Local-dependence profile

Represent the expected value of the model $f$ over the conditional distribution of $\underline{X}^{-j}$ given $X^j = z$

$$\begin{equation} g_{LD}^{f, j}(z) = E_{\underline{X}^{-j}|X^j=z}\left\{f\left(\underline{X}^{j|=z}\right)\right\} \end{equation}$$

This functions works really with if $X^j$ is a categorical variable, but if it continues we can next equation based on $N_j$ defined as the set of observations with the value of $X^j$ close to $z$.

$$\begin{equation} \hat g_{LD}^{j}(z) = \frac{1}{|N_j|} \sum_{k\in N_j} f\left(\underline{x}_k^{j| = z}\right) \end{equation}$$