The optimal CUSUM control chart with a dynamic non-random control limit and a given sampling strategy for small samples sequence.

This article proposes a performance measure to evaluate the detection performance of a control chart with a given sampling strategy for finite or small samples sequence and prove that the CUSUM control chart with dynamic non-random control limit and a given sampling strategy can be optimal under the measure. Numerical simulations and real data for an earthquake are provided to illustrate that for different sampling strategies, the CUSUM chart will have different monitoring performance in change-point detection. Among the six sampling strategies that take only a part of samples, the numerical comparing results illustrate that the uniform sampling strategy (uniformly dispersed sampling strategy) has the best monitoring effect.

Knockoff procedure for false discovery rate control in high-dimensional data streams.

Motivated by applications to root-cause identification of faults in high-dimensional data streams that may have very limited samples after faults are detected, we consider multiple testing in models for multivariate statistical process control (SPC). With quick fault detection, only small portion of data streams being out-of-control (OC) can be assumed. It is a long standing problem to identify those OC data streams while controlling the number of false discoveries. It is challenging due to the limited number of OC samples after the termination of the process when faults are detected. Although several false discovery rate (FDR) controlling methods have been proposed, people may prefer other methods for quick detection. With a recently developed method called Knockoff filtering, we propose a knockoff procedure that can combine with other fault detection methods in the sense that the knockoff procedure does not change the stopping time, but may identify another set of faults to control FDR. A theorem for the FDR control of the proposed procedure is provided. Simulation studies show that the proposed procedure can control FDR while maintaining high power. We also illustrate the performance in an application to semiconductor manufacturing processes that motivated this development.

Directional PCA for Fast Detection and Accurate Diagnosis: A Unified Framework.

Many methods for monitoring multivariate processes are built on principal component analysis (PCA), which, however, simply tells whether the process is faulty or not. In fact, there is still room for the improvement of the early detection performance by exploiting fully the information given by fault directions. To this end, this article proposes a novel directional PCA (diPCA) approach. First, by narrowing down faults to a specified direction or composite mutually orthogonal directions, diPCA can speed fault detection and facilitate accurate fault diagnosis. It also has good theoretical properties that guarantee concise control limits. Second, with appropriate fault directions, diPCA provides a unified framework for process monitoring and includes existing monitoring indices, such as Hotelling's T2 and the squared prediction error (SPE), as special cases. Third, diPCA also naturally results in a new combined monitoring statistic, which is composed of both T2 and SPE, and provides an optimal ratio of their combination. The Monte Carlo simulation results have demonstrated the power of the proposed monitoring and diagnostic methods stemming from diPCA. The proposed methods have also been implemented into the Tennessee Eastman process.