Introduction of a nonlinearity measure for principal component models

Although principal component analysis (PCA) is an important tool in standard multivariate data analysis, little interest has been devoted to assessing whether the underlying relationship within a given variable set can be described by a linear PCA model or whether nonlinear PCA must be utilized. This paper addresses this deficiency by introducing a nonlinearity measure for principal component models. The measure is based on the following two principles: (i) the range of recorded process operation is divided into smaller regions; and (ii) accuracy bounds are determined for the sum of the discarded eigenvalues. If this sum is within the accuracy bounds for each region, the process is assumed to be linear and vice versa. This procedure is automated through the use of cross-validation. Finally, the paper shows the utility of the new nonlinearity measure using two simulation studies and with data from an industrial melter process.     

A sparse PCA for nonlinear fault diagnosis and robust feature discovery of industrial processes

Pearson's correlation measure is only able to model linear dependence between random variables. Hence, conventional principal component analysis (PCA) based on Pearson's correlation measure is not suitable for application to modern industrial processes where process variables are often nonlinearly related. To address this problem, a nonparametric PCA model is proposed based on nonlinear correlation measures, including Spearman's and Kendall tau's rank correlation. These two correlation measures are also less sensitive to outliers comparing to Pearson's correlation, making the proposed PCA a robust feature extraction technique. To reveal meaningful patterns from process data, a generalized iterative deflation method is applied to the robust correlation matrix of the process data to sequentially extract a set of leading sparse pseudoeigenvectors. For online fault diagnosis, the T² and SPE statistics are computed and analyzed with respect to the subspace spanned by the extracted pseudoeigenvectors. The proposed method is applied to two industrial case studies. Its process monitoring performance is demonstrated to be superior to that of the conventional PCA and is comparable to those of Kernel PCA and kernel independent component analysis at a lower computational cost. The proposed PCA is also more robust in sparse feature extraction from contaminated process data. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1494–1513, 2016     

基于核主角的故障检测方法

基于主元分析（PCA）的统计检测方法已经被广泛应用于各种化工过程的故障检测和识别.移动主元分析（moving principal component analysis,简称MPCA）算法基于PCA,根据主元子空间的变化来判断故障是否发生.然而,基于主元分析的统计检测方法是线性方法,无法有效应用于非线性系统.因此,提出一种适合于非线性系统的故障检测方法——基于核主角(kernel principal angle,简称KPA)的故障检测方法,其基本思想与MPCA相似,主要内容包括构建特征子空间和核主角测量两部分.TE过程故障检测仿真实验证明,基于核主角的故障检测方法优于传统的多元统计检测方法（cMSPC）和MPCA.     

Nonlinear dimension reduction based neural modeling for distributed parameter processes

Many chemical processes are nonlinear distributed parameter systems with unknown uncertainties. For this class of infinite-dimensional systems, the low-order model identification from process data is very important in practice. The dimension reduction with a principal component analysis (PCA) is only a linear approximation for nonlinear problem. In this study, a nonlinear dimension reduction based low-order neural model identification approach is proposed for nonlinear distributed parameter processes. First, a nonlinear principal component analysis (NL-PCA) network is designed for the nonlinear dimension reduction, which can transform the high-dimensional spatio-temporal data into a low-dimensional time domain. Then, a neural system can be easily identified to model this low-dimensional temporal data. Finally, the spatio-temporal dynamics can be reproduced using the nonlinear time/space reconstruction. The simulations on a typical nonlinear transport-reaction process show that the proposed approach can achieve a better performance than the linear PCA based modeling approach.     

基于变量子域PCA的故障检测方法

针对工业过程监控中传统主元分析（PCA）方法没有突出局部变量信息的问题,提出一种基于变量子域PCA（variable sub-region PCA,VSR-PCA）的故障检测方法。首先使用PCA将原始数据空间分解成主元子空间（principal component subspace,PCS）和残差子空间（residual subspace,RS）,计算变量与PCS的互信息来度量两者的相关性并以此划分变量子域。然后在变量子域中计算局部T²统计量和局部SPE统计量,并通过贝叶斯推理整合所有子域的信息构造全局统计量,使得在利用所有过程信息的同时挖掘局部变量信息。在连续搅拌反应釜系统上的仿真结果表明,VSR-PCA方法具有更好的过程监控性能。     

MPCA在间歇反应过程故障诊断中的应用

将多方向主元分析(MPCA)理论应用到一个实际的PVC间歇反应过程的性能监测与故障诊断中。由于间歇反应的特点,数据具有多维性,应用传统的主元分析将使过程的统计建模与故障诊断难以实现。MPCA可将间歇过程的多维数据沿时间轨迹分割,使得多批次的数据可以在各时间序列轨迹上建立相应的PCA模型,从而完成对间歇过程的实时监视及故障诊断。     

Nonlinear process monitoring using kernel principal component analysis

In this paper, a new nonlinear process monitoring technique based on kernel principal component analysis (KPCA) is developed. KPCA has emerged in recent years as a promising method for tackling nonlinear systems. KPCA can efficiently compute principal components in high-dimensional feature spaces by means of integral operators and nonlinear kernel functions. The basic idea of KPCA is to first map the input space into a feature space via nonlinear mapping and then to compute the principal components in that feature space. In comparison to other nonlinear principal component analysis (PCA) techniques, KPCA requires only the solution of an eigenvalue problem and does not entail any nonlinear optimization. In addition, the number of principal components need not be specified prior to modeling. In this paper, a simple approach to calculating the squared prediction error (SPE) in the feature space is also suggested. Based on T² and SPE charts in the feature space, KPCA was applied to fault detection in two example systems: a simple multivariate process and the simulation benchmark of the biological wastewater treatment process. The proposed approach effectively captured the nonlinear relationship in the process variables and showed superior process monitoring performance compared to linear PCA.     

基于多尺度分析与主元分析的聚丙烯熔融指数神经元软测量预报模型

Prediction of melt index （MI）, the most important parameter in determining the product＇s grade and quality control of polypropylene produced in practical industrial processes, is studied. A novel soft-sensor model with principal component analysis （PCA）, radial basis function （RBF） networks, and multi-scale analysis （MSA） is proposed to infer the MI of manufactured products from real process variables, where PCA is carried out to select the most relevant process features and to eliminate the correlations of the input variables, MSA is introduced to a~quire much more information and to reduce the uncertainty of the system, and RBF networks are used to characterize the nonlinearity of the process. The research results show that the proposed method provides promising prediction reliability and accuracy, and supposed to have extensive application prospects in propylene polymerization processes.     

基于主元分析和核密度估计的多变量统计过程监控及在工厂聚丙烯催化剂反应器的应用

Abstract Data-driven tools, such as principal component analysis （PCA） and independent component analysis （ICA） have been applied to different benchmarks as process monitoring methods. The difference between the two methods is that the components of PCA are still dependent while ICA has no orthogonality constraint and its latentvariables are independent. Process monitoring with PCA often supposes that process data or principal components is Gaussian distribution. However, this kind of constraint cannot be satisfied by several practical processes. To ex-tend the use of PCA, a nonparametric method is added to PCA to overcome the difficulty, and kernel density estimation （KDE） is rather a good choice. Though ICA is based on non-Gaussian distribution intormation, .KDE can help in the close monitoring of the data. Methods, such as PCA, ICA, PCA.with .KDE（KPCA）, and ICA with KDE,（KICA）, are demonstrated and. compared by applying them to a practical industnal Spheripol craft polypropylene catalyzer reactor instead of a laboratory emulator.     

基于SPA相似系数的故障识别方法

传统的主元分析（PCA）相似系数法没有充分利用数据的高阶统计量等有用的过程信息,导致故障识别效果较差。针对此问题,提出一种统计量模式分析（SPA）相似系数法。该方法首先使用SPA将原始数据转换到统计量空间中,然后在统计量空间中利用PCA获取主元方向,计算主元之间的相似性识别故障。在连续搅拌反应器（CSTR）过程上的仿真结果说明提出的SPA相似系数法比传统的PCA相似系数法能更有效地识别故障。     

Hyperplane distance neighbor clustering based on local discriminant analysis for complex chemical processes monitoring

The collected training data often include both normal and faulty samples for complex chemical processes. However, some monitoring methods, such as partial least squares (PLS), principal component analysis (PCA), independent component analysis (ICA) and Fisher discriminant analysis (FDA), require fault-free data to build the normal operation model. These techniques are applicable after the preliminary step of data clustering is applied. We here propose a novel hyperplane distance neighbor clustering (HDNC) based on the local discriminant analysis (LDA) for chemical process monitoring. First, faulty samples are separated from normal ones using the HDNC method. Then, the optimal subspace for fault detection and classification can be obtained using the LDA approach. The proposed method takes the multimodality within the faulty data into account, and thus improves the capability of process monitoring significantly. The HDNC-LDA monitoring approach is applied to two simulation processes and then compared with the conventional FDA based on the K-nearest neighbor (KNN-FDA) method. The results obtained in two different scenarios demonstrate the superiority of the HDNC-LDA approach in terms of fault detection and classification accuracy.     

独立元分析方法(ICA)及其在化工过程监控和故障诊断中的应用

引　言近年来 ,多元统计过程控制 (multivariatestatisti calprocesscontrol,MSPC)作为一种基于多元统计投影理论的过程性能监控和故障诊断技术受到了学术界和工业界的广泛重视 ,并在化工生产过程中得到了成功应用[1] .MSPC中 ,人们采用主元分析方法(PCA)从过程观测数据中提取统计无关主元 ,通过构造各种信息统计量对过程运行状况进行统计分析 ,判断过程运行是否偏离了正常的操作区域并诊断引起状态偏移的原因 ,其结论成立的前提是要求观测数据服从正态分布[2 ] .然而 ,实际的工业过程数据大都不满足正态分布条件 ,传统的PCA必然导致…     

Deconstructing principal component analysis using a data reconciliation perspective

Data reconciliation (DR) and principal component analysis (PCA) are two popular data analysis techniques in process industries. Data reconciliation is used to obtain accurate and consistent estimates of variables and parameters from erroneous measurements. PCA is primarily used as a method for reducing the dimensionality of high dimensional data and as a preprocessing technique for denoising measurements. These techniques have been developed and deployed independently of each other. The primary purpose of this article is to elucidate the close relationship between these two seemingly disparate techniques. This leads to a unified framework for applying PCA and DR. Further, we show how the two techniques can be deployed together in a collaborative and consistent manner to process data. The framework has been extended to deal with partially measured systems and to incorporate partial knowledge available about the process model.     

Nonlinear dynamic process monitoring based on dynamic kernel PCA

Nonlinear dynamic process monitoring based on dynamic kernel principal component analysis (DKPCA) is proposed. The kernel functions used in kernel PCA (KPCA) are profitable for capturing nonlinear property of processes and the time-lagged data extension is suitable for describing dynamic characteristic of processes. DKPCA enables us to monitor an arbitrary process with severe nonlinearity and (or) dynamics. In this respect, it is a generalized concept of multivariate statistical monitoring approaches. A unified monitoring index combined T² with SPE is also suggested. The proposed monitoring method based on DKPCA is applied to a simulated nonlinear process and a wastewater treatment process. A comparison study of PCA, dynamic PCA, KPCA, and DKPCA is investigated in terms of type I error rate, type II error rate, and detection delay. The monitoring results confirm that the proposed methodology results in the best monitoring performance, i.e., low missing alarms and small detection delay, for all the faults.     

输入训练神经网络的维数约简算法及其在化工过程建模中的应用

Many applications of principal component analysis （PCA） can be found in dimensionality reduction. But linear PCA method is not well suitable for nonlinear chemical processes. A new PCA method based on improved input training neural network （IT-NN） is proposed for the nonlinear system modelling in this paper. Momentum factor and adaptive learning rate are introduced into learning algorithm to improve the training speed of IT-NN. Contrasting to the auto-associative neural network （ANN）, IT-NN has less hidden layers and higher training speed. The effectiveness is illustrated through a comparison of IT-NN with linear PCA and ANN with experiments. Moreover, the IT-NN is combined with RBF neural network （RBF-NN） to model the yields of ethylene and propylene in the naphtha pyrolysis system. From the illustrative example and practical application, IT-NN combined with RBF-NN is an effective method of nonlinear chemical process modelling.     

Local component based principal component analysis model for multimode process monitoring

For plant-wide processes with multiple operating conditions,the multimode feature imposes some chal-lenges to conventional monitoring techniques.Hence,to solve this problem,this paper provides a novel local component based principal component analysis(LCPCA)approach for monitoring the status of a multimode process.In LCPCA,the process prior knowledge of mode division is not required and it purely based on the process data.Firstly,LCPCA divides the processes data into multiple local components using finite Gaussian mixture model mixture(FGMM).Then,calculating the posterior probability is applied to determine each sample belonging to which local component.After that,the local component information(such as mean and standard deviation)is used to standardize each sample of local component.Finally,the standardized samples of each local component are combined to train PCA monitoring model.Based on the PCA monitoring model,two monitoring statistics T2 and SPE are used for monitoring multimode pro-cesses.Through a numerical example and the Tennessee Eastman(TE)process,the monitoring result demonstrates that LCPCA outperformed conventional PCA and LNS-PCA in the fault detection rate.     

Determination of the adequate minimum model complexity required in anaerobic bioprocesses using experimental data

BACKGROUND: Modelling of anaerobic bioprocesses requires adequate conciliation between model accuracy and complexity. Anaerobic digestion is a highly complex biocatalysed process in which organic (waste) materials are converted into biogas. Complex mathematical models of the process can be difficult to manage, while too simplistic models may be insufficient for specific applications. RESULTS: A technique based on principal component analysis (PCA) is presented to assess the most adequate complexity in anaerobic bioprocess models using preliminary data. The technique is applied to experimental data from a continuously operated pilot‐scale anaerobic digestion process treating diluted wine and to process simulation data generated by a highly complex model (ADM1). The results obtained suggest that models incorporating only four reactions can describe most of the experimental data variability. Similarly, a simplification of the highly complex modified ADM1 used to a four‐reaction model could attain equivalent simulation accuracy. A set of possible four‐reaction networks was evaluated using the PCA information and a best candidate network of reactions is obtained. CONCLUSION: The developed PCA‐based tool allows for the assessment of the adequate complexity in mathematical models of anaerobic digestion of ethanol, a process that appears describable with a high level of accuracy by considering only four biological reactions. Copyright © 2008 Society of Chemical Industry     

A local and global statistics pattern analysis method and its application to process fault identification☆

Traditional principal component analysis (PCA) is a second-order method and lacks the ability to provide higher-order representations for data variables. Recently, a statistics pattern analysis (SPA) framework has been incor-porated into PCA model to make full use of various statistics of data variables effectively. However, these methods omit the local information, which is also important for process monitoring and fault diagnosis. In this paper, a local and global statistics pattern analysis (LGSPA) method, which integrates SPA framework and locality pre-serving projections within the PCA, is proposed to utilize various statistics and preserve both local and global in-formation in the observed data. For the purpose of fault detection, two monitoring indices are constructed based on the LGSPA model. In order to identify fault variables, an improved reconstruction based contribution (IRBC) plot based on LGSPA model is proposed to locate fault variables. The RBC of various statistics of original process variables to the monitoring indices is calculated with the proposed RBC method. Based on the calculated RBC of process variables' statistics, a new contribution of process variables is built to locate fault variables. The simula-tion results on a simple six-variable system and a continuous stirred tank reactor system demonstrate that the proposed fault diagnosis method can effectively detect fault and distinguish the fault variables from normal variables.     

Fault isolation in nonlinear systems with structured partial principal component analysis and clustering analysis

Partial principal component analysis (PCA) and parity relations are proven to be useful methods in fault isolation. To overcome the limitation of applying partial PCA to nonlinear problems, a new approach utilizing clustering analysis is proposed. By dividing a partial data set into smaller subsets, one can build more accurate PCA models with fewer principal components, and isolate faults with higher precision. Simulations on a 2 × 2 nonlinear system and the Tennessee Eastman (TE) process show the advantages of using the clustered partial PCA method over other nonlinear approaches.     

Fault identification for process monitoring using kernel principal component analysis

In this research, we develop a new fault identification method for kernel principal component analysis (kernel PCA). Although it has been proved that kernel PCA is superior to linear PCA for fault detection, the fault identification method theoretically derived from the kernel PCA has not been found anywhere. Using the gradient of kernel function, we define two new statistics which represent the contribution of each variable to the monitoring statistics, Hotelling's T²and squared prediction error (SPE) of kernel PCA, respectively. The proposed statistics which have similar concept to contributions in linear PCA are directly derived from the mathematical formulation of kernel PCA and thus they are straightforward to understand. The main contribution of this work is that we firstly suggest a fault identification method especially applicable to process monitoring using kernel PCA. To demonstrate the performance, the proposed method is applied to two simulated processes, one is a simple nonlinear process and the other is a non-isothermal CSTR process. The simulation results show that the proposed method effectively identifies the source of various types of faults.