David M.J. Tax and Robert P.W. Duin. Support Vector Domain Description. Pattern Recognition Letters, 20:1191--1199, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of noise. In addition to their accuracy, a key characteristic of SVMs is their mathematical tractability and geometric interpretation. While SVMs have been widely adopted as supervised learning methods with labeled data, they have also been used for the exploration of unlabeled data (cf. [1, 8, 9]) Novelty detection and cluster analysis using SVMs are examples for learning unlabeled data. For many real world problems, the task is not to classify but to detect novel or Current address: School of Computing and Information Technology, Griffith University, Brisbane, QLD 4111, Australia. ....

....for data of all dimensions. Using SVMs, another boundary based clustering method was proposed in [1, 2] we call it SVC) This approach employs support vectors to construct cluster boundaries. Its principle is novelty detection [8] which is sometimes also called data domain description [9]. Domain description produces a description of a given set of objects. This description should cover the class of given objects, and ideally reject other possible objects in the object space. Generally, novelty detection can characterize estimating functions of the data that tell something ....

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D. M. J. Tax and R. P. W. Duin. Support vector domain description. Pattern Recognition Letters, 20(1113) :1191--1199, 1999.

....did not reach a minimum defined certainty level. The work of Vasconcelos et al. 44] shows that on a character recognition problem, this process gives much better results for rejecting novel patterns. Approaches based on support vectors have also been used in the context of novelty detection [39]. Support vectors are used to generate a minimum volume bounding hypersphere around the data. The method allows for some objects to be outside and some to be inside the sphere using slack variables. The radius of the sphere is determined using the trade off between simplicity (or volume of ....

D.M.J. Tax and R.P.W. Duin, "Support vector domain description", Pattern Recognition Letters, vol. 20, pp. 1191-1199, 1999.

....as in hierarchical clustering algorithms. Other approaches include graph theoretic methods [5] physically motivated algorithms [6] and algorithms based on density estimation [7] 2] In this paper we propose a non parametric clustering algorithm based on the support vector 1 approach [8] In [9, 10] a support vector algorithm was used to characterize the support of a high dimensional distribution. As a by product of the algorithm one can compute a set of contours which enclose the data points. These contours were interpreted by us as cluster boundaries [11] Here we discuss in detail a ....

....that are obtained by varying the scale of the Gaussian kernel. Then we proceed to discuss problems that necessitate invoking outliers in order to obtain smooth clustering boundaries. These problems include two standard benchmark examples. 2 The SVC Algorithm 2. 1 Cluster Boundaries Following [10, 9] we formulate a support vector description of a data set, that is used as the basis of our clustering algorithm. Let fx i g be a data set of N points, with IR d , the data space. Using a nonlinear transformation from to some high dimensional feature space, we look for the smallest ....

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D.M.J. Tax and R.P.W. Duin. Support vector domain description. Pattern Recognition Letters, 20:

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D.M.J. Tax and R.P.W Duin. Support vector domain description. Pattern Recognition Letters, 20(11-13):1191--1199, December 1999.

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D.M.J. Tax and R.P.W Duin. Support vector domain description. Pattern Recognition Letters, 20(1113) :1191--1199, December 1999.

.... y) K(x, y) exp( #x y# s ) 4) This Gaussian kernel contains an extra free parameter, the width parameter s in the kernel (from definition (4) For small values of s the SVDD resembles a Parzen density estimation, while for large s the original hypersphere solution is obtained [9]. As shown in [9] this parameter can be set by setting a priori the maximal allowed rejection rate of the target set, i.e. the error on the target set. Secondly, we also have the trade o# parameter C. We can define a new variable # = MC , which describes an upper bound for the fraction of ....

.... = exp( #x y# s ) 4) This Gaussian kernel contains an extra free parameter, the width parameter s in the kernel (from definition (4) For small values of s the SVDD resembles a Parzen density estimation, while for large s the original hypersphere solution is obtained [9] As shown in [9], this parameter can be set by setting a priori the maximal allowed rejection rate of the target set, i.e. the error on the target set. Secondly, we also have the trade o# parameter C. We can define a new variable # = MC , which describes an upper bound for the fraction of objects outside the ....

D.M.J. Tax and R.P.W Duin. Support vector domain description. Pattern Recognition Letters, 20(11-13):1191--1199, December 1999.

....to minimize the chance of accepting outliers. Inspired by the Support Vector Method by Vapnik (see [Vap95] or for a more simple introduction [TdRD97] one can extend this idea to determine an arbitrary shaped region in the original feature space, the Support Vector Data Description (SVDD) method [TD99]. Assume we a data set containing N data objects, fx i ; i = 1; Ng and the sphere is described by center a and radius R. To allow the possibility of outliers in the training set, the distance from x i to the center a should not be strictly smaller than R 2 , but larger distances should be ....

....data description is shown for a small arti cial banana shaped dataset. The width parameter s ranges from s = 1:0 in the left subplot to s = 25 in the right subplot. For small values of s the SVDD resembles a Parzen density estimation, while for large s the original hypersphere solution is obtained[TD99]. Note also that the number of support vectors decreases. So a more exible description than the rigid sphere description is obtained. This Gaussian kernel contains one extra free parameter, the width parameter s in the kernel (equation (5) As shown in [TD99] this parameter can be set by ....

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D.M.J. Tax and R.P.W Duin. Support vector domain description. Pattern Recognition Letters, 20(11-13):1191-1199, December 1999.

....density p(x # T ) In this paper we use a normal density, a mixture of Gaussians and the Parzen density estimation. The second type of methods fit a model to the data and compute the distance # T (x) to this model. Here we will use four simple models, the support vector data description [14], k means clustering, k center method [15] and an autoencoder neural network [6] Here a descriptive model is fitted to the data and the resemblance (or distance) to this model is used. In the SVDD a hypersphere is put around the data. By applying the kernel trick (analogous to the support vector ....

D.M.J. Tax and R.P.W Duin. Support vector domain description. Pattern Recognition Letters, 20(11-13):1191--1199, December 1999.

....accepting outliers. Inspired by the Support Vector Method by Vapnik (see (Vapnik 1995) or for a more simple introduction (Tax, de Ridder, Duin 1997) one can extend this idea to determine an arbitrary shaped region in the original feature space, the Support Vector Data Description (SVDD) method (Tax Duin 1999). Assume we have a data set containing N data objects, fx i ; i = 1; Ng and the sphere is described by center a and radius R. To allow the possibility of outliers in the training set, the distance from x i to the center a should not be strictly smaller than R 2 , but larger distances should ....

....description is obtained. In gure 1 the resulting data description is shown for a simple 2D dataset. For different values for the width parameter s in the kernel (equation (6) the resulting decision boundary is plotted. Note that the number of support vectors changes for di erent s. As shown in (Tax Duin 1999) rejection rate of the target set, i.e. the error on the target set can be estimated by the number of support vectors: E[P (error) #SV N (9) where #SV is the number of support vectors. The number of support vectors can be regulated by changing the width parameter s and therefore also the ....

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Tax, D., and Duin, R. 1999. Support vector domain description. Pattern Recognition Letters 20(11-13):1191{ 1199.

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David M.J. Tax and Robert P.W. Duin. Support Vector Domain Description. Pattern Recognition Letters, 20:1191--1199, 1999.

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D.M.J. Tax and R.P.W. Duin, "Support vector domain description", Pattern Recognition Letters, vol. 20, pp. 1191-1199, 1999b. 26

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Tax, D.M.J., Duin, R.P.W.: Support vector domain description. Pattern Recognition Letters. 20(11-13) (1999) 1191--1199.

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D.M.J. Tax and R.P.W. Duin, "Support Vector Domain Description, " Pattern Recognition Letters. vol. 20, pp. 1991-1999, 1999.

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