwin Asiribo},
  title = {Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {4},
  number = {6},
  pages = {432-437},
  doi = {10.11648/j.ajtas.20150406.13},
  url = {https://doi.org/10.11648/j.ajtas.20150406.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150406.13},
  abstract = {Multivariate statistical process control (MSPC) is the most acceptable monitoring tool for several variables, and it is advantageous when compare to the simultaneous use of univariate scheme. However, there are some disadvantages in this scheme which include identification of influential variable(s). The Mason, Young and Tracy (MYT) decomposition diagnosis is one of the approaches commonly use to identify the influential variables. This approach aid the breaking down, the overall T square value and show the individual variable contribution, while their joint contributions is also revealed. The challenges of this approach include rigorous derivation of model, computation and complexity more especially when the size of the process characteristics is large. In this research paper we extend the decomposition derivation to five variables. One hundred and twenty (120) models (decomposition partitions) are obtained from the decomposition, revealing the invariance property of the Hotelling’s T square statistic, and eighty (80) unique terms.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Hotelling’s T2 Decomposition: Approach for Five Process Characteristics in a Multivariate Statistical Process Control
AU  - Adepoju Ajibola Akeem
AU  - Abubakar Yahaya
AU  - Osebekwin Asiribo
Y1  - 2015/09/28
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150406.13
DO  - 10.11648/j.ajtas.20150406.13
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 432
EP  - 437
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20150406.13
AB  - Multivariate statistical process control (MSPC) is the most acceptable monitoring tool for several variables, and it is advantageous when compare to the simultaneous use of univariate scheme. However, there are some disadvantages in this scheme which include identification of influential variable(s). The Mason, Young and Tracy (MYT) decomposition diagnosis is one of the approaches commonly use to identify the influential variables. This approach aid the breaking down, the overall T square value and show the individual variable contribution, while their joint contributions is also revealed. The challenges of this approach include rigorous derivation of model, computation and complexity more especially when the size of the process characteristics is large. In this research paper we extend the decomposition derivation to five variables. One hundred and twenty (120) models (decomposition partitions) are obtained from the decomposition, revealing the invariance property of the Hotelling’s T square statistic, and eighty (80) unique terms.
VL  - 4
IS  - 6
ER  -

Copy | Download