Proximity scaling methods (e.g., multidimensional scaling) represent objects in a low dimensional configuration so that fitted distances between objects optimally approximate multivariate proximities. Next to finding the optimal configuration the goal is often also to assess groups of objects from the configuration. This can be difficult if the optimal configuration lacks clusteredness (coined c-clusteredness). We present Cluster Optimized Proximity Scaling (COPS), which attempts to solve this problem by finding a configuration that exhibts c-clusteredness. In COPS, a flexible scaling loss function (p-stress) is combined with an index that quantifies c-clusteredness in the solution, the OPTICS Cordillera. We present two variants of combining p-stress and Cordillera, one for finding the configuration directly and one for metaparameter selection for p-stress. The first variant is illustrated by scaling Californian counties with respect to climate change related natural hazards. We identify groups of counties with similar risk profiles and find that counties that are in high risk of drought are socially vulnerable. The second variant is illustrated by finding a clustered nonlinear representation of countries according to their history of banking crises from 1800 to 2010. (authors' abstract) / Series: Discussion Paper Series / Center for Empirical Research Methods

Accompanying the advances in computer technology is an increase collection of high dimensional data in many scientific and social studies. Sufficient dimension reduction (SDR) is a statistical method that enable us to reduce the dimension ofpredictors without loss of regression information. In this dissertation, we introduce principal asymmetric least squares (PALS) as a unified framework for linear and nonlinear sufficient dimension reduction. Classical methods such as sliced inverse regression (Li, 1991) and principal support vector machines (Li, Artemiou and Li, 2011) often do not perform well in the presence of heteroscedastic error, while our proposal addresses this limitation by synthesizing different expectile levels. Through extensive numerical studies, we demonstrate the superior performance of PALS in terms of both computation time and estimation accuracy. For the asymptotic analysis of PALS for linear sufficient dimension reduction, we develop new tools to compute the derivative of an expectation of a non-Lipschitz function. PALS is not designed to handle symmetric link function between the response and the predictors. As a remedy, we develop expectile-assisted inverse regression estimation (EA-IRE) as a unified framework for moment-based inverse regression. We propose to first estimate the expectiles through kernel expectile regression, and then carry out dimension reduction based on random projections of the regression expectiles. Several popular inverse regression methods in the literature including slice inverse regression, slice average variance estimation, and directional regression are extended under this general framework. The proposed expectile-assisted methods outperform existing moment-based dimension reduction methods in both numerical studies and an analysis of the Big Mac data. / Statistics

Proximity scaling (i.e., multidimensional scaling and related methods) is a versatile statistical method whose general idea is to reduce the multivariate complexity in a data set by employing suitable proximities between the data points and finding low-dimensional configurations where the fitted distances optimally approximate these proximities. The ultimate goal, however, is often not only to find the optimal configuration but to infer statements about the similarity of objects in the high-dimensional space based on the the similarity in the configuration. Since these two goals are somewhat at odds it can happen that the resulting optimal configuration makes inferring similarities rather difficult. In that case the solution lacks "clusteredness" in the configuration (which we call "c-clusteredness"). We present a version of proximity scaling, coined cluster optimized proximity scaling (COPS), which solves the conundrum by introducing a more clustered appearance into the configuration while adhering to the general idea of multidimensional scaling. In COPS, an arbitrary MDS loss function is parametrized by monotonic transformations and combined with an index that quantifies the c-clusteredness of the solution. This index, the OPTICS cordillera, has intuitively appealing properties with respect to measuring c-clusteredness. This combination of MDS loss and index is called "cluster optimized loss" (coploss) and is minimized to push any configuration towards a more clustered appearance. The effect of the method will be illustrated with various examples: Assessing similarities of countries based on the history of banking crises in the last 200 years, scaling Californian counties with respect to the projected effects of climate change and their social vulnerability, and preprocessing a data set of hand written digits for subsequent classification by nonlinear dimension reduction. (authors' abstract) / Series: Discussion Paper Series / Center for Empirical Research Methods

Proximity scaling (i.e., multidimensional scaling and related methods) is a versatile statistical method whose general idea is to reduce the multivariate complexity in a data set by employing suitable proximities between the data points and finding low-dimensional configurations where the fitted distances optimally approximate these proximities. The ultimate goal, however, is often not only to find the optimal configuration but to infer statements about the similarity of objects in the high-dimensional space based on the the similarity in the configuration. Since these two goals are somewhat at odds it can happen that the resulting optimal configuration makes inferring similarities rather difficult. In that case the solution lacks "clusteredness" in the configuration (which we call "c-clusteredness"). We present a version of proximity scaling, coined cluster optimized proximity scaling (COPS), which solves the conundrum by introducing a more clustered appearance into the configuration while adhering to the general idea of multidimensional scaling. In COPS, an arbitrary MDS loss function is parametrized by monotonic transformations and combined with an index that quantifies the c-clusteredness of the solution. This index, the OPTICS cordillera, has intuitively appealing properties with respect to measuring c-clusteredness. This combination of MDS loss and index is called "cluster optimized loss" (coploss) and is minimized to push any configuration towards a more clustered appearance. The effect of the method will be illustrated with various examples: Assessing similarities of countries based on the history of banking crises in the last 200 years, scaling Californian counties with respect to the projected effects of climate change and their social vulnerability, and preprocessing a data set of hand written digits for subsequent classification by nonlinear dimension reduction. (authors' abstract) / Series: Discussion Paper Series / Center for Empirical Research Methods

L'IRMd est un moyen non invasif permettant d'étudier in vivo la structure des fibres nerveuses du cerveau. Dans cette thèse, nous modélisons des données IRMd à l'aide de tenseurs d'ordre 4 (T4). Les problèmes de comparaison de groupes ou d'individu avec un groupe normal sont abordés, et résolus à l'aide d'analyses statistiques sur les T4s. Les approches utilisent des réductions non linéaires de dimension, et bénéficient des métriques non euclidiennes pour les T4s. Les statistiques sont calculées dans l'espace réduit, et permettent de quantifier la dissimilarité entre le groupe (ou l'individu) d'intérêt et le groupe de référence. Les approches proposées sont appliquées à la neuromyélite optique et aux patients atteints de locked in syndrome. Les conclusions tirées sont cohérentes avec les connaissances médicales actuelles. / DW-MRI is a non-invasive way to study in vivo the structure of nerve fibers in the brain. In this thesis, fourth order tensors (T4) were used to model DW-MRI data. In addition, the problems of group comparison or individual against a normal group were discussed and solved using statistical analysis on T4s. The approaches use nonlinear dimensional reductions, assisted by non-Euclidean metrics for T4s. The statistics are calculated in the reduced space and allow us to quantify the dissimilarity between the group (or the individual) of interest and the reference group. The proposed approaches are applied to neuromyelitis optica and patients with locked in syndrome. The derived conclusions are consistent with the current medical knowledge.