Smooth boundaries between \(N_j\) subsets

If we want smooth boundaries, we need stop assuming that all predictions have the same impact in the local-dependence profile.

To solve this we use the function \(w_i(z)\) to capture the distance between \(z\) and \(x_i^j\) based on the density function of a normal distribution with mean \(0\) and standard deviation \(s\).

\(s\) plays the role of a smoothing factor.

\[\begin{equation} w_i(z) = \phi(z - x_i^j, 0, s) \end{equation}\]

Now we just to apply the next function.

\[\begin{equation} \tilde g_{LD}^{j}(z) = \frac{1}{\sum_k w_{k}(z)} \sum_{i = 1}^n w_i(z) f\left(\underline{x}_i^{j| = z}\right) \end{equation}\]