Ordinal Trees and Random Forests: Score-Free Recursive Partitioning and Improved Ensembles

Existing ordinal trees and random forests typically use scores that are assigned to the ordered categories, which implies that a higher scale level is used. Versions of ordinal trees are proposed that take the scale level seriously and avoid the assignment of artificial scores. The construction principle is based on an investigation of the binary models that are implicitly used in parametric ordinal regression. These building blocks can be fitted by trees and combined in a similar way as in parametric models. The obtained trees use the ordinal scale level only. Since binary trees and random forests are constituent elements of the proposed trees, one can exploit the wide range of binary trees that have already been developed. A further topic is the potentially poor performance of random forests, which seems to have been neglected in the literature. Ensembles that include parametric models are proposed to obtain prediction methods that tend to perform well in a wide range of settings. The performance of the methods is evaluated empirically by using several data sets.

Bijections for Faces of the Shi and Catalan Arrangements

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in ${R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.

Secretary Problem: Graphs, Matroids and Greedoids

In the paper, the generalisation of the well-known “secretary problem” is considered. The aim of the paper is to give a generalised model in such a way that the chosen set of the possible best k elements have to be independent of all previously rejected elements. The independence is formulated using the theory of greedoids and in their special cases—matroids and antimatroids. Examples of some special cases of greedoids (uniform, graphical matroids and binary trees) are considered. Applications in cloud computing are discussed.