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As wearables provide new health-related functionalities, they can be employed in hospitals to monitor patients and notify the medical personnel regarding their status. However, in order to be approved by the medical community, wearables need to have reliable communication and high lifetime. In such scenarios, it is important to know the probability of correct notification which is affected mainly by the deployment of the wireless wearables and their energy supply. Typically, rooms in hospitals host multiple people and, thus, a clustered communication model should be adopted for more trustworthy results. Moreover, by employing wireless charging, it is possible to provide an uninterrupted operation with high reliability. Therefore, in this paper, we study the aforementioned probability in a clustered network while the wearable devices are wirelessly charged. We provide an analytical model for the wearables' ability to inform quickly the medical personnel and discuss different trade-offs via extensive simulations.

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic. NSA grant [H98230-13-1-0220] info:eu-repo/semantics/publishedVersion

We study stochastic properties of the empty space for stationary germ-grain models in R d ; in particular, we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower dimensional. We consider Poisson cluster and mixed Poisson germ-grain models, and show in several situations that more variability results in stochastically greater empty space in terms of the empty space hazard function. Furthermore, we study the asymptotic behaviour of the empty space hazard functions at 0 and at ∞.

Species spatial turnover, or β-diversity, induces a decay of community similarity with geographic distance known as the distance–decay relationship. Although this relationship is central to biodiversity and biogeography, its theoretical underpinnings remain poorly understood. Here, we develop a general framework to describe how the distance–decay relationship is influenced by population aggregation and the landscape-scale species-abundance distribution. We utilize this general framework and data from three tropical forests to show that rare species have a weak influence on distance–decay curves, and that overall similarity and rates of decay are primarily influenced by species abundances and population aggregation respectively. We illustrate the utility of the framework by deriving an exact analytical expression of the distance–decay relationship when population aggregation is characterized by the Poisson Cluster Process. Our study provides a foundation for understanding the distance–decay relationship, and for predicting and testing patterns of beta-diversity under competing theories in ecology. Ecology Letters (2008) 11: 904–917

International audience; The goal of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a fundamental integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity in proportion, and use the integral formula to derive an asymptotic expansion of the miss probability in powers of the inverse of the cache size. This enables us to quantify and improve the accuracy of the so-called Che approximation.

Cilj je ovog diplomskog rada bio proučiti Poissonov proces s klasterima i njegova osnovna svojstva. Nakon prvoga poglavlja, čitatelju su poznate definicija točkovnih procesa, mjera njihovih momenata te primjene tzv. Laplaceove transformacije i Laplaceova funkcionala točkovnoga procesa. Navedeno je i nekoliko primjera točkovnih procesa te najvažniji točkovni proces, općeniti Poissonov točkovni proces ili Poissonova slučajna mjera (PRM). Pokazano je i kako se dodavanjem nezavisne koordinate točkama poznatoga PRM-a može konstruirati novi PRM. Dalje je navedena definicija i jedna karakterizacija općenitih točkovnih procesa s klasterima. Pretpostavljamo da nam je poznat proces centara klastera. Točke tog procesa generiraju klastere koji su također točkovni procesi. Zbrajanje elemenata po klasterima čini opaženi proces. Ako su središta klastera točke Poissonova procesa, riječ je o Poissonovom procesu s klasterima. U zadnjem poglavlju bavimo se primjenom Poissonova procesa s klasterima u matematici neživotnoga osiguranja. Točku Poissonova procesa interpretiramo kao vrijeme dolaska zahtjeva za isplatu štete, a klaster koji ona uzrokuje opisuje vremena i iznose isplate tog zahtjeva. Proučavamo i chain ladder model. Na kraju se bavimo Poissonovim procesima s Poissonovim klasterima. Analiziramo njihove prve i druge momente kako bismo predvidjeli broj i ukupan iznos isplata šteta. Chain ladder model i Poissnov proces s klasterima često se koriste za procjenu pričuva, što je jedan od najvažnijih praktičnih problema u aktuarstvu. The aim of this thesis was to study the Poisson cluster process as well as its basic properties. After the first chapter the reader is familiar with the definition of point processes, their moment measures, and the application of the so called Laplace transform and the Laplace functional of the point processes. There are several examples of point processes including the most important one, the general Poisson point process or Poisson random measure (PRM). It is also shown how one can adhere an independent coordinate to the points of given PRM to construct a new PRM. In addition, there is a definition and a characterization of general point process with independent clusters. We suppose a point process of cluster centers is known. The points of the center process generate the clusters, which are also point processes. The superposition of the elements within clusters constitutes the observed process. If the cluster centers are the points of a Poisson process, we speak of a Poisson cluster process. In the final chapter we consider the applications of the Poisson cluster process in non-life insurance mathematics. A Poisson process point is interpreted as the arrival time of a claim, and the cluster that point triggers describes the times and amounts of the payment for this particular claim. We also study the chain ladder model. Lastly, we concentrate on the Poisson processes with Poisson clusters. We analyze the first and second moments of those processes in order to predict the claim number and total claim amounts. Chain ladder model and Poisson cluster process are often used to estimate reserves, which is one of the most important insurance practice problems.

Future mobile networks are converging toward heterogeneous multitier networks, where macro-, pico-, and femto-cells are randomly deployed based on user demand. A popular approach for analyzing heterogeneous networks (HetNets) is to use stochastic geometry and treat the location of BSs as points distributed according to a homogeneous Poisson point process (PPP). However, a PPP model does not provide an accurate model for the interference when nodes are clustered around highly populated areas. This motivates us to find better ways to characterize the aggregate interference when transmitting nodes are clustered following a Poisson cluster process (PCP) while taking into consideration the fact that BSs belonging to different tiers may differ in terms of transmit power, node densities, and link reliabilities. To this end, we consider K-tier HetNets and investigate the outage probability, the coverage probability, and the average achievable rate for such networks. We compare the performance of HetNets when nodes are clustered and otherwise. By comparing these two types of networks, we conclude that the fundamental difference between a PPP and a PCP is that, for a PPP, the number of simultaneously covered mobiles and the network capacity linearly increase with K. However, for a PCP, the improvements in the coverage and the capacity diminish as K grows larger, where the curves saturate at some point. Based on these observations, we determine the scenarios that jointly maximize the average achievable rate and minimize the outage probability. Qatar National Research Fund Scopus

Future mobile networks are converging towards being heterogeneous, owing to the co-existence of multi-tier networks within the same geographical area, including macro, pico- and femto-cells. The deployment of such networks is generally based on user demand, which is irregular and random, implying that the deployment of base stations (BSs) is random as well. As a result, analyzing the communication protocols over heterogeneous networks (HetNets) is very challenging. A popular approach is to use stochastic geometry and treat the location of the BSs as points distributed according to a spatial Point Process. Most of the related work on the interference modeling normally assumes homogeneous Poisson point process (PPP). This assumption holds when the nodes are uniformly distributed in space, such as sensor networks or ad-hoc networks. Due to geographical factors, it may be the case for mobile users to cluster around highly populated cities and the PPP assumption does not provide an accurate model for the interference in these conditions. This motivates us to find better ways to characterize the aggregate interference when the transmitting nodes are clustered following a Poisson Cluster Process (PCP). Furthermore, the BSs belonging to different tiers may differ in terms of the transmit power, the node densities, and their link reliabilities. To this end, we consider K-tier HetNets, where, by using the Laplace transform approach, we characterize the aggregate interference at a given destination as a heavy-tailed distribution. Using the derived distribution, we investigate the probability of outage and coverage for such networks. Due to some difficulty in obtaining closed-form expressions for these measures, we derive tight bounds and verify that through numerical examples. We also compare the performance of HetNets when the nodes are clustered and otherwise. We observe that using the PPP results in larger success probability, but using the clustered process results in a larger coverage probability. We also observe that there is an optimal intensity, i.e., number of nodes, that achieves the maximum coverage probability for the given SINR (signal-to-interference-plus-noise ratio) threshold. Qatar National Research Fund Scopus

We analyze a cognitive radio network where the primary users (PUs) and cognitive radio (CR) devices are distributed over the two-dimensional plane according to two independent homogeneous Poisson processes. Any CR that lies within the detection region of some PU switches to a different channel in order to prevent causing harmful interference at the PU. Using the concepts of stochastic geometry, we study the characteristics of the interference caused by the PUs and the CRs to a given PU. Further, these results are used to obtain tight upper and lower bounds for the success probability at the PU; defined as the probability that the signal-to-interferenceplus- noise ratio (SINR) is beyond a certain operating threshold.

Recently developed spatial capture-recapture (SCR) models represent a major advance over traditional capture-recapture (CR) models because they yield explicit estimates of animal density instead of population size within an unknown area. Furthermore, unlike nonspatial CR methods, SCR models account for heterogeneity in capture probability arising from the juxtaposition of animal activity centers and sample locations. Although the utility of SCR methods is gaining recognition, the requirement that all individuals can be uniquely identified excludes their use in many contexts. In this paper, we develop models for situations in which individual recognition is not possible, thereby allowing SCR concepts to be applied in studies of unmarked or partially marked populations. The data required for our model are spatially referenced counts made on one or more sample occasions at a collection of closely spaced sample units such that individuals can be encountered at multiple locations. Our approach includes a spatial point process for the animal activity centers and uses the spatial correlation in counts as information about the number and location of the activity centers. Camera-traps, hair snares, track plates, sound recordings, and even point counts can yield spatially correlated count data, and thus our model is widely applicable. A simulation study demonstrated that while the posterior mean exhibits frequentist bias on the order of 5-10% in small samples, the posterior mode is an accurate point estimator as long as adequate spatial correlation is present. Marking a subset of the population substantially increases posterior precision and is recommended whenever possible. We applied our model to avian point count data collected on an unmarked population of the northern parula (Parula americana) and obtained a density estimate (posterior mode) of 0.38 (95% CI: 0.19-1.64) birds/ha. Our paper challenges sampling and analytical conventions in ecology by demonstrating that neither spatial independence nor individual recognition is needed to estimate population density - rather, spatial dependence can be informative about individual distribution and density., Published in at http://dx.doi.org/10.1214/12-AOAS610 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider risk processes with non-stationary Hawkes claims arrivals, and we study the asymptotic behavior of infinite and finite horizon ruin probabilities under light-tailed conditions on the claims. Moreover, we provide asymptotically efficient simulation laws for ruin probabilities and we give numerical illustrations of the theoretical results.

In the project “statistical image analysis” of CWI we have studied some spatial point patterns that originated from biological observations. These observations were the positions of so called EGF‐receptors on the surface of human carcinoma cells. We propose a stochastic model for these point patterns. Since the EGF‐receptors appear in clusters on the cell surface, we have opted for the Poisson‐cluster‐process as the model. We estimated the three parameters in this process by means of a method described by Diggle. We also did some work in assessing the statistical reliability of our estimates. Copyright

Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented. Udgivelsesdato: SEP Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

This article concerns a simulation algorithm for unmarked and marked Hawkes processes. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work. We derive various useful measures for the error committed when using the algorithm, and we discuss various empirical results for the algorithm compared with perfect simulations. This article concerns a simulation algorithm for unmarked and marked Hawkes processes. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work. We derive various useful measures for the error committed when using the algorithm, and we discuss various empirical results for the algorithm compared with perfect simulations.

Ponencia presentada en: III Congreso de la Asociación Española de Climatología “El agua y el clima”, celebrado en Palma de Mallorca del 16 al 19 de junio de 2002. [ES]El objetivo de este trabajo es caracterizar la máxima sequía que cabe esperar en un determinado periodo de tiempo. Para ello es necesario disponer de un modelo estocástico que describa el proceso de sequias (proponemos un proceso Poisson cluster para describir la ocurrencia y tres series de variables aleatorias, Longitud, Déficit e Intensidad Máxima, para describir la severidad) y desarrollar los resultados teóricos necesarios sobre la distribución del máximo en una muestra de tamaño Poisson. [EN]This work aims to characterize the largest drought event to occur in a given period of time. A Poisson cluster process is used to model drought occurrence and three series of random variables (Length, Deficit and Maximum Intensity) to describe their severity. Some theoretical results on the distribution of the maximum in a random Poisson size sample are developed for describing the largest drought events.

We consider m-dependent random fields of bounded random vectors (generated by independent random fields) and investigate the analyticity of the cumulant generating function of sums of these random vectors. Using the Kirkwood-Salsburg equations we derive upper bounds for the cumulant generating function and prove its analyticity in a neighbourhood of zero, where the normalized bounds and the neighbourhood are independent of the number of terms in the sum, The results are applied to statistics of Poisson cluster processes and Boolean models (which have a representation in terms of an independent field) and yield probabilities of large deviations as well as Berry-Esseen results for these statistics.

The distribution of the excess process describing heights of extreme values can be approximated by the distribution of a Poisson cluster process. An estimate of the accuracy of such an approximation has been derived in [4] in terms of the Wasserstein distance. The paper presents a sharper estimate established in terms of the stronger total variation distance. We derive also a new bound to the accuracy of negative Binomial approximation to the distribution of the number of exceedances.