The SPE chart plots the sum of squares of the residuals from the principal component model. If either or the data matrix has rank less than p, then the SPE statistic is not defined and an SPE chart is not produced. The SPE statistic for observation i is denoted as

where p is the number of variables and is the ith observation for the kth variable in the error matrix, E, in the principal component model

The distribution of has been approximated in the literature under different conditions. Two methods of computing control limits for are implemented by the MVPMONITOR procedure. One method is used when the data that are used to build the principal component model consist of a single measurement per time point. The other method is used when there are multiple measurements per time point (Jensen and Solomon, 1972; Nomikos and MacGregor, 1995).
When there is a single observation at each time point, the data matrix is , with exactly one observation at each time point in the input data set. The derivation of the control limits uses the central limit theorem approach of Jensen and Solomon (1972). They begin by defining , where is the kth eigenvalue from the principal component model.
Then the quantity

is distributed as , where is defined as . The upper control limit for all is then computed by

where is the percentile of the standard normal distribution. The lower control limit is obtained similarly by using . You can specify by using the ALPHA= option in the SPECHART statement.
When there are multiple observations at a time value in an input data set, a different approximation of the SPE distribution is used to compute control limits. The approximate distribution at time i is the scaled chisquare distribution,

where and are the mean and variance, respectively, of the SPE statistics at time i. The upper control limit for all observations at time point i is computed as the percentile of the scaled chisquare distribution:

Similarly the lower control limit is computed from the percentile. You can specify by using the ALPHA= option in the SPECHART statement.
For more information about the distribution approximations, see Nomikos and MacGregor (1995) and Jackson and Mudholkar (1979).