9.2 Separating Hyperplane

• Consider a matrix X of dimensions $n*p$, and a $y_{i} \in \{-1, 1\}$. We have a new observation, $x^*$, which is a vector $x^* = (x^*_{1}...x^*_{p})^T$ which we wish to classify to one of two groups.
• We will use a separating hyperplane to classify the observation.

• We can label the blue observations as $y_{i} = 1$ and the pink observations as $y_{i} = -1$.
• Thus, a separating hyperplane has the property s.t. $\beta_{0} + \beta_{1}X_{i1} + \beta_{2}X_{i2} ... + \beta_{p}X_{ip} > 0$ if $y_{i} =1$ and $\beta_{0} + \beta_{1}X_{i1} + \beta_{2}X_{i2} ... + \beta_{p}X_{ip} < 0$ if $y_{i} = -1$.
• In other words, a separating hyperplane has the property s.t. $y_{i}(\beta_{0} + \beta_{1}X_{i1} + \beta_{2}X_{i2} ... + \beta_{p}X_{ip}) > 0$ for all $i = 1...n$.
• Consider also the magnitude of $f(x^*)$. If it is far from zero, we are confident in its classification, whereas if it is close to 0, then $x^*$ is located near the hyperplane, and we are less confident about its classification.