Herbrich, R., Graepel, T., Obermayer, K.: Regression Models for Ordinal Data: A Machine Learning Approach. Report TR 99-3, Dept. of Computer Science, Technical University of Berlin, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with the order of the attribute values in the decision nodes. The authors present repair strategies for correcting inconsistent trees in case these consistency constraints are violated, as well as an algorithm for constructing consistent trees in the rst place. Herbrich et al. 1999, Herbrich et al. 1999b] describe an algorithm based on the large margin idea known from data dependent Structural Risk Minimization [Shawe Taylor et al. 1996] The algorithm is similar to Support Vector Machines [Cortes Vapnik, 1995] They demonstrate good results on arti cial data and on a (very small) real world ....

Herbrich, R., Graepel, T., and Obermayer, K. (1999b). Regression Models for Ordinal Data: A Machine Learning Approach. Report TR 99-3, Dept. of Computer Science, Technical University of Berlin.

....di erent methods for doing this. However, if the class attribute represents a truly ordinal quantity which, by de nition, cannot be represented as a number in a meaningful way there is no principled way of devising an appropriate mapping and this procedure is necessarily ad hoc. Herbrich et al. [3] propose a strategy for ordinal classi cation that is based on a loss function between pairs of true ranks and predicted ranks. They present a corresponding algorithm similar to a support vector machine, and mention that their approach can be extended to other types of linear classi ers. Potharst ....

....input is ordered. Although machine learning algorithms for ordinal classi cation are rare, there are many statistical approaches to this problem. However, they all rely on speci c distributional assumptions for modeling the class variable and also assume a stochastic ordering of the input space [3]. The technique of generating binary dummy attributes to replace an ordered attribute can also be applied to the attributes making up the input space. Frank and Witten [2] show that this often improves performance compared to treating ordered attributes as nominal quantities. In cases where ....

R. Herbrich, T. Graepel, and K. Obermayer. Regression models for ordinal data: A machine learning approach. Technical report, TU Berlin, 1999.

....of ordinal regression can be reduced to a classification problem on pairs of objects. Therefore we call this problem also the problem of preference learning. It was shown that the Bayes optimal decision function on pairs of objects can result in a function p which is no longer transitive on X (Herbrich et al. 1999). Note also that the requirements of transitivity and asymmetry effectively reduce the space of admissible classification functions p acting on pairs of objects. The following bound on R pref (h ) gives a justification for the large margin algorithm to be presented in Section 3. A proof based ....

....effectively reduce the space of admissible classification functions p acting on pairs of objects. The following bound on R pref (h ) gives a justification for the large margin algorithm to be presented in Section 3. A proof based on a result of (Shawe Taylor et al. 1996) can be found in (Herbrich et al. 1999). Theorem 2. Assume that for a given set H of mappings from objects to ranks there exists a set F of mappings from objects to R such that for each function h 2 H there exists a function U 2 F (and vice versa) with h(x) r i , U(x) 2 [ r i Gamma1 ) r i ) 7) Let PXY be a probability ....

Herbrich, R., T. Graepel, and K. Obermayer (1999). Regression models for ordinal data: A machine learning approach.

....Vector Machines. 1 Introduction Recently, the study of classification learning has shown that the generalization error of classifiers based on real valued functions can be controlled by making use of a quantity known as the margin. Let us consider the set H k of kernel classifiers [ Weston and Herbrich, 1999 ] 1 f(x) X i=1 ff i k(x i ; x) ff 2 R : 1) Here, k is referred to as a kernel and is assumed to be symmetric and positive definite. It is known from the theory of reproducing kernel Hilbert spaces (RKHS) Whaba, 1990 ] that there exists a fixed feature space F not necessarily ....

....of Bayes optimal decision functions. 3 Billiards in Kernel Space In this section we present a BPM algorithm for estimating the Bayes point by the center of mass 4 . The 3 Note that this function need not necessarily be contained in the original set of functions. 4 For further details see [Herbrich et al. 1999a] approach utilized is similar to the method presented in [ Ruj an, 1997 ] in order to obtain the center of mass of V(S) we randomly generate points (hyperplanes in input space) and average over them. Since it is very difficult to generate hyperplanes consistent with S we average over the ....

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Ralf Herbrich, Thore Graepel, and Klaus Obermayer. Regression models for ordinal data: A machine learning approach. Technical report, TU Berlin, 1999. TR-99/03.

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Herbrich, R., Graepel, T., Obermayer, K.: Regression Models for Ordinal Data: A Machine Learning Approach. Report TR 99-3, Dept. of Computer Science, Technical University of Berlin, 1999.

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