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 $x_i^j$.