Differentiating between expanded and fault conditions using principal component analysis

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.

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.

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.

Process monitoring is an integral part of chemical process, required higher product quality and safety operation. Therefore, the objective of this paper is to ensure the suitable functioning and to improve the fault detection performance of conventional kernel Principal Components Analysis (KPCA). Thus, an online Reduced Rank KPCA (OnRR-KPCA) with adaptive model has been developed to monitor a dynamic nonlinear process. The developed method is proposed. Firstly, to extract the useful observations, from large amount of training data registered in normal operating conditions, in order to construct the reduced reference model. Secondly, to monitor the process online and update the reference model if a new useful observation is available and satisfies the condition of independencies between variables in feature space. To demonstrate the effectiveness of the OnRR-KPCA with adaptive model over the conventional KPCA and the RR-KPCA, the fault detection performances are illustrated through two examples: one using synthetic data, the second using a simulated Tennessee Eastman Process (TEP) 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.

This paper presents a novel Gabor-based kernel Principal Component Analysis (PCA) method by integrating the Gabor wavelet representation of face images and the kernel PCA method for face recognition. Gabor wavelets first derive desirable facial features characterized by spatial frequency, spatial locality, and orientation selectivity to cope with the variations due to illumination and facial expression changes. The kernel PCA method is then extended to include fractional power polynomial models for enhanced face recognition performance. A fractional power polynomial, however, does not necessarily define a kernel function, as it might not define a positive semidefinite Gram matrix. Note that the sigmoid kernels, one of the three classes of widely used kernel functions (polynomial kernels, Gaussian kernels, and sigmoid kernels), do not actually define a positive semidefinite Gram matrix either. Nevertheless, the sigmoid kernels have been successfully used in practice, such as in building support vector machines. In order to derive real kernel PCA features, we apply only those kernel PCA eigenvectors that are associated with positive eigenvalues. The feasibility of the Gabor-based kernel PCA method with fractional power polynomial models has been successfully tested on both frontal and pose-angled face recognition, using two data sets from the FERET database and the CMU PIE database, respectively. The FERET data set contains 600 frontal face images of 200 subjects, while the PIE data set consists of 680 images across five poses (left and right profiles, left and right half profiles, and frontal view) with two different facial expressions (neutral and smiling) of 68 subjects. The effectiveness of the Gabor-based kernel PCA method with fractional power polynomial models is shown in terms of both absolute performance indices and comparative performance against the PCA method, the kernel PCA method with polynomial kernels, the kernel PCA method with fractional power polynomial models, the Gabor wavelet-based PCA method, and the Gabor wavelet-based kernel PCA method with polynomial kernels.

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.

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.

Subspace methods that utilize principal component analysis (PCA) are widely used for pattern classification or detection problems. Kernel PCA (KPCA) that is an extension of PCA is also applied to subspace methods. However, its computational cost is very high since the computational cost mainly depends on the number of samples in kernel methods. Recently, subset KPCA (SKPCA) has been proposed in order to reduce its computational complexity. In this paper, we apply SKPCA to subspace methods, and compare SKPCA with KPCA using some sample selection methods. Experimental results demonstrate advantages of subspace methods using SKPCA.

Automatic ECG beat classification plays an important role in detecting cardiac disease. In this paper, we propose an automatic recognition model for ECG signals based on discrete wavelet transform (DWT), principal component analysis (PCA), kernel principal component analysis (KPCA), and adaptive kernel principal component analysis (AKPCA). We extracted different ECG features using DWT, PCA, KPCA, and AKPCA, respectively. These features were combined and used as support vector machine (SVM) input to classify the ECG. ECG records taken from the MIT-BIH arrhythmia database are selected to test the proposed method. The following five heartbeat types were classified using this method: normal beats (N), premature ventricular beats (V), right bundle branch block beats (R), left bundle branch block beats (L), and premature atrial beats (A). The sensitivity, accuracy, precision, and specificity reached 99.95%, 99.86%, 99.53%, and 99.70%, respectively. These results indicate the proposed method is reliable and efficient for ECG beat classification.

Mode changes in today's sophisticated industrial processes result in unanticipated shutdowns, which could reduce the lifespan of critical machinery and result in high maintenance costs. A technique that can identify the fault of equipment operating in various modes is needed to prevent this issue. In light of this, we suggest a novel fault detection technique that makes use of the k-nearest neighbor normalization-based weight local outlier factor (WLOF). The suggested approach carries out local normalization using neighbors to take into account potential mode changes in the normal data, and WLOF is used for fault detection. In contrast to statistical methods, such as principal component analysis (PCA) and independent component analysis (ICA), the local outlier factor (LOF) uses the density of neighbors. However, conventional LOF is greatly affected by the distance between its neighbors, performance may deteriorate when faulty data is adjacent to normal data. To improve the defect detection performance of LOF, the proposed method multiplies proportionally by the distance between each neighbor. A circulating fluidized bed boiler and a multimode numerical case were used to assess the effectiveness of the suggested method. The results of the experiments demonstrate that the suggested method performs better than conventional PCA, kernel PCA (KPCA), k-nearest neighbor (kNN), and LOF. In particular, compared to conventional techniques, the suggested method increased detection accuracy by 20%. Therefore, the proposed technique is applicable to a real process operating in multiple modes.

Summary The principal component analysis (PCA) is exists the problem of lack of probability model and the absence of higher-order statistics information. Therefore，in order to overcome its shortcomings，this paper studies the method that can overcome two disadvantages of the principal component analysis (PCA) — the probability kernel principal component analysis (PKPCA) which is based on Bayesian theory and kernel principal component analysis (KPCA). First，the sample data are mapped to the high dimensional feature space，then define probability model of the data in high-dimensional space，and finally，expectation maximization (EM) estimated is used to get the best results. This method has both the advantage of probability analysis and kernel principal component analysis (KPCA). It is able to effectively adapt to more complex reservoir conditions and can realize the non-linear probability analysis. The probability of kernel principal component analysis (PKPCA) is applied to reservoir prediction of the oilfield. The predicted results show that the method can improve the precision attribute optimization，while improving the accuracy of the forecasts of reservoir.

This paper presents a two-stage denoising method for hyper-spectral image (HSI) by combining kernel principal component analysis (KPCA) and total variation (TV). In the first stage, we use KPCA denoising to reduce spectrally uncorre-lated noise. In the second stage, the information content is largely separated from the remaining noise by means of principal component analysis (PCA). The remaining noise is then efficiently removed by fast primal-dual TV denoising in low-energy PCA channels. Experimental results on simulated and real HSIs are very encouraging.

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.