Learning models with continuous time parameter and multivariate point processes

1983 ◽  
Vol 20 (4) ◽  
pp. 884-890 ◽  
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Point Processes ◽  
Transition Probabilities ◽  
Learning Model ◽  
Time Parameter ◽  
Learning Models ◽  
Random System ◽  
Multivariate Point ◽  
Multivariate Point Processes

The concept of a learning model (or random system with complete connections) with continuous time parameter is introduced on the basis of the notion of a multivariate point process possessing an intensity. The stepwise transition probabilities in terms of the intensity are derived and a Monte Carlo method for simulating a sample is presented. By modelling the intensity process various types of learning models can be built. We propose a linear learning model which comprises the continuous-time Markov process as well as Hawkes's mutually exciting point process. We study the asymptotic behaviour of this linear model in terms of explosion or extinction and of convergence of some estimates. We close with some numerical results from computer simulations.

Learning models with continuous time parameter and multivariate point processes

1983 ◽  
Vol 20 (04) ◽  
pp. 884-890
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Point Processes ◽  
Transition Probabilities ◽  
Learning Model ◽  
Time Parameter ◽  
Learning Models ◽  
Random System ◽  
Multivariate Point ◽  
Multivariate Point Processes

The concept of a learning model (or random system with complete connections) with continuous time parameter is introduced on the basis of the notion of a multivariate point process possessing an intensity. The stepwise transition probabilities in terms of the intensity are derived and a Monte Carlo method for simulating a sample is presented. By modelling the intensity process various types of learning models can be built. We propose a linear learning model which comprises the continuous-time Markov process as well as Hawkes's mutually exciting point process. We study the asymptotic behaviour of this linear model in terms of explosion or extinction and of convergence of some estimates. We close with some numerical results from computer simulations.

Orderliness, intensities and Palm-Khinchin equations for multivariate point processes

1975 ◽  
Vol 12 (2) ◽  
pp. 383-389 ◽  
Author(s):  
D. J. Daley ◽  
R. K. Milne
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Multivariate Point ◽  
Multivariate Point Processes

Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.

Orderliness, intensities and Palm-Khinchin equations for multivariate point processes

1975 ◽  
Vol 12 (02) ◽  
pp. 383-389 ◽  
Author(s):  
D. J. Daley ◽  
R. K. Milne
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Multivariate Point ◽  
Multivariate Point Processes

Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.

Point-Process Principal Components Analysis via Geometric Optimization

2013 ◽  
Vol 25 (1) ◽  
pp. 101-122 ◽  
Author(s):  
Victor Solo ◽  
Syed Ahmed Pasha
Keyword(s):  
Principal Components Analysis ◽  
Point Process ◽  
Principal Components ◽  
Neural Coding ◽  
Point Processes ◽  
Geometric Optimization ◽  
Process Data ◽  
Components Analysis ◽  
Point Process Data ◽  
Multivariate Point

There has been a fast-growing demand for analysis tools for multivariate point-process data driven by work in neural coding and, more recently, high-frequency finance. Here we develop a true or exact (as opposed to one based on time binning) principal components analysis for preliminary processing of multivariate point processes. We provide a maximum likelihood estimator, an algorithm for maximization involving steepest ascent on two Stiefel manifolds, and novel constrained asymptotic analysis. The method is illustrated with a simulation and compared with a binning approach.

scholarly journals Likelihood Methods for Point Processes with Refractoriness

2014 ◽  
Vol 26 (2) ◽  
pp. 237-263 ◽  
Author(s):  
Luca Citi ◽  
Demba Ba ◽  
Emery N. Brown ◽  
Riccardo Barbieri
Keyword(s):  
Point Process ◽  
Continuous Time ◽  
Process Model ◽  
Point Processes ◽  
Linear Models ◽  
Continuous Time Model ◽  
Time Model ◽  
Fitting Procedures ◽  
Continuous Time Process ◽  
Conditional Intensity Function

Likelihood-based encoding models founded on point processes have received significant attention in the literature because of their ability to reveal the information encoded by spiking neural populations. We propose an approximation to the likelihood of a point-process model of neurons that holds under assumptions about the continuous time process that are physiologically reasonable for neural spike trains: the presence of a refractory period, the predictability of the conditional intensity function, and its integrability. These are properties that apply to a large class of point processes arising in applications other than neuroscience. The proposed approach has several advantages over conventional ones. In particular, one can use standard fitting procedures for generalized linear models based on iteratively reweighted least squares while improving the accuracy of the approximation to the likelihood and reducing bias in the estimation of the parameters of the underlying continuous-time model. As a result, the proposed approach can use a larger bin size to achieve the same accuracy as conventional approaches would with a smaller bin size. This is particularly important when analyzing neural data with high mean and instantaneous firing rates. We demonstrate these claims on simulated and real neural spiking activity. By allowing a substantive increase in the required bin size, our algorithm has the potential to lower the barrier to the use of point-process methods in an increasing number of applications.

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