Improving the sensitivity of statistical process monitoring of manifolds embedded in high-dimensional spaces: The truncated-Q statistic

We address the limitations of applying the well-known Q/SPE statistic for statistical process monitoring (SPM) on manifolds embedded in high-dimensional systems. Our analysis considers scenarios where the number of samples used for deriving the control limits during Phase I analysis is small regarding the number of variables to monitor, which is the prevalent situation in modern industry. After diving into the root causes of some problems associated with the implementation of SPM in large-scale processes, we propose an alternative residual statistic, called the truncated-Q statistic (Q Delta) and the associated control limit. The sensitivity of the new statistic for sensor faults is analyzed through a Monte Carlo study involving processes of different dimensionalities and correlation structures, under different testing conditions. A significant improvement in detection sensitivity is observed, especially when the dimensionality of the process is high and the faults have low magnitudes and only affect a few sensors. The practical relevance of this finding is discussed in the paper and further corroborated with a real case study from an industrial Surface Mount Technology (STM) process. The truncated-Q statistic, whose code is made available, can be readily extended to linear and non-linear manifolds embedded in high-dimensional systems.