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.

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