Approximating an AL profile

3. Replace the integral in by a summation.

Smooth

\[\begin{equation} \widetilde{g}_{AL}^{j}(z) = \sum_{k=1}^K \left[ \frac{1}{\sum_{l} w_l(z_k)} \sum_{i=1}^N w_{i}(z_k) \left\{f\left(\underline{x}_i^{j| = z_k}\right) - f\left(\underline{x}_i^{j| = z_k - \Delta}\right)\right\}\right] - \hat{c} \end{equation}\]

• \(z_k\) (\(k=0, \ldots, K\)) form a uniform grid covering the interval \((z_0,z)\) with step \(\Delta = (z-z_0)/K\)
• weight \(w_i(z_k)\) captures the distance between point \(z_k\) and observation \(x_i^j\).