On-Line Nonlinear Process Monitoring Using Kernel Principal Component Analysis and Neural Network

Principal Components Analysis (PCA) is used to classify the running condition of a machine by means of projecting the original data to the Principal Components space. However, if the data are concentrated in a nonlinear subspace, PCA will fail to work well. Kernel Principal Components Analysis (KPCA) transforms the input data from the original input space into a higher dimensional feature space with the nonlinear mapping, and then uses the nonlinear principal components to realize the classification. In this paper a case of gear fault diagnosis was studied with KPCA. The characteristic values of frequent domain from vibration signals of the gearbox under the running condition were extracted, and the KPCA method was used to classify gear crack fault. The result shows that KPCA is more effective to distinguish the state of the gear and more suitable to diagnose the gear faults in early stage.

Conventional kernel principal component analysis (KPCA) may not function well for nonlinear process monitoring, since the Gaussian assumption of the method may be violated through nonlinear and kernel transformation of the original process data. To overcome this deficiency, a statistical local approach has been incorporated into KPCA. Through this method, a new score variable which was called improved residual in the statistical local approach is constructed. This new variable approximately follows Gaussian distribution, in spite of which distribution the original data follow. Like the traditional method, two statistics can be constructed for process monitoring, with their corresponding confidence limits determined by a $${{\chi }^{2}}$$ distribution. Besides of the improvement made on KPCA, the joint local approach-KPCA method also shows superiority on detection sensitivity, especially for small faults slow changes of the process.KeywordsKernal principal component analysisNonlinear process monitoringNon-GaussianLocal approach

This paper presents a novel algorithm - robust kernel principal component analysis (robust KPCA), on the basis of the research of kernel principal component analysis (KPCA) and robust principal component analysis (RPCA). First, this algorithm sets the radius of the images of the training samples in the feature space using kernel tricks, then determines whether the samples are outliers or not, and finally analyzes the training samples which have eliminated the outliers using KPCA algorithm. The improved KPCA algorithm not only retains the non-linearity property of KPCA algorithm but also gets better robustness. Because the effects of outliers are eliminated, robust KPCA algorithm gets higher accuracy than KPCA algorithm for data analysis. The simulation experiments show that the robust KPCA algorithm developed is better than the KPCA algorithm.

Kernel principal component analysis (KPCA) is a method for performing a nonlinear form of principal component analysis (PCA). It involves nonlinear mapping of the input data onto a high dimensional feature space and performing PCA there. Through KPCA, nonlinear relations in the input space are detected as linear relations in the feature space. Furthermore, explicit computation of the mapping is not necessary in KPCA. Only dot products of the mapped data items in the feature space are required which are obtained directly from the input data using a kernel function. Moreover, control limits of traditional statistical process monitoring indices are usually determined under assumption of normal (Gaussian) distribution. Although, it is recognized that this is inappropriate in nonlinear processes, the impact of using kernel density estimation (KDE), which is a nonparametric method, to determine control limits in KPCA-based process monitoring has not been reported. This paper seeks to bridge this gap. The KPCA with KDE approach is applied to the Tennessee Eastman process for the detection of faults. The results confirm that associating KPCA with kernel density estimated control limits provides better monitoring performance than using control limits based on the normal probability density function.

Principal component analysis (PCA) is a widely used unsupervised method for dimensionality reduction. The kernelized version is called kernel principal component analysis (KPCA), which can capture the nonlinear data structure. KPCA is derived from the Gram matrix, which is not robust when outliers exist in the data. This may yield the principal axis in the feature space deviated by outliers, leading to misinterpretation of the principal components. In this paper, we propose a robust method for KPCA with a reformulation in Euclidean space to construct a robust KPCA method, where an error measurement is introduced into the loss function, and l 2,1 -regularization is added to the loss function. The idea of l 2,1 -regularization of the proposed method is motivated by sparse PCA via variable projection. However, because orthogonality is not satisfied in the proposed method, orthonormal bases are obtained by using the Gram-Schmidt orthonormalization process. In the experiments, a toy example and real data are used for outlier detection to verify the method's performance and effectiveness. In the toy example, the proposed method reduces the influence of outliers and detects more outliers than KPCA. For the real data, the proposed method improves detection in comparison to other existing methods.

Aimed at the non-stationary characteristics of rolling bearing vibration signal,a fault diagnosis method was proposed based on local-wave method and KPCA(kernel principal component analysis)-LSSVM(least squares support vector machine).Firstly,local wave decomposition was used to decompose rolling bearing vibration signal into several intrinsic mode function(IMF),whose feature energy,singular values and AR model parameters were computed as initial feature vectors.Secondly,ini tial feature vectors were mapped into a higher-dimensional space with KPCA,and the kemel principal components were extracted as new feature vectors,which used as the input of LSSVM for fault classification.The experimental results show the KPCA-LSSVM method improves LSSVM's classification performance by KPCA obtaining additional discriminative information,and has better generalization than direct LSSVM method,and can identify rolling bearing fault patterns more accurately.

The identification of between-group differences and changes in gait mechanics are useful for injury prevention. Previous studies suggest the differences in gait biomechanical variables may interact in a complex non-linear fashion rather than a simple linear fashion. A non-linear multivariate analysis technique is therefore necessary to unravel the inherent structure in the gait data. Kernel principal component analysis (KPCA) is an extension of principal component analysis (PCA), the most widely used method in this field, and can provide inside into non-linear relationships of the variables of interest. Despite the growing use of PCA in running gait research, no prior studies have evaluated the utility of KPCA in determining the between-group differences in running gait patterns. Therefore, the objective of this study was to compare the performance of KPCA and PCA in identifying differences between male and female runners and as well between young and older runners in running gait kinematics. Running kinematic data were analysed on a gender group (female, n = 50; male, n = 50) and an age group (young, n = 50; older, n = 50) for discrete variables and waveform data. The following key results were obtained: (1) the performance of the KPCA for the identification of gender and age differences in running gait kinematics increased as compared to using the linear PCA; (2) the Gaussian function often performed better than the polynomial function for these two experiments; (3) there was no consistent optimal value of either the width σ of the Gaussian kernel or the degree d of the polynomial kernel for different data types and experiments. These results suggest that non-linear features extracted by KPCA could be potentially useful in identifying and discriminating between-group differences and changes in running gait patterns, and could provide useful information about the intrinsic non-linear dynamics of running movement.

Data-driven techniques have been receiving considerable attention in the industrial process monitoring field due to their major advantages of easy implementation and less requirement for the prior knowledge and process mechanism. Principal component analysis (PCA) method is known as a popular method for monitoring and fault detection in industrial systems but as it is basically a linear method. However, most practical systems are nonlinear. To make the extension to nonlinear systems, kernel PCA (KPCA) method has been proposed for process modeling and monitoring. We present in this paper an online reduced rank optimized KPCA (RR-KPCA) technique for fault detection in order to extend the advantages of the KPCA models to online processes. Following the fault detection, the identification of the variables correlated to the fault occurred is of great importance. For this purpose, it is proposed to extend the approaches of localization by partial PCA and by elimination in the linear case to the nonlinear case, by exploiting the solution of reduction of the dimension of the kernel matrix in the feature space. The partial RR-KPCA and the elimination sensor identification (ESI-RRKPCA) are generated based on the static RR-KPCA and the online RR-KPCA methods. The idea of these approaches is to generate partial RR-KPCA models with reduced sets of variables. In other words, their goal is to generate indices of fault detection sensitive to certain faults and insensitive to others. The proposed fault isolation methods are applied for monitoring an air quality monitoring network (AIRLOR) data.

Principal component analysis (PCA) has been proven to be an efficient method in dimensionality reduction, feature extraction and pattern recognition. Kernel principal component analysis (KPCA) can be considered as a natural nonlinear generalization of PCA, which performs linear PCA in a high dimensional space implicitly by using kernel trick. However, both conventional PCA and KPCA suffer from the deficiency of being sensitive to outliers. Existing robust KPCA has to eigen-decompose the gram matrix directly in each step and is much more computationally infeasible due to the large size of the matrix when the number of training samples is large. By extending existing robust PCA algorithm using kernel methods, we present a novel robust adaptive algorithm for calculating the kernel principal components. The proposed method not only preserves the characteristic of capturing underlying nonlinear structure of KPCA but also is robust against outliers by restraining the effect of outlying samples. Compared with existing robust KPCA methods, our method is performed without having to store the kernel matrix, which can reduce significantly the storage burden. In addition, our method shows the potential of expansibility to the incremental learning version. Experimental results on synthetic data indicate that our improved algorithm is effective and promising.

The inherent physical characteristics of many real-life phenomena, including biological and physiological aspects, require adapted nonlinear tools. Moreover, the additive nature in some situations involve solutions expressed as positive combinations of data. In this paper, we propose a nonlinear feature extraction method, with a non-negativity constraint. To this end, the kernel principal component analysis is considered to define the most relevant features in the reproducing kernel Hilbert space. These features are the nonlinear principal components with high-order correlations between input variables. A pre-image technique is required to get back to the input space. With a non-negative constraint, we show that one can solve the pre-image problem efficiently, using a simple iterative scheme. Furthermore, the constrained solution contributes to the stability of the algorithm. Experimental results on event-related potentials (ERP) illustrate the efficiency of the proposed method.

Bayesian optimization of multiscale kernel principal component analysis and its application to model Gas-to-liquid (GTL) process data

Kernel principal component analysis (KPCA) has become a popular technique for process monitoring in recent years. However, the performance largely depends on kernel function, yet methods to choose an appropriate kernel function among infinite ones have only been sporadically touched in the research literatures. In this paper, a novel methodology to learn a data-dependent kernel function automatically from specific input data is proposed and the improved kernel principal component analysis is obtained through using the data-dependent kernel function in traditional KPCA. The learning procedure includes two parts: learning a kernel matrix and approximating a kernel function. The kernel matrix is learned via a manifold learning method named maximum variance unfolding (MVU) which considers underlying manifold structure to ensure that principal components are linear in kernel space. Then, a kernel function is approximated via generalized Nystrom formula. The effectiveness of the improved KPCA model is confirmed by a numerical simulation and the Tennessee Eastman (TE) process benchmark.

The article delves into the development of Statistical Fault Detection and Diagnosis Strategies for an Integrated Steel Plant (ISP) taking into account the nonlinear relationship amongst the monitored Process and Feedstock Characteristics. The strategies being devised is based on Neural Network Fitting model cum Principal Component Analysis based technique (NNF-PCA) and Kernel Principal Component Analysis (KPCA) based technique. For detection of fault(s) Hotelling T2 chart based on (Principal Component Analysis) PCA and KPCA scores were employed and an ensemble strategy amalgamating KPCA and Self Organizing Map neural network has also been proposed for the detection of the out-of-control observations or faults. The article also proposes a 2-phase Fault Diagnosis approach christened as Preliminary Diagnosis phase and Specific Diagnosis phase. The Preliminary Diagnosis phase is based on Pattern Analysis of the control chart monitoring statistic observations and the Specific Diagnosis phase is based on the employment of appropriate Fault Diagnostic Statistic. The Preliminary Diagnosis reveals the broader source of assignable cause for the onset of the fault(s) and the Specific Diagnosis reveals the relative contribution of the individual Process and Feedstock characteristics. An in-depth comparative analysis between the NNF-PCA based strategy and KPCA based strategy w.r.t. three comparative aspects and four comparative parameters were carried out with their findings being duly highlighted which revealed the slight effectiveness of the KPCA based strategy with respect to the NNF-PCA based counterpart.

Nonlinear Process Monitoring Using Dynamic Sparse Kernel Classifier

Kernel principal component analysis (KPCA) as a powerful nonlinear feature extraction method has proven as a preprocessing step for classification algorithm. A face recognition approach based on KPCA and genetic algorithms (GAs) is proposed. By the use of the polynomial functions as a kernel function in KPCA, the high order relationships can be utilized and the nonlinear principal components can be obtained. After we obtain the nonlinear principal components, we use GAs to select the optimal feature set for classification. At the recognition stage, we employed linear support vector machines (SVM) as classifier for the recognition tasks. Two face databases were used to test our algorithm and higher recognition rates were obtained which show that our algorithm is effective.