Kernel Linear Discriminant

Discriminant analysis is a statistical technique used to classify objects or cases into two or more groups based on a set of predictor variables. The aim of discriminant analysis is to find a function or set of functions that can discriminate between the groups with high accuracy.

Discriminant function

Consider a $k$-class classification dataset $D=\{(\bold x_1, y_1), (\bold x_2, y_2), \dots,(\bold x_n, y_n)\}$, where $\bold x_i \in \R^d$ and $y_i \in \{1, 2, \dots, k\}$. A discriminant function assigns an input vector $\bold x$ to one of the $k$ classes.

Generalized linear discriminant function

A generalized linear discriminant is a discriminant function that’s linear in the transformed features $\phi(\bold x)$.

Two-class formulation

When the number of classes is $2$, that is, $k=2$, the generalised linear discriminant can be formulated as a generalised linear regression model along with a threshold. In particular, considering the class labels from the set $\{-1,1\}$, the generalised linear discriminant can be defined as follows:

$${\widehat{y}}_i=g$$ ...