Sparse modeling of categorial explanatory variables

Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $L_1$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS355 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Penalized Regression with Ordinal Predictors

Ordered categorial predictors are a common case in regression modeling. In contrast to the case of ordinal response variables, ordinal predictors have been largely neglected in the literature. In this article penalized regression techniques are proposed. Based on dummy coding two types of penalization are explicitly developed; the first imposes a difference penalty, the second is a ridge type refitting procedure. A Bayesian motivation as well as alternative ways of derivation are provided. Simulation studies and real world data serve for illustration and to compare the approach to methods often seen in practice, namely linear regression on the group labels and pure dummy coding. The proposed regression techniques turn out to be highly competitive. On the basis of GLMs the concept is generalized to the case of non-normal outcomes by performing penalized likelihood estimation. The paper is a preprint of an article published in the International Statistical Review. Please use the journal version for citation

Feature Extraction in Signal Regression: A Boosting Technique for Functional Data Regression

Main objectives of feature extraction in signal regression are the improvement of accuracy of prediction on future data and identification of relevant parts of the signal. A feature extraction procedure is proposed that uses boosting techniques to select the relevant parts of the signal. The proposed blockwise boosting procedure simultaneously selects intervals in the signal’s domain and estimates the effect on the response. The blocks that are defined explicitly use the underlying metric of the signal. It is demonstrated in simulation studies and for real-world data that the proposed approach competes well with procedures like PLS, P-spline signal regression and functional data regression. The paper is a preprint of an article published in the Journal of Computational and Graphical Statistics. Please use the journal version for citation

Feature Selection and Weighting by Nearest Neighbor Ensembles

In the field of statistical discrimination nearest neighbor methods are a well known, quite simple but successful nonparametric classification tool. In higher dimensions, however, predictive power normally deteriorates. In general, if some covariates are assumed to be noise variables, variable selection is a promising approach. The paper’s main focus is on the development and evaluation of a nearest neighbor ensemble with implicit variable selection. In contrast to other nearest neighbor approaches we are not primarily interested in classification, but in estimating the (posterior) class probabilities. In simulation studies and for real world data the proposed nearest neighbor ensemble is compared to an extended forward/backward variable selection procedure for nearest neighbor classifiers, and some alternative well established classification tools (that offer probability estimates as well). Despite its simple structure, the proposed method’s performance is quite good - especially if relevant covariates can be separated from noise variables. Another advantage of the presented ensemble is the easy identification of interactions that are usually hard to detect. So not simply variable selection but rather some kind of feature selection is performed. The paper is a preprint of an article published in Chemometrics and Intelligent Laboratory Systems. Please use the journal version for citation

Regularization and Model Selection with Categorial Effect Modifiers

The case of continuous effect modifiers in varying-coefficient models has been well investigated. Categorial effect modifiers, however, have been largely neglected. In this paper a regularization technique is proposed that allows for selection of covariates and fusion of categories of categorial effect modifiers in a linear model. It is distinguished between nominal and ordinal variables, since for the latter more economic parametrizations are warranted. The proposed methods are illustrated and investigated in simulation studies and real world data evaluations. Moreover, some asymptotic properties are derived

Generalized Functional Additive Mixed Models

We propose a comprehensive framework for additive regression models for non-Gaussian functional responses, allowing for multiple (partially) nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data as well as linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. Our implementation handles functional responses from any exponential family distribution as well as many others like Beta- or scaled non-central $t$-distributions. Development is motivated by and evaluated on an application to large-scale longitudinal feeding records of pigs. Results in extensive simulation studies as well as replications of two previously published simulation studies for generalized functional mixed models demonstrate the good performance of our proposal. The approach is implemented in well-documented open source software in the "pffr()" function in R-package "refund"

Having the Second Leg At Home - Advantage in the UEFA Champions League Knockout Phase?

In soccer knockout ties which are played in a two-legged format the team having the return match at home is usually seen as advantaged. For checking this common belief, we analyzed matches of the UEFA Champions League knockout phase since 1995. It is shown that the observed differences in frequencies of winning between teams first playing away and those which are first playing at home can be completely explained by their performances on the group stage and - more importantly - by the teams' general strength

Regularization and Model Selection with Categorial Predictors and Effect Modifiers in Generalized Linear Models

We consider varying-coefficient models with categorial effect modifiers in the framework of generalized linear models. We distinguish between nominal and ordinal effect modifiers, and propose adequate Lasso-type regularization techniques that allow for (1) selection of relevant covariates, and (2) identification of coefficient functions that are actually varying with the level of a potentially effect modifying factor. We investigate the estimators’ large sample properties, and show in simulation studies that the proposed approaches perform very well for finite samples, too. Furthermore, the presented methods are compared with alternative procedures, and applied to real-world medical data

Regularization and Model Selection for Item-on-Items Regression with Applications to Food Products' Survey Data

Ordinal data are quite common in applied statistics. Although some model selection and regularization techniques for categorical predictors and ordinal response models have been developed over the past few years, less work has been done concerning ordinal-on-ordinal regression. Motivated by survey datasets on food products consisting of Likert-type items, we propose a strategy for smoothing and selection of ordinally scaled predictors in the cumulative logit model. First, the original group lasso is modified by use of difference penalties on neighbouring dummy coefficients, thus taking into account the predictors' ordinal structure. Second, a fused lasso type penalty is presented for fusion of predictor categories and factor selection. The performance of both approaches is evaluated in simulation studies, while our primary case study is a survey on the willingness to pay for luxury food products

Regularization and Model Selection with Categorial Predictors and Effect Modifiers in Generalized Linear Models

Varying-coefficient models with categorical effect modifiers are considered within the framework of generalized linear models. We distinguish between nominal and ordinal effect modifiers, and propose adequate Lasso-type regularization techniques that allow for (1) selection of relevant covariates, and (2) identification of coefficient functions that are actually varying with the level of a potentially effect modifying factor. We investigate large sample properties, and show in simulation studies that the proposed approaches perform very well for finite samples, too. In addition, the presented methods are compared with alternative procedures, and applied to real-world medical data