Jakob Raymaekers

Jakob Raymaekers, Peter J. Rousseeuw, Tim Verdonck et al.

Linear model trees are regression trees that incorporate linear models in the leaf nodes. This preserves the intuitive interpretation of decision trees and at the same time enables them to better capture linear relationships, which is hard for standard decision trees. But most existing methods for fitting linear model trees are time consuming and therefore not scalable to large data sets. In addition, they are more prone to overfitting and extrapolation issues than standard regression trees. In this paper we introduce PILOT, a new algorithm for linear model trees that is fast, regularized, stable and interpretable. PILOT trains in a greedy fashion like classic regression trees, but incorporates an $L^2$ boosting approach and a model selection rule for fitting linear models in the nodes. The abbreviation PILOT stands for $PI$ecewise $L$inear $O$rganic $T$ree, where `organic' refers to the fact that no pruning is carried out. PILOT has the same low time and space complexity as CART without its pruning. An empirical study indicates that PILOT tends to outperform standard decision trees and other linear model trees on a variety of data sets. Moreover, we prove its consistency in an additive model setting under weak assumptions. When the data is generated by a linear model, the convergence rate is polynomial.

Jakob Raymaekers, Peter J. Rousseeuw, Thomas Servotte et al.

Random forests are widely used in regression. However, the decision trees used as base learners are poor approximators of linear relationships. To address this limitation we propose RaFFLE (Random Forest Featuring Linear Extensions), a novel framework that integrates the recently developed PILOT trees (Piecewise Linear Organic Trees) as base learners within a random forest ensemble. PILOT trees combine the computational efficiency of traditional decision trees with the flexibility of linear model trees. To ensure sufficient diversity of the individual trees, we introduce an adjustable regularization parameter and use node-level feature sampling. These modifications improve the accuracy of the forest. We establish theoretical guarantees for the consistency of RaFFLE under weak conditions, and its faster convergence when the data are generated by a linear model. Empirical evaluations on 136 regression datasets demonstrate that RaFFLE outperforms the classical CART and random forest methods, the regularized linear methods Lasso and Ridge, and the state-of-the-art XGBoost algorithm, across both linear and nonlinear datasets. By balancing predictive accuracy and computational efficiency, RaFFLE proves to be a versatile tool for tackling a wide variety of regression problems.

Jakob Raymaekers, Peter J. Rousseeuw

Classification by neural nets and by tree-based methods are powerful tools of machine learning. There exist interesting visualizations of the inner workings of these and other classifiers. Here we pursue a different goal, which is to visualize the cases being classified, either in training data or in test data. An important aspect is whether a case has been classified to its given class (label) or whether the classifier wants to assign it to different class. This is reflected in the (conditional and posterior) probability of the alternative class (PAC). A high PAC indicates label bias, i.e. the possibility that the case was mislabeled. The PAC is used to construct a silhouette plot which is similar in spirit to the silhouette plot for cluster analysis (Rousseeuw, 1987). The average silhouette width can be used to compare different classifications of the same dataset. We will also draw quasi residual plots of the PAC versus a data feature, which may lead to more insight in the data. One of these data features is how far each case lies from its given class. The graphical displays are illustrated and interpreted on benchmark data sets containing images, mixed features, and tweets.

Jakob Raymaekers, Wouter Verbeke, Tim Verdonck

In many practical applications, such as fraud detection, credit risk modeling or medical decision making, classification models for assigning instances to a predefined set of classes are required to be both precise as well as interpretable. Linear modeling methods such as logistic regression are often adopted, since they offer an acceptable balance between precision and interpretability. Linear methods, however, are not well equipped to handle categorical predictors with high-cardinality or to exploit non-linear relations in the data. As a solution, data preprocessing methods such as weight-of-evidence are typically used for transforming the predictors. The binning procedure that underlies the weight-of-evidence approach, however, has been little researched and typically relies on ad-hoc or expert driven procedures. The objective in this paper, therefore, is to propose a formalized, data-driven and powerful method. To this end, we explore the discretization of continuous variables through the binning of spline functions, which allows for capturing non-linear effects in the predictor variables and yields highly interpretable predictors taking only a small number of discrete values. Moreover, we extend upon the weight-of-evidence approach and propose to estimate the proportions using shrinkage estimators. Together, this offers an improved ability to exploit both non-linear and categorical predictors for achieving increased classification precision, while maintaining interpretability of the resulting model and decreasing the risk of overfitting. We present the results of a series of experiments in a fraud detection setting, which illustrate the effectiveness of the presented approach. We facilitate reproduction of the presented results and adoption of the proposed approaches by providing both the dataset and the code for implementing the experiments and the presented approach.

Jakob Raymaekers, Ruben H. Zamar

We study a framework of regularized $K$-means methods based on direct penalization of the size of the cluster centers. Different penalization strategies are considered and compared through simulation and theoretical analysis. Based on the results, we propose HT $K$-means, which uses an $\ell_0$ penalty to induce sparsity in the variables. Different techniques for selecting the tuning parameter are discussed and compared. The proposed method stacks up favorably with the most popular regularized $K$-means methods in an extensive simulation study. Finally, HT $K$-means is applied to several real data examples. Graphical displays are presented and used in these examples to gain more insight into the datasets.