Abstract: This paper presents two of the most knowing kernel
adaptive filtering (KAF) approaches, the kernel least mean squares
and the kernel recursive least squares, in order to predict a new output
of nonlinear signal processing. Both of these methods implement a
nonlinear transfer function using kernel methods in a particular space
named reproducing kernel Hilbert space (RKHS) where the model is
a linear combination of kernel functions applied to transform the
observed data from the input space to a high dimensional feature
space of vectors, this idea known as the kernel trick. Then KAF is the
developing filters in RKHS. We use two nonlinear signal processing
problems, Mackey Glass chaotic time series prediction and nonlinear
channel equalization to figure the performance of the approaches
presented and finally to result which of them is the adapted one.
Abstract: Tikhonov regularization and reproducing kernels are the
most popular approaches to solve ill-posed problems in computational
mathematics and applications. And the Fourier multiplier operators
are an essential tool to extend some known linear transforms
in Euclidean Fourier analysis, as: Weierstrass transform, Poisson
integral, Hilbert transform, Riesz transforms, Bochner-Riesz mean
operators, partial Fourier integral, Riesz potential, Bessel potential,
etc. Using the theory of reproducing kernels, we construct a simple
and efficient representations for some class of Fourier multiplier
operators Tm on the Paley-Wiener space Hh. In addition, we give
an error estimate formula for the approximation and obtain some
convergence results as the parameters and the independent variables
approaches zero. Furthermore, using numerical quadrature integration
rules to compute single and multiple integrals, we give numerical
examples and we write explicitly the extremal function and the
corresponding Fourier multiplier operators.
Abstract: It-s known that incorporating prior knowledge into support
vector regression (SVR) can help to improve the approximation
performance. Most of researches are concerned with the incorporation
of knowledge in form of numerical relationships. Little work,
however, has been done to incorporate the prior knowledge on the
structural relationships among the variables (referred as to Structural
Prior Knowledge, SPK). This paper explores the incorporation of SPK
in SVR by constructing appropriate admissible support vector kernel
(SV kernel) based on the properties of reproducing kernel (R.K).
Three-levels specifications of SPK are studies with the corresponding
sub-levels of prior knowledge that can be considered for the method.
These include Hierarchical SPK (HSPK), Interactional SPK (ISPK)
consisting of independence, global and local interaction, Functional
SPK (FSPK) composed of exterior-FSPK and interior-FSPK. A
convenient tool for describing the SPK, namely Description Matrix
of SPK is introduced. Subsequently, a new SVR, namely Motivated
Support Vector Regression (MSVR) whose structure is motivated
in part by SPK, is proposed. Synthetic examples show that it is
possible to incorporate a wide variety of SPK and helpful to improve
the approximation performance in complex cases. The benefits of
MSVR are finally shown on a real-life military application, Air-toground
battle simulation, which shows great potential for MSVR to
the complex military applications.
Abstract: Support vector regression (SVR) has been regarded
as a state-of-the-art method for approximation and regression. The
importance of kernel function, which is so-called admissible support
vector kernel (SV kernel) in SVR, has motivated many studies
on its composition. The Gaussian kernel (RBF) is regarded as a
“best" choice of SV kernel used by non-expert in SVR, whereas
there is no evidence, except for its superior performance on some
practical applications, to prove the statement. Its well-known that
reproducing kernel (R.K) is also a SV kernel which possesses many
important properties, e.g. positive definiteness, reproducing property
and composing complex R.K by simpler ones. However, there are a
limited number of R.Ks with explicit forms and consequently few
quantitative comparison studies in practice. In this paper, two R.Ks,
i.e. SV kernels, composed by the sum and product of a translation
invariant kernel in a Sobolev space are proposed. An exploratory
study on the performance of SVR based general R.K is presented
through a systematic comparison to that of RBF using multiple
criteria and synthetic problems. The results show that the R.K is
an equivalent or even better SV kernel than RBF for the problems
with more input variables (more than 5, especially more than 10) and
higher nonlinearity.
Abstract: This paper presents a generalization kernel for gravitational
potential determination by harmonic splines. It was shown
in [10] that the gravitational potential can be approximated using a
kernel represented as a Newton integral over the real Earth body. On
the other side, the theory of geopotential approximation by harmonic
splines uses spherically oriented kernels. The purpose of this paper
is to show that in the spherical case both kernels have the same type
of representation, which leads us to conclusion that it is possible
to consider the kernel represented as a Newton integral over the real
Earth body as a kind of generalization of spherically harmonic kernels
to real geometries.
Abstract: Kernel function, which allows the formulation of nonlinear variants of any algorithm that can be cast in terms of dot products, makes the Support Vector Machines (SVM) have been successfully applied in many fields, e.g. classification and regression. The importance of kernel has motivated many studies on its composition. It-s well-known that reproducing kernel (R.K) is a useful kernel function which possesses many properties, e.g. positive definiteness, reproducing property and composing complex R.K by simple operation. There are two popular ways to compute the R.K with explicit form. One is to construct and solve a specific differential equation with boundary value whose handicap is incapable of obtaining a unified form of R.K. The other is using a piecewise integral of the Green function associated with a differential operator L. The latter benefits the computation of a R.K with a unified explicit form and theoretical analysis, whereas there are relatively later studies and fewer practical computations. In this paper, a new algorithm for computing a R.K is presented. It can obtain the unified explicit form of R.K in general reproducing kernel Hilbert space. It avoids constructing and solving the complex differential equations manually and benefits an automatic, flexible and rigorous computation for more general RKHS. In order to validate that the R.K computed by the algorithm can be used in SVM well, some illustrative examples and a comparison between R.K and Gaussian kernel (RBF) in support vector regression are presented. The result shows that the performance of R.K is close or slightly superior to that of RBF.
Abstract: It-s known that incorporating prior knowledge into support
vector regression (SVR) can help to improve the approximation
performance. Most of researches are concerned with the incorporation
of knowledge in the form of numerical relationships. Little work,
however, has been done to incorporate the prior knowledge on the
structural relationships among the variables (referred as to Structural
Prior Knowledge, SPK). This paper explores the incorporation of SPK
in SVR by constructing appropriate admissible support vector kernel
(SV kernel) based on the properties of reproducing kernel (R.K).
Three-levels specifications of SPK are studied with the corresponding
sub-levels of prior knowledge that can be considered for the method.
These include Hierarchical SPK (HSPK), Interactional SPK (ISPK)
consisting of independence, global and local interaction, Functional
SPK (FSPK) composed of exterior-FSPK and interior-FSPK. A
convenient tool for describing the SPK, namely Description Matrix
of SPK is introduced. Subsequently, a new SVR, namely Motivated
Support Vector Regression (MSVR) whose structure is motivated
in part by SPK, is proposed. Synthetic examples show that it is
possible to incorporate a wide variety of SPK and helpful to improve
the approximation performance in complex cases. The benefits of
MSVR are finally shown on a real-life military application, Air-toground
battle simulation, which shows great potential for MSVR to
the complex military applications.
Abstract: This paper investigates the inverse problem of determining
the unknown time-dependent leading coefficient in the parabolic
equation using the usual conditions of the direct problem and an additional
condition. An algorithm is developed for solving numerically
the inverse problem using the technique of space decomposition in a
reproducing kernel space. The leading coefficients can be solved by a
lower triangular linear system. Numerical experiments are presented
to show the efficiency of the proposed methods.