Partial Dependence Plot

Definition

The marginal effect one or two features have on the predicted outcome of a ML model.

A partial dependence plot can show whether the relationship between the target and a feature is linear, monotonic or more complex. For example, when applied to a linear regression model, partial dependence plots always show a linear relationship.

$S$ is a subset of features (one or two) that are independent with the others. $C$ are all other features that are not in $S$. We assume that all features in $C$ are uncorrelated with the ones in $S$.

The partial dependence is ${\hat{f}}_S\left(x_S\right)=E_{X_C}\left[\hat{f}\left(x_S,X_C\right)\right]=\int \hat{f}\left(x_S,X_C\right)d\mathbb{P}\left(X_C\right).$ We can estimate this integral using the Monte-Carlo for Integrals method. We then get the simple average over all data points where we fix the values in $S$ to the value from $x$. ${\hat{f}}_S\left(x_S\right)=\frac{1}{n}{\sum }_{i=1}^n\hat{f}\left(x_S,x_C^{(i)}\right).$

PDP-based Feature Importance

We can use the flatness of the PDP to determine the Feature Importance.

• flat → low importance
• high Variance → high importance

$I\left(x_S\right)=\sqrt{\frac{1}{K-1}{\sum }_{k=1}^K{\left({\hat{f}}_S\left(x_S^{(k)}\right)-\frac{1}{K}{\sum }_{k=1}^K{\hat{f}}_S\left(x_S^{(k)}\right)\right)}^2}$

Examples

For more than one feature you can use a Heatmap:

Pros

• intuitive
• clear interpretation
• easy to implement
• “causal” interpretation (at least in the model context, not in the real world though)

Cons

• maximum of two features
• assumption of independence with other features
  • else there might be data points that are extremely unlikely (see rug plot)
  • → use Accumulated Local Effect Plot
• heterogeneous effects might be hidden
  • → use Individual Conditional Expectation Curve
• dont show feature distribution
  • → use rug plot (see example)

In sklearn

Implemented + you can use PDPBox