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The reason that the Decision Tree does poorly here is the algorithm isn't equipped to deal with the situation you're throwing at it. You need to understand how a CART model gives its predicted output value for a continuous response. You fit a CART model to the response target, predicted by inputs category and A. You want the decision tree to learn the rule ...


This is not how decision trees work. Roughly speaking, decision tree splits data into bins (branches), conditionally on the features, and per each bin it predicts mean of the target variable. So for decision tree to predict something like identity function $y = f(y)$, you would need decision tree with the number of branches equal to the size of the data, i.e....


Same as with regular decision tree, isolation forest is not trained by directly minimizing some loss, but by using a dedicated algorithm. If you are interested in the algorithm, check the paper by it's authors, where they describe it in detail: Liu, Fei Tony, Ting, Kai Ming and Zhou, Zhi-Hua. (2008). “Isolation forest.” Data Mining. ICDM’08.

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