log-Gaussian Cox Process Model Research Articles

Fitting Log-Gaussian Cox Processes Using Generalized Additive Model Software

While log-Gaussian Cox process regression models are useful tools for modeling point patterns, they can be technically difficult to fit and require users to learn/adopt bespoke software. We show that, for suitably formatted data, we can actually fit these models using generalized additive model software, via a simple line of code, demonstrated on R by the popular mgcv package. We are able to do this because a common and computationally efficient way to fit a log-Gaussian Cox process model is to use a basis function expansion to approximate the Gaussian random field, as is provided by a generic bivariate smoother over geographic space. We further show that if basis functions are parameterized appropriately then we can estimate parameters in the spatial covariance function for the latent random field using a generalized additive model. We use simulation to show that this approach leads to model fits of comparable quality to state-of-the-art software, often more quickly. But we see the main advance from this work as lowering the technology barrier to spatial statistics for applied researchers, many of whom are already familiar with generalized additive model software.

Generalized functional linear model with a point process predictor.

Point process data have become increasingly popular these days. For example, many of the data captured in electronic health records (EHR) are in the format of point process data. It is of great interest to study the association between a point process predictor and a scalar response using generalized functional linear regression models. Various generalized functional linear regression models have been developed under different settings in the past decades. However, existing methods can only deal with functional or longitudinal predictors, not point process predictors. In this article, we propose a novel generalized functional linear regression model for a point process predictor. Our proposed model is based on the joint modeling framework, where we adopt a log-Gaussian Cox process model for the point process predictor and a generalized linear regression model for the outcome. We also develop a new algorithm for fast model estimation based on the Gaussian variational approximation method. We conduct extensive simulation studies to evaluate the performance of our proposed method and compare it to competing methods. The performance of our proposed method is further demonstrated on an EHR dataset of patients admitted into the intensive care units of the Beth Israel Deaconess Medical Center between 2001 and 2008.

Stelfi: An R package for fitting Hawkes and log-Gaussian Cox point process models.

Modelling spatial and temporal patterns in ecology is imperative to understand the complex processes inherent in ecological phenomena. Log-Gaussian Cox processes are a popular choice among ecologists to describe the spatiotemporal distribution of point-referenced data. In addition, point pattern models where events instigate others nearby (i.e., self-exciting behaviour) are becoming increasingly popular to infer the contagious nature of events (e.g., animal sightings). While there are existing R packages that facilitate fitting spatiotemporal point processes and, separately, self-exciting models, none incorporate both. We present an R package, stelfi, that fits spatiotemporal self-exciting and log-Gaussian Cox process models using Template Model Builder through a range of custom-written C++ templates. We illustrate the use of stelfi's functions fitting models to Sasquatch (bigfoot) sightings data within the USA. The structure of these data is typical of many seen in ecology studies. We show, from a temporal Hawkes process to a spatiotemporal self-exciting model, how the models offered by the package enable additional insights into the temporal and spatial progression of point pattern data. We present extensions to these well-known models that include spatiotemporal self-excitation and joint likelihood models, which are better suited to capture the complex mechanisms inherent in many ecological data. The package stelfi offers user-friendly functionality, is open source, and is available from CRAN. It offers the implementation of complex spatiotemporal point process models in R for applications even beyond the field of ecology.

Incorporating environmental heterogeneity and observation effort to predict host distribution and viral spillover from a bat reservoir.

Predicting the spatial occurrence of wildlife is a major challenge for ecology and management. In Latin America, limited knowledge of the number and locations of vampire bat roosts precludes informed allocation of measures intended to prevent rabies spillover to humans and livestock. We inferred the spatial distribution of vampire bat roosts while accounting for observation effort and environmental effects by fitting a log Gaussian Cox process model to the locations of 563 roosts in three regions of Peru. Our model explained 45% of the variance in the observed roost distribution and identified environmental drivers of roost establishment. When correcting for uneven observation effort, our model estimated a total of 2340 roosts, indicating that undetected roosts (76%) exceed known roosts (24%) by threefold. Predicted hotspots of undetected roosts in rabies-free areas revealed high-risk areas for future viral incursions. Using the predicted roost distribution to inform a spatial model of rabies spillover to livestock identified areas with disproportionate underreporting and indicated a higher rabies burden than previously recognized. We provide a transferrable approach to infer the distribution of a mostly unobserved bat reservoir that can inform strategies to prevent the re-emergence of an important zoonosis.

Extending the R number by applying hyperparameters of Log Gaussian Cox process models in an epidemiological context to provide insights into COVID-19 positivity in the City of Edinburgh and in students residing at Edinburgh University.

The impact of the COVID-19 pandemic on University students has been a topic of fiery debate and of public health research. This study demonstrates the use of a combination of spatiotemporal epidemiological models to describe the trends in COVID-19 positive cases on spatial, temporal and spatiotemporal scales. In addition, this study proposes new epidemiological metrics to describe the connectivity between observed positivity; an analogous metric to the R number in conventional epidemiology. The proposed indices, Rspatial, Rspatiotemporal and Rscaling will aim to improve the characterisation of the spread of infectious disease beyond that of the COVID-19 framework and as a result inform relevant public health policy. Apart from demonstrating the application of the novel epidemiological indices, the key findings in this study are: firstly, there were some Intermediate Zones in Edinburgh with noticeably high levels of COVID-19 positivity, and that the first outbreak during the study period was observed in Dalry and Fountainbridge. Secondly, the estimation of the distance over which the COVID-19 counts at the halls of residence are spatially correlated (or related to each other) was found to be 0.19km (0.13km to 0.27km) and is denoted by the index, Rspatial. This estimate is useful for public health policy in this setting, especially with contact tracing. Thirdly, the study indicates that the association between the surrounding community level of COVID-19 positivity (Intermediate Zones in Edinburgh) and that of the University of Edinburgh's halls of residence was not statistically significant. Fourthly, this study reveals that relatively high levels of COVID-19 positivity were observed for halls for which higher COVID-19 fines were issued (Spearman's correlation coefficient = 0.34), and separately, for halls which were non-ensuite relatively to those which were not (Spearman's correlation coefficient = 0.16). Finally, Intermediate Zones with the highest positivity were associated with student residences that experienced relatively high COVID-19 positivity (Spearman's correlation coefficient = 0.27).

On the estimation of landslide intensity, hazard and density via data-driven models

Maps that attempt to predict landslide occurrences have essentially stayed the same since 1972. In fact, most of the geo-scientific efforts have been dedicated to improve the landslide prediction ability with models that have largely increased their complexity but still have addressed the same binary classification task. In other words, even though the tools have certainly changed and improved in 50 years, the geomorphological community addressed and still mostly addresses landslide prediction via data-driven solutions by estimating whether a given slope is potentially stable or unstable. This concept corresponds to the landslide susceptibility, a paradigm that neglects how many landslides may trigger within a given slope, how large these landslides may be and what proportion of the given slope they may disrupt. The landslide intensity concept summarized how threatening a landslide or a population of landslide in a study area may be. Recently, landslide intensity has been spatially modeled as a function of how many landslides may occur per mapping unit, something, which has later been shown to closely correlate to the planimetric extent of landslides per mapping unit. In this work, we take this observation a step further, as we use the relation between landslide count and planimetric extent to generate maps that predict the aggregated size of landslides per slope, and the proportion of the slope they may affect. Our findings suggest that it may be time for the geoscientific community as a whole, to expand the research efforts beyond the use of susceptibility assessment, in favor of more informative analytical schemes. In fact, our results show that landslide susceptibility can be also reliably estimated (AUC of 0.92 and 0.91 for the goodness-of-fit and prediction skill, respectively) as part of a Log-Gaussian Cox Process model, from which the intensity expressed as count per unit (Pearson correlation coefficient of 0.91 and 0.90 for the goodness-of-fit and prediction skill, respectively) can also be derived and then converted into how large a landslide or several coalescing ones may become, once they trigger and propagate downhill. This chain of landslide intensity, hazard and density may lead to substantially improve decision-making processes related to landslide risk.

FAST METHODS FOR FITTING LOG-GAUSSIAN COX PROCESS MODELS IN ECOLOGY

FAST METHODS FOR FITTING LOG-GAUSSIAN COX PROCESS MODELS IN ECOLOGY

Spatio-temporal modeling of infectious diseases by integrating compartment and point process models.

The online version contains supplementary material available at 10.1007/s00477-022-02354-4.

Analyzing travel destinations distribution using large-scaled GPS trajectories: A spatio-temporal Log-Gaussian Cox process

Under the increasingly serious city diseases, it is very important to deeply understand the spatio-temporal correlation between travel destination and urban built environment. The spatio-temporal point process has been widely used in many specific fields to explore the evolution of events. We employ a spatio-temporal Log-Gaussian Cox process model (LGCP) to analyze travel destination, which consists of three different components: a spatial component λ(s), a temporal component μ(t), and a separable log-Gaussian stochastic intensity field exp{Υ(s,t)}, to model and predict the destination. The spatial component and temporal component are modeled by the Poisson log linear regression model combined with covariates of interest separately. Spatial covariates including land use, demographics, travel origins and road network, and temporal covariates including the periodic function, weekdays and weekend variation, are selected as influencing factors to analyze the impact of building environment on travel destinations under different combinations of spatial and temporal covariates. Then, an extensible Markov chain Monte Carlo (MCMC) algorithm is applied to simulate Gaussian random field, and the fitted LGCP model can be used to obtain the prediction distribution of Gaussian random field beyond the last time point of data observation. In the experiment, taxi destination data collected in Shenzhen city May 6 to May 23, 2019 are used to validate modeling performance. The results in experiment shows the space–time positions of destination simulated by LGCP model are very similar to those observed. This study can provide strategy for urban planners and transportation administration to conduct reasonable policy to balance travel demand and public resources.

The Geospatial Distribution of Myositis and Its Phenotypes in the United States and Associations With Roadways: Findings From a National Myositis Patient Registry.

BackgroundLittle is known about the spatial distribution of idiopathic inflammatory myopathies (IIM) in the United States (U.S.), or their geospatial associations.MethodsWe studied a national myositis patient registry, with cases diagnosed in the contiguous U.S. from 1985–2011 and comprised of dermatomyositis (DM, n = 484), polymyositis (PM, n = 358), and inclusion body myositis (IBM, n = 318) patients. To assess the association of myositis prevalence with distance from roads, we employed log-Gaussian Cox process models, offset with population density.ResultsThe U.S. IIM case distribution demonstrated a higher concentration in the Northest. DM, IBM, and cases with lung disease were more common in the East, whereas PM cases were more common in the Southeast. One area in the West and one area in the South had a significant excess in cases of DM relative to PM and of cases with lung disease relative to those without lung disease, respectively. IIM cases tended to cluster, with between-points interactions more intense in the Northeast and less in the South. There was a trend of a higher prevalence of IIM and its major phenotypes among people living within 50 m of a roadway relative to living beyond 200 m. Demographic characteristics, rural-urban commuting area, and female percentage were significantly associated with the prevalence of IIM and with major phenotypes.ConclusionsUsing a large U.S. database to evaluate the spatial distribution of IIM and its phenotypes, this study suggests clustering in some regions of the U.S. and a possible association of proximity to roadways.

How does the community COVID-19 level of risk impact on that of a care home?

ObjectivesTo model the risk of COVID-19 mortality in British care homes conditional on the community level risk.MethodsA two stage modeling process (“doubly latent”) which includes a Besag York Mollie model (BYM) and a Log Gaussian Cox Process. The BYM is adopted so as to estimate the community level risks. These are incorporated in the Log Gaussian Cox Process to estimate the impact of these risks on that in care homes.ResultsFor an increase in the risk at the community level, the number of COVID-19 related deaths in the associated care home would be increased by exp (0.833), 2. This is based on a simulated dataset. In the context of COVID-19 related deaths, this study has illustrated the estimation of the risk to care homes in the presence of background community risk. This approach will be useful in facilitating the identification of the most vulnerable care homes and in predicting risk to new care homes.ConclusionsThe modeling of two latent processes have been shown to be successfully facilitated by the use of the BYM and Log Gaussian Cox Process Models. Community COVID-19 risks impact on that of the care homes embedded in these communities.

Hierarchical log Gaussian Cox process for regeneration in uneven-aged forests

We propose a hierarchical log Gaussian Cox process (LGCP) for point patterns, where a set of points varvec{x} affects another set of points varvec{y} but not vice versa. We use the model to investigate the effect of large trees on the locations of seedlings. In the model, every point in varvec{x} has a parametric influence kernel or signal, which together form an influence field. Conditionally on the parameters, the influence field acts as a spatial covariate in the intensity of the model, and the intensity itself is a non-linear function of the parameters. Points outside the observation window may affect the influence field inside the window. We propose an edge correction to account for this missing data. The parameters of the model are estimated in a Bayesian framework using Markov chain Monte Carlo where a Laplace approximation is used for the Gaussian field of the LGCP model. The proposed model is used to analyze the effect of large trees on the success of regeneration in uneven-aged forest stands in Finland.

Combining heterogeneous spatial datasets with process-based spatial fusion models: A unifying framework

In modern spatial statistics, the structure of data has become more heterogeneous. Depending on the types of spatial data, different modeling strategies are used. For example, kriging approaches for geostatistical data; Gaussian Markov random field models for lattice data; or log Gaussian Cox process models for point-pattern data. Despite these different modeling choices, the nature of underlying data-generating (latent) processes is often the same, which can be represented by some continuous spatial surfaces. A unifying framework is introduced for process-based multivariate spatial fusion models. The framework can jointly analyze all three aforementioned types of spatial data or any combinations thereof. Moreover, the framework accommodates different likelihoods for geostatistical and lattice data. It is shown that some established approaches, such as linear models of coregionalization, can be viewed as special cases of the proposed framework. A flexible and scalable implementation using R-INLA is provided. Simulation studies confirm that the prediction of latent processes improves as one moves from univariate spatial models to multivariate spatial fusion models. The framework is illustrated via a case study using datasets from a cross-sectional study linked with a national cohort in Switzerland. The differences in underlying spatial risks between respiratory disease and lung cancer are examined in the case study.

A spatial functional count model for heterogeneity analysis in time

A spatial curve dynamical model framework is adopted for functional prediction of counts in a spatiotemporal log-Gaussian Cox process model. Our spatial functional estimation approach handles both wavelet-based heterogeneity analysis in time, and spectral analysis in space. Specifically, model fitting is achieved by minimising the information divergence or relative entropy between the multiscale model underlying the data, and the corresponding candidates in the spatial spectral domain. A simulation study is carried out within the family of log-Gaussian Spatial Autoregressive \(\ell ^{2}\)-valued processes (SAR\(\ell ^{2}\) processes) to illustrate the asymptotic properties of the proposed spatial functional estimators. We apply our modelling strategy to spatiotemporal prediction of respiratory disease mortality.

Investigating mesh‐based approximation methods for the normalization constant in the log Gaussian Cox process likelihood

The log Gaussian Cox process (LGCP) is a frequently applied method for modeling point pattern data. The normalization constant of the LGCP likelihood involves an integral over a latent field. That integral is computationally expensive, making it troublesome to perform inference with standard methods. The so‐called stochastic partial differential equation–integrated nested Laplace approximation (SPDE‐INLA) framework enables fast approximate inference for a range of hierarchical models, where a key component is to approximate the latent field by a triangulated mesh. Recent research has made it possible to fit LGCP models with this framework using an approximate integration method to compute the integral. We carefully describe several alternative variants of that approximate integration method and derive an analytical formula for the integral in question, which actually is exact under the triangular mesh assumption used by SPDE‐INLA. We compare the different integration strategies through a comprehensive simulation study and find that the analytical formula is often more accurate, but not always. Among the approximate integration methods, we recommend a simple extension to a method implemented in an R‐package for fitting LGCP models.

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.

Mapping species richness using opportunistic samples: a case study on ground-floor bryophyte species richness in the Belgian province of Limburg

In species richness studies, citizen-science surveys where participants make individual decisions regarding sampling strategies provide a cost-effective approach to collect a large amount of data. However, it is unclear to what extent the bias inherent to opportunistically collected samples may invalidate our inferences. Here, we compare spatial predictions of forest ground-floor bryophyte species richness in Limburg (Belgium), based on crowd- and expert-sourced data, where the latter are collected by adhering to a rigorous geographical randomisation and data collection protocol. We develop a log-Gaussian Cox process model to analyse the opportunistic sampling process of the crowd-sourced data and assess its sampling bias. We then fit two geostatistical Poisson models to both data-sets and compare the parameter estimates and species richness predictions. We find that the citizens had a higher propensity for locations that were close to their homes and environmentally more valuable. The estimated effects of ecological predictors and spatial species richness predictions differ strongly between the two geostatistical models. Unknown inconsistencies in the sampling process, such as unreported observer’s effort, and the lack of a hypothesis-driven study protocol can lead to the occurrence of multiple sources of sampling bias, making it difficult, if not impossible, to provide reliable inferences.

Study of the spatiotemporal correlation between sediment-related blockage events in the sewer system in Bogotá (Colombia).

The planning and scheduling of maintenance operations of large conventional sewer systems generate a complex decision-making environment due to the difficulty in the collection and analysis of the spatiotemporal information about the operational and structural condition of their components (e.g. pipes, gully pots and manholes). As such, water utilities generally carry out these operations following a corrective approach. This paper studies the impact of the spatiotemporal correlation between these failure events using Log-Gaussian Cox Process (LGCP) models. In addition, the association of failure events to physical and environmental covariates was assessed. The proposed methods were applied to analyze sediment-related blockages in the sewer system of an operative zone in Bogotá (Colombia). The results of this research allowed the identification of significant covariates that were further used to model spatiotemporal clusters with high sediment-related failure risk in sewer systems. The LGCP model proved to be more accurate in comparison to those models that build upon a fundamental assumption that a failure is equally likely to occur at any time regardless of the state of the system and the system's history of failures (i.e. a homogeneous Poisson process model).

Global multivariate point pattern models for rain type occurrence

We seek statistical methods to study the occurrence of multiple rain types observed by satellite on a global scale. The main scientific interests are to relate rainfall occurrence with various atmospheric state variables and to study the dependence between the occurrences of multiple types of rainfall (e.g. short-lived and intense vs. long-lived and weak; the heights of the rain clouds are also considered). Commonly in point process model literature, the spatial domain is assumed to be a small, and thus planar domain. We consider the log-Gaussian Cox Process (LGCP) models on the surface of a sphere and take advantage of cross-covariance models for spatial processes on a global scale to model the stochastic intensity function of the LGCP models. We present analysis results for rainfall observations from the TRMM satellite and atmospheric state variables from MERRA-2 reanalysis data over the tropical Eastern and Western Pacific Ocean, as well as over the entire tropical and subtropical ocean regions. Statistical inference is done through Monte Carlo likelihood approximation for LGCP models. We employ covariance approximation to deal with massive data.

Electric vehicle charging point placement optimisation by exploiting spatial statistics and maximal coverage location models

Electric vehicles (EVs) are increasingly considered as a promising solution to tackle climate change impacts, improve air quality, and enhance growth sustainability. This paper proposes a two-step approach for optimally deploying charging points (CPs) by bringing together spatial statistics and maximal coverage location models. CP locations are conceptualised as a spatial point pattern, driven by an underlying stochastic process, and are investigated by using a Bayesian spatial log-Gaussian Cox process model. The spatial distribution of charging demand is approximated by the predicted process intensity surface of CP locations, upon which a maximal coverage location model is formulated and solved to identify optimal CP locations. Drawing upon the large-scale urban point of interest (POI) data and other data sources, the developed method is demonstrated by exploring the deployment of CPs in London. The results show that EV charging demand is statistically significantly associated with workplace population density, travel flows, and densities of three POI categories (transport, retail and commercial). The robustness of model estimation results is assessed by running spatial point process models with a series of random subsets of the full data. Results from a policy scenario analysis suggest that with increasing numbers of charging stations to be planned, optimal CP locations gradually expand to the suburban areas of London and the marginal gains in charging demand covered decrease rapidly.