A kernel-based parametric method for conditional density estimation

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.

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.

Kernel principal component analysis (KPCA) has gained much attention for capturing nonlinear image features which is particularly important for clustering high-dimensional multi-class features. We introduce KPCA in this paper for detecting and classifying moving vehicles from its viewpoint images. The KPCA extracts non-linear features of multi-class moving vehicles by mapping input space to a higher dimensional feature space through a nonlinear map. The results provide us a clustered feature space of the car and non-car for classifying them by separating the dimensional space or eigenvector. The experimental results show the robustness of the KPCA's feature separation in our car database that lead to the cars' classification. Extended experiments on various vehicles detection have also shown the remarkable performance of the proposed method

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.

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.

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.

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.

In this paper, we are going to propose an online radial basis function (RBF) neural network algorithm without any preprocessing step. Then a kernel principal component analysis (KPCA) is coupled with the proposed online RBF neural network algorithm. Indeed, the KPCA method is used as a preprocessing step to reduce the feature dimension which fed to the RBF neural network. Reducing memory requirements of the models makes RBF neural network training efficient and fast. These two proposed algorithms are applied, with success, for identification of a mobile robot position. The simulation results present that the used sigmoid function as a kernel, compared to other kernel functions, which gives an excellent model and a minimum mean square error.

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.

We suggest two improved methods for conditional density estimation. The first is based on locally fitting a log-linear model, and is in the spirit of recent work on locally parametric techniques in density estimation. The second method is a constrained local polynomial estimator. Both methods always produce non-negative estimators. We propose an algorithm suitable for selecting the two bandwidths for either estimator. We also develop a new bootstrap test for the symmetry of conditional density functions. The proposed methods are illustrated by both simulation and application to a real data set.

Viewing to monitoring the continuous casting process, a nonlinear fault detection method based on kernel principal component analysis (KPCA) was introduced. KPCA can efficiently compute principal components in high-dimensional feature spaces by means of integral operators and nonlinear kernel functions, which 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. Based on T2 and SPE charts in feature space, principal component analysis(PCA)can be used to detect faults.. The simulation results show that the proposed approach effectively captured the nonlinear relationship in the process variables and showed superior process monitoring performance compared to linear PCA.

In this paper, the kernel principal component analysis (KPCA) is applied to perform the translation quality classification and evaluation of human-machine composite subject of scientific text. Firstly, four different translations are quantified by means of questionnaire survey according to the basic standards of Chinese-English translation. Then, the quantitative data is evaluated by Gaussian kernel function and polynomial kernel function. The results show that, on the one hand, the translation qualities of machine translation, professional translator and scientific researcher approximate to form an equilateral triangle in two-dimensional evaluation space, which indicates that the qualities of the above three translations is independent of each other in terms of evaluation space, and the translation quality of computer-aided scientific researcher is closest to the translation quality of professional translator; on the other hand, when the evaluation space dimension is reduced to one-dimensional, Gaussian kernel function can still get similar result, but polynomial kernel function gives different result when its order is greater than a certain threshold.

Kernel discriminant analysis (KDA) and the kernel principal component analysis (KPCA), which are the extension of the linear discriminant analysis (LDA) and the principal component analysis (PCA), respectively, from linear domain to nonlinear domain via the kernel trick, are two very popular nonlinear feature extraction methods. In this paper, we present a new feature extraction algorithm by combing KDA and KPCA, and then apply it to the face recognition task. The experimental results on Yale face dataset show that the proposed method can significantly improve the performance both KDA and KPCA.

This paper presents a scene classification method based on local autocorrelation of similarities with subspaces. Although conventional methods used bag-of-visual words for scene classification, superior accuracy of Kernel Principal Component Analysis (KPCA) of visual words to bag-of-visual words was reported. Here we also use KPCA of visual words to extract rich information for classification. In the original paper, all local parts mapped into subspace were integrated by summation to be robust to the order, the number, and shift of local parts. This approach discarded the effective properties for scene classification such as the relation with neighboring regions. To use them, we use Local AutoCorrelation (LAC) feature of the similarities with subspaces (outputs of KPCA of visual words). The feature has both the relation with neighboring regions and the robustness to shift of objects. The proposed method is compared with conventional scene classification methods using the same database and protocol. We demonstrate that the proposed method outperforms conventional methods.

This paper first provides a method for predicting fouling faults about flow passage of steam turbine based on kernel principal component analysis(KPCA) and least square support vector machine regression (LS-SVMR). First, KPCA is used to extract main features independent for each other from a lot of relaticve fault feature data. Afterwards, a model is established for predicting the trend of each main feature based on LS-SVMR in order to restruct feature vectors of fault classification. And then some typical fouling faults of steam turbine flow passage are identified by using SVM. Experimental results showed that the proposed method could effectively and efficiently forecast delitescent faults and typical fouling fault genres for the flow passage.