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6 Experimental Analysis of SVM-KNN-Ensemble 31 be observed that the smallest rate α = 10−1 , corresponding to a training set size of N = 262 patterns, can achieve an accuracy of up to ≈ 95% for the SVM-KNN-ensemble ENS* and also for the SVM with linear kernel. 89%) is achieved with KNN and K = 7. For the SVM approach, we can observe that a linear kernel achieves better results than an RBF-kernel and better results than KNN with K = 5 and K = 7 for training sets smaller and equal to 6−1 . While the SVM with linear kernel takes signiﬁcantly longer for training with training set sizes larger than 5−1 , it is a good recommendation for small training set sizes.
04 2 (a) 4 6 8 K 10 12 14 (b) Fig. t. training set size 5−1 , 3−1 , and 2−1 for (a) KNN and (b) ENS*. The neighborhood size has a signiﬁcant inﬂuence on the classiﬁcation error in case of the KNN classiﬁers, but the eﬀect is compensated in the ensemble. 3 Neighborhood Sizes of KNN In the following, we analyze the inﬂuence of neighborhood size K of the KNN classiﬁers and of ensemble ENS* on the recognition rate. t. diﬀerent training set sizes. 1(a), we can observe that neighborhood sizes around K = 4 to K = 6 are optimal for small training sets.
In LOO-CV, one pattern is left out for prediction based on the remaining N − 1 training patterns. The whole procedure is repeated N times. 6 Curse of Dimensionality Many machine learning methods have problems in high-dimensional data spaces. The reason is an eﬀect also know as curse of dimensionality or Hughes eﬀect. In high-dimensional data spaces, many patterns are required to cover the whole data space, and our intuition often breaks down. Hastie et al.  gives interesting arguments for this eﬀect that we review in the following.