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: 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.