Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (02) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

scholarly journals Clusterless Decoding of Position from Multiunit Activity Using a Marked Point Process Filter

2015 ◽  
Vol 27 (7) ◽  
pp. 1438-1460 ◽  
Author(s):  
Xinyi Deng ◽  
Daniel F. Liu ◽  
Kenneth Kay ◽  
Loren M. Frank ◽  
Uri T. Eden
Keyword(s):  
Real Time ◽  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Spike Sorting ◽  
Decoding Algorithm ◽  
Population Activity ◽  
Spiking Activity

Point process filters have been applied successfully to decode neural signals and track neural dynamics. Traditionally these methods assume that multiunit spiking activity has already been correctly spike-sorted. As a result, these methods are not appropriate for situations where sorting cannot be performed with high precision, such as real-time decoding for brain-computer interfaces. Because the unsupervised spike-sorting problem remains unsolved, we took an alternative approach that takes advantage of recent insights into clusterless decoding. Here we present a new point process decoding algorithm that does not require multiunit signals to be sorted into individual units. We use the theory of marked point processes to construct a function that characterizes the relationship between a covariate of interest (in this case, the location of a rat on a track) and features of the spike waveforms. In our example, we use tetrode recordings, and the marks represent a four-dimensional vector of the maximum amplitudes of the spike waveform on each of the four electrodes. In general, the marks may represent any features of the spike waveform. We then use Bayes’s rule to estimate spatial location from hippocampal neural activity. We validate our approach with a simulation study and experimental data recorded in the hippocampus of a rat moving through a linear environment. Our decoding algorithm accurately reconstructs the rat’s position from unsorted multiunit spiking activity. We then compare the quality of our decoding algorithm to that of a traditional spike-sorting and decoding algorithm. Our analyses show that the proposed decoding algorithm performs equivalent to or better than algorithms based on sorted single-unit activity. These results provide a path toward accurate real-time decoding of spiking patterns that could be used to carry out content-specific manipulations of population activity in hippocampus or elsewhere in the brain.

scholarly journals Morpho-statistical characterization of the cosmic web using marked point processes

2014 ◽  
Vol 10 (S306) ◽  
pp. 239-242
Author(s):  
Radu S. Stoica
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Probabilistic Models ◽  
Geometrical Structure ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Statistical Characterization

AbstractThe cosmic web is the intricate network of filaments outlined by the galaxies positions distribution in our Universe. One possible manner to break the complexity of such an elaborate geometrical structure is to assume it made of simple interacting objects. Under this hypothesis, the filamentary network can be considered as the realization of an object or a marked point process. These processes are probabilistic models dealing with configurations of random objects given by random points having random characteristics or marks. Here, the filamentary network is considered as the realization of such a process, with the objects being cylinders that align and connect in order to form the network. The paper presents the use of marked point processes to the detection and the characterization of the galactic filaments.

State aggregation and discrete-state Markov chains embedded in a class of point processes

1995 ◽  
Vol 32 (01) ◽  
pp. 39-51
Author(s):  
Xi-Ren Cao
Keyword(s):  
Markov Chains ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
Practical Importance ◽  
Marked Point Process ◽  
Interarrival Time ◽  
Discrete State ◽  
State Aggregation

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.

UTILITY MAXIMIZATION IN A PURE JUMP MODEL WITH PARTIAL OBSERVATION

2010 ◽  
Vol 25 (1) ◽  
pp. 29-54 ◽  
Author(s):  
Paola Tardelli
Keyword(s):  
Utility Maximization ◽  
Point Processes ◽  
Marked Point ◽  
Asset Price ◽  
Main Tool ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Underlying Event ◽  
Jump Model

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process—a marked point process having common jump times with the risky asset price process. The problem of utility maximization of terminal wealth is dealt with when the underlying event arrivals process is assumed to be unobserved by the market agents using, as the main tool, backward stochastic differential equations. The dual problem is studied. Explicit solutions in a particular case are given.

Reliability analysis of complex repairable systems by means of marked point processes

1980 ◽  
Vol 17 (1) ◽  
pp. 154-167 ◽  
Author(s):  
Peter Franken ◽  
Arnfried Streller
Keyword(s):  
Reliability Analysis ◽  
Point Processes ◽  
Marked Point ◽  
Regenerative Process ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Repairable Systems ◽  
Interval Reliability ◽  
Repair Facility ◽  
Markovian Systems

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

Power spectra of general shot noises and Hawkes point processes with a random excitation

2002 ◽  
Vol 34 (01) ◽  
pp. 205-222 ◽  
Author(s):  
P. Brémaud ◽  
L. Massoulié
Keyword(s):  
Point Process ◽  
Spectral Measure ◽  
Point Processes ◽  
Marked Point ◽  
Power Spectra ◽  
Random Excitation ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Shot Noise Process ◽  
Power Spectral

We give (i) the Cramér power spectral measure of the general shot noise process with random excitation and non-Poisson stationary driving point processes and (ii) the Bartlett power spectral measure of the self-exciting Hawkes point process with random excitation, also called the Hawkes branching point process with random fertility rate. The latter is obtained via the isometry formula for integrals with respect to the canonical martingale measure associated with a marked point process.

Reliability analysis of complex repairable systems by means of marked point processes

1980 ◽  
Vol 17 (01) ◽  
pp. 154-167 ◽  
Author(s):  
Peter Franken ◽  
Arnfried Streller
Keyword(s):  
Reliability Analysis ◽  
Point Processes ◽  
Marked Point ◽  
Regenerative Process ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Repairable Systems ◽  
Interval Reliability ◽  
Repair Facility ◽  
Markovian Systems

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

1997 ◽  
Vol 29 (04) ◽  
pp. 1039-1059
Author(s):  
Vinod Sharma
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Discrete Time Queues ◽  
Time Queues ◽  
Finite State

Recently, Asmussen and Koole (Journal of Applied Probability 30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

State aggregation and discrete-state Markov chains embedded in a class of point processes

1995 ◽  
Vol 32 (1) ◽  
pp. 39-51
Author(s):  
Xi-Ren Cao
Keyword(s):  
Markov Chains ◽  
Poisson Process ◽  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
Practical Importance ◽  
Marked Point Process ◽  
Interarrival Time ◽  
Discrete State ◽  
State Aggregation

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.

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