Method

Remember, one-dimensional CP profile for all possible values $z$ of the explanatory variable $j$ based the interest observation $\underline{x}_*$.

$$h^{j}_{\underline{x}_*}(z) \equiv f\left(\underline{x}_*^{j|=z}\right).$$

$vip_{CP}^j(\underline{x}_*)$ is the expected absolute deviation of the CP profile $h^{j}_{\underline{x}_*}()$ from the model’s prediction $f(\underline{x}_*)$, computed over the distribution $g^j(z)$ of the $j$-th explanatory variable.

$$vip_{CP}^j(\underline{x}_*) = \int_{\mathcal R} |h^{j}_{\underline{x}_*}(z) - f(\underline{x}_*)| g^j(z)dz=E_{X^j}\left\{|h^{j}_{\underline{x}_*}(X^j) - f(\underline{x}_*)|\right\}.$$