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By Phillips H. B.

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SVM relies on the complexity of the hypothesis space and empirical error (a measure of how well the model fits the training data). VapnikChervonenkis (VC) theory proves that a VC bound on the risk exists. VC is a measure of the complexity of the hypothesis space. The VC dimension of a hypothesis H relates to the maximum number of points that can be shattered by H. H shatters N points, if H correctly separates all the positive instances from the negative ones. In other words, the VC capacity is equal to the number of training points N that the model can separate into 2N different labels.

30 Chapter 2 ■ Machine Learning and Knowledge Discovery Figure 2-4. The percentage of variance of a principal component transform of a dataset with 160 features reduced to 59 components Principal component analysis (PCA) is a widely used analytic technique that identifies patterns to reduce the dimensions of the dataset without significant loss of information. The goal of PCA is to project a high-dimensional feature space into a smaller subset to decrease computational cost. PCA computes new features, called principal components (PCs), which are uncorrelated linear combinations of the original features projected in the direction of greater variability.

As the complexity index increases, the error reaches its minimum for the optimal model indexed h* before it starts increasing again. For high model indices, the structure starts adapting its learning model to the training data which results in an overfitting that reduces the training error value and increases the model VC however, at the expense of a deterioration in the test error. underfitting best model overfitting error test error VC (confidence term) training error/empirical error h (model index) structure Figure 3-2.

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