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).