# Multivariate Statistical Process Control Using Enhanced Bottleneck Neural Network
## 1 Introduction  
Recently, increasing sensor availability in process monitoring has led to higher demands on the ability to early detect and identify any sensor faults, especially when the monitoring procedure is based on the information obtained from many sensors. Thus, monitoring or control using accurate measurements is very useful to enhance process performance and improve production quality. In the literature, multivariate statistical tools have been widely used to enhance process monitoring [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. Among the popular multivariate statistical process control, the traditional principal component analysis (PCA) and the independent component analysis (ICA) [10; 12; 13] are the most early and major methods often used in monitoring, which serve as reference of the desired process behaviour and against which new data can be compared ([PERSON] and [PERSON], 2007). However, recent modificationof them, such as multiscale PCA, robust NLPCA, kernel PCA, and ICA-PCA combination are used in the literature ([PERSON], 1998; [PERSON] et al., 2007, [PERSON] and [PERSON], 2005). Generally, traditional industrial process monitoring relies on the Gaussian assumption of mean \(\mu\) and variance \(\sigma\) which leads to obtaining a confidence limit as constant threshold. However, in practice, the process variables follow approximately mixture Gaussian distributions \((\mu_{j},\sigma_{j})\) due to process nonlinearity, which gives a multimodal behaviour; therefore, an adaptive confidence limit (ACL) is expected to improve the process performance. In this context, we propose a robust process monitoring strategy based on Gaussian mixture model (GMM) to extract multiple normal operating modes characterized by \(m\) Gaussian components \((\mu_{j},\sigma_{j})\) during normal conditions ([PERSON] et al., 2015; [PERSON] et al., 2015). In this work, an enhanced bottleneck neural network (EBNN) is used to estimate the monitored measurements and to classify them according to their probability rates of belonging to various operating modes. Compared to the classical auto-associative neural networks (AANNs) that can be used only for mapping inputs to outputs, the proposed EBNN is not limited to mapping the outputs from input dataset, but also provides a supervised classification of the monitored variables. Then, the kernel density estimation has been adopted to calculate the local confidence limits of the various operating modes, ([PERSON] et al., 2015; [PERSON] et al., 2015; [PERSON] et al., 2015). Subsequently, an ACL is obtained through a weighted sum of the probability rates of each normal mode estimated by the \(m\) neurons of the EBNN output layer dedicated to classification and the \(m\) Local Confidence Limits \(LCL_{j}^{(KDE)}\) of each local \(SPEj\). The obtained ACL is associated with the global \(SPE(k)\), which is able to significantly improve the monitoring performance, reduce the false alarms, and detect the onset of process deviations earlier compared to other traditional methods. Traditional contribution plots to the squared prediction error (SPE) are used to isolate the defective sensor. In order to overcome the drawback of the contribution plots, we have successfully proposed a variant of the sensor validity index (SVI) method, which is based on the elimination of the fault on the SPE by reconstructing the faulty variable, the Figure 1 shows a monitoring cycle of the proposed strategy. The paper is organised as follows: Section 2 introduces the multivariate monitoring tools, including the Gaussian mixture model for calculating the probability rates of normal operating modes, which are used as targets to train the EBNN. Section 3 is dedicated to the case study, in which the effectiveness of the proposed process monitoring strategy is investigated, where we discuss and assess the different results obtained from the used process monitoring tools which are tested on benchmark simulation model No.