Space-time Stochastic Processes Research Articles

The cost of fear: Impact of violence risk on child health during conflict

The fear of exposure to conflict events often triggers changes in the behavior of economic agents even before/without any manifestation of violence in a given area. It generates a treatment status (exposure to the adverse effects of conflict) that goes beyond violence incidence. This paper develops a new approach to capture such treatment. Violence is modeled as a space–time stochastic process with an unknown underlying distribution that is backed out of the observed pattern of conflict events. A new risk measure is built from this density and used to evaluate the impact of conflict on child health using data from Ivory Coast and Uganda. The empirical evidence suggests that conflict is a local public bad, with cohorts of children exposed to high risk of violence equally suffering major health setbacks even when this risk does not materialize in violent events around them.

Drone delivery systems: job assignment and dimensioning

This article studies how to dimension and control at the system level a fleet of autonomous aerial vehicles delivering goods from depots to customers. Customer requests (jobs) arrive according to a space-time stochastic process. We compute a lower bound for the infrastructure expenditure required to achieve a certain expected delivery time. It is shown that job assignment policies can exhibit a tipping point behavior: One vehicle makes the difference between almost optimal delivery time and instability. This phenomenon calls for careful dimensioning of the system. We thus demonstrate the trade-off between financial costs and service quality. We propose a policy that assigns each incoming job to the vehicle that will do the job faster than other ones, seeking to minimize the overall workload in the system in the long term. This policy is scalable with the number of depots and vehicles, performs optimal in low load, and works well up to high loads. Simulations suggest that it stabilizes the system for any load if the number of vehicles per depot is sufficient.

A Note on Continuous Spatial-Temporal Dynamics of Stochastic Processes

We propose a general form to analyze the space-time interdependency of continuous space-time stochastic processes. We present a new space-time approach based on the intensity function of the underlying point process. These formulations can be, to some extent, analytically solved to obtain explicit formulae of interest. We define a general function that controls the space-time interaction and allows for closed forms depending on the particular choice of several mathematical tools playing a role in this interaction function. In particular, we make use of copulas and Laplace transforms to provide interesting examples of the dynamics of the random intensity function and, in turn, of the number of points contained in a given region.

Space–time dependence dynamics for birth–death point processes

This paper analyses the space–time interdependency of a spatially explicit birth–death process based on the intensity function. Based on intensity functions, these formulations can be, to some extent, analytically solved to obtain the explicit formulae of, for instance, the total point population size contained in the unit torus at equilibrium. The definition of continuous space–time processes based on point intensities opens up new promising lines of research to analyse ecological dynamics: our spatially explicit birth–death process can be easily expanded to mimic other realistic ecological scenarios. Note that although space–time stochastic processes are (generally) intractable, theoretical development of their corresponding intensity function provides useful insights into these complex dynamics. Hence, the analytical analysis of the point intensity provides a complementary method to simulation-based analyses of complex space–time processes.

On Random and Gibbsian Particle Motions for Point Processes Evolving in Space and Time

This article presents an analysis of space-time interdependencies of spatial point processes considering random and deterministic Gibbsian point motions caused by repulsion effects between particles. Two deterministic models of Gibbsian motions are considered by formulating a constant (i.e., Strauss-like) and a linear interaction motion functions. Given that theoretical development of continuous space-time stochastic processes are mathematically intractable, we have mainly based our analysis on numerical simulations. Our results suggest that to fully understand such complex dynamics, the analysis of purely spatial patterns should be combined with their interactions in the space-time domain. Otherwise, analysis of pure spacial patterns may not fully explain the real mechanism generating such dynamical configurations. We highlight that adding movement to sedentary points opens new areas of application and research to study biological phenomena, where particles not only evolve through time but also can change spatial positions in terms of their neighbor locations.

On soft and hard particle motions for stochastic marked point processes

This article analyses the space–time interdependency of marked point processes considering marked point random and deterministic motions. In this context, marked points can move randomly or/and deterministically as a result of their own growing status or/and the growing status of their neighbours. A continuous stochastic process for the generation of space–time patterns of moving marked points is formulated and illustrated. Three deterministic models of collision of moving and growing hard particles are also developed. We consider deterministic particle motion promoted by: (a) particle's own growth when touching other neighbours; (b) the growth of touching neighbours; and (c) a combination of these two collision dynamics. Moreover, we extend this combined deterministic model into its counterpart stochastic approach. Our analysis suggests that the generating mechanism of the space–time stochastic process may not become fully reflected in snap-shot analyses of purely spatial patterns when assuming random motion of marked points. Thus, the understanding of such complex dynamics may involve not only the analysis of purely spatial patterns but also their interaction in the space–time domain.

Modelling Forest Dynamics: A Perspective from Point Process Methods

This paper reviews the main applications of (marked) point process theory in forestry including functions to analyse spatial variability and the main (marked) point process models. Although correlation functions do describe spatial variability at distinct range of scale, they are clearly restricted to the analysis of few dominant species since they are based on pairwise analysis. This has over-simplified the spatial analysis of complex forest dynamics involving "large" number of species. Moreover, although process models can reproduce, to some extent, real forest spatial patterns of trees, the biological forest-ecological interpretation of the resulting spatial structures is difficult since these models usually lack of biological realism. This problem gains in strength as usually most of these point process models are defined in terms of purely spatial relationships, though in real life, forest develop through time. We thus aim to discuss the applicability of such formulations to analyse and simulate "real" forest dynamics and unwrap their shortcomes. We present a unified approach of modern spatially explicit forest growth models. Finally, we focus on a continuous space-time stochastic process as an alternative approach to generate marked point patterns evolving through space and time.

Estimating integrals of stochastic processes using space-time data

Consider a space–time stochastic process $Z_ t(x)=S(x) + \xi_t(x)$ where $S(x)$ is a signal process defined on $\mathbb{R}^d$ and $\xi_t(x)$ represents measurement errors at time $t$. For a known measurable function $v(x)$ on $\mathbb{R}^d$ and a fixed cube $ D\subset \mathbb{R}^d$, this paper proposes a linear estimator for the stochastic integral $\int_D v(x)S(x)dx$ based on space–time observations $\{Z_t(x_i); i = 1,\ldots,n; t=1,\ldots,T\}$. Under mild conditions, the asymptotic properties of the mean squared error of the estimator are derived as the spatial distance between spatial sampling locations tends to zero and as time $T$ increases to infinity. Central limit theorems for the estimation error are also studied.

Stochastic evolution equations and related measure processes

Basic results on stochastic differential equations in Hilbert and Banach space, linear stochastic evolution equations and some classes of nonlinear stochastic evolution equations are reviewed. The emphasis is on equations relevant to the study of spacetime stochastic processes. In particular the class of measure processes, the continuous analogs of spacetime population processes, is studied in detail.

Stochastic evolution equations

A class of space-time stochastic processes that arise as solutions of stochastic evolution equations is discussed. By analogy with Itô stochastic differential equations, a stochastic evolution equation is given by the formal equation ∂X ∂t = AX + f(X)W where A is a positive self-adjoint operator and W is a space-time white noise. Existence and continuity results are obtained for equations of the foregoing type when f is a Lipschitz mapping. In addition, an equation is obtained for the covariance function of the solution.