\varphi_\alpha (\mathbb{P}(Y=0|X\in A_j), \mathbb{P}(Y=1|X\in A_j))$. The impurity of node $A_j$ measures the class homogeneity in $A_j$, but does not depend on the predictive class label $z_j$. Let $\tilde{z}_j =\mathbbm{1}_{\{\mathbb{P}(Y=1|X\in A_j)\ge1/(1+\alpha)\}}$ denote the dominant class label in $A_j$ under weight $\alpha$. We define a signed impurity taking into account $z_j$ as $$\tilde{I}_\alpha(A_j, \mathbb{P}) = \mathbbm{1}_{\{z_j=\tilde{z}_j\}} I_\alpha(A_j, \mathbb{P}) + \mathbbm{1}_{\{z_j\ne\tilde{z}_j\}} (1-I_\alpha(A_j, \mathbb{P})).$$ If the predictive class label $z_j$ matches the dominant class label $\tilde{z}_j$ in node $A_j$, the signed impurity of node $A_j$ is equal to the impurity of node $A_j$ and is no greater than $1/2$. Otherwise, an extra penalty is applied. Taking a weighted average of the signed impurities across the leaf nodes, one obtains the tree impurity and signed tree impurity.

\[tree impurity\] Let $T$ be a tree and $A_1, A_2, \ldots A_m$ be all the leaf nodes of this tree. Up-weighting the minority class by $\alpha$, the tree impurity of $T$ is $$I_\alpha(T, \mathbb{P}) = \sum_{j=1}^m \mathbb{P}_\alpha(X\in A_j) I_\alpha(A_j, \mathbb{P}),$$ where $\mathbb{P}_\alpha(X\in A_j)$ is the weighted version of $\mathbb{P}$ defined in section 2.1.

\[signed tree impurity\] Under the notation of Definition \[tree impurity\], the signed tree impurity is $\tilde{I}_\alpha(T, \mathbb{P}) = \sum_{j=1}^m \mathbb{P}_\alpha(X\in A_j) \tilde{I}_\alpha(A_j, \mathbb{P}).$

SVR-Tree Classifiers
