Data depth and floating body

Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth

Generative linear mixture modelling.

For multivariate data with a low–dimensional latent structure, a novel approach to linear dimension reduction based on Gaussian mixture models is pro- posed. A generative model is assumed for the data, where the mixture centres (or ‘mass points’) are positioned along lines or planes spanned through the data cloud. All involved parameters are estimated simultaneously through the EM al- gorithm, requiring an additional iteration within each M-step. Data points can be projected onto the low–dimensional space by taking the posterior mean over the estimated mass points. The compressed data can then be used for further pro- cessing, for instance as a low–dimensional predictor in a multivariate regression problem

A Structured Domain is Responsible for Bundle Selection of Myosin X

Primer pla, contrapicat, d'un edifici d'habitatges al carrer Avinyó, 24. Consta de: planta baixa i quatre plantes pis. Els seus balcons, escassament volats, s'allunyen del que és habitual en les edificacions d'aquest tipus, dins el S. XVIII

Halfspace depth for general measures: The ray basis theorem and its consequences

The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under minimal assumptions on the underlying measure. The proof applies not only to the median, but also to other trimmed regions. Motivated by the technical development of the amended ray basis theorem, we specify connections between the trimmed regions, floating bodies, and additional equi-affine convex sets related to the depth. As a consequence, minimal conditions for the strict monotonicity of the depth are obtained. Applications to the computation of the depth and robust estimation are outlined

Another look at halfspace depth: Flag halfspaces with applications

The halfspace depth is a well studied tool of nonparametric statistics in multivariate spaces, naturally inducing a multivariate generalisation of quantiles. The halfspace depth of a point with respect to a measure is defined as the infimum mass of closed halfspaces that contain the given point. In general, a closed halfspace that attains that infimum does not have to exist. We introduce a flag halfspace - an intermediary between a closed halfspace and its interior. We demonstrate that the halfspace depth can be equivalently formulated also in terms of flag halfspaces, and that there always exists a flag halfspace whose boundary passes through any given point $x$, and has mass exactly equal to the halfspace depth of $x$. Flag halfspaces allow us to derive theoretical results regarding the halfspace depth without the need to differentiate absolutely continuous measures from measures containing atoms, as was frequently done previously. The notion of flag halfspaces is used to state results on the dimensionality of the halfspace median set for random samples. We prove that under mild conditions, the dimension of the sample halfspace median set of $d$-variate data cannot be $d-1$, and that for $d=2$ the sample halfspace median set must be either a two-dimensional convex polygon, or a data point. The latter result guarantees that the computational algorithm for the sample halfspace median form the R package TukeyRegion is exact also in the case when the median set is less-than-full-dimensional in dimension $d=2$

Hloubka funkcionálních dat

Hĺbková funkcia (resp. funkcionál) je moderný neparametrický nástroj štatistickej analýzy (konečnorozmerných) dát s množstvom praktických aplikácií. V práci sa zameriame na možnosti rozšírenia konceptu hĺbky na prípad funkcionálnych dát. V prípade konečnorozmerných funkcionálnych dát využijeme izomorfizmus priestoru funkcií a konečnorozmerného euklidovského priestoru, čo nám umožní zaviesť indukované hĺbky funkcionálnych dát. Dokážeme tvrdenie o vlastnostiach indukovaných hĺbok a na príkladoch si ukážeme možnosti a obmedzenia ich praktického použitia. Ďalej popíšeme a na jednoduchých príkladoch ukážeme výhody aj nevýhody zavedených hĺbkových funkcionálov používaných v literatúre (Fraimanových-Munizovej hĺbok a pásových hĺbok). Na odstránenie najväčšej vyvstávajúcej nevýhody známych hĺbok pre funkcionálne dáta zavedieme novú, K-pásovú hĺbku založenú na rozšírení inferencie zo spojitých na hladké funkcie. Odvodíme niekoľko dôležitých vlastností a na záverečnej simulačnej štúdií ukážeme na príklade riadenej klasifikácie funkcionálnych dát praktickú výhodnosť nového prístupu oproti predchádzajúcim. Na záver porovnáme výpočetnú náročnosť všetkých predstavených hĺbkových funkcionálov.The depth function (functional) is a modern nonparametric statistical analysis tool for (finite-dimensional) data with lots of practical applications. In the present work we focus on the possibilities of the extension of the depth concept onto a functional data case. In the case of finite-dimensional functional data the isomorphism between the functional space and the finite-dimensional Euclidean space will be utilized in order to introduce the induced functional data depths. A theorem about induced depths' properties will be proven and on several examples the possibilities and restraints of it's practical applications will be shown. Moreover, we describe and demonstrate the advantages and disadvantages of the established depth functionals used in the literature (Fraiman-Muniz depths and band depths). In order to facilitate the outcoming drawbacks of known depths, we propose new, K-band depth based on the inference extension from continuous to smooth functions. Several important properties of the K-band depth will be derived. On a final supervised classification simulation study the reasonability of practical use of the new approach will be shown. As a conclusion, the computational complexity of all presented depth functionals will be compared.Department of Probability and Mathematical StatisticsKatedra pravděpodobnosti a matematické statistikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult