Na rozdiel od distribúcií na reálnej osi, vo viacrozmerných priestoroch neexistuje jednoznačne prijímaná definícia symetrie rozdelenia. Niekoľko rôznych prístupov kategorizujú Zuo a Serfling (2000), z čoho neskôr vychádza Serfling (2006) pri uvádzaní týchto definícií do štatistickej literatúry. V príspevku preskúmame niektoré tvrdenia Zua a Serflinga (2000) a ukážeme, že najzaujímavejšie dôkazy v ich článku nie sú úplné. Pri ďalšom skúmaní týchto problémov narazíme na neznámy dôkaz Funkovej charakterizácie symetrie konvexných telies - problému, formulovaného v roku 1913, ktorý bol vyriešený až v roku 1970. Pokiaľ vieme, jedná sa o prvý elementárny dôkaz tohto významného tvrdenia v literatúre.
Depth has become a quite popular concept in functional data analysis. In the talk we discuss its general framework. We show that most known functional depths can be classified into few groups, within which they share similar theoretical properties. We focus on uniform consistency results for the sample versions of these functionals, and demonstrate that some well-known approaches to depth assessment are hardly theoretically adequate.
In nonparametric statistics the concept of quantiles is of paramount importance. In multivariate spaces, however, quantiles cannot be defined directly, due to the lack of natural ordering of points. In the talk we focus on one possible solution to this problem, using a tool called data depth. Depth is a function that quantifies the "centrality" of points, with respect to a given probability distribution. Points with high depth values form the "inner" quantile regions of the distribution; points of low depth lie on the outskirts of the data cloud. We discuss approaches to the definition of data depth, and illustrate these in a series of simple examples. The applications of this methodology include data visualisation,(robust) estimation, classification, clustering, or outlier detection for multivariate, high-dimensional, and even functional (infinite-dimensional) datasets.
A series of informal talks about the diverse aspects of functional data analysis. The workshop is open to everyone.
It is not unusual in cases where a body is discovered that it is necessary to determine a time of death or more formally a post mortem interval (PMI). Forensic entomology can be used to estimate this PMI by examining evidence obtained from the body from insect larvae growth. In this talk, we propose a method to estimate the hatching time of larvae (or maggots) based on their lengths, the temperature profile at the crime scene and experimental data on larval development. This requires the estimation of a time-dependent growth curve from experiments where larvae have been exposed to a relatively small number of constant temperature profiles. Since the temperature influences the developmental speed, a crucial steps is the time alignment of the curves at different temperatures. We then propose a model for time varying temperature profiles based on the local growth rate estimated from the experimental data. This allows us to find out the most likely hatching time for a sample of larvae from the crime scene. We explore via simulations the robustness of the method to errors in the estimated temperature profile and apply it to the data from two criminal cases from the United Kingdom. Asymptotic properties are also provided for the estimators of the growth curves and the hatching time.
Joint work with D. Pigoli, J.A.D. Aston, A. Mazumder, C. Richards and M.J.R. Hall.
In spatial statistics it is a custom in the last decades to use functional summary statistics rather than numerical ones to describe observed data or theoretical properties of a model. This makes statistical inference of such data challenging and functional data analysis comes in. In this talk we will review the concept of envelope tests that has been developed in recent years and has become popular in the field of spatial statistic. The envelope test deals with the situation where we have observed a single function (as is common in this field) and we want to decide if it is extreme or not extreme with respect to a given population of functions. The advantage of the envelope test is that it provides a graphical interpretation and indicates the reason of (possible) rejection. This is important in applications as it facilitates coming up with interpretations and new hypotheses.
In the broad framework of functional data analysis, the aim of this contribution is to introduce a functional regression framework for modelling space-time geochemical measurements. The motivating data set includes monthly measurements of potassium chloride pH taken from the site near Brno, Czech Republic. Sampling locations were selected with the purpose of testing if the site can be divided in two parts, agricultural and forest soil, according to its chemical properties. We suggest treating measurements as functions of time distributed in space and propose a function-on-scalar spatial regression model to describe the temporal distribution of the geochemical elements. To test for the possible differences between the two soil types, we propose a non-parametric functional testing procedure. The inference cannot be done directly on the original observations due to their dependency on spatial coordinates, instead it is performed on the residuals of the spatial functional model. Several regression models were fit to the data in order to derive spatially independent and thus permutable residuals. The proposed methodology will be demonstrated on the available geochemical dataset and geological interpretation of the results will be given.
Joint work with A. Menafoglio, A. Pini and E. Fišerová.
Sleep is a dynamical process, which plays important role in our lives. Its structure, quality and length influence humans' daily behaviour, affectivity, mood and also health. Current studies dealing with relationship between sleep structure and humans' physiological state or well-being measures are based on the extraction of one-dimensional sleep characteristics and their correlation with variables representing daily life behaviour. However, we hypothesise that this can lead to the loss of important information about the sleep dynamics. The Probabilistic sleep model (PSM) provides an alternative way of sleep modelling. The PSM operates on three-second long time segments of the EEG signal for each of which probability values of their relationship to one of sleep states - called microstates - is computed. Considering the probability values of a sleep microstate as a function of time we obtain a sleep probabilistic curve. In this presentation we provide an overview of chosen techniques of the functional data analysis, which may be useful in the sleep structure analysis and its relationships with measures representing daily behaviour (subjectively scored sleep quality, physiological factors, cognitive tests). We mainly focus on the functional cluster analysis, time alignment of curves and multilevel functional principal component analysis.
Joint work with R. Rosipal.
In many situations, the shape of functional observations is an important feature that must be taken into account in statistical analysis. The information about the shape properties can be extracted from the derivatives of the sample trajectories. Though, this approach can be applied only if the curves are regular and smooth, and the derivatives must be estimated. We present a simple alternative to this methodology based on simultaneous evaluation of multivariate projections of the data. This technique does not require smoothness or continuity, yet provides fine recognition of shape traits of the curves. The idea is illustrated on - but not limited to - functional data depth.
A series of informal talks of the members of the research team on their past and present research activities, and general research interests. The workshop is open to everyone, no registration is needed.
Point processes are stochastic models for random collections of points, i.e. in a fixed observation window the positions as well as the number of observed points is random. The points may represent e.g. positions of trees of a given species in a forest, home addresses of patiens with a certain disease, nuclei of specific cells in a tissue etc. We discuss some basic characteristics of the point processes that describe their distribution and show how they can be used to test statistical hypotheses about the observed data. Simulation-based (Monte-Carlo) tests are often used in this context. Our attention will focus mainly on the case where the null hypothesis is not specific enough to enable simulations from the null model and we will discuss the stochastic reconstruction approach - it can be used instead of simulations to provide independent replicates (reconstructions, not copies) of the data to be used in the Monte-Carlo tests.
We discuss Ramsey numbers of ordered hypergraphs. That is, for an ordered k-uniform hypergraph H, we ask for the minimum positive integer N such that every 2-coloring of the complete ordered k-uniform hypergraph on N vertices contains a monochromatic copy of H. We consider several estimates on ordered Ramsey numbers for various classes of ordered hypergraphs and we show that some of them admit natural geometric interpretations that yield new results for various extremal problems in discrete geometry. In particular, we mention connections between estimating ordered Ramsey numbers of monotone paths and the Erdos-Szekeres theorem.
How well can a convex body be approximated by a polytope? This is a fundamental question in convex geometry, also in view of applications in many other areas of mathematics and related fields. It often involves side conditions like a prescribed number of vertices and a requirement that the body contains the polytope or vice versa. Accuracy of approximation is often measured in the symmetric difference metric but other metrics can and have been considered. We will present several results, mostly related to approximation by "random polytopes". We will introduce floating bodies and an affine invariant from affine differential geometry associated to them, the affine surface area. This affine invariant appears naturally as an important ingredient in approximation questions.
We establish asymptotic results for weighted floating bodies of polytopes, which show connections between the weighted volume of a polytope and its flags. This connection is established by introducing the notion of flag simplices, which translate between the metric and combinatorial structure.