Multivariate log Gaussian Cox processes are flexible models for multivariate point patterns. However, they have so far only been applied in bivariate cases. In this paper we move beyond the bivariate case in order to model multispecies point patterns of tree locations. In particular we address the problems of identifying parsimonious models and of extracting biologically relevant information from the fitted models. The latent multivariate Gaussian field is decomposed into components given in terms of random fields common to all species and components which are species specific. This allows a decomposition of variance that can be used to quantify to which extent the spatial variation of a species is governed by common respectively species specific factors. Crossvalidation is used to select the number of common latent fields in order to obtain a suitable trade-off between parsimony and fit of the data. The selected number of common latent fields provides an index of complexity of the multivariate covariance structure. Hierarchical clustering is used to identify groups of species with similar patterns of dependence on the common latent fields.

Spatial Cox point processes is a natural framework for quantifying the various sources of variation governing the spatial distribution of rain forest trees. We introduce a general criterion for variance decomposition for spatial Cox processes and apply it to specific Cox process models with additive or log linear random intensity functions. We moreover consider a new and flexible class of pair correlation function models given in terms of normal variance mixture covariance functions. The proposed methodology is applied to point pattern data sets of locations of tropical rain forest trees.

Summary Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering not accounted for by the available covariates, likelihood based inference becomes computationally cumbersome due to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practise is solved numerically. The derivation of the optimal estimating function has close similarities to the derivation of quasi-likelihood for standard data sets. The approximate solution is further equivalent to a quasi-likelihood score for binary spatial data. We therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient.

The novel coronavirus disease (COVID-19) has spread rapidly across the world in a short period of time and with a heterogeneous pattern. Understanding the underlying temporal and spatial dynamics in the spread of COVID-19 can result in informed and timely public health policies. In this paper, we use a spatio-temporal stochastic model to explain the temporal and spatial variations in the daily number of new confirmed cases in Spain, Italy and Germany from late February 2020 to mid January 2021. Using a hierarchical Bayesian framework, we found that the temporal trends of the epidemic in the three countries rapidly reached their peaks and slowly started to decline at the beginning of April and then increased and reached their second maximum in the middle of November. However decline and increase of the temporal trend seems to show different patterns in Spain, Italy and Germany.

BACKGROUNDIdentification of factors in the choice of suicide methods is important in understanding the phenomenon.OBJECTIVESWe aimed to quantify the effect of gender, age, living area, education level and marital status on the choice of suicide method among residents of Kermanshah province in the west of Iran.DESIGNA cross-sectional study of all completed suicides from March 2006 to September 2013.SETTINGKermanshah Province, IranMETHODSData were extracted from suicide forms in the electronic files of the Forensic Medicine Organization. A total of 1901 (1138 men), suicide cases were identified. After preliminary analysis, a multinomial logistic model was fitted to the data to test and quantify the impact of each influential factor on the choice of suicide method. The relative risk of each suicide method over hanging as the reference method was estimated by calculating relative-risk ratios from the multinomial logistic model.MAIN OUTCOME MEASURESRelative risk of suicide by self-immolation, drug and toxic poisoning and firearms.RESULTSWe found that women are at a higher relative risk than men for suicide by self-immolation, intentional drug poisoning and toxic poisoning. The relative risk of suicide by self-immolation and intentional drug poisoning was higher for urban residents and young individuals. On the other hand, men and rural residents were at higher relative risk of suicide by firearm.CONCLUSIONIn Kermanshah province, the impact of rapid social changes on women and the availability of firearms in rural areas and drugs in urban households require more attention in any suicide prevention planning.LIMITATIONSThe lack of data prevented analysis of factors that may be more influential in choosing suicide.

The epidemic-type aftershock sequence (ETAS) model is the most widely used statistical model to describe earthquake catalogs. ETAS is an R package for fitting the space-time ETAS model to an earthquake catalog using the stochastic declustering approach introduced by Zhuang, Ogata, and Vere-Jones (2002). The package provides two classes and several functions to facilitate data preparation, model fitting and some simple diagnostic checks. The present paper is a description of the package and illustrates modeling earthquake data using the space-time ETAS model.where c > 0 and p > 1 are unknown parameters.is the PDF of the occurrence location of a triggered event generated from an event of magnitude m i occurring at the location (x i , y i ). It is assumed that the probability distribution of the location of a triggered event depends on the magnitude and the location of its direct main shock. The radially symmetric density functionis usually considered for this distribution, where D > 0, γ > 0 and q > 1 are unknown parameters.The ETAS model assumes that each event is either a background (spontaneous) event or triggered by a previous event (Ogata 1998). The background events are generated by a Poisson process with intensityũ(x, y), which is stationary in time. The relaxing coefficient coefficient µ inũ(x, y) = µu(x, y) is introduced in order to fasten the convergence of the model Journal

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

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