Accumulated-local profile

Suppose that we know \(f()\), then we can calculate calculate the Partial Derivative base on any dependent variable \(X^j\) at a specific point \(\underline{u}\) to describe the local effect (change) of the model due to \(X^j\).

\[ q^j(\underline{u})=\left\{ \frac{\partial f(\underline{x})}{\partial x^j} \right\}_{\underline{x}=\underline{u}}. \]

And define the accumulated-local (AL) profile which measures the accumulated effect of changing \(X^j\) from \(z_0\) (near the lower bound of \(X^j\)) to \(z\), while averaging out the effects of other variables \(\underline{X}^{-j}\).

\[\begin{equation} g_{AL}^{j}(z) = \int_{z_0}^z \left[E_{\underline{X}^{-j}|X^j=v}\left\{ q^j(\underline{X}^{j|=v}) \right\}\right] dv + c \end{equation}\]

Averaging of the local effects allows avoiding the issue of capturing the effect of other variables in the profile for a particular variable in additive models (without interactions).