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Motivated by the general ability of cross-validation to reduce overfitting and mean square error, we develop a cross-validation-based statistical theory for general point processes. It is based on the combination of two novel concepts for general point processes: cross-validation and prediction errors. Our cross-validation approach uses thinning to split a point process/pattern into pairs of training and validation sets, while our prediction errors measure discrepancy between two point processes. The new statistical approach, which may be used to model different distributional characteristics, exploits the prediction errors to measure how well a given model predicts validation sets using associated training sets. Having indicated that our new framework generalizes many existing statistical approaches, we then establish different theoretical properties for it, including large sample properties. We further recognize that nonparametric intensity estimation is an instance of Papangelou conditional intensity estimation, which we exploit to apply our new statistical theory to kernel intensity estimation. Using independent thinning-based cross-validation, we numerically show that the new approach substantially outperforms the state-of-the-art in bandwidth selection. Finally, we carry out intensity estimation for a dataset in forestry and a dataset in neurology.

In the literature on spatial point processes, there is an emerging challenge in studying marked point processes with points being labelled by functions. In this paper, we focus on point processes living on linear networks and, from distinct points of view, propose several mark summary characteristics that are of great use in studying the average association and dispersion of the function-valued marks. Through a simulation study, we evaluate the performance of our proposed mark summary characteristics, both when marks are independent and when some sort of spatial dependence is evident among them. Finally, we employ our proposed mark summary characteristics to study the spatial structure of urban cycling profiles in Vancouver, Canada.

Within the applications of spatial point processes, it is increasingly becoming common that events are labelled by marks, prompting an exploration beyond the spatial distribution of events by incorporating the marks in the undertaken analysis. In this paper, we first consider marked spatial point processes in R², where marks are either integer-valued, real-valued, or object-valued, and review the state-of-the-art to analyze the spatial structure and type of interaction/correlation between marks. More specifically, we review cross/dot-type summary characteristics, mark-weighted summary characteristics, various mark correlation functions, and frequency domain approaches. We also propose novel cross/dot-type higher-order summary characteristics, mark-weighted summary characteristics, and mark correlation functions for marked point processes on linear networks. Through a simulation study, we show that ignoring the underlying network gives rise to erroneous conclusions about the interaction/correlation between marks. Finally, we consider two applications: the locations of two different proteins on the membranes of cells infected with the influenza virus and the locations of public trees along the street network of Vancouver, Canada, where trees are labelled by their diameters at breast height.

We consider several cases where one needs to estimate the (first-order) intensity function for different spatial point patterns on some state spaces such as. R2 and linear networks. Furthermore, we consider spatial point patterns on irregular windows, replicated point patterns, time-ordered point patterns, and trajectories. The use of various kernel- and Voronoi-based intensity estimators in R is largely discussed through several applications, including wildfires, (street) crimes, active fires, mineral flotation, war data, and taxi movements.

We are grateful to all discussants for their invaluable comments, suggestions, questions, and contributions to our article. We have attentively reviewed all discussions with keen interest. In this rejoinder, our objective is to address and engage with all points raised by the discussants in a comprehensive and considerate manner. Consistently, we identify the discussants, in alphabetical order, as follows: CJK for Cronie, Jansson, and Konstantinou, DS for Stoyan, GP for Grabarnik and Pommerening, MRS for Myllymäki, Rajala, and Särkkä, and MCvL for van Lieshout throughout this rejoinder.

We propose novel second/higher-order summary statistics for inhomogeneous spatio-temporal point processes when the spatial locations are limited to a linear network. More specifically, letting the spatial distance between events be measured by a regular distance metric, appropriate forms of 𝐾- and 𝐽-functions are introduced, and their theoretical relationships are studied. The theoretical forms of our proposed summary statistics are investigated under homogeneity, Poissonness, and independent thinning. Moreover, non-parametric estimators are derived, facilitating the use of our proposed summary statistics to study the spatio-temporal dependence between events. Through simulation studies, we demonstrate that our proposed 𝐽-function effectively identifies spatio-temporal clustering, inhibition, and randomness. Finally, we examine spatio-temporal dependencies for street crimes in Valencia, Spain, and traffic accidents in New York, USA.

Detecting change-points in multivariate settings is usually carried out by analyzing all marginals either independently, via univariate methods, or jointly, through multivariate approaches. The former discards any inherent dependencies between different marginals and the latter may suffer from domination/masking among different change-points of distinct marginals. As a remedy, we propose an approach which groups marginals with similar temporal behaviors, and then performs group-wise multivariate change-point detection. Our approach groups marginals based on hierarchical clustering using distances which adjust for inherent dependencies. Through a simulation study we show that our approach, by preventing domination/masking, significantly enhances the general performance of the employed multivariate change-point detection method. Finally, we apply our approach to two datasets: (i) Land Surface Temperature in Spain, during the years 2000–2021, and (ii) The WikiLeaks Afghan War Diary data.