Adaptive process monitoring using efficient recursive PCA and moving window PCA algorithms

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Abstract

In process monitoring, principal component analysis (PCA) is a very popular method and has found wide applications. Conventionally, a fixed PCA model is used for monitoring. This paper presents the use of both recursive PCA (RPCA) and moving window PCA (MWPCA) to online update the PCA model and its corresponding control limits for monitoring statistics. An efficient algorithm is derived based on rank-one matrix update of the covariance matrix, which is tailored for RPCA and MWPCA computations. By the proposed method, the performance of process monitoring can be improved in two aspects. First, more consistent PCA model and control limits for monitoring statistics are resulted because of the increasing number of normal observations for modeling. Second, for parameter-varying processes, when natural drifting behavior or changing of operation region is acceptable, more reasonable PCA model and control limits for monitoring statistics are obtained in an adaptive manner. Simulation results have shown the effectiveness of the proposed approaches compared to the conventional PCA and RPCA methods.

Introduction

Statistical process monitoring (SPM) is a widely used technique for fault diagnosis of chemical processes to improve process quality and productivity. The principal component analysis (PCA) is a most popular method for this purpose (MacGregor and Kourti, 1995, Wise and Gallagher, 1996). The basic strategy of PCA is to discard the noise and collinearity between process variables, while preserving the most important information of the original data set. To use PCA for process monitoring, a PCA model is first established based on collected process data under normal operating. Then, the control limits of monitoring statistics (e.g. T2, Q) are calculated and thus the process can be online monitored by these statistics (Jackson, 1991).

Many successful applications of PCA for process monitoring have been reported in the literature (MacGregor and Kourti, 1995, Raich and Cinar, 1995). Despite its tremendous success, conventional PCA-based monitoring technique has a few major drawbacks. One is that normal operating data may be insufficient when the process monitoring is started. Since the confidence limits of monitoring statistics are obtained from a statistical manner, the number of normal observations would be the more the better. It is thus desirable that the new observation, once found normal, is augmented into the normal data set to modify the confidence limits, making them more consistent. The other drawback is the inability to deal with processes with time-varying parameters, where it interprets the natural changes in the process as fault. The PCA-based monitoring with fixed-model may lead to numerous false alarms. Thus, updating the PCA model to make it more representative of the current process status is necessary.

Constructing the PCA model requires computation of singular value decomposition (SVD) or eigen-decomposition (Jackson, 1991), so the computational load is usually heavy and not practical for online applications. Therefore, a recursive PCA (RPCA) algorithm is preferred to update the PCA model efficiently. Once a new observation becomes available, it offers efficient computation by updating the PCA model using the previous model rather than completely building it from the whole data set. Wold (1994) proposed the use of exponentially weighted moving average (EWMA) filter for updating of PCA and partial least squares (PLS) models. Li et al. (2000) proposed two recursive PCA algorithms for sample-wise and block-wise recursions. Notice that the data on which the PCA model is updated is ever-growing. Such recursive approaches have also been applied in adaptive statistical process control (Choi et al., 2006, He and Yang, 2008, Jin et al., 2006).

However, most industrial processes are time-varying, so that the older samples are not representative of the current process status. Thus, RPCA may be difficult to implement in practice because it leads to a reduction in the speed of model adaptation as the data size increases. Although a forgetting factor can be introduced to down-weight older samples, the selection of this factor is difficult without a priori knowledge of likely conditions (Wang et al., 2005). As a result, recursion with a window sliding along the data, i.e. including the newest sample and excluding the oldest one, is more appropriate for time-varying processes. This adaptation approach is so-call as moving window PCA (MWPCA). Wang et al. (2005) proposed a fast moving window PCA scheme for process monitoring, where only the recursive update of the correlation matrix was presented but the efficient algorithm for updating PCA model was not addressed. Recently, Liu et al. (2009) proposed the moving window kernel PCA for adaptive monitoring of nonlinear process. Although the MWPCA approach can be used for monitoring time-varying processes, it will encounter difficulty in the case of only limited samples available to initialize the monitoring procedure.

Motivated by the difficulties encountered when RPCA or MWPCA is solely implemented for process monitoring, the study reported in this paper focuses on developing a novel adaptive monitoring scheme for time-varying processes by taking advantage of both RPCA and MWPCA. Efficient algorithms for RPCA and MWPCA to online update the PCA model is derived based on rank-one matrix update of the covariance matrix (Erdogus et al., 2004), and these algorithms can be implemented for the sample-wise and moving window recursions. Consequently, a complete system for online adaptive process monitoring, which combines the RPCA and MWPCA algorithms, is proposed.

The paper is organized as follows. In Section 2, the preliminaries about the conventional PCA and its drawbacks are provided. The efficient algorithms to recursively update the PCA model for RPCA and MWPCA are illustrated in Sections 3 Sample-wise recursive PCA algorithm, 4 Moving Window PCA algorithm, respectively. Section 5 presents the complete adaptive process monitoring procedures. The performance of the proposed monitoring scheme is demonstrated through simulation examples in Section 6. Finally, conclusions are drawn in Section 7.

Section snippets

Conventional PCA-based process monitoring

PCA involves the decomposition of a data matrix XN×M, which contains N regular-sampled observations of M process variables and is typically mean-centered, into a transformed subspace of reduced dimension. This subspace is defined by the span of a chosen subset of the eigenvectors of the covariance or correlation matrix associated with X. Each chosen eigenvector, or principal component (PC), captures the maximum amount of variability in the data in an ordered fashion. In other words, the first

Sample-wise recursive PCA algorithm

During the past decades, recursive eigen-decomposition technique has become an interested subject in the field of signal processing. In this study, the recursive eigen-decomposition algorithm based on first-order perturbation (FOP) that first proposed by Champagne (1994) and modified by Erdogus et al. (2004) is applied to estimate the corresponding eigenvalues and eigenvectors recursively. The main advantages of this procedure are computationally efficient and easy to be implemented.

FOP-based

Moving Window PCA algorithm

When slow and natural process changes occur in the processes, it is suitable to update the PCA model by a moving window because the old data cannot represent the current status of the process. That is, the newest sample is augmented to the data matrix and the oldest sample is discarded, keeping a fixed number of samples in the data matrix (i.e. fixed window size).

Let the kth data matrix with window size N be Xk=xkN+1xkN+2xkT, and the next data matrix would be Xk+1=xkN+2xkN+3xk+1T. These

Adaptive process monitoring procedures

With the presented recursive RPCA and MWPCA algorithms, an adaptive process monitoring scheme can be implemented in real-time. A sufficient large number of normal observations necessary to obtain consistent thresholds for monitoring statistics, denoted as Nt, has to be prescribed preliminarily. The selection of Nt can depend on the number of process variable and the range of operation region. When the initial size of normal observations is less than Nt, RPCA is applied for process monitoring.

Simulation examples

The proposed adaptive process monitoring schemes is now applied to simulation examples to demonstrate its effectiveness.

Conclusions

A new adaptive process monitoring method which combines the recursive PCA and moving window PCA has been proposed. The RPCA is used for collecting more normal operating data to obtain a consistent PCA model. On the other hand, the MWPCA can update the PCA model to adapt for normal process changes such as drifting. The number of PCs and the confidence limits for process monitoring are also calculated recursively. Efficient recursive algorithms for both RPCA and MWPCA, which significantly reduce

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