Ex. 3.24

Ex. 3.24

LAR directions. Using the notation around equation (3.55) on page 74, show that the LAR direction makes an equal angle with each of the predictors in \(\mathcal{A}_k\).

Soln. 3.24

By definition of LAR we have

\[\begin{eqnarray} \bX_{\mathcal{A}_k}^Tu_k &=&\bX_{\mathcal{A}_k}^T(\bX_{\mathcal{A}_k}\delta_k)\non\\ &=&\bX_{\mathcal{A}_k}^T\bX_{\mathcal{A}_k}(\bX_{\mathcal{A}_k}^T\bX_{\mathcal{A}_k})^{-1}\bX_{\mathcal{A}_k}^Tr_k\non\\ &=&\bX_{\mathcal{A}_k}^Tr_k.\non \end{eqnarray}\]

By procedures of LAR, a new predictor \(x_{j'}\) is added when the absolute value of \(x_{j'}^Tr\) equals that of \(x_j^Tr\) for all predictors \(x_j\in \mathcal{A}_k\), we know the direction \(u_k\) makes an equal angle with all predictors in \(\mathcal{A}_k\).