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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.

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.

Crime has both varying patterns in space, related to features of the environment, economy and policing, and patterns in time arising from criminal behaviour, such as retaliation. Serious crimes may also be presaged by minor crimes of disorder. We demonstrate that these spatial and temporal patterns are generally confounded, requiring analyses to take both into account, and propose a spatiotemporal self-exciting point process model that incorporates spatial features, near repeat and retaliation effects, and triggering. We develop inference methods and diagnostic tools, such as residual maps, for this model, and through extensive simulation and crime data obtained from Pittsburgh, Pennsylvania, demonstrate its properties and usefulness.

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.

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.

Crime risk assessment needs tackling complex interrelationships between stochastic and deterministic components of spatio-temporal models. Criminal phenomena can be modeled using spatio-temporal point patterns of certain criminal data, and here we pay attention to the stochastic models of log-Gaussian Cox processes (LGCP) and self-exciting Hawkes processes (SEHP). We provide a comprehensive modeling strategy, combining both processes, noting that: (a) an LGCP facilitates the incorporation of first-order information through spatial and temporal deterministic components and second-order information through a stochastic component, and (b) a SEHP provides sufficient flexibility to incorporate various components in the background subprocess. To account for crime risk assessment, the deterministic components of the LGCP were estimated using a generalized linear model (GLM) for the temporal part, and a generalized additive model with B-splines for the highly nonlinear spatial covariates. In addition, the background rate components of the SEHP were estimated by a non-parametric stochastic reconstruction technique that includes a temporal periodicity, a separable spatial component, a long-term trend, and a semi-parametric method for the relaxation coefficients. MCMC-MALA and maximum likelihood were used for inference in both the LGCP and SEHP processes. We analyze crime events from the city of Riobamba (Ecuador), and with a complementary use of both stochastic point process models, we are able to assess the risk of crime, and provide reliable forecasts for weeks ahead.

Determinantal Point Processes (DPPs) are popular models for point processes with repulsion. They appear in numerous contexts, from physics to graph theory, and display appealing theoretical properties. On the more practical side of things, since DPPs tend to select sets of points that are some distance apart (repulsion), they have been advocated as a way of producing random subsets with high diversity. DPPs come in two variants: fixed-size and varying-size. A sample from a varying-size DPP is a subset of random cardinality, while in fixed-size “k-DPPs” the cardinality is fixed. The latter makes more sense in many applications, but unfortunately their computational properties are less attractive, since, among other things, inclusion probabilities are harder to compute. In this work, we show that as the size of the ground set grows, k-DPPs and DPPs become equivalent, in the sense that fixed-order inclusion probabilities converge. As a by-product, we obtain saddlepoint formulas for inclusion probabilities in k-DPPs. These turn out to be extremely accurate, and suffer less from numerical difficulties than exact methods do. Our results also suggest that k-DPPs and DPPs also have equivalent maximum likelihood estimators. Finally, we obtain results on asymptotic approximations of elementary symmetric polynomials which may be of independent interest.

We analyse the spatio-temporal distribution of visitors' stops by touristic attractions in Palermo (Italy), using theory of stochastic point processes living on linear networks. We first propose an inhomogeneous Poisson point process model with a separable parametric spatio-temporal first-order intensity. We account for the spatial interaction among points on the given network, fitting a Gibbs point process model with mixed effects for the purely spatial component. This allows us to study first-order and second-order properties of the point pattern, accounting both for the spatio-temporal clustering and interaction and for the spatio-temporal scale at which they operate. Due to the strong degree of clustering in the data, we then formulate a more complex model, fitting a spatio-temporal log-Gaussian Cox process to the point process on the linear network, addressing the problem of the choice of the most appropriate distance metric.