Contribution Plots

One way to diagnose the behavior of out-of-control points in multivariate control charts is to use contribution plots (Miller, Swanson, and Heckler, 1998). These plots tell you which variables contribute to the distance between the points in an SPE or $T^2$ chart and the sample mean of the data.

A contribution plot is a bar chart of the contributions of the process variables to the statistic. For the ith SPE statistic, the contribution of the kth variable is the kth entry of the vector $\mb {e}_ i$, which is computed as

\[  \mb {e}_ i = \mb {x}_ i \left( \bI - \bP _ j \bP _ j^{\prime } \right)  \]

where $\mb {e}_ i$ is the vector of errors from the principal component model for observation i and $\mb {x}_ i$ is the ith observation. The contributions to the ith $T^2$ statistic are computed in the same way as the entries of the vector

\[  \mb {T}^2_{i} = \mb {x}_ i \bP _ j \bL ^{-1} \bP _ j^{\prime }  \]

where $\mb {P}_ j$ is the matrix of the first j eigenvectors and $\mb {L}$ is the diagonal matrix of the first j eigenvalues.