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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.

It is well-known that real data often contain outliers. The term outlier typically refers to a case, that is, a row of the $n \times d$ data matrix. In recent times a different type has come into focus, the cellwise outliers. These are suspicious cells (entries) that can occur anywhere in the data matrix. Even a relatively small proportion of outlying cells can contaminate over half the rows, which is a problem for rowwise robust methods. In this article we discuss the challenges posed by cellwise outliers, and some methods developed so far to deal with them. We obtain new results on cellwise breakdown values for location, covariance and regression. We also propose a cellwise robust method for correspondence analysis, with real data illustrations. The paper concludes by formulating some points for debate.

Outliers contaminating data sets are a challenge to statistical estimators. Even a small fraction of outlying observations can heavily influence most classical statistical methods. In this paper we propose generalized spherical principal component analysis, a new robust version of principal component analysis that is based on the generalized spatial sign covariance matrix. Supporting theoretical properties of the proposed method including influence functions, breakdown values and asymptotic efficiencies are studied, and a simulation study is conducted to compare our new method to existing methods. We also propose an adjustment of the generalized spatial sign covariance matrix to achieve better Fisher consistency properties. We illustrate that generalized spherical principal component analysis, depending on a chosen radial function, has both great robustness and efficiency properties in addition to a low computational cost.

Passing-Bablok regression is a standard tool for method and assay comparison studies thanks to its place in industry guidelines such as CLSI. Unfortunately, its computational cost is high as a naive approach requires O(n2) time. This makes it impossible to compute the Passing-Bablok regression estimator on large datasets. Additionally, even on smaller datasets it can be difficult to perform bootstrap-based inference. We introduce the first quasilinear time algorithm for the equivariant Passing-Bablok estimator. In contrast to the naive algorithm, our algorithm runs in O(n log(n)) expected time using O(n) space, allowing for its application to much larger data sets. Additionally, we introduce a fast estimator for the variance of the Passing-Bablok slope and discuss statistical inference based on bootstrap and this variance estimate. Finally, we propose a diagnostic plot to identify influential points in Passing-Bablok regression. The superior performance of the proposed methods is illustrated on real data examples of clinical method comparison studies.

The usual Minimum Covariance Determinant (MCD) estimator of a covariance matrix is robust against casewise outliers. These are cases (that is, rows of the data matrix) that behave differently from the majority of cases, raising suspicion that they might belong to a different population. On the other hand, cellwise outliers are individual cells in the data matrix. When a row contains one or more outlying cells, the other cells in the same row still contain useful information that we wish to preserve. We propose a cellwise robust version of the MCD method, called cellMCD. Its main building blocks are observed likelihood and a sparsity penalty on the number of flagged cellwise outliers. It possesses good breakdown properties. We construct a fast algorithm for cellMCD based on concentration steps (C-steps) that always lower the objective. The method performs well in simulations with cellwise outliers, and has high finite-sample efficiency on clean data. It is illustrated on real data with visualizations of the results.

We propose a data-analytic method for detecting cellwise outliers. Given a robust covariance matrix, outlying cells (entries) in a row are found by the cellHandler technique which combines lasso regression with a stepwise application of constructed cutoff values. The penalty term of the lasso has a physical interpretation as the total distance that suspicious cells need to move in order to bring their row into the fold. For estimating a cellwise robust covariance matrix we construct a detection-imputation method which alternates between flagging outlying cells and updating the covariance matrix as in the EM algorithm. The proposed methods are illustrated by simulations and on real data about volatile organic compounds in children.

This is an invited comment on the discussion paper "The power of monitoring: how to make the most of a contaminated multivariate sample" by A. Cerioli, M. Riani, A. Atkinson and A. Corbellini that will appear in the journal Statistical Methods & Applications.

Surface, image and video data can be considered as functional data with a bivariate domain. To detect outlying surfaces or images, a new method is proposed based on the mean and the variability of the degree of outlyingness at each grid point. A rule is constructed to flag the outliers in the resulting functional outlier map. Heatmaps of their outlyingness indicate the regions which are most deviating from the regular surfaces. The method is applied to fluorescence excitation-emission spectra after fitting a PARAFAC model, to MRI image data which are augmented with their gradients, and to video surveillance data.