Kernel Principal Component Analysis for Identification of Between-Group Differences and Changes in Running Gait Patterns

An SVM-like framework provides a novel way to learn linear principal component analysis (PCA). Actually it is a weighted PCA and leads to a semi-definite optimization problem (SDP). In this paper, we learn linear and nonlinear PCA with linear programming problems, which are easy to be solved and can obtain the unique global solution. Moreover, two algorithms for learning linear and nonlinear PCA are constructed, and all principal components can be obtained. To verify the performance of the proposed method, a series of experiments on artificial datasets and UCI benchmark datasets are accomplished. Simulation results demonstrate that the proposed method can compete with or outperform the standard PCA and kernel PCA (KPCA) in generalization ability but with much less memory and time consuming.

In kernel methods, choosing a suitable kernel is in dispensable for favorable results. No well-founded methods, however, have been established in general for unsupervised learning. We focus on kernel Princ ipal Component Analysis (kernel PCA), which is a nonlinear extension of principal component analysis and ha s been used electively for extracting nonlinear featu res and reducing dimensionality. As a kernel method , kernel PCA also suffers from the problem of kernel choice. Although cross-validation is a popular meth od for choosing hyperparameters, it is not applicable straightforwardly to choose a kernel in kernel PCA because of the incomparable norms given by differen t kernels. It is important, thus, to develop a well founded method for choosing a kernel in kernel PCA. This study proposes a method for choosing hyperparameters in kernel PCA (kernel and the number of components) based on cross-validation for the comparable reconstruction errors of pre-images in t he original space. The experimental results on synthesized and real-world datasets demonstrate tha t the proposed method successfully selects an appropriate kernel and the number of components in kernel PCA in terms of visualization and classification errors on the principal components. The results imply that the proposed method enables automatic design of hyperparameters in kernel PCA.

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.

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.

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 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.

A kernel based nonlinear subspace projection (KNSP) method is proposed for reduction of hyperspectral image dimensionality. This method involves three steps: subspace partition of full data space, feature extraction based on kernel principal component analysis (KPCA) in subspace and feature selection based on class separability criterion. The main merit of the proposed method is that it is more suitable for feature extraction than linear principal component analysis (PCA) and segmented principal component: transformation (SPCT), in particular, when hyperspectral data have nonlinear characteristics. In order to testify the effectiveness of the KNSP method for reduction of hyperspectral image dimensionality, hyperspectral image classification is performed on AVIRIS data. Experimental results show that when the hyperspectral dimensionality is reduced to a few features, the average classification accuracy of the new method is higher than those of PCA and SPCT methods.

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.

Besides the performance and range requirements, the breakthrough of electromobility depends crucially on the safety of battery systems. Cell faults that can lead to thermal runaway of the energy storage reduce customer acceptance. Thermal runaways are often preceded by an Internal Short Circuit (ISC). Thus, there is a necessity for a method that is able to detect the formation of ISCs, before a significant temperature rise is observed and the risk of thermal propagation arises. To achieve this, a data-driven method that enables early detection of soft ISCs is presented. The unsupervised machine-learning method is based on linear Principal Component Analysis (PCA) and nonlinear Kernel PCA (KPCA). Since the method only requires fault-free voltage measurements for training, it is directly applicable in conventional battery systems. The nonlinear KPCA is thoroughly compared with the linear PCA using experimental data. The data originates from a module consisting of twelve automotive cells. While the linear method has advantages in computational complexity, the nonlinear method detects ISCs earlier due to its high sensitivity and specificity.

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.

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.

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.

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.

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.