On goodness‐of‐fit testing for self‐exciting point processes

In this paper we propagate a large deviations approach for proving limit theory for (generally) multivariate time series with heavy tails. We make this notion precise by introducing regularly varying time series. We provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin. We study a variety of such functionals, including large deviations of random walks, their suprema, the ruin functional, and further derive weak limit theory for maxima, point processes, cluster functionals and the tail empirical process. One of the main results of this paper concerns bounds for the ruin probability in various heavy-tailed models including GARCH, stochastic volatility models and solutions to stochastic recurrence equations. 1. Preliminaries and basic motivation In the last decades, a lot of efforts has been put into the understanding of limit theory for dependent sequences, including Markov chains (Meyn and Tweedie [42]), weakly dependent sequences (Dedecker et al. [21]), long-range dependent sequences (Doukhan et al. [23], Samorodnitsky [54]), empirical processes (Dehling et al. [22]) and more general structures (Eberlein and Taqqu [25]), to name a few references. A smaller part of the theory was devoted to limit theory under extremal dependence for point processes, maxima, partial sums, tail empirical processes. Resnick [49, 50] started a systematic study of the relations between the convergence of point processes, sums and maxima, see also Resnick [51] for a recent account. He advocated the use of multivariate regular variation as a flexible tool to describe heavy-tail phenomena combined with advanced continuous mapping techniques. For example, maxima and sums are understood as functionals acting on an underlying point process, if the point process converges these functionals converge as well and their limits are described in terms of the points of the limiting point process. Davis and Hsing [13] recognized the power of this approach for limit theory of point processes, maxima, sums, and large deviations for dependent regularly varying processes, i.e., stationary sequences whose finite-dimensional distributions are regularly varying with the same index. Before [13], limit theory for particular regularly varying stationary sequences was studied for the sample mean, maxima, sample autocovariance and autocorrelation functions of linear and bilinear processes with iid regularly varying noise and extreme value theory was considered for regularly varying ARCH processes and solutions to stochastic recurrence equation, see Rootz\'en [53], Davis and 1991 Mathematics Subject Classification. Primary 60F10, 60G70, secondary 60F05. Key words and phrases. Large deviation principle, regularly varying processes, central limit theorem, ruin probabilities, GARCH.

First we put together basic definitions and fundamental facts and results from the theory of (un)marked point processes defined on Euclidean spaces \({\mathbb{R}}^{d}\). We introduce the notion random marked point process together with the concept of Palm distributions in a rigorous way followed by the definitions of factorial moment and cumulant measures and characteristics related with them. In the second part we define a variety of estimators of second-order characteristics and other so-called summary statistics of stationary point processes based on observations on a “convex averaging sequence” of windows \(\{W_{n},\,n \in \mathbb{N}\}\). Although all these (mostly edge-corrected) estimators make sense for fixed bounded windows our main issue is to study their behaviour when W n grows unboundedly as n → ∞. The first problem of large-domain statistics is to find conditions ensuring strong or at least mean-square consistency as n → ∞ under ergodicity or other mild mixing conditions put on the underlying point process. The third part contains weak convergence results obtained by exhausting strong mixing conditions or even m-dependence of spatial random fields generated by Poisson-based point processes. To illustrate the usefulness of asymptotic methods we give two Kolmogorov–Smirnov-type tests based on K-functions to check complete spatial randomness of a given point pattern in \({\mathbb{R}}^{d}\).

Influence of an exclusion radius around each nucleus on the microstructure and transformation kinetics

In this paper we address the problem of the Bayesian de-convolution of a widely spread class of processes, filtered point processes, whose underlying point process is a self-excited point process. In order to achieve this de-convolution, we perform powerful stochastic algorithm, the Markov chains Monte Carlo (MCMC), which despite their power have not been yet widely used in signal processing.

We propose a doubly stochastic point process for modeling traffic data. The traffic intensity is modeled as a self-similar process and is generated by applying an inverse orthogonal wavelet transform to a sequence of independent random sequences, having different variances at different scales. The underlying point process is characterized by a fractal renewal point process of a dimension less than one. The proposed model is intrinsically able to synthesize a point process characterized by arrivals packed into sparsely located clusters separated by occasionally very long interarrival times. This behavior is often encountered on real traffic data and it deserves particular attention because is often the main responsible for packet losses and thus directly affects the network performance. The model is validated by comparing the packet loss rate of a queueing buffer element driven by real and simulated traffic.

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.

The authors propose a new type of scan statistic to test for the presence of space‐time clusters in point processes data, when the goal is to identify and evaluate the statistical significance of localized clusters. Their method is based only on point patterns for cases; it does not require any specific knowledge of the underlying population. The authors propose to scan the three‐dimensional space with a score test statistic under the null hypothesis that the underlying point process is an inhomogeneous Poisson point process with space and time separable intensity. The alternative is that there are one or more localized space‐time clusters. Their method has been implemented in a computationally efficient way so that it can be applied routinely. They illustrate their method with space‐time crime data from Belo Horizonte, a Brazilian city, in addition to presenting a Monte Carlo study to analyze the power of their new test.

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.

Summary 1Considering that the spatial pattern of trees is a footprint of the biological processes that drive their dynamics, increasing work has been undertaken to analyse spatial patterns and fit spatial point processes to them. When diameter is taken into account, the underlying point process is a marked point process. The question then is how to correctly model the dependence between the mark and location, and the different patterns at the different scales, to gain understanding of the underlying biological processes. 2The data used comes from the Paracou rain forest in French Guiana (5°15′ N, 52°55′ W). The spatial pattern of trees in this forest exhibits regularity at small distances (c. 6 m) and clustering at larger distances (c. 30 m). The pattern is linked to diameter, with a shift from clustering to regularity as trees grow. 3Two models of spatial pattern are used. The first one is pattern-driven in the sense that it breaks down the observed pattern into a mixture of regular, clustered and random contributions. The second one is process-based and uses a simple individual-based space-dependent model of forest dynamics as a simulation algorithm. It is obtained as the limit of this spatio-temporal model when time tends to infinity. 4Both models are consistent with the observed pattern, with a better fit provided by the second model. Moreover this second model provides a biological interpretation of the observed pattern in terms of forest dynamics. However the first model's implementation is simpler (both for simulation and for parameter estimation). 5Synthesis. Modelling the spatial pattern of plants using spatial point processes gives insights into the biological processes that drive their dynamics. It allows the reconstruction of their dynamics given only a snapshot of plant locations. Very few solutions exist to model complex marked spatial patterns when point location and mark are dependent. We defined and compared two point processes that successfully modelled the spatial pattern of trees in a rain forest with interaction between tree location and tree size. Both models highlight competition (either symmetric or asymmetric) as a driving process towards regularity. The second model further reveals a self-organizing dynamic with a feedback effect of competition.

We discuss Bayesian modelling and computational methods in analysis of indirectly observed spatial point processes. The context involves noisy measurements on an underlying point process that provide indirect and noisy data on locations of point outcomes. We are interested in problems in which the spatial intensity function may be highly heterogenous, and so is modelled via flexible nonparametric Bayesian mixture models. Analysis aims to estimate the underlying intensity function and the abundance of realized but unobserved points. Our motivating applications involve immunological studies of multiple fluorescent intensity images in sections of lymphatic tissue where the point processes represent geographical configurations of cells. We are interested in estimating intensity functions and cell abundance for each of a series of such data sets to facilitate comparisons of outcomes at different times and with respect to differing experimental conditions. The analysis is heavily computational, utilizing recently introduced MCMC approaches for spatial point process mixtures and extending them to the broader new context here of unobserved outcomes. Further, our example applications are problems in which the individual objects of interest are not simply points, but rather small groups of pixels; this implies a need to work at an aggregate pixel region level and we develop the resulting novel methodology for this. Two examples with with immunofluorescence histology data demonstrate the models and computational methodology.

Statistical methodology for analysing patterns of points on a network of lines, such as road traffic accident locations, often assumes that the underlying point process is “stationary” or “correlation‐stationary.” However, such processes appear to be rare. In this paper, popular procedures for constructing a point process are adapted to linear networks: many of the resulting models are no longer stationary when distance is measured by the shortest path in the network. This undermines the rationale for popular statistical methods such as the K‐function and pair correlation function. Alternative strategies are proposed, such as replacing the shortest‐path distance by another metric on the network. Copyright © 2017 John Wiley & Sons, Ltd.

We propose a novel alternative to case-control sampling for the estimation of individual-level risk in spatial epidemiology. Our approach uses weighted estimating equations to estimate regression parameters in the intensity function of an inhomogeneous spatial point process, when information on risk-factors is available at the individual level for cases, but only at a spatially aggregated level for the population at risk. We develop data-driven methods to select the weights used in the estimating equations and show through simulation that the choice of weights can have a major impact on efficiency of estimation. We develop a formal test to detect non-Poisson behavior in the underlying point process and assess the performance of the test using simulations of Poisson and Poisson cluster point processes. We apply our methods to data on the spatial distribution of childhood meningococcal disease cases in Merseyside, U.K. between 1981 and 2007.

The meta distribution (MD) of the signal-to-interference ratio (SIR) provides fine-grained reliability performance in wireless networks modeled by point processes. In particular, for an ergodic point process, the SIR MD yields the distribution of the per-link reliability for a target SIR. Here, we reveal that the SIR MD has a second important application, which is rate control. Specifically, we calculate the distribution of the SIR threshold (equivalently, the distribution of the transmission rate) that guarantees each link a target reliability and show its connection to the distribution of the per-link reliability. This connection also permits an approximate calculation of the SIR MD when only partial (local) information about the underlying point process is available.

The study of double-stranded RNA unwinding by helicases is a problem of basic scientific interest. One such example is provided by studies on the hepatitis C virus (HCV) NS3 helicase using single molecule mechanical experiments. HCV currently infects nearly 3% of the world population and NS3 is a protein essential for viral genome replication. The objective of this study is to model the RNA unwinding mechanism based on previously published data and study its characteristics and their dependence on force, ATP and NS3 protein concentration. In this work, RNA unwinding by NS3 helicase is hypothesized to occur in a series of discrete steps and the steps themselves occurring in accordance with an underlying point process. A point process driven change point model is employed to model the RNA unwinding mechanism. The results are in large agreement with findings in previous studies. A gamma distribution based renewal process was found to model well the point process that drives the unwinding mechanism. The analysis suggests that the periods of constant extension observed during NS3 activity can indeed be classified into pauses and subpauses and that each depend on the ATP concentration. The step size is independent of external factors and seems to have a median value of 11.37 base pairs. The steps themselves are composed of a number of substeps with an average of about 4 substeps per step and an average substep size of about 3.7 base pairs. An interesting finding pertains to the stepping velocity. Our analysis indicates that stepping velocity may be of two kinds- a low and a high velocity.

We present the construction and characterization of novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single channel equivalent</i> representations of a Poisson Point Process (PPP) whose points follow multiple channel laws with respect to the origin. Two variants are presented. The first, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Type 1</i> Equivalent Process (EPP), allows for IID fading while maintaining equivalence in terms of path loss and fading with respect to a single channel law. The second, called the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Type 2 EPP</i> , also maintains equivalence with respect to a single channel law, but requires the fading distribution to be dependent on the underlying point process. The Type 1 EPP only exists in certain settings, while the Type 2 EPP always exists.