We propose a tree-based algorithm (μCART) for classification and regression problems in the context of functional data analysis, which allows to leverage measure learning and multiple splitting rules at the node level, with the objective of reducing error while retaining the interpretability of a tree. For each internal node, our main contribution is the idea of learning a weighted functional space by means of constrained convex optimization, which is then used to extract multiple weighted integral features from the functional predictors, in order to determine the binary split. The approach is designed to manage multiple functional predictors and/or responses, by defining suitable splitting rules and loss functions that can depend on the specific problem and can also be combined with additional scalar and categorical predictors, as the tree is grown with the original greedy CART algorithm. We focus on the case of scalar-valued functional predictors defined on unidimensional domains and illustrate the effectiveness of our method in both classification and regression tasks, through a simulation study and four real-world applications.

中文翻译：

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测量功能数据的诱导分类和回归树

我们提出了一种基于树的算法（μCART），用于在功能数据分析的背景下进行分类和回归问题，它允许在节点级别利用度量学习和多个拆分规则，以减少错误，同时保持数据的可解释性。一颗树。对于每个内部节点，我们的主要贡献是学习加权函数的想法空间通过约束凸优化，然后用于从功能预测器中提取多个加权积分特征，以确定二元分割。该方法旨在通过定义合适的拆分规则和损失函数来管理多个功能预测变量和/或响应，这些拆分规则和损失函数可以取决于特定问题，也可以与其他标量和分类预测变量结合，因为树是在原始贪婪的情况下生长的购物车算法。我们专注于在一维域上定义的标量值功能预测器的情况，并通过模拟研究和四个实际应用来说明我们的方法在分类和回归任务中的有效性。