Specialized Control Charts

Multivariate Control Charts

See SHWT2 in the SAS/QC Sample Library

In many industrial applications, the output of a process characterized by p variables that are measured simultaneously. Independent variables can be charted individually, but if the variables are correlated, a multivariate chart is needed to determine whether the process is in control.

Many types of multivariate control charts have been proposed; refer to Alt (1985) for an overview. Denote the i^th measurement on the j^th variable as X_ij for i = 1,2, ... ,n, where n is the number of measurements, and j = 1,2, ... ,p. Standard practice is to construct a chart for a statistic T²_i of the form

$T^2_i = (X_i - {\bar{X}}_n)^' S^{-1}_n (X_i - {\bar{X}}_n)$

where

${\bar{X}}_j = \frac{1}n \sum_{i=1}^n X_{ij}, & X_i = [ X_{i1} \ X_{i2} \ \vdots... ..., & {\bar{X}}_n = [ {\bar{X}}_{1} \ {\bar{X}}_{2} \ \vdots \ {\bar{X}}_{p} ]$

and

$S_n = \frac{1}{n-1} \sum_{i=1}^n (X_i - {\bar{X}}_n) (X_i - {\bar{X}}_n)^'$

It is assumed that X_i has a p-dimensional multivariate normal distribution with mean vector ${\mu} = (\mu_1 \mu_2 ... \mu_p)^'$ and covariance matrix ${\Sigma}$ for i = 1,2, ... ,n. Depending on the assumptions made about the parameters, a $\chi^2$ , Hotelling T², or beta distribution is used for T²_i, and the percentiles of this distribution yield the control limits for the multivariate chart.

In this example, a multivariate control chart is constructed using a beta distribution for T²_i. The beta distribution is appropriate when the data are individual measurements (rather than subgrouped measurements) and when ${\mu}$ and ${\Sigma}$ are estimated from the data being charted. In other words, this example illustrates a start-up phase chart where the control limits are determined from the data being charted.

Calculating the Chart Statistic

Examining the Principal Component Contributions

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