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Class: ClassificationSVM
Classification edge for support vector machine classifiers by resubstitution
The edge is the weighted mean of the classification margins.
The weights are the prior class probabilities. If you supply weights, then the software normalizes them to sum to the prior probabilities in the respective classes. The software uses the renormalized weights to compute the weighted mean.
One way to choose among multiple classifiers, e.g., to perform feature selection, is to choose the classifier that yields the highest edge.
The classification margins are, for each observation, the difference between the score for the true class and maximal score for the false classes. Provided that they are on the same scale, margins serve as a classification confidence measure, i.e., among multiple classifiers, those that yield larger margins are better [2].
The SVM score for classifying observation x is the signed distance from x to the decision boundary ranging from -∞ to +∞. A positive score for a class indicates that x is predicted to be in that class, a negative score indicates otherwise.
The score is also the numerical, predicted response for x, $$f(x)$$, computed by the trained SVM classification function
$$f(x)={\displaystyle \sum _{j=1}^{n}{\alpha}_{j}}{y}_{j}G({x}_{j},x)+b,$$
where $$({\alpha}_{1},\mathrm{...},{\alpha}_{n},b)$$ are the estimated SVM parameters, $$G({x}_{j},x)$$ is the dot product in the predictor space between x and the support vectors, and the sum includes the training set observations.
For binary classification, the software defines the margin for observation j, m_{j}, as
$${m}_{j}=2{y}_{j}f({x}_{j}),$$
where y_{j} ∊ {-1,1}, and f(x_{j}) is the predicted score of observation j for the positive class. However, the literature commonly uses m_{j} = y_{j}f(x_{j}) to define the margin.
[1] Christianini, N., and J. C. Shawe-Taylor. An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge, UK: Cambridge University Press, 2000.
[2] Hu, Q., X. Che, L. Zhang, and D. Yu. "Feature Evaluation and Selection Based on Neighborhood Soft Margin." Neurocomputing. Vol. 73, 2010, pp. 2114–2124.
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