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

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.

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Marked point processes as limits of Markovian arrival streams

Journal of Applied Probability ◽

10.1017/s0021900200117371 ◽

1993 ◽

Vol 30 (02) ◽

pp. 365-372 ◽

Cited By ~ 19

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.

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

Journal of Applied Probability ◽

10.1017/s0021900200102554 ◽

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.

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Marked point processes as limits of Markovian arrival streams

Journal of Applied Probability ◽

10.2307/3214845 ◽

1993 ◽

Vol 30 (2) ◽

pp. 365-372 ◽

Cited By ~ 114

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.

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

Neural Computation ◽

10.1162/neco_a_00744 ◽

2015 ◽

Vol 27 (7) ◽

pp. 1438-1460 ◽

Cited By ~ 34

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.

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

Journal of Applied Probability ◽

10.2307/3214919 ◽

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

Proceedings of the International Astronomical Union ◽

10.1017/s1743921314010709 ◽

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.

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RISK-NEUTRAL MEASURES AND PRICING FOR A PURE JUMP PRICE PROCESS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964809990131 ◽

2009 ◽

Vol 24 (1) ◽

pp. 47-76 ◽

Cited By ~ 2

Author(s):

Anna Gerardi ◽

Paola Tardelli

Keyword(s):

Point Processes ◽

Marked Point ◽

Asset Price ◽

Risky Asset ◽

Marked Point Processes ◽

Marked Point Process ◽

Martingale Measure ◽

Price Process ◽

Risk Neutral Measures

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, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

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PRICING FOR GEOMETRIC MARKED POINT PROCESSES UNDER PARTIAL INFORMATION: ENTROPY APPROACH

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024909005191 ◽

2009 ◽

Vol 12 (02) ◽

pp. 179-207 ◽

Cited By ~ 9

Author(s):

CLAUDIA CECI ◽

ANNA GERARDI

Keyword(s):

Stock Price ◽

Point Processes ◽

Partial Information ◽

Marked Point ◽

Asset Price ◽

Marked Point Processes ◽

Marked Point Process ◽

Martingale Measure ◽

Minimal Entropy Martingale Measure ◽

Risk Neutral Measures

The problem of the arbitrage-free pricing of a European contingent claim B is considered in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described through a marked point process Y, whose local characteristics depend on some unobservable jump diffusion process X. The processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. Risk-neutral measures are characterized and in particular, the minimal entropy martingale measure is studied. The problem of pricing under restricted information is discussed, and the arbitrage-free price of the claim B w.r.t. the minimal entropy martingale measure is computed by using filtering techniques.

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Connected-Tube MPP Model for Unsupervised 3D Fiber Detection

Electronic Imaging ◽

10.2352/issn.2470-1173.2020.14.coimg-305 ◽

2020 ◽

Vol 2020 (14) ◽

pp. 305-1-305-6

Author(s):

Tianyu Li ◽

Camilo G. Aguilar ◽

Ronald F. Agyei ◽

Imad A. Hanhan ◽

Michael D. Sangid ◽

...

Keyword(s):

Composite Material ◽

Point Process ◽

Marked Point ◽

Experimental Results ◽

Marked Point Process ◽

Fiber Reinforced Composite ◽

Reinforced Composite ◽

Proposed Model ◽

Cylinder Model ◽

Reinforced Composite Material

In this paper, we extend our previous 2D connected-tube marked point process (MPP) model to a 3D connected-tube MPP model for fiber detection. In the 3D case, a tube is represented by a cylinder model with two spherical areas at its ends. The spherical area is used to define connection priors that encourage connection of tubes that belong to the same fiber. Since each long fiber can be fitted by a series of connected short tubes, the proposed model is capable of detecting curved long tubes. We present experimental results on fiber-reinforced composite material images to show the performance of our method.

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