On estimation of intrinsic volume densities of stationary random closed sets via parallel sets in the plane

summary:A method of estimation of intrinsic volume densities for stationary random closed sets in $\mathbb{R}^d$ based on estimating volumes of tiny collars has been introduced in T. Mrkvička and J. Rataj, On estimation of intrinsic volume densities of stationary random closed sets, Stoch. Proc. Appl. 118 (2008), 2, 213-231. In this note, a stronger asymptotic consistency is proved in dimension 2. The implementation of the method is discussed in detail. An important step is the determination of dilation radii in the discrete approximation, which differs from the standard techniques used for measuring parallel sets in image analysis. A method of reducing the bias is proposed and tested on simulated data

Quick inference for log Gaussian Cox processes with non-stationary underlying random fields

For point patterns observed in natura, spatial heterogeneity is more the rule than the exception. In numerous applications, this can be mathematically handled by the flexible class of log Gaussian Cox processes (LGCPs); in brief, a LGCP is a Cox process driven by an underlying log Gaussian random field (log GRF). This allows the representation of point aggregation, point vacuum and intermediate situations, with more or less rapid transitions between these different states depending on the properties of GRF. Very often, the covariance function of the GRF is assumed to be stationary. In this article, we give two examples where the sizes (that is, the number of points) and the spatial extents of point clusters are allowed to vary in space. To tackle such features, we propose parametric and semiparametric models of non-stationary LGCPs where the non-stationarity is included in both the mean function and the covariance function of the GRF. Thus, in contrast to most other work on inhomogeneous LGCPs, second-order intensity-reweighted stationarity is not satisfied and the usual two step procedure for parameter estimation based on e.g. composite likelihood does not easily apply. Instead we propose a fast three step procedure based on composite likelihood. We apply our modelling and estimation framework to analyse datasets dealing with fish aggregation in a reservoir and with dispersal of biological particles

Rendering Complex 3D Scenes

Tato práce se zabývá problémem zobrazování rozsáhlých a obsahově velmi bohatých 3D scén, které jsou běžné např. pro moderní počítačové hry. Cílem práce je vytvoření tzv. datově řízeného zobrazovacího systému, který na základě popisu scény bude schopen sám scénu správně zobrazovat. Popis scény přitom musí být velmi jednoduchý tak, aby jej mohli vytvářet i lidé bez hlubších znalostí programování. První část této práce se zaměřuje především na navržení způsobu popisu scény a jeho následného využití při zobrazování scény. Druhá část práce se následně zabývá již vlastním využitím navrženého popisu scény při implementaci zobrazovacího systému.This thesis deals with representation of large and complex 3D scenes which are usually used by modern computer games. Main aim is design and implementation of data driven rendering system. Proper rendering is directed (driven) by scene description. This description is also designed with respect to scene creators whose typically do not have deep knowledge of programming languages in contrast to game programming developers. First part is focused on design of efficient scene description and its possible applications at scene rendering. Second part is focused on proper system implementation. Finally, consequently important system optimizations are mentioned too.

False discovery rate envelope for functional test statistics

False discovery rate (FDR) is a common way to control the number of false discoveries in multiple testing. In this paper, the focus is on functional test statistics which are discretized into $m$ highly correlated hypotheses and thus resampling based methods are investigated. The aim is to find a graphical envelope that detects the outcomes of all individual hypotheses by a simple rule: the hypothesis is rejected if and only if the empirical test statistic is outside of the envelope. Such an envelope offers a straightforward interpretation of the test results similarly as in global envelope testing recently developed for controlling the family-wise error rate. Two different algorithms are developed to fulfill this aim. The proposed algorithms are adaptive single threshold procedures which include the estimation of the true null hypotheses. The new methods are illustrated by two real data examples

GET: Global envelopes in R

This work describes the R package GET that implements global envelopes for a general set of $d$-dimensional vectors $T$ in various applications. A $100(1-\alpha)$% global envelope is a band bounded by two vectors such that the probability that $T$ falls outside this envelope in any of the $d$ points is equal to $\alpha$. Global means that the probability is controlled simultaneously for all the $d$ elements of the vectors. The global envelopes can be employed for central regions of functional or multivariate data, for graphical Monte Carlo and permutation tests where the test statistic is multivariate or functional, and for global confidence and prediction bands. Intrinsic graphical interpretation property is introduced for global envelopes, and the global envelopes included in the GET package that have the property are described. Examples of different uses of global envelopes and their implementation in the GET package are presented, including global envelopes for single and several one- or two-dimensional functions, Monte Carlo goodness-of-fit tests for simple and composite hypotheses, comparison of distributions, graphical functional analysis of variance (ANOVA), and general linear model (GLM), and confidence bands in polynomial regression

Nonparametric testing of the dependence structure among points-marks-covariates in spatial point patterns

We investigate the problem of testing the hypothesis of independence between a covariate and the marks in a marked point process. This would be rather straightforward if the (unmarked) process of points was independent of the covariate and the marks. In practice, however, such an assumption is questionable, and possible preferential sampling effects (dependence between the point process and the covariate and/or the marks) may lead to incorrect conclusions. Hence we propose to investigate the complete dependence structure in the triangle points-marks-covariates together. We take advantage of the recent development of the nonparametric random shift methods, namely the new variance correction approach, and propose tests of the null hypothesis of independence between the marks and the covariate, and also between the points and the covariate. We present a detailed simulation study showing the performance of the methods, and provide two theorems establishing the appropriate form of the correction factors for the variance correction. Finally, we illustrate the use of the proposed methods in two real applications

Global quantile regression

Quantile regression is used to study effects of covariates on a particular quantile of the data distribution. Here we are interested in the question whether a covariate has any effect on the entire data distribution, i.e., on any of the quantiles. To this end, we treat all the quantiles simultaneously and consider global tests for the existence of the covariate effect in the presence of nuisance covariates. This global quantile regression can be used as the extension of linear regression or as the extension of distribution comparison in the sense of Kolmogorov-Smirnov test. The proposed method is based on pointwise coefficients, permutations and global envelope tests. The global envelope test serves as the multiple test adjustment procedure under the control of the family-wise error rate and provides the graphical interpretation which automatically shows the quantiles or the levels of categorical covariate responsible for the rejection. The Freedman-Lane permutation strategy showed liberality of the test for extreme quantiles, therefore we propose four alternatives that work well even for extreme quantiles and are suitable in different conditions. We present a simulation study to inspect the performance of these strategies, and we apply the chosen strategies to two data examples.Comment: 44 pages, 12 figure

A New Functional Clustering Method with Combined Dissimilarity Sources and Graphical Interpretation

Clustering is an essential task in functional data analysis. In this study, we propose a framework for a clustering procedure based on functional rankings or depth. Our methods naturally combine various types of between-cluster variation equally, which caters to various discriminative sources of functional data; for example, they combine raw data with transformed data or various components of multivariate functional data with their covariance. Our methods also enhance the clustering results with a visualization tool that allows intrinsic graphical interpretation. Finally, our methods are model-free and nonparametric and hence are robust to heavy-tailed distribution or potential outliers. The implementation and performance of the proposed methods are illustrated with a simulation study and applied to three real-world applications

Estimation of intersection intensity in a Poisson process of segments

summary:The minimum variance unbiased estimator of the intensity of intersections is found for stationary Poisson process of segments with parameterized distribution of primary grain with known and unknown parameters. The minimum variance unbiased estimators are compared with commonly used estimators