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Nowadays advanced sensing technology enables real-time data collection of key variables during manufacturing, known as multi-channel profiles. These data facilitate in-process monitoring and anomaly detection, which have been extensively studied in recent years. However, most studies treat each profile as a whole, e.g., a high-dimensional vector or function, and construct monitoring schemes accordingly. As a result, these methods cannot be implemented until the entire profile has been obtained, leading to long detection delay especially if anomalies occur in early sensing points of the profile. In addition, they require that profiles of different samples have the same time length and feature location, yet additional time-warping operation for real misaligned samples may weaken the anomaly patterns. To address these problems, this article proposes an in-profile monitoring (INPOM) control chart, which not only gives the feasibility of detecting anomalies inside the profile, but also can handle the misalignment problem of different samples. In particular, our INPOM scheme is built upon state space model (SSM). To better describe the clustered between-profile correlation and avoid overfitting, SSM is extended to a regularized SSM (RSSM), where regularizations are imposed as prior information and expectation maximization algorithm is integrated for posterior maximization to efficiently learn the model parameters. Furthermore, a monitoring statistic based on one-step-ahead prediction error of RSSM is constructed for INPOM control chart. Thorough numerical studies and real case studies demonstrate the effectiveness and applicability of our proposed RSSM-INPOM framework.
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