Table 2.

Variable Importance

For the m-th tree ${\widehat{f}}_{A,m}^{\ast }$, m ∈ (1, 2, …, M*), in the embedded tree model, do steps a) – c). Select the corresponding m-th out-of-bag (OOB) data which consists of the data not selected in the m-th bootstrap sample. Drop OOB data down the fitted tree ${\widehat{f}}_{A,m}^{\ast }$ and calculate prediction mean squared error, MSEA,m. For each variable $j\in \mathcal{P}\backslash {\mathcal{P}}_A^d$, do the following: Randomly permute the values of the jth variable X(j) in the OOB data. Drop permuted OOB data down the fitted tree ${\widehat{f}}_{A,m}^{\ast }$, and calculate the permuted mean squared error, ${\mathit{PMSE}}_{A,m}^j$. For $j\in {\mathcal{P}}_A^d,{\widehat{VI}}_A(j)=0$. For variable $j\in \mathcal{P}\backslash {\mathcal{P}}_A^d$, average over M* measurements to get the variable importance measure: $${\widehat{VI}}_A(j)=\frac{{\sum }_{m=1}^{M^{\ast }}{\mathit{PMSE}}_{A,m}^j}{{\sum }_{m=1}^{M^{\ast }}{\mathit{MSE}}_{A,m}}-1.$$