Markov Point Processes Research Articles

Nearest-neighbour Markov point processes on graphs with Euclidean edges

Abstract We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

Going beyond Poisson processes: a new statistical framework in neuronal modeling and data analysis

Cortical spike trains in vivo often show a high irregularity, reflected by a coefficient of variation (CV) close to one [1]. Such irregular single neuron spiking has consequently been associated with Poisson processes, and many models involving neuronal spike trains inherited this ideology. This viewpoint is further supported by the Palm-Khintchine theorem, which states that a superposition of a large number of renewal processes with very small intensity behaves like a Poisson process. It was demonstrated, however, that this theorem doesn't always apply to the superposition of neuronal spike trains [2-4]. Moreover, Poisson processes lack the temporal properties observed in population responses to input modulation either via a stimulus in vivo [5] or via electrical stimulation in vitro [6]. In this study, we report on new techniques to deal with non-Possonian aspects of stationary neuronal spike trains, as well as non-equilibrium population responses [7], based on Markov point processes (MPP), commonly known as continuous time Markov chains (CTMC). We compute the interspike interval (ISI) distribution by algebraically solving the first passage time problem for MPP neuron models, and compute the transient population responses with a similar technique. The same technique is used to compute exact cross-correlation functions for a shared input paradigm [8]. We advertise MPPs as a new powerful framework in neural network modeling and neural data analysis with many possible applications.

Markov Models of Neuronal Populations: a Reduction of Integrate-and-Fire Dynamics

Event Abstract Back to Event Markov Models of Neuronal Populations: a Reduction of Integrate-and-Fire Dynamics Taskin Deniz1, 2*, Moritz Deger1, 2 and Stefan Rotter1, 2 1 Bernstein Center Freiburg, Germany 2 Faculty of Biology, Freiburg University, Germany Neuronal populations are capable of responding to transient input much faster than single neurons can. Neuronal codes that exploit this fact are thought to be more robust than a code based on stationary firing rates. Here, we studied populations of unconnected leaky integrate-and-fire (LIF) neurons in the fluctuation driven regime, and compared them to ensembles of Markov Point processes (MPP). In particular, we were interested in characterising the transient responses to input mean modulation vs. input variance modulation, and finding the correspondence between the two models. Markov Point Processes are very attractive for neuronal modelling because many aspects of them are analytically tractable. Acknowledgements Funding by the German Ministry of Education and Research ( BCCN Freiburg, grant 01GQ0420 and BFNT Freiburg*Tubingen, grant 01GQ0830 ) is acknowledged.

MODELLING THE POSITION OF CELL PROFILES ALLOWING FOR BOTH INHOMOGENEITY AND INTERACTION

It is of interest to consider models for point processes that allow for interaction between the points as well as for inhomogeneity in the intensity of the points. Markov point process models are very useful to describe point interaction and can also be used to describe inhomogeneity. A particular type of inhomogeneous Markov point processes obtained by transforming a homogeneous Markov point process will be considered. The position of cell proles in a 2D section of the mucous membrane in the stomach of a rat will be examined using this model.

INHOMOGENEITY IN SPATIAL COX POINT PROCESSES – LOCATION DEPENDENT THINNING IS NOT THE ONLY OPTION

In the literature on point processes the by far most popular option for introducing inhomogeneity into a point process model is the location dependent thinning (resulting in a second-order intensity-reweighted stationary point process). This produces a very tractable model and there are several fast estimation procedures available. Nevertheless, this model dilutes the interaction (or the geometrical structure) of the original homogeneous model in a special way. When concerning the Markov point processes several alternative inhomogeneous models were suggested and investigated in the literature. But it is not so for the Cox point processes, the canonical models for clustered point patterns. In the contribution we discuss several other options how to define inhomogeneous Cox point process models that result in point patterns with different types of geometric structure. We further investigate the possible parameter estimation procedures for such models.

Markov point processes for modeling of spatial forest patterns in Amazonia derived from interferometric height

The spatial distribution of very large trees in primary Amazon forest is studied with an indicative data set. Very large trees with height larger than 30 m are shown to be highly influential on forest structure, ecology and biomass regime. In particular, they account for a large portion of total above-ground biomass. Their spatial patterns are extracted from airborne SAR data, namely from a digital model of interferometric forest height, by an approach of local maximum filtering. The spatial point patterns describing the distribution of very large trees in the forest within three sample blocks of 100 ha each are modeled by a series of Markov point process models. These models are fitted and assessed by standard spatial statistical methodology. Spatial distribution is regular, and interaction decreases with distance; very large trees are shown to exert repulsive interaction with their neighboring very large trees. The significance of these results for approaches of quantitative forest assessment in primary forests in the Brazilian Amazon is discussed.

Statistical Inference for Transformation Inhomogeneous Point Processes

Abstract. Statistical inference for exponential inhomogeneous Markov point processes by transformation is discussed. It is argued that the inhomogeneity parameter can be estimated, using a partial likelihood based on an inhomogeneous Poisson point process. The inhomogeneity parameter can thereby be estimated without taking the interaction into account, which simplifies the statistical analysis considerably. Data analysis and simulation experiments support the results.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Non‐parametric confidence intervals for correlations in nearest‐neighbour Markov point processes

AbstractWe study bootstrap confidence intervals for correlation functions in nearest‐neighbour Markov point processes, where the neighbours are characterized by an interaction of bounded radius r. In forestry statistics, the points are tree locations belonging to a region (forest) A, and the marks are qualitative or quantitative tree variables, such as tree species, the stem diameter, crown length or tree height. Estimating and analysing correlation functions between locations and marks, cross‐correlations between different species and their marks, is typically a key step in statistical interpretation of mapped data sets from a forest stand. In order to define the original sample, we propose coding schemes, which are fixed and divide the observed region A of the point process into regular, conditionally independent subregions {Bυ} located at Euclidean distance d ≥ 2r. Bootstrap confidence intervals are then obtained directly, by considering kernel density estimates from all subregions {Bυ} as conditionally independent replicates. Copyright © 2002 John Wiley & Sons, Ltd.

Clustering methods based on variational analysis in the space of measures

We formulate clustering as a minimisation problem in the space of measures by modelling the cluster centres as a Poisson process with unknown intensity function.We derive a Ward-type clustering criterion which, under the Poisson assumption, can easily be evaluated explicitly in terms of the intensity function. We show that asymptotically, i.e. for increasing total intensity, the optimal intensity function is proportional to a dimension-dependent power of the density of the observations. For fixed finite total intensity, no explicit solution seems available. However, the Ward-type criterion to be minimised is convex in the intensity function, so that the steepest descent method of Molchanov and Zuyev (2001) can be used to approximate the global minimum. It turns out that the gradient is similar in form to the functional to be optimised. If we discretise over a grid, the steepest descent algorithm at each iteration step increases the current intensity function at those points where the gradient is minimal at the expense of regions with a large gradient value. The algorithm is applied to a toy one-dimensional example, a simulation from a popular spatial cluster model and a real-life dataset from Strauss (1975) concerning the positions of redwood seedlings. Finally, we discuss the relative merits of our approach compared to classical hierarchical and partition clustering techniques as well as to modern model based clustering methods using Markov point processes and mixture distributions.

Inhomogeneous Markov Point Processes by Transformation

We construct parametrized models for point processes, allowing for both inhomogeneity and interaction. The inhomogeneity is obtained by applying parametrized transformations to homogeneous Markov point processes. An interesting model class, which can be constructed by this transformation approach, is that of exponential inhomogeneous Markov point processes. Statistical inference for such processes is discussed in some detail.

Markov interacting component processes

A generalization of Markov point processes is introduced in which interactions occur between connected components of the point pattern. A version of the Hammersley-Clifford characterization theorem is proved which states that a point process is a Markov interacting component process if and only if its density function is a product of interaction terms associated with cliques of connected components. Integrability and superpositional properties of the processes are shown and a pairwise interaction example is used for detailed exploration.

Markov interacting component processes

A generalization of Markov point processes is introduced in which interactions occur between connected components of the point pattern. A version of the Hammersley-Clifford characterization theorem is proved which states that a point process is a Markov interacting component process if and only if its density function is a product of interaction terms associated with cliques of connected components. Integrability and superpositional properties of the processes are shown and a pairwise interaction example is used for detailed exploration.

Directed Markov Point Processes as Limits of Partially Ordered Markov Models

In this paper, we consider spatial point processes and investigate members of a subclass of the Markov point processes, termed the directed Markov point processes (DMPPs), whose joint distribution can be written in closed form and, as a consequence, its parameters can be estimated directly. Furthermore, we show how the DMPPs can be simulated rapidly using a one-pass algorithm. A subclass of Markov random fields on a finite lattice, called partially ordered Markov models (POMMs), has analogous structure to that of DMPPs. In this paper, we show that DMPPs are the limits of auto-Poisson and auto-logistic POMMs. These and other results reveal a close link between inference and simulation for DMPPs and POMMs.

On connected component Markov point processes

We note some interesting properties of the class of point processes which are Markov with respect to the ‘connected component’ relation. Results in the literature imply that this class is closed under random translation and independent cluster generation with almost surely non-empty clusters. We further prove that it is closed under superposition. A wide range of examples is also given.

On connected component Markov point processes

We note some interesting properties of the class of point processes which are Markov with respect to the ‘connected component’ relation. Results in the literature imply that this class is closed under random translation and independent cluster generation with almost surely non-empty clusters. We further prove that it is closed under superposition. A wide range of examples is also given.

On the variance to mean ratio for random variables from Markov chains and point processes

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

Transformations of Markov point processes

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On the MLE for a spatial point pattern

Parameter estimation for a two-dimensional point pattern is difficult because most of the available stochastic models have intractable likelihoods ([2]). An exception is the class of Gibbs or Markov point processes ([1], [5]), where the likelihood typically forms an exponential family and is given explicitly up to a normalising constant. However, the latter is not known analytically, so parameter estimates must be based on approximations ([3], [6], [7]). In this paper we present comparisons amongst the different techniques available in the literature to obtain an approximation of the maximum likelihood estimate (MLE). Two stochastic methods are specifically illustrated: a Newton-Raphson algorithm ([7]) and the Robbins-Monro procedure ([8]). We use a very simple point process model, the Strauss process ([4]), to test and compare those approximations.