G. Louppe, L. Wehenkel, A. Sutera, P. Geurts. Understanding Variable Importances in Forests of Randomized Trees. Advances in Neural Information Processing Systems, 2013.

Louppe et al. discuss the Mean Decrease Impurity (MDI) variable importance measure and derive several theoretical properties. Given a forest of $T$ randomized trees, each expecting $D$-dimensional input $x = (x_1, \ldots, x_D)$, the Mean Decrease Impurity variable importance for dimension $d$ is computed as:

$MDI(x_d) = \frac{1}{T} \sum_{t = 1}^T \sum_{s(v) = x_n} \frac{N_v}{N} \Delta i(v)$(1)

where $N$ is the total number of examples, $N_v$ is the number of examples reaching inner node $v$ and $\Delta i(v)$ is the impurity decrease achieved at node $v$. Here, the second sum in equation (1) runs over all inner nodes $v$ in tree $t$ where feature dimension $d$ is selected as split feature.

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