Principal Components Method in Control Charts Analysis

Abstract

Shewhart control charts are successfully used to control a multi-parameter technological process, provided there is no correlation between controlled parameters. In this case, the space of dispersion of allowable values of the resulting vector $\overline{{\varvec{x}}}$ components is a hyper-parallelepiped with pairs of opposite sides corresponding to the upper and lower allowable deviations of measured process parameters. If there is a correlation between these parameters, the real area of acceptable scattering is a hyper-ellipsoid, axes of which are inclined with respect to the axes of the hyper-parallelepiped. In this case, the use of Shewhart charts leads to methodological erroneous decisions. Another control tool in the presence of correlation is the Hotelling chart, which can be successfully used to assess the quality of a multidimensional process. However, it should be noted that the Hotelling criterion itself allows assessing the state of the process as a whole, without highlighting the cause of its disorder. The Hotelling chart does not show which indicator directly (or the combined influence of indicators) is associated with a process violation. It is possible to radically solve the problem of controlling a multidimensional process by the use of principal components method. This method is based on applying a linear transformation of the resulting vector, which makes it possible to proceed to an independent analysis for each component without distorting the original relation of correlated data. In addition, the principal component method projects a multidimensional resulting vector into the space of components of a lower dimension. Most often, as practice has shown, the variation of the resulting vector can be explained by only two or three components. This allows to build control charts.

The article describes in detail the method of constructing control charts based on principal components. Evaluation of the effectiveness of the application of the method is carried out on simulated data, which are close to the measurement results obtained during the control of a real technological process. The results show that the proposed method is effective for controlling a multi-parameter technological process in the presence of a correlated parameters.