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Based on the Nyström approximation and the primal-dual formulation of the least squares support vector machines, it becomes possible to apply a nonlinear model to a large scale regression problem. This is done by using a sparse approximation of the nonlinear mapping induced by the kernel matrix, with an active selection of support vectors based on quadratic Renyi entropy criteria. The methodology is applied to the case of load forecasting as an example of a real-life large scale problem in industry. The forecasting performance, over ten different load series, shows satisfactory results when the sparse representation is built with less than 3% of the available sample.  相似文献
2.
The quantile regression problem is considered by learning schemes based on ? 1—regularization and Gaussian kernels. The purpose of this paper is to present concentration estimates for the algorithms. Our analysis shows that the convergence behavior of ? 1—quantile regression with Gaussian kernels is almost the same as that of the RKHS-based learning schemes. Furthermore, the previous analysis for kernel-based quantile regression usually requires that the output sample values are uniformly bounded, which excludes the common case with Gaussian noise. Our error analysis presented in this paper can give satisfactory convergence rates even for unbounded sampling processes. Besides, numerical experiments are given which support the theoretical results.  相似文献
3.
Kernel Based Regression (KBR) minimizes a convex risk over a possibly infinite dimensional reproducing kernel Hilbert space. Recently, it was shown that KBR with a least squares loss function may have some undesirable properties from a robustness point of view: even very small amounts of outliers can dramatically affect the estimates. KBR with other loss functions is more robust, but often gives rise to more complicated computations (e.g. for Huber or logistic losses). In classical statistics robustness is often improved by reweighting the original estimate. In this paper we provide a theoretical framework for reweighted Least Squares KBR (LS-KBR) and analyze its robustness. Some important differences are found with respect to linear regression, indicating that LS-KBR with a bounded kernel is much more suited for reweighting. In two special cases our results can be translated into practical guidelines for a good choice of weights, providing robustness as well as fast convergence. In particular a logistic weight function seems an appropriate choice, not only to downweight outliers, but also to improve performance at heavy tailed distributions. For the latter some heuristic arguments are given comparing concepts from robustness and stability.  相似文献
4.
We propose a framework for the visualization of directed networks relying on the eigenfunctions of the magnetic Laplacian, called here Magnetic Eigenmaps. The magnetic Laplacian is a complex deformation of the well-known combinatorial Laplacian. Features such as density of links and directionality patterns are revealed by plotting the phases of the first magnetic eigenvectors. An interpretation of the magnetic eigenvectors is given in connection with the angular synchronization problem. Illustrations of our method are given for both artificial and real networks.  相似文献
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