Let Z be an n × 2 matrix with the ith row equal to (1, 0) if gi = 1 and (0, 1) if gi = 0 (i = 1, . . ., n). The optimal scores are assumed to be mutually orthogonal and normalized with respect to an inner product. Thus, the minimization of (1) is subject to the constraint N−1‖ZΘ‖2 = 1, where Θ = [θ(0) θ(1)]T is a 2 × 1 vector of the optimal scores. Hastie et al [16] state that the minimization of this constrained optimization problem leads to estimates of η that are proportional to the discriminant variables (ie, the discriminant function) in LDA. In particular, they propose the following algorithm for the estimation of the LDA functions Choose an initial score matrix Θ0 satisfying Θ0TDpΘ0=I, where Dp = ZTZ/n. Let Θ0∗=ZΘ0.Let X be the n × p matrix with ith row Xi. Fit a linear regression model of Θ0∗ on X, yielding fitted values Θ^. Let f^(X) be the vector of fitted regression functions.Obtain the eigenvector matrix Φ of Θ0∗TΘ^; the optimal scores are then Θ* = Θ0Φ.Define fopt(x)=ΦTf^(x).
