9.12 More than Two Classes

• The concept of separating hyperplanes does not extend naturally to more than two classes, but there are some ways around this.
• A one-versus-one approach constructs $K \choose 2$ SVMs, where $K$ is the number of classes. An observation is classified to each of the $K \choose 2$ classes, and the number of times it appears in each class is counted.
• The $k^\text{th}$ class might be coded as +1 versus the $(k')^\text{th}$ class is coded as -1.
• The data point is classified to the class for which it was most often assigned in the pairwise classifications.
• Another option is one-versus-all classification. This can be useful when there are a lot of classes.
• $K$ SVMs are fitted, and one of the K classes to the remaining $K-1$ classes.
• $\beta_{0k}...\beta_{pk}$ denotes the parameters that results from constructing an SVM comparing the $k$th class (coded as +1) to the other classes (-1).
• Assign test observation $x^*$ to the class $k$ for which $\beta_{0k} + ... + \beta_{pk}x^*_{p}$ is largest.