483 research outputs found
Solving Support Vector Machines in Reproducing Kernel Banach Spaces with Positive Definite Functions
In this paper we solve support vector machines in reproducing kernel Banach
spaces with reproducing kernels defined on nonsymmetric domains instead of the
traditional methods in reproducing kernel Hilbert spaces. Using the
orthogonality of semi-inner-products, we can obtain the explicit
representations of the dual (normalized-duality-mapping) elements of support
vector machine solutions. In addition, we can introduce the reproduction
property in a generalized native space by Fourier transform techniques such
that it becomes a reproducing kernel Banach space, which can be even embedded
into Sobolev spaces, and its reproducing kernel is set up by the related
positive definite function. The representations of the optimal solutions of
support vector machines (regularized empirical risks) in these reproducing
kernel Banach spaces are formulated explicitly in terms of positive definite
functions, and their finite numbers of coefficients can be computed by fixed
point iteration. We also give some typical examples of reproducing kernel
Banach spaces induced by Mat\'ern functions (Sobolev splines) so that their
support vector machine solutions are well computable as the classical
algorithms. Moreover, each of their reproducing bases includes information from
multiple training data points. The concept of reproducing kernel Banach spaces
offers us a new numerical tool for solving support vector machines.Comment: 26 page
Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method
In this paper we present the theoretical framework needed to justify the use
of a kernel-based collocation method (meshfree approximation method) to
estimate the solution of high-dimensional stochastic partial differential
equations (SPDEs). Using an implicit time stepping scheme, we transform
stochastic parabolic equations into stochastic elliptic equations. Our main
attention is concentrated on the numerical solution of the elliptic equations
at each time step. The estimator of the solution of the elliptic equations is
given as a linear combination of reproducing kernels derived from the
differential and boundary operators of the SPDE centered at collocation points
to be chosen by the user. The random expansion coefficients are computed by
solving a random system of linear equations. Numerical experiments demonstrate
the feasibility of the method.Comment: Updated Version in International Journal of Computer Mathematics,
Closed to Ye's Doctoral Thesi
Intraday Load Forecasts with Uncertainty
We provide a comprehensive framework for forecasting five minute load using Gaussian processes with a positive definite kernel specifically designed for load forecasts. Gaussian processes are probabilistic, enabling us to draw samples from a posterior distribution and provide rigorous uncertainty estimates to complement the point forecast, an important benefit for forecast consumers. As part of the modeling process, we discuss various methods for dimension reduction and explore their use in effectively incorporating weather data to the load forecast. We provide guidance for every step of the modeling process, from model construction through optimization and model combination. We provide results on data from the largest deregulated wholesale U.S. electricity market for various periods in 2018. The process is transparent, mathematically motivated, and reproducible. The resulting model provides a probability density of five minute forecasts for 24 h
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