arXiv subjects

Dietrich Stoyan

Publications and source records attributed to Dietrich Stoyan.

10 recordsLinked to original sources

arXiv2026

The distributions of convex body functionals of the Poisson polyhedron

This note presents new findings on the distribution of the volume and surface area of both the Poisson polyhedron and the Crofton polyhedron. It also considers the distribution of the area of the Poisson polygon and the Crofton polygon in the plane. This note motivates the modeling of the distributions of these convex body functionals as mixture distributions and the essential inclusion of the generalized gamma distribution therein as mixture components. Finally, model parameters are estimated and models are selected based on extensive stochastic simulations of these random polytopes. This includes that for the first time, this note gives very accurate estimates of the side number probabilities for both the Poisson polyhedron and the Crofton polyhedron, as these are used as mixing weights in the distribution models.

math.PR↗

arXiv2023

Statistics did not prove that the Huanan Seafood Wholesale Market was the early epicenter of the COVID-19 pandemic

In a recent prominent study Worobey et al.\ (2022, Science, 377, pp.\ 951--9) purported to demonstrate statistically that the Huanan Seafood Wholesale Market was the epicenter of the early COVID-19 epidemic. We show that this statistical conclusion is invalid on two grounds: (1) The assumption that a centroid of early case locations or another simply constructed point is the origin of an epidemic is unproved. (2) A Monte Carlo test used to conclude that no other location than the seafood market can be the origin is flawed. Hence, the question of the origin of the pandemic has not been answered by their statistical analysis.

stat.AP↗

arXiv2023

Cutting a Cake Is Not Always a 'Piece of Cake': A Closer Look at the Foundations of Cake-Cutting Through the Lens of Measure Theory

Cake-cutting is a playful name for the fair division of a heterogeneous, divisible good among agents, a well-studied problem at the intersection of mathematics, economics, and artificial intelligence. The cake-cutting literature is rich and edifying. However, different model assumptions are made in its many papers, in particular regarding the set of allowed pieces of cake that are to be distributed among the agents and regarding the agents' valuation functions by which they measure these pieces. We survey the commonly used definitions in the cake-cutting literature, highlight their strengths and weaknesses, and make some recommendations on what definitions could be most reasonably used when looking through the lens of measure theory.

cs.GT↗

arXiv2016

Continuity Assumptions in Cake-Cutting

In important papers on cake-cutting -- one of the key areas in fair division and resource allocation -- the measure-theoretical fundamentals are not fully correctly given. It is not clear (i) which family of sets should be taken for the pieces of cake, (ii) which set-functions should be used for evaluating the pieces, and (iii) which is the relationship between various continuity properties appearing in cake-cutting. We show that probably the best choice for the familiy of subsets of $[0,1]$ is the Borel $σ$-algebra and for the set-function any `sliceable' Borel measure. At least in dimension one it does not make sense to work with only finitely additive contents on finite unions of intervals. For the continuity property we see two possibilities. The weaker is the traditional divisibility property, which is equivalent to being atom-free. The stronger is simply absolute continuity with respect to Lebesgue measure. We also consider the case of a base set (cake or pie) more general than $[0,1]$.

cs.DS↗

arXiv2016

Heuristic Approximations for Closed Networks: A Case Study in Open-pit Mining

We investigate a fundamental model from open-pit mining, which is a cyclic system consisting of a shovel, traveling loaded, unloading facility, and traveling back empty. The interaction of these subsystem determines the capacity of the shovel, which is the fundamental quantity of interest. To determine this capacity one needs the stationary probability that the shovel is idle. Because an exact analysis of the performance of the system is out of reach, besides of simulations there are various approximation algorithms proposed in the literature which stem from computer science and can be characterized as general purpose algorithms. We propose for solving the special problem under mining conditions an extremely simple algorithm. Comparison with several general purpose algorithms shows that for realistic situations the special algorithm outperforms the precision of the general purpose algorithms. This holds even if these general purpose candidates incorporate more details of the underlying models than our simple algorithm, which works on a strongly reduced model. The comparison and assessment is done with extensive simulations on a level of detail which the general purpose algorithms are able to cover.

math.OC↗

arXiv2014

Cuboidal Dice and Gibbs Distributions

What are the face-probabilities of a cuboidal die, i.e. a die with different side-lengths? This paper introduces a model for these probabilities based on a Gibbs distribution. Experimental data produced in this work and drawn from the literature support the Gibbs model. The experiments also reveal that the physical conditions, such as the quality of the surface onto which the dice are dropped, can affect the face-probabilities. In the Gibbs model, those variations are condensed in a single parameter, adjustable to the physical conditions.

math-ph↗

arXiv2013

Deviation test construction and power comparison for marked spatial point patterns

The deviation test belong to core tools in point process statistics, where hypotheses are typically tested considering differences between an empirical summary function and its expectation under the null hypothesis, which depend on a distance variable r. This test is a classical device to overcome the multiple comparison problem which appears since the functional differences have to be considered for a range of distances r simultaneously. The test has three basic ingredients: (i) choice of a suitable summary function, (ii) transformation of the summary function or scaling of the differences, and (iii) calculation of a global deviation measure. We consider in detail the construction of such tests both for stationary and finite point processes and show by two toy examples and a simulation study for the case of the random labelling hypothesis that the points (i) and (ii) have great influence on the power of the tests.

stat.ME↗

arXiv2010

Measuring galaxy segregation using the mark connection function

(abridged) The clustering properties of galaxies belonging to different luminosity ranges or having different morphological types are different. These characteristics or `marks' permit to understand the galaxy catalogs that carry all this information as realizations of marked point processes. Many attempts have been presented to quantify the dependence of the clustering of galaxies on their inner properties. The present paper summarizes methods on spatial marked statistics used in cosmology to disentangle luminosity, colour or morphological segregation and introduces a new one in this context, the mark connection function. The methods used here are the partial correlation functions, including the cross-correlation function, the normalised mark correlation function, the mark variogram and the mark connection function. All these methods are applied to a volume-limited sample drawn from the 2dFGRS, using the spectral type as the mark. We show the virtues of each method to provide information about the clustering properties of each population, the dependence of the clustering on the marks, the similarity of the marks as a function of the pair distances, and the way to characterise the spatial correlation between the marks. We demonstrate by means of these statistics that passive galaxies exhibit stronger spatial correlation than active galaxies at small scales (r <20 Mpc/h). The mark connection function, introduced here, is particularly useful for understanding the spatial correlation between the marks.

astro-ph.CO↗

arXiv2002

Point field models for the galaxy point pattern. Modelling the singularity of the two-point correlation function

There is empirical evidence that the two-point correlation function of the galaxy distribution follows, for small scales, reasonably well a power-law expression $ξ(r)\propto r^{-γ}$ with $γ$ between 1.5 and 1.9. Nevertheless, most of the point field models suggested in the literature do not have this property. This paper presents a new class of models, which is produced by modifying point fields commonly used in cosmology to mimic the galaxy distribution, but where $γ=2$ is too large. The points are independently and randomly shifted, leading to the desired reduction of the value of $γ$.

astro-ph↗

arXiv1999

Comparing estimators of the galaxy correlation function

We present a systematic comparison of some usual estimators of the 2--point correlation function, some of them currently used in Cosmology, others extensively employed in the field of the statistical analysis of point processes. At small scales, it is known that the correlation function follows reasonably well a power--law expression $ξ(r) \propto r^{-γ}$. The accurate determination of the exponent $γ$ (the order of the pole) depends on the estimator used for $ξ(r)$; on the other hand, its behavior at large scale gives information on a possible trend to homogeneity. We study the concept, the possible bias, the dependence on random samples and the errors of each estimator. Errors are computed by means of artificial catalogues of Cox processes for which the analytical expression of the correlation function is known. We also introduce a new method for extracting simulated galaxy samples from cosmological simulations.

astro-ph↗