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This paper treats functional marked point processes (FMPPs), which are defined as marked point processes where the marks are random elements in some (Polish) function space. Such marks may represent, for example, spatial paths or functions of time. To be able to consider, for example, multivariate FMPPs, we also attach an additional, Euclidean, mark to each point. We indicate how the FMPP framework quite naturally connects the point process framework with both the functional data analysis framework and the geostatistical framework. We further show that various existing stochastic models fit well into the FMPP framework. To be able to carry out nonparametric statistical analyses for FMPPs, we study characteristics such as product densities and Palm distributions, which are the building blocks for many summary statistics. We proceed to defining a new family of summary statistics, so-called weighted marked reduced moment measures, together with their nonparametric estimators, in order to study features of the functional marks. We further show how other summary statistics may be obtained as special cases of these summary statistics. We finally apply these tools to analyse population structures, such as demographic evolution and sex ratio over time, in Spanish provinces.

As a workaround for the lack of transitive transformations on linear network structures, which are required to consider different notions of distributional invariance, including stationarity, we introduce the notions of pseudostationarity and intensity reweighted moment pseudostationarity for point processes on linear networks. Moreover, using arbitrary so-called regular linear network distances, e.g. the Euclidean and the shortest-path distance, we further propose geometrically corrected versions of different higher-order summary statistics, including the inhomogeneous empty space function, the inhomogeneous nearest neighbour distance distribution function and the inhomogeneous J-function. Such summary statistics detect interactions of order higher than two. We also discuss their nonparametric estimators and through a simulation study, considering models with different types of spatial interaction and different networks, we study the performance of our proposed summary statistics by means of envelopes. Our summary statistic estimators manage to capture clustering, regularity as well as Poisson process independence. Finally, we make use of our new summary statistics to analyse two different datasets: motor vehicle traffic accidents and spiderwebs.

In order to optimally utilise the resources of a country’s prehospital care system, i.e. ambulance service(s),it is crucial that one is able to spatio-temporally forecast hot-spots, i.e. spatial regions and periods with anincreased risk of seeing a call to the emergency number 112 which results in the dispatch of an ambulance.Such forecasts allow the dispatcher to make strategic decisions regarding e.g. the fleet size and where todirect unoccupied ambulances. In addition, simulations based on forecasts may serve as the startingpoint for different optimal routing strategies. Although the associated data typically comes in the form ofspatio-temporal point patterns, point process based modelling attempts in the literature has been scarce.In this paper, we study a unique set of Swedish spatio-temporal ambulance call data, which consists ofthe spatial (gps) locations of the dispatch addresses and the associated days of occurrence of the calls.The spatial study region is given by the four northernmost regions of Sweden and the study period isJanuary 1, 2014 to December 31, 2018. Motivated by the non-infectious disease nature of the data, wehere employ log-Gaussian Cox processes (LGCPs) for the spatio-temporal modelling and forecasting ofthe calls. To this end, we propose a K-means based bandwidth selection method for the kernel estimationof the spatial component of the separable spatio-temporal intensity function. The temporal componentof the intensity function is modelled by means of Poisson regression, using different calendar covariates,and the spatio-temporal random field component of the random intensity of the LGCP is fitted usingsimulation via the Metropolis-adjusted Langevin algorithm. A study of the spatio-temporal dynamics ofthe data shows that a hot-spot can be found in the south eastern part of the study region, where mostpeople in the region live and our fitted model/forecasts manage to capture this behaviour quite well. Thefitted temporal component of the intensity functions reveals that there is a significant association betweenthe expected number of calls and the day of the week as well as the season of the year. In addition,non-parametric second-order spatio-temporal summary statistic estimates indicate that LGCPs seem tobe reasonable models for the data. Finally, we find that the fitted forecasts generate simulated futurespatial event patterns which quite well resemble the actual future data.

Aside from reviewing different intensity estimation schemes for point processes on linear networks, this paper introduces two Voronoi-based intensity estimation approaches for spatio-temporal linear network point processes. The first is a separable estimator, which is obtained as a scaled product of a resample-smoothed Voronoi intensity estimator on the linear network in question and another one on the time domain. The second one, which we refer to as a pseudo-separable resample-smoothed Voronoi intensity estimator, uses a slightly different thinning strategy. Through a simulation study we show that the latter performs slightly better than the former. We finally apply the latter estimator to a spatio-temporal traffic accident point pattern.

The development of compressive sensing in recent years has given much attention to sparse signal recovery. In sparse signal recovery, spike and slab priors are playing a key role in inducing sparsity. The use of such priors, however, results in non-convex and mixed integer programming problems. Most of the existing algorithms to solve non-convex and mixed integer programming problems involve either simplifying assumptions, relaxations or high computational expenses. In this paper, we propose a new adaptive alternating direction method of multipliers (AADMM) algorithm to directly solve the suggested non-convex and mixed integer programming problem. The algorithm is based on the one-to-one mapping property of the support and non-zero element of the signal. At each step of the algorithm, we update the support by either adding an index to it or removing an index from it and use the alternating direction method of multipliers to recover the signal corresponding to the updated support. Moreover, as opposed to the competing “adaptive sparsity matching pursuit” and “alternating direction method of multipliers” methods our algorithm can solve non-convex problems directly. Experiments on synthetic data and real-world images demonstrated that the proposed AADMM algorithm provides superior performance and is computationally cheaper than the recently developed iterative convex refinement (ICR) and adaptive matching pursuit (AMP) algorithms.

Climate extremes have profound impacts on key socio-economic sectors such as agriculture. In a changing climate context, characterised by an intensification of these extremes and where the population is expected to grow, exposure and vulnerability must be accurately assessed. However, most risk assessments analyse extremes independently, thus potentially being overconfident in the resilience of the socio-economic sectors. Here, we propose a novel approach to defining and characterising concurrent climate extremes (i.e. extremes occurring within a specific temporal lag), which is able toidentify spatio-temporal dependences without making any strict assumptions. the method is applied to large-scale heat stress and drought events in the key wheat producing regions of the world, as these extremes can cause serious yield losses and thus trigger market shocks. Wheat regions likely to haveconcurrent extremes (heat stress and drought events) are identified, as well as regions independent ofeach other or inhibiting each other in terms of these extreme events. this tool may be integrated in all risk assessments but could also be used to explore global climate teleconnections.

This paper treats functional marked point processes (FMPPs), which are defined as marked point processes where the marks are random elements in some (Polish) function space. Such marks may represent e.g. spatial paths or functions of time. To be able to consider e.g. multivariate FMPPs, we also attach an additionally, Euclidean, mark to each point. We indicate how the FMPP framework quite naturally connects the point process framework with both the functional data analysis framework and the geostatistical framework; in particular we define spatio-temporal geostatistical marking for point processes. We further show that various existing stochastic models fit well into the FMPP framework, in particular marked point processes with real valued marks. To be able to carry out non-parametric statistical analyses for functional marked point patterns, we study characteristics such as product densities and Palm distributions, which are the building blocks for summary statistics such as marked inhomogeneous J-functions and our so-called K-functionals. We finally apply these statistical tools to analyse a few different functional marked point patterns.

Voronoi estimators are non-parametric and adaptive estimators of the intensity of a point process. The intensity estimate at a given location is equal to the reciprocal of the size of the Voronoi/Dirichlet cell containing that location. Their major drawback is that they tend to paradoxically under-smooth the data in regions where the point density of the observed point pattern is high, and over-smooth where the point density is low. To remedy this behaviour, we propose to apply an additional smoothing operation to the Voronoi estimator, based on resampling the point pattern by independent random thinning. Through a simulation study we show that our resample-smoothing technique improves the estimation substantially. In addition, we study statistical properties such as unbiasedness and variance, and propose a rule-of-thumb and a data-driven cross-validation approach to choose the amount of smoothing to apply. Finally we apply our proposed intensity estimation scheme to two datasets: locations of pine saplings (planar point pattern) and motor vehicle traffic accidents (linear network point pattern).

To analyse interactions in marked spatio-temporal point processes (MSTPPs), we introduce marked second-order reduced moment measures and K-functions for inhomogeneous second-order intensity reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark-based classifications of the points, are depending on the specific mark space and mark reference measure chosen. Unbiased and consistent minus-sampling estimators are derived for all statistics considered and a test for random labelling is indicated. In addition, we treat Voronoi intensity estimators for MSTPPs. These new statistics are finally employed to analyse an Andaman sea earthquake data set.