doc. RNDr. Kateřina Helisová, Ph.D.

The preocular tear film is critically important for maintaining healthy ocular surface. In lagophthalmos, increased evaporation and tear film instability can occur. The level of tear matrix metalloproteinase 9 (MMP-9) is considered as a possible marker of ocular surface damage and inflammation. The aim of this study was to evaluate the possible usefulness of measuring tear film levels of MMP-9 in patients with lagophthalmos. Sixteen adult patients with unilateral lagophthalmos due to cerebellopontine angle mass surgery were included. Basic clinical examination including tear film osmolarity, degree of lagophthalmos, ocular surface sensitivity testing, corneal fluorescein staining, and tear break-up time (TBUT) were performed. Furthermore, tear MMP-9 quantification was performed and the values from lagophthalmic and contralateral healthy eye were compared. Possible correlations between tear MMP-9 levels and other parameters were analyzed. The Oxford score was higher in lagophthalmic eyes in comparison to healthy eyes. TBUT and corneal sensitivity were lower in lagophthalmic eyes. There was no difference in osmolarity between the two groups. Tear MMP-9 values were higher in lagophthalmic eyes. A higher MMP-9 value was associated with an increase in ocular surface fluorescein staining and a decrease of TBUT in lagophthalmic eyes. Tear MMP-9 may be used for monitoring ocular surface damage, contribute to early detection of inflammation progression and facilitate treatment adjustments.

The paper concerns a method for assessing similarity of realisations of random sets based on a construction of their morphological skeletons and a consequent covering of the realisations by unions of the so-called maximal discs. Since the realisations are considered to be binary images, the skeletons together with the corresponding discs can be viewed as realisations of marked point processes with specific properties. A special function for such marked point processes is defined. This function is analogous to the mark-weighted K-function. The function is then used for comparison of given realisations. More precisely, a random sample of the functions is taken from the realisations and the equality in distribution of the functions is tested by an envelope test and by a kernel test. The described procedure is illustrated on a simulation study with the aim to distinguish between realisations coming from different processes and to determine similarity of realisations coming from the same processes.

The paper concerns an overview of recently developed methods for comparing two realisations of random sets. The methods are based on deriving a sample of functions from each realisation, which describe the realisation shape, and consequent testing of equality of their probability distributions. The methods are briefly described, their advantages and disadvantages are mentioned and the conclusions are justified by a simulation study.

This paper concerns a method of testing the equality of distributions of random convex compact sets. The main theoretical result involves a construction of a metric on the space of distributions of random convex compact sets. We obtain it by using the theory of n-distances and the redefined characteristic function of random convex compact set. We propose an approximation of the metric through its finite-dimensional counterparts. This result leads to a new statistical test for testing the equality of distributions of two random convex compact sets. Consequently, we show a heuristic approach how to determine whether two realisations of random sets that can be approximated by a union of identically distributed random convex compact sets come from the same underlying process using the constructed test. Each procedure is justified by an extensive simulation study and the heuristic method for comparing random sets using their convex compact counterparts is moreover applied to real data concerning histological images of two different types of mammary tissue.

The paper concerns a new statistical method for assessing dissimilarity of two random sets based on one realisation of each of them. The method focuses on shapes of the components of the random sets, namely on the curvature of their boundaries together with the ratios of their perimeters and areas. Theoretical background is introduced and then, the method is described, justified by a simulation study and applied to real data of two different types of tissue -mammary cancer and mastopathy.

Článek je přehledem dosud zkoumaných a publikovaných metodrozlišování dvou realizací náhodných množin ve smyslu rozhodnutí, zda jsousi realizace (ne)podobné svými základními rysy. Metody se zaměřují na různápojetí podobnosti realizací, přičemž všechny jsou založené na dvou náhod-ných výběrech funkcí – po jednom z každé realizace – popisujících specifickérysy komponent. Shoda pravděpodobnostních rozdělení těchto funkcí je paktestována a realizace jsou považovány za podobné, jestliže shoda není zamít-nuta. Metody jsou zde stručně popsány a numericky ilustrovány na simulačnístudii. Na závěr je pak uvedeno srovnání výhod a nevýhod jednotlivých me-tod.

A statistical analysis of submissions (forms, emails, letters etc.) to municipalities in the Czech Republic is presented. The aim is to describe the behaviour of different types of the submissions (electronic, personal, sent by post etc.), provide a model of dependencies among them and study the influence of changes in laws in the Czech Republic to them. Methods for multiple time series are chosen as the main tool, namely linear and correlation analysis, and methods for detecting change points are used. Further, the number of submissions per month is modelled by gamma distribution. The obtained results are briefly commented and explained.

In recent years random sets were recognized as a valuable tool in modelling different processes from fields like biology, biomedicine or material sciences. Nevertheless, the full potential of applications has not still been reached and one of the main problems in advancement is the usual inability to correctly differentiate between underlying processes generating real world realisations. This paper presents a measure of dissimilarity of stationary and isotropic random sets through a heuristic based on convex compact approximations, support functions and envelope tests. The choice is justified through simulation studies of common random models like Boolean and Quermass-interaction processes.

The paper concerns an extension of random disc Quermass-interaction process, i.e. the model of discs with mutual interactions, to the process of interacting objects of more general shapes. Based on the results for the random disc process and the process with polygonal grains, theoretical results for the generalized process are derived. Further, a simulation method, its advantages and the corresponding complications are described, and some examples are introduced. Finally, a short comparison to the random disc process is given.

In the recent years, space-time point processes were recognised asa valuable tool in modelling different random events in fields such as biology,medicine, material sciences or economy. Nevertheless, new applications occurfrequently. This paper concerns the use of the space-time point processesfor modelling of submissions to municipalities in the Czech Republic. Thepositions and times of submissions are considered to be coordinates in thesubset of Euclidean space, so they form a space-time point pattern whichis to be analysed. Both continuous and discrete approaches to statisticalinference and modelling are employed and their advantages and disadvantagesare described. Finally, the most appropriate model is chosen, fitted to the dataand its suitability is justified through classical methods of spatial statisticsusing mainly simulations.

Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper. Copyright (c) 2015John Wiley & Sons, Ltd.

Many objects studied in biology, medicine or material sciences create spatial formations of random shape in which we can observe mutual interactions among those objects. In order to analyse the data composed of such patterns, we use the methods of spatial statistics. Recently, extended random-disc Quermass-interaction process was studied, simulated and consequently statistically analysed using MCMC maximum likelihood method (MCMC MLE). However, this analysis brought some problems. First, it was quite time-consuming, secondly, in some special cases, the parameter estimates may undervalue the real parameter values. In this paper, we describe how we can solve these problems by dimension reduction.

The Quermass-interaction model allows to generalize the classical germ-grain Boolean model in adding a morphological interaction between the grains. It enables to model random structures with specific morphologies, which are unlikely to be generated from a Boolean model. The Quermass-interaction model depends in particular on an intensity parameter, which is impossible to estimate from classical likelihood or pseudo-likelihood approaches because the number of points is not observable from a germ-grain set. In this paper, we present a procedure based on the Takacs–Fiksel method, which is able to estimate all parameters of the Quermass-interaction model, including the intensity. An intensive simulation study is conducted to assess the efficiency of the procedure and to provide practical recommendations. It also illustrates that the estimation of the intensity parameter is crucial in order to identify the model. The Quermass-interaction model is finally fitted by our method to P. Diggle’s heather data set.

There are many probabilistic models describing random items and behaviour observed on the surface of a material (particle orientations, mutual particle interactions etc.). While some of these processes can be statistically analysed using known basic methods and consequently easily simulated (e.g. Voronoi tessellation), other ones require more difficult statistical and simulating methods using more sophisticated mathematical tools like MCMC simulations (e.g. Quermass-interaction process), random marked sets (e.g. characterization of grain boundary structure) etc. This contribution concerns newly developed mathematical methods of describing, statistical analyses and simulations of different material structures.

A space-time random set is defined and methods of its parameters estimation are investigated. The evolution in discrete time is described by a state-space model. The observed output is a planar union of interacting discs given by a probability density with respect to a reference Poisson process of discs. The state vector is to be estimated together with auxiliary parameters of transitions caused by a random walk. Three methods of parameters estimation are involved, first of which is the maximum likelihood estimation (MLE) for individual outputs at fixed times. In the space-time model the state vector can be estimated by the particle filter (PF), where MLE serves to the estimation of auxiliary parameters. In the present paper the aim is to compare MLE and PF with particle Markov chain Monte Carlo (PMCMC). From the group of PMCMC methods we use specially the particle marginal Metropolis-Hastings (PMMH) algorithm which updates simultaneously the state vector and the auxiliary parameters. A simulation study is presented in which all estimators are compared by means of the integrated mean square error. New data are then simulated repeatedly from the model with parameters estimated by PMMH and the fit with the original model is quantified by means of the spherical contact distribution function.

The ultrafine-grained microstructure of pure copper processed by equal-channel angular pressing and its temperature-induced changes were evaluated in order to characterize heterogeneous distribution of fine- and larger-sized grains in the microstructure. ECAP was conducted at room temperature with a die that had an internal angle of 90° between the two parts of the channel. The subsequent extrusion passes were performed by route B C up to 8 ECAP passes and tested under constant load. Creep test was performed on the samples processed by 8 ECAP passes in tension at 373 K under an applied stress 260 MPa. The analyses of microstructure were performed by 3 dimensional electron-back scatter diffraction (3D EBSD) technique. The volume characteristics of microstructure and its inhomogeneity were evaluated and the relationships microstructure/creep behaviour of UFG copper was discussed.

Space-time models in stochastic geometry are used in many applications. Mostly these are models of space-time point processes. A second frequent situation are growth models of random sets. The present chapter aims to present more general models. It has two parts according to whether the time is considered to be discrete or continuous. In the discrete-time case we focus on state-space models and the use of Monte Carlo methods for the inference of model parameters. Two applications to real situations are presented: a) evaluation of a neurophysiological experiment, b) models of interacting discs. In the continuous-time case we discuss space-time Lévy-driven Cox processes with different second-order structures. Besides the wellknown separable models, models with separable kernels are considered. Moreover fully nonseparable models based on ambit processes are introduced. Inference for the models based on second-order statistics is developed.

The contribution concerns methods of dimension reduction in extended Quermass-interaction process. It mentions problems with procedures of estimating parameters in this process and describes how the dimension reduction solves them.

A spatio-temporal random set parametric model is defined based on the union of interacting discs. There are two types of parameters: those of the spatial part of the model and those of the state space model for temporal evolution. The simulation of the random set is available using a Markov chain Monte Carlo algorithm. Integral-geometric characteristics are evaluated and serve as an input to parameter estimation. We compare an MCMC maximum likelihood estimator with a particle filter estimator in a simulation study by drawing their temporal evolution and globally by means of the integrated mean square error. Interpretations of parameters and possible applications are discussed.

The paper concerns likelihood inference for a random set modelled by a germ-grain model, where the grains form a disc process and the individual grains are unobservable. The minimal sufficient statistic of the model density (with respect to a Boolean model) depends on various geometric properties of the random set. We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analyzing Peter Diggle's heather dataset, where we discuss the results of simulation-based maximum likelihood inference and the effect of specifying different processes.

Consider a random set given by a union of interacting discs described by a probability density with respect to a given Boolean model. The contribution concerns two methods for estimating the parameters of the density - MCMC maximum likelihood and Takacs-Fiksel method using integral characterisation of Gibbs process. Since both these methods are computationally complicated, it is shown how can the application of the power tesselation make the computations faster.

The contribution presents statistical analysis of data given by a digital image of heather growth. The bushes are modeled as a process of interacting discs. Two different ways of estimating the parameters of the model are compared.

Příspěvek se zabývá modelem náhodné množiny dané konečným sjednocením kruhů, mezi nimiž se vyskytují vzájemné interakce. Tento model je popsán hustotou pravděpodobnosti vzhledem k Booleovskému modelu, která závisí na geometrických charakteristikách (např. plocha, obvod apod.) dané množiny. Popsány jsou zde hlavně základní teoretické vlastnosti modelu, metody odhadu parametrů hustoty, testování jejich významnosti a kontroly vhodnosti modelu.

The power tessellation turns out to be particularly useful in connection to a flexible class of random set models specified by an underlying process of interacting discs. We discuss how to simulate these models and calculate various geometric characteristics of power tessellations, where new relations for the characteristics are established. The proposed model is fitted to a heather dataset.