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.

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