Statistical Characterization of Electromagnetic Fields Scattered by Poisson Point Process Distributed PEC cylinders

The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

We consider the Voronoi tessellation based on a homogeneous Poisson point process in an Euclidean space. For a geometric characteristic of the cells (e.g., the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us to compute efficiently the values of the extremal index and the cluster size probabilities by simulation for various geometric characteristics. The extension to the Poisson–Delaunay tessellation is also discussed.

Spatial Poisson process models are popular partly because of their mathematical tractability based on a simple and powerful calculus whose main ingredients, besides Campbell’s formula, are the covariance formula and the exponential formula (also know as Campbell’s second formula). These formulas extend immediately to marked Poisson processes with independent (or location dependent) IID marks as a consequence of the fact that such marked Poisson processes can be described as unmarked Poisson processes in a higher-dimensional space. This slight modification of point of view will be applied to the study of the point processes obtained by meas of elementary transformations of a basic Poisson process, such as thinning, translation or clustering. All the results concerning Poisson processes find straightforward extensions concerning Cox processes. The calculus of spatial marked Poisson or Cox processes will be applied to a particular type of stochastic geometry (featuring the Boolean model) and to the exact sampling of cluster point processes.

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places agrain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

Spectral efficiency analysis of distributed millimeter wave massive MIMO system using spatial point processes

Continuum percolation and stochastic epidemic models on Poisson and Ginibre point processes

We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of such coverage models are $k$-coverage in the Boolean model (coverage by at least $k$ grains) and SINR-coverage (coverage if the signal-to-interference-and-noise ratio is large). In particular, we answer in affirmative the hypothesis of existence of phase transition in the percolation of $k$-faces in the \v{C}ech simplicial complex (called also clique percolation) on point processes which cluster less than the Poisson process. We also construct a Cox point process, which is "more clustered" than the Poisson point process and whose Boolean model percolates for arbitrarily small radius. This shows that clustering (at least, as detected by our specific tools) does not always "worsen" percolation, as well as that upper-bounding this clustering by a Poisson process is a consequential assumption for the phase transition to hold.

Cellular networks have a great transformation during the last decade. The traditional cellular network system is often overlaid with well-deployed BSs of proper power. However, more and more complex elements such as picocells, femtocells and relays have been introduced to cellular network system. BSs in these elements are unplanned, user-installed and randomly deployed. All of these things compose heterogeneous cellular networks (HCNs). In order to evaluate the performance of HCNs, people usually model the distribution of BSs in HCNs as spatial stochastic point process model. One of the most popular spatial models for these HCNs is Poisson Point Process (PPP) model. However the PPP model is not exact for some cases where BSs in PPP model may be too close to each other. In this paper, we propose a more reasonable and practical model called Matern-like point process (MLPP) model which imposes a certain minimal distance between any two BSs to overcome this problem. Simulations show that the coverage probability of the proposed model noticeably outperforms the PPP model. This improvement depends on the value of minimal distance. We also evaluate the system efficiency defined by the ratio of coverage probability and total power. The results show that the MLPP model has a significant improvement compared to PPP model.

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that use persistence diagrams in statistical inference, a full Bayesian treatment remains absent. This paper, relying on the theory of point processes, presents a generalized Bayesian framework for inference with persistence diagrams relying on a substitution likelihood argument. In essence, we model persistence diagrams as Poisson point processes with prior intensities and compute posterior intensities by adopting techniques from the theory of marked point processes. We then propose a family of conjugate prior intensities via Gaussian mixtures to obtain a closed form of the posterior intensity. Finally, we demonstrate the utility of this generalized Bayesian framework with a classification problem in materials science using Bayes factors.

Non-Poisson Point Processes Motivation In Chapters 5–7 we see how Poisson network models often give tractable results, and how random propagation effects such as fading and shadowing can render a network to appear more Poisson. Nevertheless, the Poisson model may still not be appropriate for certain network layouts. For example, although the Poisson process statistically does not exhibit either repulsion or clustering, it is possible for the clusters or voids of points to exist in network layouts. Appropriate point processes In Chapter 3 we see just a small selection of the possible spatial point processes. To choose appropriate point processes for network models, researchers have statistically analyzed networks located in various cities to demonstrate that other point processes may be more appropriate, although of course such analyses can be only as good as the data. But when developing stochastic geometry models of cellular networks, the number of suitable and tractable point processes quickly becomes limited. In terms of the SINR for a single user, there are two important and necessary features of a potential point process. One is the knowledge of the Palm distribution, as covered in Section 3.2.4, which immediately limits the choice of possible point processes. The other is being able to write down a calculable expression (at least, numerically) of the Laplace functional of the point process. In short, although there is a rich range of possible point processes, the bulk of them unfortunately lack the necessary properties that make them tractable in our setting, which partly explains why the clear majority of network models are based on the Poisson point process. Consequently, we restrict ourselves to results for point processes from two general families that exhibit repulsion and clustering, respectively, determinantal processes and a specific shot noise Cox process. Choice of the base station and propagation effects In Section 5.1.3, we see the two different model assumptions for how a user at the typical location (the origin) chooses the serving base station under the Poisson model.

Original ArticleComputer simulation in 3D of a phase transformation nucleated by simple sequential inhibition process

Assuming a connected base station at the point of reference (or origin) in the tier of interest, we derive an expression for the downlink probability of coverage over a heterogeneous network, wherein the base station locations result from different point processes, such as Poisson point and Poisson cluster processes. Numerical results show increasing the base station density in each tier lowers the coverage probability, but not as much as increasing the coverage threshold does. We also provide lower and upper bounds for the coverage probability in a two-tier heterogeneous network modeled with Poisson point and cluster processes, and evaluate the effect on those bounds when changing the various parameter values. These results are of potential use for future cellular heterogeneous network designs.

In Section 2.1 we look at σ-finite point processes with particular emphasis laid on Poisson processes having σ-finite intensity measures. Point processes having σ-finite intensity measures build a subclass of the class of σ-finite point processes. Moreover, it will be shown that homogeneous Poisson processes—introduced in Example 1.4.1—are equal, in distribution, to Poisson processes based on partial sums of i.i.d. exponential r.v.’s.

It is well known that one can map certain properties of random matrices, fermionic gases,and zeros of the Riemann zeta function to a unique point process on the real line . Here we analytically provide exact generalizations of such a point process ind-dimensional Euclidean space for any d, which are special cases of determinantal processes. In particular, we obtain then-particle correlationfunctions for any n, which completely specify the point processes in . We also demonstrate that spin-polarized fermionic systems in have these same n-particle correlation functions in each dimension. The point processes for anyd are shown to be hyperuniform, i.e., infinite wavelength densityfluctuations vanish, and the structure factor (or power spectrum)S(k) has a non-analytic behavior at the origin given byS(k)∼|k| (). The latter result implies that the pair correlation functiong2(r) tends to unity for large pair distances with a decay rate that is controlled by the power law1/rd+1, which is a well-known property of bosonic ground states and more recently has beenshown to characterize maximally random jammed sphere packings. We graphically displayone-and two-dimensional realizations of the point processes in order to vividlyreveal their ‘repulsive’ nature. Indeed, we show that the point processes can becharacterized by an effective ‘hard core’ diameter that grows like the square root ofd. The nearest-neighbor distribution functions for these point processes arealso evaluated and rigorously bounded. Among other results, this analysisreveals that the probability of finding a large spherical cavity of radiusr indimension d behaves like a Poisson point process but in dimensiond+1, i.e., this probabilityis given by exp[−κ(d)rd+1] for large r andfinite d, whereκ(d) is a positived-dependent constant.We also show that as d increases, the point process behaves effectively like a spherepacking with a coverage fraction of space that is no denser than1/2d. This coverage fraction has a special significance in the study of sphere packings inhigh-dimensional Euclidean spaces.