Mark correlations: relating physical properties to spatial distributions

Mark correlations provide a systematic approach to look at objects both distributed in space and bearing intrinsic information, for instance on physical properties. The interplay of the objects' properties (marks) with the spatial clustering is of vivid interest for many applications; are, e.g., galaxies with high luminosities more strongly clustered than dim ones? Do neighbored pores in a sandstone have similar sizes? How does the shape of impact craters on a planet depend on the geological surface properties? In this article, we give an introduction into the appropriate mathematical framework to deal with such questions, i.e. the theory of marked point processes. After having clarified the notion of segregation effects, we define universal test quantities applicable to realizations of a marked point processes. We show their power using concrete data sets in analyzing the luminosity-dependence of the galaxy clustering, the alignment of dark matter halos in gravitational $N$-body simulations, the morphology- and diameter-dependence of the Martian crater distribution and the size correlations of pores in sandstone. In order to understand our data in more detail, we discuss the Boolean depletion model, the random field model and the Cox random field model. The first model describes depletion effects in the distribution of Martian craters and pores in sandstone, whereas the last one accounts at least qualitatively for the observed luminosity-dependence of the galaxy clustering.Comment: 35 pages, 12 figures. to be published in Lecture Notes of Physics, second Wuppertal conference "Spatial statistics and statistical physics

Reconstruction thresholds on regular trees

We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2x2 transition matrix P. We say that "reconstruction is possible" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\to\infty$. Adapting a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_{11}=0$. The latter case is closely related to the "hard-core model" from statistical physics; a corollary of our results is that, for the hard-core model on the (k+1)-regular tree with activity $\lambda=1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k.Comment: 12 page

Copula-based orderings of multivariate dependence

In this paper I investigate the problem of defining a multivariate dependence ordering. First, I provide a characterization of the concordance dependence ordering between multivariate random vectors with fixed margins. Central to the characterization is a multivariate generalization of a well-known bivariate elementary dependence increasing rearrangement. Second, to order multivariate random vectors with non- fixed margins, I impose a scale invariance principle which leads to a copula-based concordance dependence ordering. Finally, a wide family of copula-based measures of dependence is characterized to which Spearmanís rank correlation coefficient belongs.copula, concordance ordering, dependence measures, dependence orderings, multivariate stochastic dominance, supermodular ordering

Comparing hard and soft prior bounds in geophysical inverse problems

In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about X sub E is essential if y is to supply useful limits for z. The one exception occurs when all the prediction functionals are linear combinations of the data functionals. Two forms of prior information are compared: a soft bound on X sub E is a probability distribution p sub x on X which describeds the observer's opinion about where X sub E is likely to be in X; a hard bound on X sub E is an inequality Q sub x(X sub E, X sub E) is equal to or less than 1, where Q sub x is a positive definite quadratic form on X. A hard bound Q sub x can be softened to many different probability distributions p sub x, but all these p sub x's carry much new information about X sub E which is absent from Q sub x, and some information which contradicts Q sub x. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound p sub x. If that probability distribution was obtained by softening a hard prior bound Q sub x, rather than by objective statistical inference independent of y, then p sub x contains so much unsupported new information absent from Q sub x that conclusions about z obtained with SI or BI would seen to be suspect

Global permutation tests for multivariate ordinal data: alternatives, test statistics, and the null dilemma

We discuss two-sample global permutation tests for sets of multivariate ordinal data in possibly high-dimensional setups, motivated by the analysis of data collected by means of the World Health Organisation's International Classification of Functioning, Disability and Health. The tests do not require any modelling of the multivariate dependence structure. Specifically, we consider testing for marginal inhomogeneity and direction-independent marginal order. Max-T test statistics are known to lead to good power against alternatives with few strong individual effects. We propose test statistics that can be seen as their counterparts for alternatives with many weak individual effects. Permutation tests are valid only if the two multivariate distributions are identical under the null hypothesis. By means of simulations, we examine the practical impact of violations of this exchangeability condition. Our simulations suggest that theoretically invalid permutation tests can still be 'practically valid'. In particular, they suggest that the degree of the permutation procedure's failure may be considered as a function of the difference in group-specific covariance matrices, the proportion between group sizes, the number of variables in the set, the test statistic used, and the number of levels per variable

High-SIR Transmission Capacity of Wireless Networks with General Fading and Node Distribution

In many wireless systems, interference is the main performance-limiting factor, and is primarily dictated by the locations of concurrent transmitters. In many earlier works, the locations of the transmitters is often modeled as a Poisson point process for analytical tractability. While analytically convenient, the PPP only accurately models networks whose nodes are placed independently and use ALOHA as the channel access protocol, which preserves the independence. Correlations between transmitter locations in non-Poisson networks, which model intelligent access protocols, makes the outage analysis extremely difficult. In this paper, we take an alternative approach and focus on an asymptotic regime where the density of interferers $\eta$ goes to 0. We prove for general node distributions and fading statistics that the success probability \p \sim 1-\gamma \eta^{\kappa} for $\eta \rightarrow 0$, and provide values of $\gamma$ and $\kappa$ for a number of important special cases. We show that $\kappa$ is lower bounded by 1 and upper bounded by a value that depends on the path loss exponent and the fading. This new analytical framework is then used to characterize the transmission capacity of a very general class of networks, defined as the maximum spatial density of active links given an outage constraint.Comment: Submitted to IEEE Trans. Info Theory special issu

Cell shape analysis of random tessellations based on Minkowski tensors

To which degree are shape indices of individual cells of a tessellation characteristic for the stochastic process that generates them? Within the context of stochastic geometry and the physics of disordered materials, this corresponds to the question of relationships between different stochastic models. In the context of image analysis of synthetic and biological materials, this question is central to the problem of inferring information about formation processes from spatial measurements of resulting random structures. We address this question by a theory-based simulation study of shape indices derived from Minkowski tensors for a variety of tessellation models. We focus on the relationship between two indices: an isoperimetric ratio of the empirical averages of cell volume and area and the cell elongation quantified by eigenvalue ratios of interfacial Minkowski tensors. Simulation data for these quantities, as well as for distributions thereof and for correlations of cell shape and volume, are presented for Voronoi mosaics of the Poisson point process, determinantal and permanental point processes, and Gibbs hard-core and random sequential absorption processes as well as for Laguerre tessellations of polydisperse spheres and STIT- and Poisson hyperplane tessellations. These data are complemented by mechanically stable crystalline sphere and disordered ellipsoid packings and area-minimising foam models. We find that shape indices of individual cells are not sufficient to unambiguously identify the generating process even amongst this limited set of processes. However, we identify significant differences of the shape indices between many of these tessellation models. Given a realization of a tessellation, these shape indices can narrow the choice of possible generating processes, providing a powerful tool which can be further strengthened by density-resolved volume-shape correlations.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jense

Heat conductivity from molecular chaos hypothesis in locally confined billiard systems

We study the transport properties of a large class of locally confined Hamiltonian systems, in which neighboring particles interact through hard core elastic collisions. When these collisions become rare and the systems large, we derive a Boltzmann-like equation for the evolution of the probability densities. We solve this equation in the linear regime and compute the heat conductivity from a Green-Kubo formula. The validity of our approach is demonstated by comparing our predictions to the results of numerical simulations performed on a new class of high-dimensional defocusing chaotic billiards.Comment: 4 pages, 2 color figure