6 Support Vector Machines in R the linear kernel implementing the simplest of all kernel functions k(x,x)=(x,x (15) the Gaussian Radial Basis Function (RBF)kernel k(x,x')=exp(-allx-xl2) (16) the polynomial kernel (x,x)=(scale.(x)+offset)degree (17) the hyperbolic tangent kernel (x,x)=tanh (scale.(x,x+offset) (18) the Bessel function of the first kind kernel k(x,x')= Bessel+1)(allx-x'll) (n-t'l)-n(v+1) (19) the Laplace Radial Basis Function(RBF)kenrel k(x,x)=exp(-allx-x'll) (20) the ANOVA radial basis kernel k(x,x)） ep-oe-ry】 (21) . the linear splines kernel in one dimension ,）=1+min(c,)-专'(min(e,y+ams,r 3 (22) and for the multidimensional case k(x,x')=II=1k(,). The Gaussian and Laplace RBF and Bessel kernels are general-purpose kernels used when there is no prior knowledge about the data.The linear kernel is useful when dealing with large sparse data vectors as is usually the case in text categorization.The polynomial kernel is popular in image processing and the sigmoid kernel is mainly used as a proxy for neural networks.The splines and ANOVA RBF kernels typically perform well in regression problems. 2.5.Software Support vector machines are currently used in a wide range of fields,from bioinformatics to astrophysics.Thus,the existence of many SVM software packages comes as little surprise. Most existing software is written in C or C++,such as the award winning libsvm(Chang and Lin 2001),which provides a robust and fast SVM implementation and produces state of the6 Support Vector Machines in R • the linear kernel implementing the simplest of all kernel functions k(x, x 0 ) = hx, x 0 i (15) • the Gaussian Radial Basis Function (RBF) kernel k(x, x 0 ) = exp(−σkx − x 0 k 2 ) (16) • the polynomial kernel k(x, x 0 ) = ￾ scale · hx, x 0 i + offset
degree (17) • the hyperbolic tangent kernel k(x, x 0 ) = tanh ￾ scale · hx, x 0 i + offset
(18) • the Bessel function of the first kind kernel k(x, x 0 ) = Besseln (ν+1)(σkx − x 0k) (kx − x 0k)−n(ν+1) (19) • the Laplace Radial Basis Function (RBF) kenrel k(x, x 0 ) = exp(−σkx − x 0 k) (20) • the ANOVA radial basis kernel k(x, x 0 ) = Xn k=1 exp(−σ(x k − x 0k ) 2 ) !d (21) • the linear splines kernel in one dimension k(x, x0 ) = 1 + xx0 min(x, x0 ) − x + x 0 2 (min(x, x0 ) 2 + (min(x, x0 ) 3 ) 3 (22) and for the multidimensional case k(x, x 0 ) = Qn k=1 k(x k , x0k ). The Gaussian and Laplace RBF and Bessel kernels are general-purpose kernels used when there is no prior knowledge about the data. The linear kernel is useful when dealing with large sparse data vectors as is usually the case in text categorization. The polynomial kernel is popular in image processing and the sigmoid kernel is mainly used as a proxy for neural networks. The splines and ANOVA RBF kernels typically perform well in regression problems. 2.5. Software Support vector machines are currently used in a wide range of fields, from bioinformatics to astrophysics. Thus, the existence of many SVM software packages comes as little surprise. Most existing software is written in C or C++, such as the award winning libsvm (Chang and Lin 2001), which provides a robust and fast SVM implementation and produces state of the