Spatial mixture modelling for unobserved point processes: examples in immunofluorescence histology

This paper presents a statistically based method for spotting target words in documents. The crux of the method is the representation of a word by a spatial (planar) point process evolving on a regular lattice of coordinate pairs. This is accomplished by extracting the coordinate pairs, i.e. pixel locations, where the binary bitmap values of the word are non-zero. With this representation the word is completely determined by the spatial intensity function, i.e. the unnormalized spatial probability density function, associated with the extracted set of coordinate pairs. In this work, we use a finite number of moments of the intensity function to characterize the word. Location and scale invariance is obtained by transforming the coordinate pairs to have zero mean and unit variance. Finally, optimal detection strategies are applied to the moments to make the decision. 1.

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.

Networks and point processes provide flexible tools for representing and modeling complex dependencies in data arising from various social and physical domains. Graphs, or networks, encode relational dependencies between entities, while point processes characterize temporal or spatial interactions among events.<br>In the first part of this dissertation, we consider dynamic network data (such as communication networks) in which links connecting pairs of nodes appear continuously over time. We propose latent space point process models to capture two different aspects of the data: (i) communication occurs at a higher rate between individuals with similar latent attributes (i.e., homophily); and (ii) individuals tend to reciprocate communications from others, but in a varied manner. Our framework marries ideas from point process models, including Poisson and Hawkes processes, with ideas from latent space models of static networks. We evaluate our models on several real-world datasets and show that a dual latent space model, which accounts for heterogeneity in both homophily and reciprocity, significantly improves performance in various link prediction and network embedding tasks.<br>In the second part of this dissertation, we develop nonparametric goodness-of-fit tests for discrete distributions and point processes that contain intractable normalization constants, providing the first generally applicable and computationally feasible approaches under those circumstances. Specifically, we propose and characterize Stein operators for discrete distributions, and construct a general Stein operator for point processes using the Papangelou conditional intensity function. Based on the proposed Stein operators, we establish kernelized Stein discrepancy measures for discrete distributions and point processes, which enable us to develop nonparametric goodness-of-fit tests for un-normalized density/intensity functions. We apply the kernelized Stein discrepancy tests to discrete distributions (including network models) as well as temporal and spatial point processes. Our experiments demonstrate that the proposed tests typically outperform two-sample tests based on the maximum mean discrepancy, which, unlike our goodness-of-fit tests, assume the availability of exact samples from the null model.<br>

Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case-control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica54, 329-350) concerning estimation of the second-order properties of a nonstationary spatial point process. First, we show how case-control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second-order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.

Point processes in time have a wide range of applications that include the claims arrival process in insurance or the analysis of queues in operations research. Due to advances in technology, such samples of point processes are increasingly encountered. A key object of interest is the local intensity function. It has a straightforward interpretation that allows to understand and explore point process data. We consider functional approaches for point processes, where one has a sample of repeated realizations of the point process. This situation is inherently connected with Cox processes, where the intensity functions of the replications are modeled as random functions. Here we study a situation where one records covariates for each replication of the process, such as the daily temperature for bike rentals. For modeling point processes as responses with vector covariates as predictors we propose a novel regression approach for the intensity function that is intrinsically nonparametric. While the intensity function of a point process that is only observed once on a fixed domain cannot be identified, we show how covariates and repeated observations of the process can be utilized to make consistent estimation possible, and we also derive asymptotic rates of convergence without invoking parametric assumptions.

This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.

The K-function is arguably the most important functional summary statistic for spatial point processes. It is used extensively for goodness-of-fit testing and in connection with minimum contrast estimation for parametric spatial point process models. It is thus pertinent to understand the asymptotic properties of estimates of the K-function. In this paper we derive the functional asymptotic distribution for the K-function estimator. Contrary to previous papers on functional convergence we consider the case of an inhomogeneous intensity function. We moreover handle the fact that practical K-function estimators rely on plugging in an estimate of the intensity function. This removes two serious limitations of the existing literature.

A point process is a collection of positional variables in a given space, while a marked point process (MPP) is basically a point process that measures some additional feature or value at each point. Existing applications such as financial market models and hospitalization models have demonstrated that MPPS are suitable for modeling random events with meaningful values. A big challenge is to study the relationship between different types of events, or how different types of markers affect each other. While Hawkes procedures are an effective and efficient way to model point processes with internal influences, inferential methods of intensity functions can be expensive, especially for multi-dimensional processes. Graph convolutional Network (GCN) is a powerful neural network designed to directly process graphs. A GCN-based model is presented to study potential patterns in the marking process and to predict the location of future events without estimating the Hawkes intensity function. A real data set of water pipe leakage and rupture records over the last 50 years was used in the experiment, modeled as a marker point process, where leakage and rupture are two marker forms. Experimental results on synthetic data sets and real water pipe data sets show that the proposed model is superior to recent state-of-the-art methods.

Point processes are a useful mathematical tool for describing events over time, and so there are many recent approaches for representing and learning them. One notable open question is how to precisely describe the flexibility of point process models and whether there exists a general model that can represent all point processes. Our work bridges this gap. Focusing on the widely used event intensity function representation of point processes, we provide a proof that a class of learnable functions can universally approximate any valid intensity function. The proof connects the well known Stone-Weierstrass Theorem for function approximation, the uniform density of non-negative continuous functions using a transfer functions, the formulation of the parameters of a piece-wise continuous functions as a dynamic system, and a recurrent neural network implementation for capturing the dynamics. Using these insights, we design and implement UNIPoint, a novel neural point process model, using recurrent neural networks to parameterise sums of basis function upon each event. Evaluations on synthetic and real world datasets show that this simpler representation performs better than Hawkes process variants and more complex neural network-based approaches. We expect this result will provide a practical basis for selecting and tuning models, as well as furthering theoretical work on representational complexity and learnability.

The pair correlation function is a fundamental spatial point process\ncharacteristic that, given the intensity function, determines second order\nmoments of the point process. Non-parametric estimation of the pair correlation\nfunction is a typical initial step of a statistical analysis of a spatial point\npattern. Kernel estimators are popular but especially for clustered point\npatterns suffer from bias for small spatial lags. In this paper we introduce a\nnew orthogonal series estimator. The new estimator is consistent and\nasymptotically normal according to our theoretical and simulation results. Our\nsimulations further show that the new estimator can outperform the kernel\nestimators in particular for Poisson and clustered point processes.\n

The projection method obtains non-trivial point processes from higher-dimensional Poisson point processes by constructing a random subset of the higher-dimensional space and projecting the points of the Poisson process lying in that set onto the lower-dimensional region. This paper presents a review of this method related to spatial point processes as well as some examples of its applications. The results presented here are known for sometime but were not published before. Also, we present a backward construction of general spatial pure-birth processes and spatial birth and death processes based on the projection method that leads to a perfect simulation scheme for some Gibbs distributions in compact regions.

We propose a new bandwidth selection method for kernel estimators of spatial point process intensity functions. The method is based on an optimality criterion motivated by the Campbell formula applied to the reciprocal intensity function. The new method is fully nonparametric, does not require knowl- edge of higher-order moments, and is not restricted to a specific class of point process. Our approach is computationally straightforward and does not require numerical approximation of integrals.

The probability hypothesis density (PHD) filter, which is used for multi-target tracking based on sensor measurements, relies on the propagation of the first-order moment, or intensity function, of a point process. This algorithm assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. In this paper, we construct a second-order PHD filter based on determinantal point processes which are able to model repulsion between targets. Such processes are characterized by their first- and second-order moments, which allows the algorithm to propagate variance and covariance information in addition to first-order target count estimates. Our approach relies on posterior moment formulas for the estimation of a general hidden point process after a thinning operation and a superposition with a Poisson point process, and on suitable approximation formulas in the determinantal point process setting. The repulsive properties of determinantal point processes apply to the modeling of negative correlation between distinct measurement domains. Monte Carlo simulations with correlation estimates are provided.

The inhomogeneous Poisson point process is a common model for time series of discrete, stochastic events. When an event from a point process is detected, it may trigger a random dead time in the detector, during which subsequent events will fail to be detected. It can be difficult or impossible to obtain a closed-form expression for the distribution of intervals between detections, even when the rate function (often referred to as the intensity function) and the dead-time distribution are given. Here, a method is presented to numerically compute the interval distribution expected for any arbitrary inhomogeneous Poisson point process modified by dead times drawn from any arbitrary distribution. In neuroscience, such a point process is used to model trains of neuronal spikes triggered by the detection of excitatory events while the neuron is not refractory. The assumptions of the method are that the process is observed over a finite observation window and that the detector is not in a dead state at the start of the observation window. Simulations are used to verify the method for several example point processes. The method should be useful for modeling and understanding the relationships between the rate functions and interval distributions of the event and detection processes, and how these relationships depend on the dead-time distribution.

Point pattern data are ubiquitous in both the natural environment and human activities. Spatial and spatio-temporal point processes are widely utilized to model the occurrence of random events across space and time. These models aim to uncover the underlying patterns of such events by analyzing environmental variables that influence their probabilities, and by testing, estimating and simulating spatial or spatio-temporal correlations. Despite significant advances in point process theory and methodology in the last decades, classical approaches still require continuous refinement and extension. This need arises from the distinctive characteristics of point pattern data in practical applications, as well as the increasing amount and complexity of data available in contemporary research. This thesis explores spatial and spatio-temporal modelling in the context of fire risk prediction in the Netherlands. Fire occurrences exhibit unique features that pose specific challenges in point process modelling. Four critical research questions are identified based on real applications of point process methods to the data. To address these challenges, we developed novel statistical and machine learning approaches, proposing targeted solutions that advance the state-of-the-art in the research domain.<br/><br/>