Functional summary statistics for point processes on the sphere with an application to determinantal point processes

We study point processes on $\mathbb S^d$, the $d$-dimensional unit sphere $\mathbb S^d$, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case $d=2$. The first part studies reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley's and inhomogeneous $K$-functions. The second part partly discusses the appealing properties of determinantal point process (DPP) models on the sphere and partly considers the application of functional summary statistics to DPPs. In fact DPPs exhibit repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on the sphere with aggregation on the large scale and regularity on the small scale. We conclude with a discussion on future work on statistics for spatial point processes on the sphere

Score, Pseudo-Score and Residual Diagnostics for Spatial Point Process Models

We develop new tools for formal inference and informal model validation in the analysis of spatial point pattern data. The score test is generalized to a "pseudo-score" test derived from Besag's pseudo-likelihood, and to a class of diagnostics based on point process residuals. The results lend theoretical support to the established practice of using functional summary statistics, such as Ripley's $K$-function, when testing for complete spatial randomness; and they provide new tools such as the compensator of the $K$-function for testing other fitted models. The results also support localization methods such as the scan statistic and smoothed residual plots. Software for computing the diagnostics is provided.Comment: Published in at http://dx.doi.org/10.1214/11-STS367 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Modeling Heterogeneous Wireless Networks Using Non-Poisson Point Processes

Future wireless networks are required to support 1000 times higher data rate, than the current LTE standard. In order to meet the ever increasing demand, it is inevitable that, future wireless networks will have to develop seamless interconnection between multiple technologies. A manifestation of this idea is the collaboration among different types of network tiers such as macro and small cells, leading to the so-called heterogeneous networks (HetNets). Researchers have used stochastic geometry to analyze such networks and understand their real potential. Unsurprisingly, it has been revealed that interference has a detrimental effect on performance, especially if not modeled properly. Interference can be correlated in space and/or time, which has been overlooked in the past. For instance, it is normally assumed that the nodes are located completely independent of each other and follow a homogeneous Poisson point process (PPP), which is not necessarily true in real networks since the node locations are spatially dependent. In addition, the interference correlation created by correlated stochastic processes has mostly been ignored. To this end, we take a different approach in modeling the interference where we use non-PPP, as well as we study the impact of spatial and temporal correlation on the performance of HetNets. To illustrate the impact of correlation on performance, we consider three case studies from real-life scenarios. Specifically, we use massive multiple-input multiple-output (MIMO) to understand the impact of spatial correlation; we use the random medium access protocol to examine the temporal correlation; and we use cooperative relay networks to illustrate the spatial-temporal correlation. We present several numerical examples through which we demonstrate the impact of various correlation types on the performance of HetNets.Comment: Submitted to IEEE Communications Magazin

Stochastic Multipath Model for the In-Room Radio Channel based on Room Electromagnetics

We propose a stochastic multipath model for the received signal for the case where the transmitter and receiver, both with directive antennas, are situated in the same rectangular room. This scenario is known to produce channel impulse responses with a gradual specular-to-diffused transition in delay. Mirror source theory predicts the arrival rate to be quadratic in delay, inversely proportional to room volume and proportional to the product of the antenna beam coverage fractions. We approximate the mirror source positions by a homogeneous spatial Poisson point process and their gain as complex random variables with the same second moment. The multipath delays in the resulting model form an inhomogeneous Poisson point process which enables derivation of the characteristic functional, power/kurtosis delay spectra, and the distribution of order statistics of the arrival delays in closed form. We find that the proposed model matches the mirror source model well in terms of power delay spectrum, kurtosis delay spectrum, order statistics, and prediction of mean delay and rms delay spread. The constant rate model, assumed in e.g. the Saleh-Valenzuela model, is unable to reproduce the same effects.Comment: 14 pages, Manuscript Submitted to IEEE Transaction on Antennas and Propagatio

Intracell interference characterization and cluster interference for D2D communication

The homogeneous spatial Poisson point process (SPPP) is widely used for spatial modeling of mobile terminals (MTs). This process is characterized by a homogeneous distribution, complete spatial independence, and constant intensity measure. However, it is intuitive to understand that the locations of MTs are neither homogeneous, due to inhomogeneous terrain, nor independent, due to homophilic relations. Moreover, the intensity is not constant due to mobility. Therefore, assuming an SPPP for spatial modeling is too simplistic, especially for modeling realistic emerging device-centric frameworks such as device-to-device (D2D) communication. In this paper, assuming inhomogeneity, positive spatial correlation, and random intensity measure, we propose a doubly stochastic Poisson process, a generalization of the homogeneous SPPP, to model D2D communication. To this end, we assume a permanental Cox process (PCP) and propose a novel Euler-Characteristic-based approach to approximate the nearest-neighbor distribution function. We also propose a threshold and spatial distances from an excursion set of a chi-square random field as interference control parameters for different cluster sizes. The spatial distance of the clusters is incorporated into a Laplace functional of a PCP to analyze the average coverage probability of a cellular user. A closed-form approximation of the spatial summary statistics is in good agreement with empirical results, and its comparison with an SPPP authenticates the correlation modeling of D2D nodes