scholarly journals Valuations for Spike Train Prediction

2008 ◽  
Vol 20 (3) ◽  
pp. 644-667 ◽  
Author(s):  
Vladimir Itskov ◽  
Carina Curto ◽  
Kenneth D. Harris
Keyword(s):  
Experimental Data ◽  
Spike Train ◽  
Computational Efficiency ◽  
Euclidean Distance ◽  
Intensity Function ◽  
Probabilistic Interpretation ◽  
Conditional Intensity ◽  
Prediction Quality ◽  
Log Likelihood ◽  
Conditional Intensity Function

The ultimate product of an electrophysiology experiment is often a decision on which biological hypothesis or model best explains the observed data. We outline a paradigm designed for comparison of different models, which we refer to as spike train prediction. A key ingredient of this paradigm is a prediction quality valuation that estimates how close a predicted conditional intensity function is to an actual observed spike train. Although a valuation based on log likelihood (L) is most natural, it has various complications in this context. We propose that a quadratic valuation (Q) can be used as an alternative to L. Q shares some important theoretical properties with L, including consistency, and the two valuations perform similarly on simulated and experimental data. Moreover, Q is more robust than L, and optimization with Q can dramatically improve computational efficiency. We illustrate the utility of Q for comparing models of peer prediction, where it can be computed directly from cross-correlograms. Although Q does not have a straightforward probabilistic interpretation, Q is essentially given by Euclidean distance.

Measuring the association of a time series and a point process

1982 ◽  
Vol 19 (3) ◽  
pp. 597-608 ◽  
Author(s):  
J. S. Willie
Keyword(s):  
Time Series ◽  
Point Process ◽  
Intensity Function ◽  
Asymptotic Distributions ◽  
The Other ◽  
Stationary Case ◽  
Conditional Intensity ◽  
A Value ◽  
Conditional Intensity Function ◽  
Ordinary Time

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.

Exploratory analysis of earthquake clusters by likelihood-based trigger models

2001 ◽  
Vol 38 (A) ◽  
pp. 202-212 ◽  
Author(s):  
Yosihiko Ogata
Keyword(s):  
Maximum Likelihood ◽  
Point Process ◽  
Process Model ◽  
Intensity Function ◽  
Exploratory Analysis ◽  
Maximum Likelihood Estimates ◽  
Conditional Intensity ◽  
Point Process Model ◽  
Conditional Intensity Function ◽  
Earthquake Clusters

The paper considers the superposition of modified Omori functions as a conditional intensity function for a point process model used in the exploratory analysis of earthquake clusters. For the examples discussed, the maximum likelihood estimates converge well starting from appropriate initial values even though the number of parameters estimated can be large (though never larger than the number of observations). Three datasets are subjected to different analyses, showing the use of the model to discover and study individual clustering features.

A Computationally Efficient Method for Nonparametric Modeling of Neural Spiking Activity with Point Processes

2010 ◽  
Vol 22 (8) ◽  
pp. 2002-2030 ◽  
Author(s):  
Todd P. Coleman ◽  
Sridevi S. Sarma
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Intensity Function ◽  
Process Models ◽  
Conditional Intensity ◽  
Computationally Efficient ◽  
Neural Data ◽  
Neural Spiking ◽  
Point Process Models ◽  
Conditional Intensity Function

Point-process models have been shown to be useful in characterizing neural spiking activity as a function of extrinsic and intrinsic factors. Most point-process models of neural activity are parametric, as they are often efficiently computable. However, if the actual point process does not lie in the assumed parametric class of functions, misleading inferences can arise. Nonparametric methods are attractive due to fewer assumptions, but computation in general grows with the size of the data. We propose a computationally efficient method for nonparametric maximum likelihood estimation when the conditional intensity function, which characterizes the point process in its entirety, is assumed to be a Lipschitz continuous function but otherwise arbitrary. We show that by exploiting much structure, the problem becomes efficiently solvable. We next demonstrate a model selection procedure to estimate the Lipshitz parameter from data, akin to the minimum description length principle and demonstrate consistency of our estimator under appropriate assumptions. Finally, we illustrate the effectiveness of our method with simulated neural spiking data, goldfish retinal ganglion neural data, and activity recorded in CA1 hippocampal neurons from an awake behaving rat. For the simulated data set, our method uncovers a more compact representation of the conditional intensity function when it exists. For the goldfish and rat neural data sets, we show that our nonparametric method gives a superior absolute goodness-of-fit measure used for point processes than the most common parametric and splines-based approaches.

scholarly journals Kernel estimation and display of a five-dimensional conditional intensity function

2010 ◽  
Vol 17 (2) ◽  
pp. 237-244 ◽  
Author(s):  
G. Adelfio
Keyword(s):  
Kernel Estimation ◽  
Intensity Function ◽  
Second Order ◽  
High Dimensional ◽  
General Version ◽  
Conditional Intensity ◽  
Nonparametric Smoothing ◽  
Conditional Intensity Function

Abstract. The aim of this paper is to find a convenient and effective method of displaying some second order properties in a neighbourhood of a selected point of the process. The used techniques are based on very general high-dimensional nonparametric smoothing developed to define a more general version of the conditional intensity function introduced in earlier earthquake studies by Vere-Jones (1978).

scholarly journals Stochastic Models for the Infectivity Function in an Infinite Population of Susceptible Individuals

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Viswanathan Arunachalam ◽  
Liliana Blanco
Keyword(s):  
Stochastic Models ◽  
Intensity Function ◽  
The Other ◽  
Statistical Characteristics ◽  
Model Parameters ◽  
Conditional Intensity ◽  
Transient Behaviour ◽  
Infinite Population ◽  
External Sources ◽  
Conditional Intensity Function

Two stochastic models to study the course of the transient behaviour of the total infectivity present in an infinite population of susceptible individuals are developed. The conditional intensity function of the contagion comprises two components: one is due to the external sources only and the other is the contribution of each of the infected persons which is nonstationary in nature. The statistical characteristics of the number of infected individuals at any time are explicitly obtained. Estimation of the model parameters is also indicated.

A Point Process Framework for Relating Neural Spiking Activity to Spiking History, Neural Ensemble, and Extrinsic Covariate Effects

2005 ◽  
Vol 93 (2) ◽  
pp. 1074-1089 ◽  
Author(s):  
Wilson Truccolo ◽  
Uri T. Eden ◽  
Matthew R. Fellows ◽  
John P. Donoghue ◽  
Emery N. Brown
Keyword(s):  
Point Process ◽  
Discrete Time ◽  
Likelihood Function ◽  
Intensity Function ◽  
Relative Importance ◽  
Conditional Intensity ◽  
Process Framework ◽  
Neural Spiking ◽  
Conditional Intensity Function ◽  
Spiking Activity

Multiple factors simultaneously affect the spiking activity of individual neurons. Determining the effects and relative importance of these factors is a challenging problem in neurophysiology. We propose a statistical framework based on the point process likelihood function to relate a neuron's spiking probability to three typical covariates: the neuron's own spiking history, concurrent ensemble activity, and extrinsic covariates such as stimuli or behavior. The framework uses parametric models of the conditional intensity function to define a neuron's spiking probability in terms of the covariates. The discrete time likelihood function for point processes is used to carry out model fitting and model analysis. We show that, by modeling the logarithm of the conditional intensity function as a linear combination of functions of the covariates, the discrete time point process likelihood function is readily analyzed in the generalized linear model (GLM) framework. We illustrate our approach for both GLM and non-GLM likelihood functions using simulated data and multivariate single-unit activity data simultaneously recorded from the motor cortex of a monkey performing a visuomotor pursuit-tracking task. The point process framework provides a flexible, computationally efficient approach for maximum likelihood estimation, goodness-of-fit assessment, residual analysis, model selection, and neural decoding. The framework thus allows for the formulation and analysis of point process models of neural spiking activity that readily capture the simultaneous effects of multiple covariates and enables the assessment of their relative importance.

Measuring the association of a time series and a point process

1982 ◽  
Vol 19 (03) ◽  
pp. 597-608
Author(s):  
J. S. Willie
Keyword(s):  
Time Series ◽  
Point Process ◽  
Intensity Function ◽  
Asymptotic Distributions ◽  
The Other ◽  
Stationary Case ◽  
Conditional Intensity ◽  
A Value ◽  
Conditional Intensity Function ◽  
Ordinary Time

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ 11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ 11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.

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