{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T09:15:08Z","timestamp":1783070108799,"version":"3.54.6"},"reference-count":30,"publisher":"MDPI AG","issue":"13","license":[{"start":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:00:00Z","timestamp":1783036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Youth Top Talent Project of Beijing Polytechnic University","award":["2023R003-JFQB"],"award-info":[{"award-number":["2023R003-JFQB"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Sensor-based process monitoring often involves both process variables and quality-related variables. These variables are usually used together to detect faults and to evaluate their effects on process quality or performance. However, most existing monitoring methods still rely on squared reconstruction residuals. This treatment assumes a fixed residual structure and may be insufficient for nonlinear industrial processes. In practice, residual variances may vary with operating conditions. Residuals from different variables may also be correlated. To address this problem, this paper proposes a probabilistic residual modeling method for process\u2013quality fault detection. The method retains the latent-variable structure of deep variational canonical correlation analysis. It further introduces conditional residual distributions for the process side and the quality side. These distributions are parameterized by the latent operating state inferred from sensor measurements. Residual negative log-likelihoods are then used as monitoring statistics. In this way, residual abnormality is evaluated under the current operating condition. The proposed method is verified on a three-phase flow facility and a continuous stirred tank reactor process. Compared with PLS, CCA, DCCA, and DVCCA, the proposed method improves the detection of process-side disturbances and provides clearer separation between process-related and quality-related abnormal responses. Quantitatively, in the TPFF air line blockage case, the process-side statistic Jx achieved an FDR of 82.02% with an FAR of 1.52%, compared with an FDR of 52.02% obtained by the corresponding DVCCA statistic SPEx. In the TPFF open direct bypass case, Jx and Jy achieved FDRs of 90.87% and 91.07%, respectively, with FARs of 0.00%. In the CSTR coolant-temperature sensor-bias case, Jx achieved an FDR of 88.29% with an FAR of 0.00%, while Jy remained below its control limit, supporting process\u2013quality fault discrimination.<\/jats:p>","DOI":"10.3390\/s26134201","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T07:59:47Z","timestamp":1783065587000},"page":"4201","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Probabilistic Residual Modeling for Sensor-Based Process\u2013Quality Fault Detection in Industrial Systems"],"prefix":"10.3390","volume":"26","author":[{"given":"Lirong","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Aeronautical Engineering, Beijing Polytechnic University, No.9, Liangshuihe 1st Street, Beijing Economic and Development Zone, Beijing 100176, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xianwen","family":"Bao","sequence":"additional","affiliation":[{"name":"School of Aeronautical Engineering, Beijing Polytechnic University, No.9, Liangshuihe 1st Street, Beijing Economic and Development Zone, Beijing 100176, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6418","DOI":"10.1109\/TIE.2014.2301773","article-title":"A review on basic data-driven approaches for industrial process monitoring","volume":"61","author":"Yin","year":"2014","journal-title":"IEEE Trans. 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