### Abstract

A process-oriented basis representation can be used to express multivariate quality vectors as linear combinations of fault patterns, plus a residual. Monitoring the estimated coefficients of the linear relationship is especially useful when the quality vector contains measurements of the same unit at different locations on a part or other types of profile data. To use process-oriented methods for monitoring changes in the mean of the quality vector, one needs to identify whether the effects occur only as special causes or also as common causes of variation. The calculation of process-oriented model coefficients is shown for each case. In general, the coefficients must be computed by weighted least squares, but we show that, in some circumstances, the ordinary least squares estimates are equivalent. In such cases, charting the proposed U^{2} statistic is equivalent to charting a T^{2} statistic computed from the process-oriented coefficients, making the process-oriented statistical process control (SPC) statistic optimal in the sense of being most powerful for detecting mean shifts in the process-oriented space. When there are fewer cause-related patterns than the number of elements in the quality vector, the process-oriented basis is incomplete. In this case, the SPC methods are applied in a subspace of the original quality vector space. For some practical examples, it is shown that the process-oriented basis representation approach yields substantially better average run-length performance compared with the usual T^{2} chart applied to the original quality vectors.

Original language | English (US) |
---|---|

Pages (from-to) | 159-172 |

Number of pages | 14 |

Journal | Journal of Quality Technology |

Volume | 39 |

Issue number | 2 |

State | Published - Mar 2007 |

### Fingerprint

### Keywords

- Control charts
- Multivariate quality control
- Profile monitoring
- SPC
- Statistical process control

### ASJC Scopus subject areas

- Industrial and Manufacturing Engineering
- Statistics and Probability
- Management Science and Operations Research

### Cite this

*Journal of Quality Technology*,

*39*(2), 159-172.

**Optimal monitoring of multivariate data for fault patterns.** / Runger, George; Barton, Russell R.; Del Castillo, Enrique; Woodall, William H.

Research output: Contribution to journal › Article

*Journal of Quality Technology*, vol. 39, no. 2, pp. 159-172.

}

TY - JOUR

T1 - Optimal monitoring of multivariate data for fault patterns

AU - Runger, George

AU - Barton, Russell R.

AU - Del Castillo, Enrique

AU - Woodall, William H.

PY - 2007/3

Y1 - 2007/3

N2 - A process-oriented basis representation can be used to express multivariate quality vectors as linear combinations of fault patterns, plus a residual. Monitoring the estimated coefficients of the linear relationship is especially useful when the quality vector contains measurements of the same unit at different locations on a part or other types of profile data. To use process-oriented methods for monitoring changes in the mean of the quality vector, one needs to identify whether the effects occur only as special causes or also as common causes of variation. The calculation of process-oriented model coefficients is shown for each case. In general, the coefficients must be computed by weighted least squares, but we show that, in some circumstances, the ordinary least squares estimates are equivalent. In such cases, charting the proposed U2 statistic is equivalent to charting a T2 statistic computed from the process-oriented coefficients, making the process-oriented statistical process control (SPC) statistic optimal in the sense of being most powerful for detecting mean shifts in the process-oriented space. When there are fewer cause-related patterns than the number of elements in the quality vector, the process-oriented basis is incomplete. In this case, the SPC methods are applied in a subspace of the original quality vector space. For some practical examples, it is shown that the process-oriented basis representation approach yields substantially better average run-length performance compared with the usual T2 chart applied to the original quality vectors.

AB - A process-oriented basis representation can be used to express multivariate quality vectors as linear combinations of fault patterns, plus a residual. Monitoring the estimated coefficients of the linear relationship is especially useful when the quality vector contains measurements of the same unit at different locations on a part or other types of profile data. To use process-oriented methods for monitoring changes in the mean of the quality vector, one needs to identify whether the effects occur only as special causes or also as common causes of variation. The calculation of process-oriented model coefficients is shown for each case. In general, the coefficients must be computed by weighted least squares, but we show that, in some circumstances, the ordinary least squares estimates are equivalent. In such cases, charting the proposed U2 statistic is equivalent to charting a T2 statistic computed from the process-oriented coefficients, making the process-oriented statistical process control (SPC) statistic optimal in the sense of being most powerful for detecting mean shifts in the process-oriented space. When there are fewer cause-related patterns than the number of elements in the quality vector, the process-oriented basis is incomplete. In this case, the SPC methods are applied in a subspace of the original quality vector space. For some practical examples, it is shown that the process-oriented basis representation approach yields substantially better average run-length performance compared with the usual T2 chart applied to the original quality vectors.

KW - Control charts

KW - Multivariate quality control

KW - Profile monitoring

KW - SPC

KW - Statistical process control

UR - http://www.scopus.com/inward/record.url?scp=34047235199&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34047235199&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:34047235199

VL - 39

SP - 159

EP - 172

JO - Journal of Quality Technology

JF - Journal of Quality Technology

SN - 0022-4065

IS - 2

ER -