The MVPMODEL Procedure
Classical 
Charts
Classical charts are defined as follows. Assume that there are n observations for p variables, denoted by 
, where 
is a p-dimensional vector. The statistic for observation i is
![\[ T^2_ i = \left(\mb {X}_ i - \bar{\mb {X}}_ n \right)^{\prime } \mb {S}^{-1} \left(\mb {X}_ i - \bar{\mb {X}}_ n \right) \]](images/qcug_mvpmodel0010.png){kind=small} |
where
![\[ \begin{array}{lcr} \bar{X}_ j = \frac{1}{n} \sum _{i=1}^ n X_{ij}~ ~ , & \mb {X}_ i = \left[ \begin{array}{c} X_{i1} \\ X_{i2} \\ \vdots \\ X_{ip} \end{array} \right] , & \bar{\mb {X}}_ n = \left[ \begin{array}{c} \bar{X}_{1} \\ \bar{X}_{2} \\ \vdots \\ \bar{X}_{p} \end{array} \right] \end{array} \]](images/qcug_mvpmodel0011.png) |
and
![\[ \mb {S} = \frac{1}{n-1} \sum _{i=1}^ n \left(\mb {X}_ i - \bar{\mb {X}}_ n\right) \left(\mb {X}_ i - \bar{\mb {X}}_ n\right)^{\prime } \]](images/qcug_mvpmodel0012.png){kind=small} |
For purposes of deriving control limits for the chart, it is assumed that has a p-dimensional multivariate normal distribution with mean vector 
and covariance matrix 
for 
. The classical formulation of the chart does not involve a principal component model for the data, and it bases the computation of on the sample covariance matrix 
. See Alt (1985) for theoretical details and the section Multivariate Control Charts for an example.
A classical chart is equivalent to a 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.