Friedman’s H-statistic

If there is no interaction we can decompose the PDP into $P{D}_{jk}\left(x_j,x_k\right)=P{D}_j\left(x_j\right)+P{D}_k\left(x_k\right)$ and the prediction function into $\hat{f}(x)=P{D}_j\left(x_j\right)+P{D}_{-j}\left(x_{-j}\right).$ Here $-j$ means all features that are not $j$.

The two statistics are $H_{jk}^2=\frac{{\sum }_{i=1}^n{\left[P{D}_{jk}\left(x_j^{(i)},x_k^{(i)}\right)-P{D}_j\left(x_j^{(i)}\right)-P{D}_k\left(x_k^{(i)}\right)\right]}^2}{{\sum }_{i=1}^{n}P{D}_{jk}^2\left(x_j^{(i)},x_k^{(i)}\right)}$ (interaction of feature $j$ with $k$) and $H_j^2=\frac{{\sum }_{i=1}^n{\left[\hat{f}\left(x^{(i)}\right)-P{D}_j\left(x_j^{(i)}\right)-P{D}_{-j}\left(x_{-j}^{(i)}\right)\right]}^2}{{\sum }_{i=1}^n{\hat{f}}^2\left(x^{(i)}\right)}$ (interaction of feature $j$ with all other features).

The statistic is expensive to evaluate, it takes $2n^2$ and $3n^2$ calls to the model.

Examples

Pros

• underlying theory
• meaningful interpretation
• dimensionless (comparable across features and models)
• detects all interactions
• arbitrarily high interactions

Cons

• computationally expensive
• estimates have variance
• no test for strong enough interaction
• same as PDP → Correlation can lead to high values