Journal of Statistical Software 5 2.2.Novelty detection SVMs have also been extended to deal with the problem of novelty detection (or one-class classification;see Scholkopf,Platt,Shawe-Taylor,Smola,and Williamson 1999;Tax and Duin 1999),where essentially an SVM detects outliers in a data set.SVM novelty detection works by creating a spherical decision boundary around a set of data points by a set of support vectors describing the sphere's boundary.The primal optimization problem for support vector novelty detection is the following: minimize tw,5,)=w2-p+s mv i=1 subject to (Φ(x),w〉+b≥p-E(i=1,.,m) (12) 5≥0(i=1,.,m): The v parameter is used to control the volume of the sphere and consequently the number of outliers found.The value of v sets an upper bound on the fraction of outliers found in the data. 2.3.Regression By using a different loss function called the c-insensitive loss function lly-f()le=max{0,lly- f(x)-e},SVMs can also perform regression.This loss function ignores errors that are smaller than a certain threshold e>0 thus creating a tube around the true output.The primal becomes: minimize 9P+2G+ =1 subject to (Φ(x),w)+b)-≤e- (13) -(④(x),w〉+b)≤e- (14) 接≥0(i=1,.,m) We can estimate the accuracy of SVM regression by computing the scale parameter of a Laplacian distribution on the residuals s=y-f(c),where f(x)is the estimated decision function (Lin and Weng 2004). The dual problems of the various classification,regression and novelty detection SVM formu- lations can be found in the Appendix. 2.4.Kernel functions As seen before,the kernel functions return the inner product between two points in a suitable feature space,thus defining a notion of similarity,with little computational cost even in very high-dimensional spaces.Kernels commonly used with kernel methods and SVMs in particular include the following:Journal of Statistical Software 5 2.2. Novelty detection SVMs have also been extended to deal with the problem of novelty detection (or one-class classification; see Sch¨olkopf, Platt, Shawe-Taylor, Smola, and Williamson 1999; Tax and Duin 1999), where essentially an SVM detects outliers in a data set. SVM novelty detection works by creating a spherical decision boundary around a set of data points by a set of support vectors describing the sphere’s boundary. The primal optimization problem for support vector novelty detection is the following: minimize t(w, ξ, ρ) = 1 2 kwk 2 − ρ + 1 mν Xm i=1 ξi subject to hΦ(xi), wi + b ≥ ρ − ξi (i = 1, . . . , m) (12) ξi ≥ 0 (i = 1, . . . , m). The ν parameter is used to control the volume of the sphere and consequently the number of outliers found. The value of ν sets an upper bound on the fraction of outliers found in the data. 2.3. Regression By using a different loss function called the -insensitive loss function ky−f(x)k = max{0, ky− f(x)k − }, SVMs can also perform regression. This loss function ignores errors that are smaller than a certain threshold > 0 thus creating a tube around the true output. The primal becomes: minimize t(w, ξ) = 1 2 kwk 2 + C m Xm i=1 (ξi + ξ ∗ i ) subject to (hΦ(xi), wi + b) − yi ≤ − ξi (13) yi − (hΦ(xi), wi + b) ≤ − ξ ∗ i (14) ξ ∗ i ≥ 0 (i = 1, . . . , m) We can estimate the accuracy of SVM regression by computing the scale parameter of a Laplacian distribution on the residuals ζ = y − f(x), where f(x) is the estimated decision function (Lin and Weng 2004). The dual problems of the various classification, regression and novelty detection SVM formulations can be found in the Appendix. 2.4. Kernel functions As seen before, the kernel functions return the inner product between two points in a suitable feature space, thus defining a notion of similarity, with little computational cost even in very high-dimensional spaces. Kernels commonly used with kernel methods and SVMs in particular include the following: