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Let 𝓟 be a simple, stationary point process on ℝ d having fast decay of correlations, that is, its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let 𝓟 n := 𝓟 ∩ Wn be its restriction to windows W n : = [ − 1 2 n 1 / d , 1 2 n 1 / d ] d ⊂ ℝ d . We consider the statistic H n ξ : = Σ x ∈ P n ξ ( x , P n ) where ξ(x,𝓟 n ) denotes a score function representing the interaction of x with respect to 𝓟 n . When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics and central limit theorems for H n ξ and, more generally, for statistics of the re-scaled, possibly signed, ξ-weighted point measures μ n ξ : = Σ x ∈ P n ξ ( x , P n ) δ n − 1 / d x , as Wn ↑ ℝ d . This gives the limit theory for nonlinear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model and total edge length of the k-nearest neighbors graph) of α-determinantal point processes (for −1/α ∈ ℕ) having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [Ann. Probab. 30 (2002) 171–187] to nonlinear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ ℕ) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [Comm. Math. Phys. 310 (2012) 75–98] and Shirai and Takahashi [J. Funct. Anal. 205 (2003) 414–463], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [Stochastic Process. Appl. 56 (1995) 321–335; Statist. Probab. Lett. 36 (1997) 299–306] to show the fast decay of the correlations of ξ-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing, and consequently yields the asymptotic normality of μ n ξ via an extension of the cumulant method.

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.

Although the importance of edaphic factors and habitat structure for plant growth and survival is known, both are often neglected in favor of climatic drivers when investigating the spatial patterns of plant species and diversity. Yet, especially in mountain ecosystems with complex topography, missing edaphic and habitat components may be detrimental for a sound understanding of biodiversity distribution. Here, we compare the relative importance of climate, soil and land cover variables when predicting the distributions of 2,616 vascular plant species in the European Alps, representing approximately two-thirds of all European flora. Using presence-only data, we built point-process models (PPMs) to relate species observations to different combinations of covariates. We evaluated the PPMs through block cross-validations and assessed the independent contributions of climate, soil, and land cover covariates to predict plant species distributions using an innovative predictive partitioning approach. We found climate to be the most influential driver of spatial patterns in plant species with a relative influence of ~58.5% across all species, with decreasing importance from low to high elevations. Soil (~20.1%) and land cover (~21.4%), overall, were less influential than climate, but increased in importance along the elevation gradient. Furthermore, land cover showed strong local effects in lowlands, while the contribution of soil stabilized at mid-elevations. The decreasing influence of climate with elevation is explained by increasing endemism, and the fact that climate becomes more homogeneous as habitat diversity declines at higher altitudes. In contrast, soil predictors were found to follow the opposite trend. Additionally, at low elevations, human-mediated land cover effects appear to reduce the importance of climate predictors. We conclude that soil and land cover are, like climate, principal drivers of plant species distribution in the European Alps. While disentangling their effects remains a challenge, future studies can benefit markedly by including soil and land cover effects when predicting species distributions.

A Hawkes process is also known under the name of a self-exciting point process and has numerous applications throughout science and engineering. We derive the statistical estimation (maximum likelihood estimation) and goodness-of-fit (mainly graphical) for multivariate Hawkes processes with possibly dependent marks. As an application, we analyze two data sets from finance.

For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point u with $K(u,u)>0$ so that, almost surely, $X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference $\\xi_u: = X\\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of $\\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.

The paper is concerned with parameter estimation for inhomogeneous spatial point processes with a regression model for the intensity function and tractable second-order properties (K-function). Regression parameters are estimated by using a Poisson likelihood score estimating function and in the second step minimum contrast estimation is applied for the residual clustering parameters. Asymptotic normality of parameter estimates is established under certain mixing conditions and we exemplify how the results may be applied in ecological studies of rainforests.

Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on a non-homogeneous alternating renewal process in which interval generation is assumed to be time-dependent, and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to the homogeneous approach through a number of illustrative examples.

Spatial capture–recapture (SCR) is now routinely used for estimating abundance and density of wildlife populations. A standard SCR model includes sub-models for the distribution of individual activity centers (ACs) and for individual detections conditional on the locations of these ACs. Both sub-models can be expressed as point processes taking place in continuous space, but there is a lack of accessible and efficient tools to fit such models in a Bayesian paradigm. Here, we describe a set of custom functions and distributions to achieve this. Our work allows for more efficient model fitting with spatial covariates on population density, offers the option to fit SCR models using the semi-complete data likelihood (SCDL) approach instead of data augmentation, and better reflects the spatially continuous detection process in SCR studies that use area searches. In addition, the SCDL approach is more efficient than data augmentation for simple SCR models while losing its advantages for more complicated models that account for spatial variation in either population density or detection. We present the model formulation, test it with simulations, quantify computational efficiency gains, and conclude with a real-life example using non-invasive genetic sampling data for an elusive large carnivore, the wolverine (Gulo gulo) in Norway. area search, binomial point process, continuous sampling, NIMBLE, non-invasive genetic sampling, Poisson point process, spatial capture–recapture, wolverine

We introduce a broad family of generalised self-exciting point processes with CIR-type intensities, and we develop associated algorithms for their exact simulation. The underlying models are extensions of the classical Hawkes process, which already has numerous applications in modelling the arrival of events with clustering or contagion effect in finance, economics, and many other fields. Interestingly, we find that the CIR-type intensity, together with its point process, can be sequentially decomposed into simple random variables, which immediately leads to a very efficient simulation scheme. Our algorithms are also pretty accurate and flexible. They can be easily extended to further incorporate externally excited jumps, or, to a multidimensional framework. Some typical numerical examples and comparisons with other well-known schemes are reported in detail. In addition, a simple application for modelling a portfolio loss process is presented. The online appendix is available at https://doi.org/10.1287/opre.2017.1640

We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance, and insurance. We provide an application of this process to credit risk, and a simulation algorithm for further industrial implementation and statistical analysis.