The MVPMODEL Procedure

Classical $T^2$ Charts

Classical $T^2$ charts are defined as follows. Assume that there are n observations for p variables, denoted by $\mb {X}_1,\ldots ,\mb {X}_ n$, where $\mb {X}_ i$ is a p-dimensional vector. The $T^2$ statistic for observation i is

\[  T^2_ i = \left(\mb {X}_ i - \bar{\mb {X}}_ n \right)^{\prime } \mb {S}^{-1} \left(\mb {X}_ i - \bar{\mb {X}}_ n \right)  \]


\[  \begin{array}{lcr} \bar{X}_ j = \frac{1}{n} \sum _{i=1}^ n X_{ij}~ ~ , &  \mb {X}_ i = \left[ \begin{array}{c} X_{i1} \\ X_{i2} \\ \vdots \\ X_{ip} \end{array} \right] , &  \bar{\mb {X}}_ n = \left[ \begin{array}{c} \bar{X}_{1} \\ \bar{X}_{2} \\ \vdots \\ \bar{X}_{p} \end{array} \right] \end{array}  \]


\[  \mb {S} = \frac{1}{n-1} \sum _{i=1}^ n \left(\mb {X}_ i - \bar{\mb {X}}_ n\right) \left(\mb {X}_ i - \bar{\mb {X}}_ n\right)^{\prime }  \]

For purposes of deriving control limits for the $T^2$ chart, it is assumed that $\mb {X}_ i$ has a p-dimensional multivariate normal distribution with mean vector $\bmu =\left(\mu _1,\mu _2,\cdots ,\mu _ p\right)^\prime $ and covariance matrix $\bSigma $ for $i=1,2,\ldots ,n$. The classical formulation of the $T^2$ chart does not involve a principal component model for the data, and it bases the computation of $T^2$ on the sample covariance matrix $\bS $. See Alt (1985) for theoretical details and the section Multivariate Control Charts for an example.

A classical $T^2$ chart is equivalent to a $T^2$ chart based on a full principal component model (with p components), as discussed in the section Relationship of Principal Components to Multivariate Control Charts. See Example 12.2 for more information.