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Jesper Møller

Publications and source records attributed to Jesper Møller.

At least 37 records · Page 2Linked to original sources
arXiv2019

Structured space-sphere point processes and $K$-functions

This paper concerns space-sphere point processes, that is, point processes on the product space of $\mathbb R^d$ (the $d$-dimensional Euclidean space) and $\mathbb S^k$ (the $k$-dimen\-sional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere $K$-function which is a natural extension of the inhomogeneous $K$-function for point processes on $\mathbb R^d$ to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere $K$-function is shown to be proportional to the product of the inhomogeneous spatial and spherical $K$-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The usefulness of the space-sphere $K$-function is illustrated for real and simulated datasets with varying dimensions $d$ and $k$.

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arXiv2019

Quick inference for log Gaussian Cox processes with non-stationary underlying random fields

For point patterns observed in natura, spatial heterogeneity is more the rule than the exception. In numerous applications, this can be mathematically handled by the flexible class of log Gaussian Cox processes (LGCPs); in brief, a LGCP is a Cox process driven by an underlying log Gaussian random field (log GRF). This allows the representation of point aggregation, point vacuum and intermediate situations, with more or less rapid transitions between these different states depending on the properties of GRF. Very often, the covariance function of the GRF is assumed to be stationary. In this article, we give two examples where the sizes (that is, the number of points) and the spatial extents of point clusters are allowed to vary in space. To tackle such features, we propose parametric and semiparametric models of non-stationary LGCPs where the non-stationarity is included in both the mean function and the covariance function of the GRF. Thus, in contrast to most other work on inhomogeneous LGCPs, second-order intensity-reweighted stationarity is not satisfied and the usual two step procedure for parameter estimation based on e.g. composite likelihood does not easily apply. Instead we propose a fast three step procedure based on composite likelihood. We apply our modelling and estimation framework to analyse datasets dealing with fish aggregation in a reservoir and with dispersal of biological particles.

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arXiv2019

Isotropic covariance functions on graphs and their edges

We develop parametric classes of covariance functions on linear networks and their extension to graphs with Euclidean edges, i.e., graphs with edges viewed as line segments or more general sets with a coordinate system allowing us to consider points on the graph which are vertices or points on an edge. Our covariance functions are defined on the vertices and edge points of these graphs and are isotropic in the sense that they depend only on the geodesic distance or on a new metric called the resistance metric (which extends the classical resistance metric developed in electrical network theory on the vertices of a graph to the continuum of edge points). We discuss the advantages of using the resistance metric in comparison with the geodesic metric as well as the restrictions these metrics impose on the investigated covariance functions. In particular, many of the commonly used isotropic covariance functions in the spatial statistics literature (the power exponential, Mat{é}rn, generalized Cauchy, and Dagum classes) are shown to be valid with respect to the resistance metric for any graph with Euclidean edges, whilst they are only valid with respect to the geodesic metric in more special cases.

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arXiv2019

Pair correlation functions and limiting distributions of iterated cluster point processes

We consider a Markov chain of point processes such that each state is a super position of an independent cluster process with the previous state as its centre process together with some independent noise process. The model extends earlier work by Felsenstein and Shimatani describing a reproducing population. We discuss when closed term expressions of the first and second order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

math.PR↗
arXiv2018

Log Gaussian Cox processes on the sphere

A log Gaussian Cox process (LGCP) is a doubly stochastic construction consisting of a Poisson point process with a random log-intensity given by a Gaussian random field. Statistical methodology have mainly been developed for LGCPs defined in the $d$-dimensional Euclidean space. This paper concerns the case of LGCPs on the $d$-dimensional sphere, with $d=2$ of primary interest. We discuss the existence problem of such LGCPs, provide sufficient existence conditions, and establish further useful theoretical properties. The results are applied for the description of sky positions of galaxies, in comparison with previous analysis based on a Thomas process, using simple estimation procedures and making a careful model checking. We account for inhomogeneity in our models, and as the model checking is based on a thinning procedure which produces homogeneous/isotropic LGCPs, we discuss its sensitivity.

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arXiv2017

Investigations of the effects of random sampling patterns on the stability of generalized sampling

We investigate how the choice of spatial point process for generating random sampling patterns affects the numerical stability of non-uniform generalized sampling between Fourier bases and Daubechies scaling functions. Specifically, we consider binomial, Poisson and determinantal point processes and demonstrate that the more regular point patterns from the determinantal point process are superior.

stat.AP↗
arXiv2017

The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications

We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The appendix includes additional methods and examples, together with technical details. The R-code used for all examples is available at http://people.math.aau.dk/~christophe/Rcode.zip.

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arXiv2016

Some recent developments in statistics for spatial point patterns

This paper reviews developments in statistics for spatial point processes obtained within roughly the last decade. These developments include new classes of spatial point process models such as determinantal point processes, models incorporating both regularity and aggregation, and models where points are randomly distributed around latent geometric structures. Regarding parametric inference the main focus is on various types of estimating functions derived from so-called innovation measures. Optimality of such estimating functions is discussed as well as computational issues. Maximum likelihood inference for determinantal point processes and Bayesian inference are briefly considered too. Concerning non-parametric inference, we consider extensions of functional summary statistics to the case of inhomogeneous point processes as well as new approaches to simulation based inference.

stat.ME↗
arXiv2016

The cylindrical K-function and Poisson line cluster point processes

Analyzing point patterns with linear structures has recently been of interest in e.g. neuroscience and geography. To detect anisotropy in such cases, we introduce a functional summary statistic, called the cylindrical $K$-function, since it is a directional $K$-function whose structuring element is a cylinder. Further we introduce a class of anisotropic Cox point processes, called Poisson line cluster point processes. The points of such a process are random displacements of Poisson point processes defined on the lines of a Poisson line process. Parameter estimation based on moment methods or Bayesian inference for this model is discussed when the underlying Poisson line process and the cluster memberships are treated as hidden processes. To illustrate the methodologies, we analyze a two and a three-dimensional point pattern data set. The 3D data set is of particular interest as it relates to the minicolumn hypothesis in neuroscience, claiming that pyramidal and other brain cells have a columnar arrangement perpendicular to the pial surface of the brain.

math.ST↗
arXiv2016

Determinantal point process models on the sphere

We consider determinantal point processes on the $d$-dimensional unit sphere $\mathbb S^d$. These are finite point processes exhibiting repulsiveness and with moment properties determined by a certain determinant whose entries are specified by a so-called kernel which we assume is a complex covariance function defined on $\mathbb S^d\times\mathbb S^d$. We review the appealing properties of such processes, including their specific moment properties, density expressions and simulation procedures. Particularly, we characterize and construct isotropic DPPs models on $\mathbb{S}^d$, where it becomes essential to specify the eigenvalues and eigenfunctions in a spectral representation for the kernel, and we figure out how repulsive isotropic DPPs can be. Moreover, we discuss the shortcomings of adapting existing models for isotropic covariance functions and consider strategies for developing new models, including a useful spectral approach.

stat.ME↗
arXiv2016

A tutorial on Palm distributions for spatial point processes

This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes , we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally we discuss some examples of Palm distributions for specific models and some applications.

math.ST↗
arXiv2016

Functional summary statistics for point processes on the sphere with an application to determinantal point processes

We study point processes on $\mathbb S^d$, the $d$-dimensional unit sphere $\mathbb S^d$, considering both the isotropic and the anisotropic case, and focusing mostly on the spherical case $d=2$. The first part studies reduced Palm distributions and functional summary statistics, including nearest neighbour functions, empty space functions, and Ripley's and inhomogeneous $K$-functions. The second part partly discusses the appealing properties of determinantal point process (DPP) models on the sphere and partly considers the application of functional summary statistics to DPPs. In fact DPPs exhibit repulsiveness, but we also use them together with certain dependent thinnings when constructing point process models on the sphere with aggregation on the large scale and regularity on the small scale. We conclude with a discussion on future work on statistics for spatial point processes on the sphere.

stat.ME↗
arXiv2016

Palm distributions for log Gaussian Cox processes

This paper establishes a remarkable result regarding Palmdistributions for a log Gaussian Cox process: the reduced Palmdistribution for a log Gaussian Cox process is itself a log Gaussian Coxprocess which only differs from the original log Gaussian Cox processin the intensity function. This new result is used to study functionalsummaries for log Gaussian Cox processes.

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arXiv2015

Modelling aggregation on the large scale and regularity on the small scale in spatial point pattern datasets

We consider a dependent thinning of a regular point process with the aim of obtaining aggregation on the large scale and regularity on the small scale in the resulting target point process of retained points. Various parametric models for the underlying processes are suggested and the properties of the target point process are studied. Simulation and inference procedures are discussed when a realization of the target point process is observed, depending on whether the thinned points are observed or not. The paper extends previous work by Dietrich Stoyan on interrupted point processes.

stat.ME↗
arXiv2014

Fingerprint Analysis with Marked Point Processes

We present a framework for fingerprint matching based on marked point process models. An efficient Monte Carlo algorithm is developed to calculate the marginal likelihood ratio for the hypothesis that two observed prints originate from the same finger against the hypothesis that they originate from different fingers. Our model achieves good performance on an NIST-FBI fingerprint database of 258 matched fingerprint pairs.

stat.ME↗
arXiv2014

Variational approach for spatial point process intensity estimation

We introduce a new variational estimator for the intensity function of an inhomogeneous spatial point process with points in the $d$-dimensional Euclidean space and observed within a bounded region. The variational estimator applies in a simple and general setting when the intensity function is assumed to be of log-linear form $β+{θ}^{\top}z(u)$ where $z$ is a spatial covariate function and the focus is on estimating ${θ}$. The variational estimator is very simple to implement and quicker than alternative estimation procedures. We establish its strong consistency and asymptotic normality. We also discuss its finite-sample properties in comparison with the maximum first order composite likelihood estimator when considering various inhomogeneous spatial point process models and dimensions as well as settings were $z$ is completely or only partially known.

math.ST↗
arXiv2014

Determinantal point process models and statistical inference : Extended version

Statistical models and methods for determinantal point processes (DPPs) seem largely unexplored. We demonstrate that DPPs provide useful models for the description of spatial point pattern datasets where nearby points repel each other. Such data are usually modelled by Gibbs point processes, where the likelihood and moment expressions are intractable and simulations are time consuming. We exploit the appealing probabilistic properties of DPPs to develop parametric models, where the likelihood and moment expressions can be easily evaluated and realizations can be quickly simulated. We discuss how statistical inference is conducted using the likelihood or moment properties of DPP models, and we provide freely available software for simulation and statistical inference.

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