Approximating an AL profile

1. Replace the derivative with a finite difference of the CP profiles to remove the effect of all variables other than \(X^j\) (works for models without interactions).

\[ q^j(\underline{x}^{j|=v})dv \approx f(\underline{x}^{j|=v+dv})-f(\underline{x}^{j|=v}) \]

2. Consider a partition of the range of observed values \(x_{i}^j\) of variable \(X^j\) into \(K\) intervals.

\[ N_j(k)=\left(z_{k-1}^j,z_k^j\right] \qquad (k=1,\ldots,K) \]