Using principal component analysis in process performance for multivariate data
Introduction
The capability indices, Cp, Cpk, and Cpm, are widely used to evaluate process performance based on a single engineering specification [1], [2], [3], [4]. However, process monitoring problems with several related variables of interest, such as automatic inspection procedure which makes it relatively easy to measure many parameters on each unit of product manufactured [3], are particularly important today. That is, the capability analysis usually involves more than one engineering specification and the multivariate statistical technique can be used to analyze related quality characteristics. Kotz and Johnson [4] reviewed the multivariate process capability indices thoroughly for assessing multivariate process data. The multivariate process capability indices can be obtained from:
- 1.
The ratio of a tolerance region to a process region. This approach is similar to the conventional univariate capability index such as Cp=(USL−LSL)/6σ. The detailed discussion can be found in [2], [4], [5].
- 2.
The probability of the nonconforming product. The function of the multivariate probability distribution is used to compute the probability of the nonconforming product [7], [8].
- 3.
Other approaches consisting of loss functions [2], and vector representations [9], [10].
Section snippets
Applying principal component analysis
This section discusses the application of principal component analysis and the procedure for component extraction. Assume that X is a m×n sample data matrix, where m denotes the number of product quality characteristics observed from a part and n represents the number of parts being measured. Also, X̄ represents the sample mean of the observations, which is an m-vector value, and S, a nonsingular m×m symmetric matrix, represents the covariance between observations. Engineering specifications
Multivariate capability index
Once the quality measurements have been obtained, multivariate normality should be examined prior to applying the capability index analysis. Mardia et al. [17] showed that the normality of multivariate data can be validated using a univariate analog. A function, MVMMT, of International Mathematics and Statistics Language (IMSL) computes Mardia’s multivariate measurements for p values of the multivariate skewness and kurtosis [18]. These measurements are then used to examine multivariate
Demonstration
In order to demonstrate the proposed methodology, the data from the literature or real-world cases is used. Five examples were used to illustrate the implementation of two, three, and six variables in which they were either multivariate normal or non-normal data.
Example 1: process data was collected from a fabric production line. The basic weight (BW) and the loss on ignition (LOI) of the product were the monitored attributes. Nineteen observations were taken. The specification limits for both
Conclusions
Process capability is calculated from the collected data of a process, and can be used to evaluate the process performance. The process capability can be depicted by three indices, Cp, Cpk, and Cpm. Currently, these process capability indices can only be applied to univariate data. However, most of time, the product quality has to be measured in several characteristics. That is, it is common to deal with multivariate data while measuring process performance. When these variables are correlated
Acknowledgements
The authors wish to gratefully acknowledge the referees of this paper who helped to clarify and improve the presentation. We would like to thank Professor Norma F. Hubele and Dr F. P. Lawrence for their many helpful suggestions pertaining to the content and presentation of this study.
References (21)
Process capability indices
Journal of Quality Technology
(1986)- et al.
Distributional and inferential properties of process capability indices
Journal of Quality Technology
(1992) Introduction to statistical quality control
(1996)- et al.
Process capability indices
(1993) - et al.
A multivariate measure of process capability
Journal of Modeling and Simulation
(1991) - et al.
A note on multivariate capability indices
Journal of Applied Statistics
(1993) - Wierda SJ. A multivariate process capability index. ASQC Quality Congress Transactions...
A multivariate process capability index over a rectangular solid tolerance zone
Statistica Sinica
(1994)- et al.
A multivariate process capability vector
- Shahriari H. A contribution to multivariate statistical process control. Unpublished Ph.D. dissertation. Arizona State...
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