Weighted Reconstruction-Based Contribution for Improved Fault Diagnosis Haipeng Xu, † Fan Yang, † Hao Ye,* ,† Weichang Li, ‡ Peng Xu, ‡ and Adam K. Usadi ‡ † Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Automation, Tsinghua University, Beijing 100084, China ‡ Corporate Strategic Research, ExxonMobil Research and Engineering Company, 1545 Route 22 East, Annandale, New Jersey 08801, United States ABSTRACT: In this article, a new fault diagnosis method is proposed based on weighted reconstruction-based contribution (WRBC) analysis. The new method reduces fault smearing and, therefore, improves diagnosis accuracy. The current RBC analysis finds the sensor directions and amplitudes that, if removed, would minimize certain quadratic fault indices. However, the estimate of the RBC coefficient along a certain fault direction is typically subject to contamination by the effects of other directions. This is because the RBC kernel matrices reside in a subspace onto which the projections of data from various orthogonal fault directions become collinear, which leads to smearing of the corresponding fault coefficient estimates. This is especially the case for faults involving several sensor variables. Motivated by this observation, we propose to first filter the test data by a set of orthogonalized fault directions and then perform the reconstruction-based contribution calculation. This weighted RBC analysis method can reduce fault coefficient smearing and, therefore, adaptively improve diagnosis accuracy. The filter coefficient allows a flexible tradeoff between fault smearing across fault directions and bias contribution from normal data. These results are demonstrated through numerical experiments with both single-sensor and multisensor faults. 1. INTRODUCTION Multivariate statistical fault detection and diagnosis have been extensively studied in industrial process monitoring applications. Principal component analysis (PCA) and its variants constitute one of the most prevailing classes of methods. 1-4 PCA-based techniques typically involve the partitioning of data into a principal and residual subspaces. Two fault detection indices, namely, the T 2 statistic and the squared prediction error (SPE), are then derived from data projection onto each of these two subspaces. SPE measures the amount of faulty variations from the residual subspace, whereas the T 2 statistic captures the faulty variations from the principal subspace. Fault detection decisions are made by comparing the value of the SPE and/or T 2 indices against certain threshold levels. However, both the SPE and T 2 are fault detection indices; neither explicitly provides any diagnosis information. A number of fault diagnosis techniques have been reported in the literature. In a recent review, Alcala and Qin 5 summarized various types of fault diagnosis methods, including the genera- lized contribution plot and the reconstruction-based contribu- tion analysis. The generalized contribution plot analysis computes either complete or partial data space decomposition to derive sensor directions that make dominant contributions to the computed fault indices. Reconstruction-based contribu- tion (RBC) analysis, recently proposed by Alcala and Qin, 6 finds sensor fault directions and amplitudes that, if removed, would minimize the associated fault detection index value. A reconstruction-based fault diagnosis method was applied to continuous processes by Li et al. 7 Alcala and Qin 8 also combined kernel PCA and RBC for both fault detection and diagnosis. Furthermore, output-relevant fault diagnosis based on RBC was developed by Li et al. 9 with the Tennessee Eastman Process as an example. Despite these differences, a common assumption in both types of methods is that variables with larger contributions to the fault detection indices are more likely to be the true faulty variables. Therefore, accurate estimations of the contribution magnitudes along a set of hypothesized fault direction becomes essential in both cases. As pointed out by Alcala and Qin, 6 general contribution-plot-based methods often lead to smeared estimates of fault magnitudes, even in the case of a single-sensor fault. In this article, an example is used first to show that RBC can still lead to fault amplitude smearing, especially with multisensor faults. We then point out that this smearing effect is intrinsically related to data projection onto the kernel matrix in the cor- responding fault index. Simply put, when viewed from the subspace, the components of two orthogonal fault directions would become collinear and not entirely differentiable. Then, an improved fault diagnosis method is proposed by introducing a weighting matrix into the RBC analysis; this new method is called the weighted reconstruction-based contribution (WRBC) analysis. By introducing a weighting matrix and working along a set of eigen-fault directions, we demonstrate that, when transformed back to the sensor-wise coordinate system, the fault coefficients can be estimated with significantly reduced smearing. The rest of this article is organized as follows: Section 2 provides the problem formulation and a brief review of the PCA- based fault detection method. In section 3, after introducing the notations of reconstruction-based contribution analysis, we point out the limitations of the existing RBC method. In section 4, the Received: March 14, 2012 Revised: June 13, 2013 Accepted: June 19, 2013 Published: June 19, 2013 Article pubs.acs.org/IECR © 2013 American Chemical Society 9858 dx.doi.org/10.1021/ie300679e | Ind. Eng. Chem. Res. 2013, 52, 9858-9870