Upper bounds for spatial point process approximations

We consider the behavior of spatial point processes when subjected to a class of linear transformations indexed by a variable T. It was shown in Ellis [Adv. in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the transformed processes behave approximately like Poisson processes for large T. In this article, under very similar assumptions, explicit upper bounds are given for the d_2-distance between the corresponding point process distributions. A number of related results, and applications to kernel density estimation and long range dependence testing are also presented. The main results are proved by applying a generalized Stein-Chen method to discretized versions of the point processes.Comment: Published at http://dx.doi.org/10.1214/105051604000000684 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stein's method and Poisson process approximation for a class of Wasserstein metrics

Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ161 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Bounds for the probability generating functional of a Gibbs point process

We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics like the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity and higher order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.Comment: 15 page

Gibbs point process approximation: Total variation bounds using Stein's method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.Comment: Published in at http://dx.doi.org/10.1214/13-AOP895 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A new metric between distributions of point processes

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric $\bar{d}_1$ that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about $\bar{d}_1$ and its induced Wasserstein metric $\bar{d}_2$ for point process distributions are given, including examples of useful $\bar{d}_1$-Lipschitz continuous functions, $\bar{d}_2$ upper bounds for Poisson process approximation, and $\bar{d}_2$ upper and lower bounds between distributions of point processes of i.i.d. points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of $\bar{d}_1$ in applications.Comment: 20 pages, 2 figure

Distance maps between Japanese kanji characters based on hierarchical optimal transport

We introduce a general framework for assigning distances between kanji based on their dissimilarity. What we mean by this term may depend on the concrete application. The only assumption we make is that the dissimilarity between two kanji is adequately expressed as a weighted mean of penalties obtained from matching nested structures of components in an optimal way. For the cost of matching, we suggest a number of modules that can be freely combined or replaced with other modules, including the relative unbalanced ink transport between registered components, the distance between the transformations required for registration, and the difference in prespecified labels. We give a concrete example of a kanji distance function obtained in this way as a proof of concept. Based on this function, we produce 2D kanji maps by multidimensional scaling and a table of 100 randomly selected J\=oj\=o kanji with their 16 nearest neighbors. Our kanji distance functions can be used to help Japanese learners from non-CJK backgrounds acquire kanji literacy. In addition, they may assist editors of kanji dictionaries in presenting their materials and may serve in text processing and optical character recognition systems for assessing the likelihood of errors.Comment: 24 pages, 5 figure