We introduce the Integrated Tsallis Combination (ITC), a hybrid impurity measure for decision tree learning that combines normalized Tsallis entropy with an exponential polarization component. While many existing measures sacrifice theoretical soundness for computational efficiency or vice versa, ITC provides a mathematically principled framework that balances both aspects. The core innovation lies in the complementarity between Tsallis entropy's information-theoretic foundations and the polarization component's sensitivity to distributional asymmetry. We establish key theoretical properties-concavity under explicit parameter conditions, proper boundary conditions, and connections to classical measures-and provide a rigorous justification for the hybridization strategy. Through an extensive comparative evaluation on seven benchmark datasets comparing 23 impurity measures with five-fold repetition, we show that simple parametric measures (Tsallis $α=0.5$) achieve the highest average accuracy ($91.17\%$), while ITC variants yield competitive results ($88.38-89.16\%$) with strong theoretical guarantees. Statistical analysis (Friedman test: $χ^2=3.89$, $p=0.692$) reveals no significant global differences among top performers, indicating practical equivalence for many applications. ITC's value resides in its solid theoretical grounding-proven concavity under suitable conditions, flexible parameterization ($α$, $β$, $γ$), and computational efficiency $O(K)$-making it a rigorous, generalizable alternative when theoretical guarantees are paramount. We provide guidelines for measure selection based on application priorities and release an open-source implementation to foster reproducibility and further research.

Ordinal Classification (OC) is a machine learning field that addresses classification tasks where the labels exhibit a natural order. Unlike nominal classification, which treats all classes as equally distinct, OC takes the ordinal relationship into account, producing more accurate and relevant results. This is particularly critical in applications where the magnitude of classification errors has implications. Despite this, OC problems are often tackled using nominal methods, leading to suboptimal solutions. Although decision trees are one of the most popular classification approaches, ordinal tree-based approaches have received less attention when compared to other classifiers. This work conducts an experimental study of tree-based methodologies specifically designed to capture ordinal relationships. A comprehensive survey of ordinal splitting criteria is provided, standardising the notations used in the literature for clarity. Three ordinal splitting criteria, Ordinal Gini (OGini), Weighted Information Gain (WIG), and Ranking Impurity (RI), are compared to the nominal counterparts of the first two (Gini and information gain), by incorporating them into a decision tree classifier. An extensive repository considering 45 publicly available OC datasets is presented, supporting the first experimental comparison of ordinal and nominal splitting criteria using well-known OC evaluation metrics. Statistical analysis of the results highlights OGini as the most effective ordinal splitting criterion to date. Source code, datasets, and results are made available to the research community.

Variable importance measures are often used to highlight key predictor variables in tree-based ensemble models. However, there has generally been a lack of a theoretical understanding of these measures. In this paper, the authors study the theoretical properties of the Mean Decrease Impurity (MDI) importance measures (such as the Gini importance). Through most of the paper they use the Shanon entropy as the impurity measure but show that the theorems and results are applicable to any impurity measure. They begin with an asymptotic analysis of totally randomized fully-developed tree ensembles learned using an infinitely large ensemble.

Decision Tree is a well understood Machine Learning model that is based on minimizing impurities in the internal nodes. The most common impurity measures are Shannon entropy and Gini impurity. These impurity measures are insensitive to the order of training data and hence the final tree obtained is invariant to any permutation of the data. This leads to a serious limitation in modeling data instances that have order dependencies. In this work, we propose the use of Effort-To-Compress (ETC) - a complexity measure, for the first time, as an impurity measure. Unlike Shannon entropy and Gini impurity, structural impurity based on ETC is able to capture order dependencies in the data, thus obtaining potentially different decision trees for different permutations of the same data instances (Permutation Decision Trees). We then introduce the notion of Permutation Bagging achieved using permutation decision trees without the need for random feature selection and sub-sampling. We compare the performance of the proposed permutation bagged decision trees with Random Forests. Our model does not assume that the data instances are independent and identically distributed. Potential applications include scenarios where a temporal order present in the data instances is to be respected.

Recently, decision trees (DT) have been used as an explainable representation of controllers (a.k.a. strategies, policies, schedulers). Although they are often very efficient and produce small and understandable controllers for discrete systems, complex continuous dynamics still pose a challenge. In particular, when the relationships between variables take more complex forms, such as polynomials, they cannot be obtained using the available DT learning procedures. In contrast, support vector machines provide a more powerful representation, capable of discovering many such relationships, but not in an explainable form. Therefore, we suggest to combine the two frameworks in order to obtain an understandable representation over richer, domain-relevant algebraic predicates. We demonstrate and evaluate the proposed method experimentally on established benchmarks.

Random forests have been widely used for their ability to provide so-called importance measures, which give insight at a global (per dataset) level on the relevance of input variables to predict a certain output. On the other hand, methods based on Shapley values have been introduced to refine the analysis of feature relevance in tree-based models to a local (per instance) level. In this context, we first show that the global Mean Decrease of Impurity (MDI) variable importance scores correspond to Shapley values under some conditions. Then, we derive a local MDI importance measure of variable relevance, which has a very natural connection with the global MDI measure and can be related to a new notion of local feature relevance. We further link local MDI importances with Shapley values and discuss them in the light of related measures from the literature. The measures are illustrated through experiments on several classification and regression problems.

Recent advances have shown how decision trees are apt data structures for concisely representing strategies (or controllers) satisfying various objectives. Moreover, they also make the strategy more explainable. The recent tool dtControl had provided pipelines with tools supporting strategy synthesis for hybrid systems, such as SCOTS and Uppaal Stratego. We present dtControl 2.0, a new version with several fundamentally novel features. Most importantly, the user can now provide domain knowledge to be exploited in the decision tree learning process and can also interactively steer the process based on the dynamically provided information. To this end, we also provide a graphical user interface. It allows for inspection and re-computation of parts of the result, suggesting as well as receiving advice on predicates, and visual simulation of the decision-making process. Besides, we interface model checkers of probabilistic systems, namely Storm and PRISM and provide dedicated support for categorical enumeration-type state variables. Consequently, the controllers are more explainable and smaller.