PROC MVPMODEL: Classical T-Square Charts

Classical T-Square Charts

Classical $\text{[math]}$ charts are defined as follows. Assume that there are $\text{[math]}$ observations for $\text{[math]}$ variables, denoted by $\text{[math]}$ , where $\text{[math]}$ is a $\text{[math]}$ -dimensional vector. The $\text{[math]}$ statistic for observation $\text{[math]}$ is

$\text{[math]}$

where

$\text{[math]}$

and

$\text{[math]}$

For purposes of deriving control limits for the $\text{[math]}$ chart, it is assumed that $\text{[math]}$ has a $\text{[math]}$ -dimensional multivariate normal distribution with mean vector $\text{[math]}$ and covariance matrix $\text{[math]}$ for $\text{[math]}$ . The classical formulation of the $\text{[math]}$ chart does not involve a principal components model for the data and bases the computation of the $\text{[math]}$ on the sample covariance matrix $\text{[math]}$ . Refer to Alt (1985) for theoretical details and to the section Multivariate Control Charts for an example.

A classical $\text{[math]}$ chart is equivalent to a $\text{[math]}$ chart based on a full principal components model (with $\text{[math]}$ components) as discussed in the section Relationship of Principal Components to Multivariate Control Charts. See Example 10.2 for more discussion.

Note: This procedure is experimental.