Spatio-temporal Point Patterns Research Articles

Nonparametric second-order estimation for spatiotemporal point patterns.

Many existing methodologies for analyzing spatiotemporal point patterns are developed based on the assumption of stationarity in both space and time for the second-order intensity or pair correlation. In practice, however, such an assumption often lacks validity or proves to be unrealistic. In this paper, we propose a novel and flexible nonparametric approach for estimating the second-order characteristics of spatiotemporal point processes, accommodating non-stationary temporal correlations. Our proposed method employs kernel smoothing and effectively accounts for spatial and temporal correlations differently. Under a spatially increasing-domain asymptotic framework, we establish consistency of the proposed estimators, which can be constructed using different first-order intensity estimators to enhance practicality. Simulation results reveal that our method, in comparison with existing approaches, significantly improves statistical efficiency. An application to a COVID-19 dataset further illustrates the flexibility and interpretability of our procedure.

Trivariate Kernel Density Estimation of Spatiotemporal Crime Events with Case Study for Lithuania

The paper presents the results of the investigation of the applicability of spatiotemporal kernel density estimation (KDE) methods for density mapping of violent crime in Lithuania. Spatiotemporal crime research helps to understand and control specific types of crime, thereby contributing to Sustainable Development Goals. The target dataset contained 135,989 records of the events registered by the police of Lithuania from 2015–2018 that were classified as violent. The research focused on choosing appropriate KDE functions and their parameters for modeling the spatiotemporal point pattern of this particular type of crime. The aim was to estimate density, mass, and intensity function(s) so that they can be used in further confirmatory spatial modeling. The application-driven objective was to obtain reliable and practically interpretable KDE surfaces of crime events. Several options for improving and extending the investigated KDE methods are demonstrated.

Locally weighted minimum contrast estimation for spatio-temporal log-Gaussian Cox processes

A local version of spatio-temporal log-Gaussian Cox processes is proposed by using Local Indicators of Spatio-Temporal Association (LISTA) functions plugged into the minimum contrast procedure, to obtain space as well as time-varying parameters. The new procedure resorts to the joint minimum contrast fitting method to estimate the set of second-order parameters. This approach has the advantage of being suitable in both separable and non-separable parametric specifications of the correlation function of the underlying Gaussian Random Field. Simulation studies to assess the performance of the proposed fitting procedure are presented, and an application to seismic spatio-temporal point pattern data is shown.

A mechanistic spatio-temporal modeling of COVID-19 data.

Understanding the evolution of an epidemic is essential to implement timely and efficient preventive measures. The availability of epidemiological data at a fine spatio-temporal scale is both novel and highly useful in this regard. Indeed, having geocoded data at the case level opens the door to analyze the spread of the disease on an individual basis, allowing the detection of specific outbreaks or, in general, of some interactions between cases that are not observable if aggregated data are used. Point processes are the natural tool to perform such analyses. We analyze a spatio-temporal point pattern of Coronavirus disease 2019 (COVID-19) cases detected in Valencia (Spain) during the first 11 months (February 2020 to January 2021) of the pandemic. In particular, we propose a mechanistic spatio-temporal model for the first-order intensity function of the point process. This model includes separate estimates of the overall temporal and spatial intensities of the model and a spatio-temporal interaction term. For the latter, while similar studies have considered different forms of this term solely based on the physical distances between the events, we have also incorporated mobility data to better capture the characteristics of human populations. The results suggest that there has only been a mild level of spatio-temporal interaction between cases in the study area, which to a large extent corresponds to people living in the same residential location. Extending our proposed model to larger areas could help us gain knowledge on the propagation of COVID-19 across cities with high mobility levels.

Non-parametric comparative analysis of the spatiotemporal pattern of human-caused and natural wildfires in Galicia

Background Wildfire is a major environmental threat worldwide and climate change is expected to increase its severity. Galicia has suffered high wildfire incidence during the last decades, most wildfires being from arson, in contrast with the low rate of natural wildfires. Aim This work aims to characterise the spatiotemporal dynamics of human-caused and natural fires in Galicia. Methods We apply first- and second-order non-parametric inference to spatiotemporal wildfire point patterns. Key results The distribution of natural wildfires remained stable over years, with high incidence in summer and in the eastern area of Galicia. Arson wildfires had aggregated patterns, with strong interaction between outbreaks and fires, and their distribution varied both over and within years, with high incidence shifting between the southern and western areas, and high hazard in early spring and late summer. Negligence wildfire patterns showed short-distance aggregation, but large-distance aggregation between outbreaks and fires; their spatial distribution also varied between and within years. Conclusions Different models and covariates are required to predict the hazard from each wildfire type. Natural fires are linked to meteorological and environmental factors, whereas socioeconomic covariates are crucial in human-caused wildfires. Implications These results are the basis for the future development of predictive spatiotemporal point process models for human-caused wildfires.

Testing the first-order separability hypothesis for spatio-temporal point patterns

First-order separability of a spatio-temporal point process plays a fundamental role in the analysis of spatio-temporal point pattern data. While it is often a convenient assumption that simplifies the analysis greatly, existing non-separable structures should be accounted for in the model construction. Three different tests are proposed to investigate this hypothesis as a step of preliminary data analysis. The first two tests are exact or asymptotically exact for Poisson processes. The first test based on permutations and global envelopes allows one to detect at which spatial and temporal locations or lags the data deviate from the null hypothesis. The second test is a simple and computationally cheap χ2-test. The third test is based on stochastic reconstruction method and can be generally applied for non-Poisson processes. The performance of the first two tests is studied in a simulation study for Poisson and non-Poisson models. The third test is applied to the real data of the UK 2001 epidemic foot and mouth disease.1

Marked spatio-temporal point patterns, periodicity analysis and earthquakes: an analytical extension including hypocenter depth

The longitude, latitude and depth of the hypocenter in 3-D space and the date and time of rupture makes an earthquake a “point” in a spatio-temporal point pattern, observed over a region and months, years or decades. The magnitude of earthquakes marks the point pattern, as would hypocenter depth do if only the longitude and latitude of epicenters were used for location in 2-D space. Stochastic declustering, based on a space–time ETAS model (ETAS: epidemic-type aftershock sequence), is a procedure that can be applied in the preliminary stage of an earthquake catalog data analysis. Stochastic declustering procedures have underlying assumptions, such as the time independence of the background intensity function whether the spatial framework is 2-D or 3-D, and a separate treatment of hypocenter depth from longitude and latitude when the spatial framework is 3-D. Cyclical processes in the Earth, including tides and seasonal surface water loads, can introduce periodic behavior in earthquake occurrence and related variables. The effects of ETAS-based 2-D and 3-D declustering on the outcome of periodicity analyses performed from the resulting earthquake data catalogs are studied. The research objectives and statistical challenges include the detection of periodicities for hypocenter depth in addition to monthly earthquake number, and the risk of missing observations for hypocenter depth when the monthly earthquake number after declustering is zero. A version of the method of multi-frequential periodogram analysis (MFPA) that allows for missing observations in the input temporal series is presented in detail, and applied to hypocenter depth (monthly mean and median) for central and northern California from January 2006 to December 2014. The results obtained for the 2-D and 3-D declustered catalogs are compared with those for the original catalog for this region. A semiannual periodicity in hypocenter depth is detected for the original and 2-D declustered catalogs, and fitted with the goal of relating it to periodicities found in the time series of monthly earthquake numbers. Using these results for central and northern California earthquakes, some of the assumptions on the intensity function of the spatio-temporal point process in stochastic declustering are discussed and future research perspectives are proposed.

Large-scale modelling and forecasting of ambulance calls in northern Sweden using spatio-temporal log-Gaussian Cox processes

Although ambulance call data typically come in the form of spatio-temporal point patterns, point process-based modelling approaches presented in the literature are scarce. In this paper, we study a unique set of Swedish spatio-temporal ambulance call data, which consist of the spatial (GPS) locations of the calls (within the four northernmost regions of Sweden) and the associated days of occurrence of the calls (January 1, 2014–December 31, 2018). Motivated by the nature of the data, we here employ log-Gaussian Cox processes (LGCPs) for the spatio-temporal modelling and forecasting of the calls. To this end, we propose a K-means clustering based bandwidth selection method for the kernel estimation of the spatial component of the separable spatio-temporal intensity function. The temporal component of 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 using the Metropolis-adjusted Langevin algorithm. Spatial hot-spots have been found in the south-eastern part of the study region, where most people in the region live and our fitted model/forecasts manage to capture this behaviour quite well. Also, there is a significant association between the expected number of calls and the day-of-the-week, and the season-of-the-year. A non-parametric second-order analysis indicates that LGCPs seem to be reasonable models for the data. Finally, we find that the fitted forecasts generate simulated future spatial event patterns which quite well resemble the actual future data.

First- and Second-Order Characteristics of Spatio-Temporal Point Processes on Linear Networks

We present several characteristics for spatio-temporal point patterns when the spatial locations are restricted to a linear network. A nonparametric kernel-based intensity estimator is proposed to highlight the concentration of events within the network and time, either jointly or separately. We also provide second-order characteristics for spatio-temporal point patterns on linear networks such as K-function and pair correlation function to analyze the type of interaction between points. They are independent of network’s geometry and have known values for Poisson point processes. Finally, we consider some applications to traffic accidents and demonstrate our findings by analyzing datasets of Houston (United States), Medellín (Colombia), and Eastbourne (United Kingdom). Supplementary materials for this article are available online.

Scalable high-resolution forecasting of sparse spatiotemporal events with kernel methods: A winning solution to the NIJ “Real-Time Crime Forecasting Challenge”

We propose a generic spatiotemporal event forecasting method which we developed for the National Institute of Justice’s (NIJ) Real-Time Crime Forecasting Challenge (National Institute of Justice (2017)). Our method is a spatiotemporal forecasting model combining scalable randomized Reproducing Kernel Hilbert Space (RKHS) methods for approximating Gaussian processes with autoregressive smoothing kernels in a regularized supervised learning framework. While the smoothing kernels capture the two main approaches in current use in the field of crime forecasting, kernel density estimation (KDE) and self-exciting point process (SEPP) models, the RKHS component of the model can be understood as an approximation to the popular log-Gaussian Cox Process model. For inference, we discretize the spatiotemporal point pattern and learn a log-intensity function using the Poisson likelihood and highly efficient gradient-based optimization methods. Model hyperparameters including quality of RKHS approximation, spatial and temporal kernel lengthscales, number of autoregressive lags and bandwidths for smoothing kernels as well as cell shape, size and rotation, were learned using cross validation. Resulting predictions significantly exceeded baseline KDE estimates and SEPP models for sparse events.

Optimizing and accelerating space–time Ripley ’s K function based on Apache Spark for distributed spatiotemporal point pattern analysis

With increasing point of interest (POI) datasets available with fine-grained spatial and temporal attributes, space–time Ripley’s K function has been regarded as a powerful approach to analyze spatiotemporal point process. However, space–time Ripley’s K function is computationally intensive for point-wise distance comparisons, edge correction and simulations for significance testing. Parallel computing technologies like OpenMP, MPI and CUDA have been leveraged to accelerate the K function, and related experiments have demonstrated the substantial acceleration. Nevertheless, previous works have not extended optimization of Ripley’s K function from space dimension to space–time dimension. Without sophisticated spatiotemporal query and partitioning mechanisms, extra computational overhead can be problematic. Meanwhile, these researches were limited by the restricted scalability and relative expensive programming cost of parallel frameworks and impeded their applications for large POI dataset and Ripley’s K function variations. This paper presents a distributed computing method to accelerate space–time Ripley’s K function upon state-of-the-art distributed computing framework Apache Spark, and four strategies are adopted to simplify calculation procedures and accelerate distributed computing respectively: (1) spatiotemporal index based on R-tree is utilized to retrieve potential spatiotemporally neighboring points with less distance comparison; (2) spatiotemporal edge correction weights are reused by 2-tier cache to reduce repetitive computation in L value estimation and simulations; (3) spatiotemporal partitioning using KDB-tree is adopted to decrease ghost buffer redundancy in partitions and support near-balanced distributed processing; (4) customized serialization with compact representations of spatiotemporal objects and indexes is developed to lower the cost of data transmission. Based on the optimized method, a web-based visual analytics framework prototype has been developed. Experiments prove the feasibility and time efficiency of the proposed method, and also demonstrate its value on promoting applications of space–time Ripley’s K function in ecology, geography, sociology, economics, urban transportation and other fields.

Analysis of Tornado Reports Through Replicated Spatiotemporal Point Patterns

SummaryUnderstanding the spatiotemporal distribution of tornado events is increasingly imperative, not only because of the natural phenomenon itself and its tremendous complexity but also because we can potentially reduce the risks that they entail. In particular, the US regions are particularly susceptible to tornadoes and they are the focus and motivation of our statistical analysis. Tornado reports can be treated as spatiotemporal point patterns, and we develop some methods for the analysis of replicated spatiotemporal patterns to identify significant structural differences between cold and warm seasons along the years. We extend some existing spatial techniques to the spatiotemporal context to test the null hypothesis that two (or more) observed spatiotemporal point patterns with replications are realizations of point processes that have the same second-order descriptors. In particular, we develop a non-parametric test to approximate the null distribution of the test statistics. We present intensive simulation studies that demonstrate the validity and power of our test and apply our methods to the motivating problem of tornadoes.

Modeling Bronchiolitis Incidence Proportions in the Presence of Spatio-Temporal Uncertainty

Bronchiolitis (inflammation of the lower respiratory tract) in infants is primarily due to viral infection and is the single most common cause of infant hospitalization in the United States. To increase epidemiological understanding of bronchiolitis (and, subsequently, develop better prevention strategies), this research analyzes data on infant bronchiolitis cases from the U.S. Military Health System between the years 2003–2013 in Norfolk, Virginia, USA. For privacy reasons, child home addresses, birth dates, and diagnosis dates were randomized (jittered) creating spatio-temporal uncertainty in the geographic location and timing of bronchiolitis incidents. Using spatio-temporal point patterns, we created a modeling strategy that accounts for the jittering to estimate and quantify the uncertainty for the incidence proportion (IP) of bronchiolitis. Additionally, we regress the IP onto key covariates including pollution where we adequately account for uncertainty in the pollution levels (i.e., covariate uncertainty) using a land use regression model. Our analysis results indicate that the IP is positively associated with sulfur dioxide and population density. Further, we demonstrate how scientific conclusions may change if various sources of uncertainty (either spatio-temporal or covariate uncertainty) are not accounted for. Code submitted with this article was checked by an Associate Editor for Reproducibility and is available as an online supplement.

Velocities for spatio-temporal point patterns

Point patterns gathered over space and time are receiving increasing attention in the literature. Examples include incidence of disease events, incidence of insurgent activity events, or incidence of crime events. Point pattern models can attempt to explain these events. Here, a log Gaussian Cox process specification is used to learn about the behavior of the intensity over space and time.Our contribution is to expand inference by introducing the notion of the velocity of a point pattern. We develop a velocity at any location and time within the region and period of study. These velocities are associated with the evolution of the intensity driving the spatio-temporal point pattern, where this intensity is a realization of a stochastic process. Working with directional derivative processes, we are able to develop derivatives in arbitrary directions in space as well as derivatives in time. The ratio of the latter to the former provides a velocity in that direction at that location and time, i.e., speed of change in intensity in that direction. This velocity can be interpreted in terms of speed of change in chance for an event. The magnitude and direction of the minimum velocity provides the slowest speed and direction of change in chance for an event.We use a sparse Gaussian process model approximation to expedite the demanding computation for model fitting and gradient calculation. We illustrate our methodology with a simulation for proof of concept and with a spatio-temporal point pattern of theft events in San Francisco, California in 2012.

Celebrating the 25th International Conference of The International Environmetrics Society

Celebrating the 25th International Conference of The International Environmetrics Society

Hybrid Indexing for Parallel Analysis of Spatiotemporal Point Patterns

GIScience 2016 Short Paper Proceedings Hybrid Indexing for Parallel Analysis of Spatiotemporal Point Patterns Alexander Hohl 1 , Irene Casas 2 , Eric M. Delmelle 1 , Wenwu Tang 1 University of North Carolina at Charlotte, Department of Geography and Earth Sciences and Center for Applied GIScience, Charlotte, NC, 28223, USA. Email: {ahohl; Eric.Delmelle; WenwuTang}@uncc.edu Louisiana Tech University, Department of Social Sciences, Ruston, LA, 71272, USA. Email: icasas@latech.edu Abstract High-performance parallel computing outperforms desktop workstations for computationally demanding problem solving. Domain decomposition and spatial indexing are widely used to accelerate spatial search. A single index method for spatiotemporal data processing lacks retrieval efficiency for massive computation. Combining multiple indexing methods to a hybrid spatiotemporal index holds potential for addressing this data retrieval challenge. We perform adaptive octree decomposition of the spatiotemporal domain and build local k-d trees to accelerate nearest neighbour search for space-time kernel density estimation (STKDE). Our parallel implementation reaches substantial speedup compared to sequential processing. The hybrid index outperforms octree decomposition alone, especially at lower-levels of parallelization. This approach facilitates finer-scale computation, enabling us to discover patterns that would be hidden otherwise. 1. Introduction Many algorithms for analyzing spatiotemporal data are computationally intensive because they often rely on extensive nearest-neighbor (NN) search. Spatial indexing methods, including R-trees and quadtrees, have long been used to accelerate spatial queries (Lu and Ooi 1993). Further, parallel computing offers the capacity for efficient processing of intensive analyses. Efficient algorithms for range queries and k nearest neighbor (kNN) queries have been developed within parallel communication models, like Hadoop (Agarwal et al. 2016). Hering (2013) showed that performance of in-memory k-d trees is best for intermediate number of dimensions (6-13). In addition, kNN search was implemented using graphics processing units (GPU) (see Liang et al. 2010; Sismanis et al. 2012), where distance calculation and sorting are parallelized. Alternatively, the strategy of spatial domain decomposition is used for distributing the resulting subdomains to multiple computing resources for concurrent processing (Wilkinson and Allen 1999). However, including time in geographic models complicates requirements for spatial indexing, resulting in low retrieval efficiency (Gu 2011). Merging multiple indexing methods to form hybrid spatiotemporal indices was recently proposed by Azri et al. (2013). In this study, we perform octree-based recursive decomposition of the space-time domain for parallel computation of STKDE. Using k-d tree indexing within octree leaf nodes (Liu et al. 2008) we accelerate kNN search for STKDE. 2. Methods 2.1 Data We used a dataset of dengue fever cases in Cali, Colombia for years 2011 and 2012. Each of the 11,168 records holds x- and y-coordinates and a timestamp (Delmelle et al. 2013). The rectangular envelope of the dataset spans 15,000m * 20,000m * 727days. 2.2 Space-Time Kernel Density Estimation STKDE results in a 3D volume where each voxel (volumetric pixel) is assigned a density estimate based on the surrounding datapoints. Specifically:

Spatio-temporal analysis of fox rabies cases in Germany 2005-2006.

Aiming to achieve new insights into rabies dynamics, this paper is the first to investigate fox rabies in Germany from a space-time pattern perspective. Based on a locally restricted dataset covering a fourteen month period, our findings indicate a strongly aggregated spatiotemporal point pattern resulting from an inhomogeneous stochastic process. In contrast to spatial or temporal approaches or cellular automata, our analysis focuses on the disease dynamics in time and space in a continuous time domain. Our findings confirm existing theories regarding fox rabies control highlighting the potential risk of urban areas and the need for effective rabies vaccination.

Estimating Second-Order Characteristics of Inhomogeneous Spatio-Temporal Point Processes

Non-parametric estimates of the K-function and the pair correlation function play a fundamental role for exploratory and explanatory analysis of spatial and spatio-temporal point patterns. These estimates usually require information from outside of the study region, resulting to the so-called edge effects which have to be corrected. They also depend on first-order characteristics, which have to be estimated in practice. In this paper, we extend classical edge correction methods to the spatio-temporal setting and compare the performance of the related estimators for stationary/non-stationary and/or isotropic/anisotropic point patterns. Further, we explore the influence of the estimated intensity function on these estimators.

Quantification of annual wildfire risk; A spatio-temporal point process approach.

Policy responses for local and global firemanagement depend heavily on the proper understanding of the fire extent as well as its spatio-temporal variation across any given study area. Annual fire risk maps are important tools for such policy responses, supporting strategic decisions such as location-allocation of equipment and human resources. Here, we define risk of fire in the narrow sense as the probability of its occurrence without addressing the loss component. In this paper, we study the spatio-temporal point patterns of wildfires and model them by a log Gaussian Cox processes. Themean of predictive distribution of randomintensity function is used in the narrow sense, as the annual fire risk map for next year.

Spatio-temporal heterogeneity: concepts and analyses

Foreword Preface 1. Conceptual introduction 2. Spatio-temporal stochastic processes: definitions and properties 3. Heterogeneity analysis of spatial point patterns 4. Heterogeneity analysis of temporal and spatio-temporal point patterns 5. Heterogeneity analysis of point patterns: process modeling and summary 6. Heterogeneity analysis of time series 7. Heterogeneity analysis of spatial surface patterns 8. Heterogeneity analysis of spatio-temporal surface patterns 9. Sampling and study design aspects in heterogeneity analysis of surface patterns 10. Conclusions Index.