Superposition Of Point Processes Research Articles

Variational Bayes approach for classification of points in superpositions of point processes

We investigate the problem of classifying superpositions of spatial point processes. In particular, we are interested in realizations formed as a superposition of a regular point process and a Poisson point process. The aim is to decide which of the two processes each point belongs to. Recently, a Markov chain Monte Carlo (MCMC) approach was suggested by Redenbach et al. (2015), which however, is computationally heavy. In this paper, we will introduce a method based on variational Bayes approximation and compare its performance to the performance of a slightly refined version of the MCMC approach.

Internet traffic modelling: from superposition to scaling

Internet traffic at various tiers of service providers is essentially a superposition or active mixture of traffic from various sources. Statistical properties of this superposition and a resulting phenomenon of scaling are important for network performance (queuing), traffic engineering (routing) and network dimensioning (bandwidth provisioning). In this article, the authors study the process of superposition and scaling jointly in a non-asymptotic framework so as to better understand the point process nature of cumulative input traffic process arriving at telecommunication devices (e.g., switches, routers). The authors further assess the scaling dynamics of the structural components (packets, flows and sessions) of the cumulative input process and their relation with superposition of point processes. Classical and new results are discussed with their applicability in access and core networks. The authors propose that renewal theory-based approximate point process models, that is, Pareto renewal process superposition and Weibull renewal process superposition can model the similar second-order scaling, as observed in traffic data of access and backbone core networks, respectively.

Effective neuronal refractoriness dominates the statistics of superimposed spike trains

The pooled spike trains of populations of neurons are typically modeled as Poisson processes [2]. It is known, though, that the superposition of point processes is a Poisson process if and only if all components are Poisson processes [3]. However, neocortical neurons spike more regularly [1]. Partly this is because they often have a refractory period, but also because the membrane potential is hyperpolarized after each spike, as illustrated in Figure ​Figure1A.1A. Here we analyze neuronal spike trains recorded intracellularly in vivo from rat somatosensory cortex. We match them with a Poisson process with dead-time [4], which is the simplest model of neuronal activity that incorporates refractory effects. The dead-time here models the effective refractoriness of the neuron, which can be larger than the refractory period due to channel kinetics alone. From the spike train recordings we construct independent superpositions (see Figure ​Figure1B)1B) and compare their statistics to our analytical results for the model processes. We find that the effective refractoriness of the neurons dominates the second-order statistics of the superposition spike trains. We uncover profound statistical differences as compared to Poisson processes, which considerably affect the dynamics of the membrane potential of neurons that receive such superpositions, as we further show in numerical simulations (see also [5]). Figure 1 A: Membrane potential trajectories of a simulated neocortical neuron. After each spike, the potential has to charge up until spikes can be initiated by input fluctuations, leading to an effective refractoriness. Green line shows the mean subthreshold ...

ON THE CONVERGENCE OF SUPERPOSITIONS OF POINT PROCESSES

In this work, we give conditions for the superposition of independent point processes to converge to a nonhomogeneous twodimensional Poisson process. Using our results, not only that of Grigelionis (1963) and Weissman (1975), but also some limiting results related to thinned point processes can be obtained. Hence this establishes the connection between thinnings and superpositions of point processes.

Superposition models of the discharge patterns of units in the lower auditory system.

Superposition of point processes has often been suggested as an abstract model for the generation of neural discharge patterns due to its simplicity (input spike trains are simply merged and then retransmitted by the model neuron). The properties of the superposition of two renewal processes are examined in relation to the properties of its components. A satisfactory condition is found so that the superposition of two renewal processes possesses negative serial dependence of intervent intervals. However, the imposition of a dead time of approximately the same duration as that of the components on their superposition process is shown to remove much of this dependence. Thus, the adequacy of the simple superposition of renewal processes as a model for discharge patterns from regions that are typified by negative serial dependence (such as the lateral superior olive) may be limited, but may suffice for the discharges of primary-like units.

Serial dependence in a superposition of renewal processes

Superposition of point processes has often been suggested [I. Gath, Math. Biosci. 22, 211–222 (1974); C. E. Molnar and R. R. Pfeiffer, Proc. IEEE 56, 993–1004 (1968)] as a model for neural discharge patterns due to its intuitive appeal (input spike trains are simply merged and then retransmitted), and the tendency for successive interspike intervals in the superposition to be negatively correlated. Discharge patterns in the lateral superior olive (LSO) of the cat often display such negative serial dependence. The properties of the superposition of two identical renewal processes are examined in relation to the properties of its components. Components with increasing intensity functions are proven to yield a superposition process having negative serial dependence. However, the imposition of a dead time on a superposition process is shown to remove much of the serial dependence. For this reason, it is suggested that superposition of renewal processes is not a good model for discharge patterns having both a dead time and significant serial dependence, such as LSO discharge patterns.

Convergence of superpositions of point processes in the space D[0, 1

Convergence of superpositions of point processes in the space D[0, 1

On Stationarity and Superposition of Point Processes

This paper applies ideas from random set theory to simple point processes. We show stationarity of the hitting distributions suffices for the strict stationarity of a simple point process, but that in general all forms of stationarity differ. We compare and contrast the superposition operations of summation for random measures and union for random sets, specialized to point processes. Finally we consider completely random sets and their factors.

On the Superposition of Point Processes

Summary Given point processes N1, …, Nm their superposition is the point process N defined by N(t) = N1(t) + … + Nm(t), t ≥ 0. An equivalent description of the system (N1, …, Nm) is by the process (Xn, Tn) where the Tn are the points of N, and Xn = k if and only if Tn is a point of Nk. The use of (X, T) process enables one to study the dependence of N1, …, Nm. Necessary and sufficient conditions are obtained for N, …, Nm to be independent, and for the superposition N to be a renewal process. For example, if N1 and N2 are renewal processes and X is independent of N, then N is a renewal process only if X is a homogeneous Markov chain.