Types Of Point Processes Research Articles

A Markovian arrival stream approach to stochastic gene expression in cells.

We analyse a generalisation of the stochastic gene expression model studied recently in Fromion et al. (SIAM J Appl Math 73:195-211, 2013) and Robert (Probab Surv 16:277-332, 2019) that keeps track of the production of both mRNA and protein molecules, using techniques from the theory of point processes, as well as ideas from the theory of matrix-analytic methods. Here, both the activity of a gene and the creation of mRNA are modelled with an arbitrary Markovian Arrival Process governed by finitely many phases, and each mRNA molecule during its lifetime gives rise to protein molecules in accordance with a Poisson process. This modification is important, as Markovian Arrival Processes can be used to approximate many types of point processes on the nonnegative real line, meaning this framework allows us to further relax our assumptions on the overall process of transcription.

Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques

Spatial statistics is a powerful tool for analyzing data that are illustrated as points or positions in a regular or non-regular state space. Techniques that are proposed to investigate the spatial association between neighboring positions are based on the point process analysis. One of the main goals is to simulate real data positions (such as antenna base stations) using the type of point process that most closely matches the data. Spatial patterns could be detailed describing the observed positions and appropriate models were proposed to simulate these patterns. A common model to simulate spatial patterns is the Poisson point process. In this work analyses of the Poisson point process—as well as modified types such as inhibition point process and determinantal Poisson point process—are presented with simulated data close to the true data (i.e., antenna base station positions). Investigation of the spatial variation of the data led us to the spatial association between positions by applying Ripley’s K-functions and L-Function.

A probabilistic model to describe the interference produced by VSAT MF‐TDMA networks

SummaryThis paper develops a mathematical model to describe the statistical behavior of the interference produced by VSAT/MF‐TDMA networks. The model is used to assess the interference produced by the uplinks of a VSAT/MF‐TDMA network into links of a network that uses a neighboring satellite. In the proposed model, analytical expressions were developed to account for the effects of variations in transmitting powers, antenna sizes, and transmitting antenna pointing errors. The earth station locations are modeled by a 2‐dimensional Poisson point process. The model is general enough to accommodate other types of point processes and can be applied to situations involving service areas containing multiple types of earth station geographical distribution. Numerical results obtained with the proposed model are compared with those based on the actual parameter values (eg, earth station locations, antenna sizes, and transmitting powers), which were provided by a Brazilian satellite operator.

Marked point process for modelling seismic activity (case study in Sumatra and Java)

Earthquake is a natural phenomenon that is random, irregular in space and time. Until now the forecast of earthquake occurrence at a location is still difficult to be estimated so that the development of earthquake forecast methodology is still carried out both from seismology aspect and stochastic aspect. To explain the random nature phenomena, both in space and time, a point process approach can be used. There are two types of point processes: temporal point process and spatial point process. The temporal point process relates to events observed over time as a sequence of time, whereas the spatial point process describes the location of objects in two or three dimensional spaces. The points on the point process can be labelled with additional information called marks. A marked point process can be considered as a pair (x, m) where x is the point of location and m is the mark attached to the point of that location. This study aims to model marked point process indexed by time on earthquake data in Sumatra Island and Java Island. This model can be used to analyse seismic activity through its intensity function by considering the history process up to time before t. Based on data obtained from U.S. Geological Survey from 1973 to 2017 with magnitude threshold 5, we obtained maximum likelihood estimate for parameters of the intensity function. The estimation of model parameters shows that the seismic activity in Sumatra Island is greater than Java Island.

Determining the largest Lyapunov exponent of chaotic dynamics from sequences of interspike intervals contaminated by noise

We discuss abilities of quantifying low-dimensional chaotic oscillations at the input of two threshold models from the output sequences of interspike intervals in the presence of noise. We propose a modification of the standard approach for computing the largest Lyapunov exponent from a time series that verifies the performed estimations for noisy data. We consider features of its application to different types of point processes.

Asymptotics and Approximation of the SIR Distribution in General Cellular Networks

It has recently been observed that the SIR distributions of a variety of cellular network models and transmission techniques look very similar in shape. As a result, they are well approximated by a simple horizontal shift (or gain) of the distribution of the most tractable model, the Poisson point process (PPP). To study and explain this behavior, this paper focuses on general single-tier network models with nearest-base station association and studies the asymptotic gain both at 0 and at infinity. We show that the gain at 0 is determined by the so-called mean interference-to-signal ratio (MISR) between the PPP and the network model under consideration, while the gain at infinity is determined by the expected fading-to-interference ratio (EFIR). The analysis of the MISR is based on a novel type of point process, the so-called relative distance process, which is a one-dimensional point process on the unit interval [0,1] that fully determines the SIR. A comparison of the gains at 0 and infinity shows that the gain at 0 indeed provides an excellent approximation for the entire SIR distribution. Moreover, the gain is mostly a function of the network geometry and barely depends on the path loss exponent and the fading. The results are illustrated using several examples of repulsive point processes.

Influences of premotoneuronal command statistics on the scaling of motor output variability during isometric plantar flexion

This study focuses on neuromuscular mechanisms behind ankle torque and EMG variability during a maintained isometric plantar flexion contraction. Experimentally obtained torque standard deviation (SD) and soleus, medial gastrocnemius, and lateral gastrocnemius EMG envelope mean and SD increased with mean torque for a wide range of torque levels. Computer simulations were performed on a biophysically-based neuromuscular model of the triceps surae consisting of premotoneuronal spike trains (the global input, GI) driving the motoneuron pools of the soleus, medial gastrocnemius, and lateral gastrocnemius muscles, which activate their respective muscle units. Two types of point processes were adopted to represent the statistics of the GI: Poisson and Gamma. Simulations showed a better agreement with experimental results when the GI was modeled by Gamma point processes having lower orders (higher variability) for higher target torques. At the same time, the simulations reproduced well the experimental data of EMG envelope mean and SD as a function of mean plantar flexion torque, for the three muscles. These results suggest that the experimentally found relations between torque-EMG variability as a function of mean plantar flexion torque level depend not only on the intrinsic properties of the motoneuron pools and the muscle units innervated, but also on the increasing variability of the premotoneuronal GI spike trains when their mean rates increase to command a higher plantar flexion torque level. The simulations also provided information on spike train statistics of several hundred motoneurons that compose the triceps surae, providing a wide picture of the associated mechanisms behind torque and EMG variability.

A Third Order Point Process Characteristic

Second order characteristics, in particular Ripley's K‐function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T(r) of r‐close triples of points per unit area. Equivalently this is the expected number of r‐close point pairs in an r‐neighbourhood of the typical point. Various estimators for this function are proposed and compared, and we give an explicit formula for the isotropic edge correction. The theoretical value of T seems to be unobtainable for most point process models apart from the homogeneous Poisson process. However, simulation studies show that the function T discriminates well between different types of point processes. In particular it detects a clear difference between the Poisson process and the Baddeley–Silverman cell process whereas the K‐functions for these processes coincide.

A third order point process characteristic

Second order characteristics, in particular Ripley's K-function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T(r) of r-close triples of points per unit area. Equivalently this is the expected number of r-close point pairs in an r-neighbourhood of the typical point. Various estimators for this function are proposed and compared, and we give an explicit formula for the isotropic edge correction. The theoretical value of T seems to be unobtainable for most point process models apart from the homogeneous Poisson process. However, simulation studies show that the function T discriminates well between different types of point processes. In particular it detects a clear difference between the Poisson process and the Baddeley- Silverman cell process whereas the K-functions for these processes coincide.

Bayesian Sequential Hypothesis Testing

We consider the Bayesian sequential detection problem for general observation processes with a continuous time parameter. Primary emphasis is placed on first exit policies and their generic optimality. A new geometric formulation and solution to the existence and uniqueness of the optimal first exit policy will be given as well as an explicit const ructive algorithm for its computation. We will apply these results to the problem of Bayesian sequential detection for diffusion-type observations although our methods and results generalize to problems of optimal stopping and decision for partially observed Markov chains where the observation processes a re of the diffusion or point process type. These results will appear elsewhere.