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396
Generalized weighted Chinese restaurant processes for species sampling mixture models
 Statistica Sinica
, 2003
"... Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conj ..."
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Cited by 53 (8 self)
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Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its twoparameter extension, the PitmanYor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, twoparameter PoissonDirichlet process, prediction rule, random probability measure, species sampling sequence.
The Gaussian mixture probability hypothesis density filter
 IEEE Trans. SP
, 2006
"... Abstract — A new recursive algorithm is proposed for jointly estimating the timevarying number of targets and their states from a sequence of observation sets in the presence of data association uncertainty, detection uncertainty, noise and false alarms. The approach involves modelling the respecti ..."
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Cited by 51 (8 self)
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Abstract — A new recursive algorithm is proposed for jointly estimating the timevarying number of targets and their states from a sequence of observation sets in the presence of data association uncertainty, detection uncertainty, noise and false alarms. The approach involves modelling the respective collections of targets and measurements as random finite sets and applying the probability hypothesis density (PHD) recursion to propagate the posterior intensity, which is a first order statistic of the random finite set of targets, in time. At present, there is no closed form solution to the PHD recursion. This work shows that under linear, Gaussian assumptions on the target dynamics and birth process, the posterior intensity at any time step is a Gaussian mixture. More importantly, closed form recursions for propagating the means, covariances and weights of the constituent Gaussian components of the posterior intensity are derived. The proposed algorithm combines these recursions with a strategy for managing the number of Gaussian components to increase efficiency. This algorithm is extended to accommodate mildly nonlinear target dynamics using approximation strategies from the extended and unscented Kalman filters. Index Terms — Multitarget tracking, optimal filtering, point
A flowbased model for Internet backbone traffic
 in Proc. of ACM SIGCOMM Internet Measurement Workshop
, 2002
"... Our goal is to design a traffic model for uncongested IP backbone links that is simple enough to be used in network operation, and that is protocol and application agnostic in order to be as general as possible. The proposed solution is to model the traffic at the flow level by a Poisson shotnoise ..."
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Cited by 48 (6 self)
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Our goal is to design a traffic model for uncongested IP backbone links that is simple enough to be used in network operation, and that is protocol and application agnostic in order to be as general as possible. The proposed solution is to model the traffic at the flow level by a Poisson shotnoise process. In our model, a flow is a generic notion that must be able to capture the characteristics of any kind of data stream. We analyze the accuracy of the model with real traffic traces collected on the Sprint IP backbone network. Despite its simplicity, our model provides a good approximation of the real traffic observed in the backbone and of its variation. Finally, we discuss three applications of our model to network design and management.
Internet traffic tends toward Poisson and independent as the load increases, 2002. 14 Additional figures Mean results of the legitimate connection completion rates when using the fixedthreshold strategy and when using threshold timeout and linear timeout
"... Abstract — Network devices put packets on an Internet link, and multiplex, or superpose, the packets from different active connections. Extensive empirical and theoretical studies of packet traffic variables — arrivals, sizes, and packet counts — demonstrate that the number of active connections has ..."
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Cited by 47 (3 self)
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Abstract — Network devices put packets on an Internet link, and multiplex, or superpose, the packets from different active connections. Extensive empirical and theoretical studies of packet traffic variables — arrivals, sizes, and packet counts — demonstrate that the number of active connections has a dramatic effect on traffic characteristics. At low connection loads on an uncongested link — that is, with little or no queueing on the linkinput router — the traffic variables are longrange dependent, creating burstiness: large variation in the traffic bit rate. As the load increases, the laws of superposition of marked point processes push the arrivals toward Poisson, the sizes toward independence, and reduces the variability of the counts relative to the mean. This begins a reduction in the burstiness; in network parlance, there are multiplexing gains. Once the connection load is sufficiently large, the network begins pushing back on the attraction to Poisson and independence by causing queueing on the linkinput router. But if the link speed is high enough, the traffic can get quite close to Poisson and independence before the pushback begins in force; while some of the statistical properties are changed in this highspeed case, the pushback does not resurrect the burstiness. These results reverse the commonlyheld presumption that Internet traffic is everywhere bursty and that multiplexing gains do not occur. Very simple statistical time series models — fractional sumdifference (FSD) models — describe the statistical variability of the traffic variables and their change toward Poisson and independence before significant queueing sets in, and can be used to generate openloop packet arrivals and sizes for simulation studies. Both science and engineering are affected. The magnitude of multiplexing needs to become part of the fundamental scientific framework that guides the study of Internet
Arcsine laws and interval partitions derived from a stable subordinator
 Proc. London Math. Soc
, 1992
"... Le"vy discovered that the fraction of time a standard onedimensional Brownian motion B spends positive before time t has arcsine distribution, both for / a fixed time when B, #0 almost surely, and for / an inverse local time, when B, = 0 almost surely. This identity in distribution is extended fro ..."
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Cited by 44 (25 self)
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Le"vy discovered that the fraction of time a standard onedimensional Brownian motion B spends positive before time t has arcsine distribution, both for / a fixed time when B, #0 almost surely, and for / an inverse local time, when B, = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 1.
Stochastic Geometry and Architecture of Communication Networks
 J. Telecommunication Systems
, 1995
"... This paper proposes a new approach for communication networks planning; this approach is based on stochastic geometry. We first summarize the state of the art in this domain, together with its economic implications, before sketching the main expectations of the proposed method. The main probabilisti ..."
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Cited by 43 (6 self)
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This paper proposes a new approach for communication networks planning; this approach is based on stochastic geometry. We first summarize the state of the art in this domain, together with its economic implications, before sketching the main expectations of the proposed method. The main probabilistic tools are point processes and stochastic geometry. We show how several performance evaluation and optimization problems within this framework can actually be posed and solved by computing the mathematical expectation of certain functionals of point processes. We mainly analyze models based on Poisson point processes, for which analytical formulae can often be obtained, although more complex models can also be analyzed, for instance via simulation.
Estimating a StateSpace Model from Point Process Observations
, 2003
"... A widely used signal processing paradigm is the statespace model. The statespace model is defined by two equations: an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time. Inspired by neu ..."
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Cited by 39 (4 self)
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A widely used signal processing paradigm is the statespace model. The statespace model is defined by two equations: an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time. Inspired by neurophysiology experiments in which neural spiking activity is induced by an implicit (latent) stimulus, we develop an algorithm to estimate a statespace model observed through point process measurements. We represent the latent process modulating the neural spiking activity as a gaussian autoregressive model driven by an external stimulus. Given the latent process, neural spiking activity is characterized as a general point process defined by its conditional intensity function. We develop an approximate expectationmaximization (EM) algorithm to estimate the unobservable statespace process, its parameters, and the parameters of the point process. The EM algorithm combines a point process recursive nonlinear filter algorithm, the fixed interval smoothing algorithm, and the statespace covariance algorithm to compute the complete data log likelihood efficiently. We use a KolmogorovSmirnov test based on the timerescaling theorem to evaluate agreement between the model and point process data. We illustrate the model with two simulated data examples: an ensemble of Poisson neurons driven by a common stimulus and a single neuron whose conditional intensity function is approximated as a local Bernoulli process.
A Poisson Limit for Buffer Overflow Probabilities
 in Proceedings of IEEE INFOCOM
, 2002
"... Abstract — A key criterion in the design of highspeed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumpti ..."
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Cited by 39 (1 self)
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Abstract — A key criterion in the design of highspeed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumptions on the input processes we show that the steady state probability of the buffer content exceeding a threshold � � tends to the corresponding probability assuming Poisson input processes. We verify the assumptions for a large class of longrange dependent sources commonly used to model data traffic. Our results show that with superposition, significant multiplexing gains can be achieved for even smaller buffers than suggested by previous results, which consider Ç Ò buffer size. Moreover, simulations show that for realistic values of the exceedance probability and moderate utilisations, convergence to the Poisson limit takes place at reasonable values of the number of sources superposed. This is particularly relevant for highspeed networks in which the cost of highspeed memory is significant. Keywords—Longrange dependence, overflow probability, Poisson limit, heavy tails, point processes, multiplexing.
Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks
"... Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accoun ..."
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Cited by 39 (6 self)
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Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communicationtheoretic results accounting for the network’s geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs – including point process theory, percolation theory, and probabilistic combinatorics – have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.
On a Voronoi Aggregative Process Related to a Bivariate Poisson Process
 Adv. in Appl. Probab
, 1996
"... We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0particles within a typical Voronoi Π1cell. We find the first and the second moments of these variables ..."
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Cited by 34 (4 self)
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We consider two independent homogeneous Poisson processes Π0 and Π1 in the plane with intensities λ0 and λ1, respectively. We study additive functionals of the set of Π0particles within a typical Voronoi Π1cell. We find the first and the second moments of these variables as well as upper and lower bounds on their distribution functions, implying an exponential asymptotic behavior of their tails. Explicit formulae are given for the number and the sum of distances from Π0particles to the nucleus within a typical Voronoi Π1cell.