Chapter 21 How PD, CD and AL Profiles are different and which to choose

In previous chapters we introduced different was to calculate model level explainers for feature effects. A natural question is how these approaches are different and which one should we choose.

An example that illustrate differences between these approaches is presented in Figure @ref{accumulatedLocalEffects}. Here we have a model \(f(x_1, x_2) = x_1*x_2 + x_2\) and what is important features are correlated \(x_1 \sim U[-1,1]\) and \(x_2 = x_1\).

We have 8 points for which we calculated instance level profiles.

\(x_1\) \(x_2\)
-1 -1
-0.71 -0.71
-0.43 -0.43
-0.14 -0.14
0.14 0.14
0.43 0.43
0.71 0.71
1 1

Panel A) shows Ceteris Paribus for 8 data points, the feature \(x_1\) is on the OX axis while \(f\) is on the OY. Panel B) shows Partial Dependency Profiles calculated as an average from CP profiles.

\[ g_{PD}^{f,1}(z) = E[z*x^2 + x^2] = 0 \] Panel C) shows Conditional Dependency Profiles calculated as an average from conditional CP profiles. In the figure the conditioning is calculated in four bins, but knowing the formula for \(f\) we can calculated it directly as.

\[ g_{CD}^{f,1}(z) = E[X^1*X^2 + X^2 | X^1 = z] = z^2+z \]

Panel D) shows Accumulated Local Effects calculated as accumulated changes in conditional CP profiles. In the figure the conditioning is calculated in four bins, but knowing the formula for \(f\) we can calculated it directly as.

\[ g_{AL}^{f,1}(z) = \int_{z_0}^z E\left[\frac{\partial (X^1*X^2 + X^2)}{\partial x_1}|X^1 = v\right] dv = \int_{z_0}^z E\left[X^2|X^1 = v\right] dv = \frac{z^2 -1 }{2}, \]

(fig:accumulatedLocalEffects) Differences between Partial Dependency, Marginal and Accumulated Local Effects profiles. Panel A) shows Ceteris Paribus Profiles for 8 points. Panel B) shows Partial Dependency profiles, i.e. an average out of these profiles. Panel C shows Marginal profiles, i.e. an average from profiles similar to the point that is being explained. Panel D shows Accumulated Local Effects, i.e. effect curve that takes into account only changes in the Ceteris Paribus Profiles.

Figure 21.1: (fig:accumulatedLocalEffects) Differences between Partial Dependency, Marginal and Accumulated Local Effects profiles. Panel A) shows Ceteris Paribus Profiles for 8 points. Panel B) shows Partial Dependency profiles, i.e. an average out of these profiles. Panel C shows Marginal profiles, i.e. an average from profiles similar to the point that is being explained. Panel D shows Accumulated Local Effects, i.e. effect curve that takes into account only changes in the Ceteris Paribus Profiles.

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