Accumulated Local Effect Plot

Definition

Describes how features influence the prediction of a ML model on average. ALE plots are faster and unbiased alternative to PDPs. ALE plots are more meaningful in the case of correlated features.

The computation of a PDP for a feature that is strongly correlated with other features involves averaging predictions of artificial data instances that are unlikely in reality.

To get rid of unlikely data points we can use a Marginal Plot where we only look at the conditional distribution (bin around a value). This however still includes the effects of correlated features. E.g. lets say we have a feature that does not influence the target but it is strongly correlated to a different feature that does influcence it. Now when we create a Marginal Plot for that feature, we would still see an effect even though it doesn’t exist.

So we want to remove the correlation. We do this by only looking at a small interval and calculating the gradients (differences) in this small interval. In the end we integrate over these gradients to get the actual values.

1. Divide features into intervals
2. For each interval
   1. For each datapoint in interval
      1. Calculate prediction difference of lower and upper limit
   2. Average prediction differences
3. Integrate prediction differences
4. Center

$$\begin{array}{rl}{\hat{f}}_{S,ALE}\left(x_S\right)& ={\int }_{z_{0,S}}^{x_S}{E}_{X_C\mid X_S=x_S}\left[{\hat{f}}^S\left(X_s,X_c\right)\mid X_S=z_S\right]d{z}_S-\text{ constant }\\ & ={\int }_{z_{0,S}}^{x_S}\left({\int }_{x_C}{\hat{f}}^S\left(z_s,X_c\right)d\mathbb{P}\left(X_C\mid X_S=z_S\right)d\right)d{z}_S-\mathrm{constant}\end{array}$$

We can estimate these values

${\hat{\tilde{f}}}_{j,ALE}(x)={\sum }_{k=1}^{k_j(x)}\frac{1}{n_j(k)}{\sum }_{i:x_j^{(i)}\in N_j(k)}\left[\hat{f}\left(z_{k,j},x_{-j}^{(i)}\right)-\hat{f}\left(z_{k-1,j},x_{-j}^{(i)}\right)\right]$ ${\hat{f}}_{j,ALE}(x)={\hat{\tilde{f}}}_{j,ALE}(x)-\frac{1}{n}{\sum }_{i=1}^n{\hat{\tilde{f}}}_{j,ALE}\left(x_j^{(i)}\right)$

For Feature Interaction we can use a Heatmap again.

Compare this to the Partial Dependence Plot which is not able to capture the interactions.

To get the total effect it makes sense to look at the PDP, for ALE you have to calculate the total effect from the interaction and the individual effects.

Pros

• unbiased
• faster to compute
• clear interpretation
• centered at zero
• feature interactions

Cons

• interpretation of effect across intervals is not possible
• can become shakey (how big should intervals be)
• not ICE
• implementation is complex
• feature interaction heatmap hard to interpret
• still not working for strong Correlations