Modelling rankings in R: the PlackettLuce package

This paper presents the R package PlackettLuce, which implements a generalization of the Plackett-Luce model for rankings data. The generalization accommodates both ties (of arbitrary order) and partial rankings (complete rankings of subsets of items). By default, the implementation adds a set of pseudo-comparisons with a hypothetical item, ensuring that the underlying network of wins and losses between items is always strongly connected. In this way, the worth of each item always has a finite maximum likelihood estimate, with finite standard error. The use of pseudo-comparisons also has a regularization effect, shrinking the estimated parameters towards equal item worth. In addition to standard methods for model summary, PlackettLuce provides a method to compute quasi standard errors for the item parameters. This provides the basis for comparison intervals that do not change with the choice of identifiability constraint placed on the item parameters. Finally, the package provides a method for model-based partitioning using covariates whose values vary between rankings, enabling the identification of subgroups of judges or settings that have different item worths. The features of the package are demonstrated through application to classic and novel data sets.Comment: In v2: review of software implementing alternative models to Plackett-Luce; comparison of algorithms provided by the PlackettLuce package; further examples of rankings where the underlying win-loss network is not strongly connected. In addition, general editing to improve organisation and clarity. In v3: corrected headings Table 4, minor edit

Selection of Ordinally Scaled Independent Variables

Ordinal categorial variables are a common case in regression modeling. Although the case of ordinal response variables has been well investigated, less work has been done concerning ordinal predictors. This article deals with the selection of ordinally scaled independent variables in the classical linear model, where the ordinal structure is taken into account by use of a difference penalty on adjacent dummy coefficients. It is shown how the Group Lasso can be used for the selection of ordinal predictors, and an alternative blockwise Boosting procedure is proposed. Emphasis is placed on the application of the presented methods to the (Comprehensive) ICF Core Set for chronic widespread pain. The paper is a preprint of an article accepted for publication in the Journal of the Royal Statistical Society Series C (Applied Statistics). Please use the journal version for citation

Non-Global Logarithms, Factorization, and the Soft Substructure of Jets

An outstanding problem in QCD and jet physics is the factorization and resummation of logarithms that arise due to phase space constraints, so-called non-global logarithms (NGLs). In this paper, we show that NGLs can be factorized and resummed down to an unresolved infrared scale by making sufficiently many measurements on a jet or other restricted phase space region. Resummation is accomplished by renormalization group evolution of the objects in the factorization theorem and anomalous dimensions can be calculated to any perturbative accuracy and with any number of colors. To connect with the NGLs of more inclusive measurements, we present a novel perturbative expansion which is controlled by the volume of the allowed phase space for unresolved emissions. Arbitrary accuracy can be obtained by making more and more measurements so to resolve lower and lower scales. We find that even a minimal number of measurements produces agreement with Monte Carlo methods for leading-logarithmic resummation of NGLs at the sub-percent level over the full dynamical range relevant for the Large Hadron Collider. We also discuss other applications of our factorization theorem to soft jet dynamics and how to extend to higher-order accuracy.Comment: 46 pages + appendices, 10 figures. v2: added current figures 4 and 5, as well as corrected several typos in appendices. v3: corrected some typos, added current figure 9, and added more discussion of fixed-order versus dressed gluon expansions. v4: fixed an error in numerics of two-dressed gluon; corrected figure 8, modified comparison to BMS. Conclusions unchanged. v5: fixed minor typ

Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of computing the worst-case error. A general algorithm is presented and explicit expressions for base~2 are given. To obtain an efficient component-by-component construction algorithm we exploit the structure of the underlying cyclic group. We compare our new higher order multivariate quadrature rules to existing quadrature rules based on higher order digital nets by computing their worst-case error. These numerical results show that the higher order polynomial lattice rules improve upon the known constructions of quasi-Monte Carlo rules based on higher order digital nets