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 \(z\).

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