The MVPMONITOR Procedure

Computing $T^2$ Control Limits

The control limits for the $T^2$ chart are the same for all the $T^2$ statistics on the chart. The control limits are computed based on one of the following distributions:

  • a beta distribution

    \[ T^2_ i \sim \frac{(n-1)^2}{n} B \left( \frac{j}{2}, \frac{n-j-1}{2} \right) \qquad j \geq 2,\, n \geq j+1 \]

  • a $\chi ^2$ distribution

    \[ T^2_ i \sim \chi ^2(j) \qquad j \geq 2,\, n \geq j+1 \]

  • an F distribution

    \[ T^2_ i \sim \frac{j(n+1)(n-1)}{n(n-j)} F( j, n-j ) \qquad j \geq 2,\, n \geq j+1 \]

where i is the observation, j is the number of principal components in the model, and n is the number of observations used to build the principal component model.

The upper control limit is computed as the $(1-\frac{\alpha }{2})$ quantile of the distribution, and the lower control limit is computed as the $\frac{\alpha }{2}$ quantile. You can specify the ALPHA= option in the TSQUARECHART statement to specify $\alpha $.

You can specify the LIMITDIST= option in the TSQUARECHART statement to select the distribution that is used to compute the control limits. A beta distribution is used by default. See Tracy, Young, and Mason (1992) for a discussion of the conditions under which each distribution is applicable.

See the section Computing the $T^2$ and SPE Statistics for details of computing the $T^2$ statistic based on a principal component model.