A Fully Non-separable Log-Gaussian Cox Process to Model Forest Fires

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described bynth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

This article presents a methodology for modeling aggregated disease incidence data with the spatially continuous log-Gaussian Cox process. Statistical models for spatially aggregated disease incidence data usually assign the same relative risk to all individuals in the same reporting region (census areas or postal regions). A further assumption that the relative risks in two regions are independent given their neighbor's risks (the Markov assumption) makes the commonly used Besag-York-Mollié model computationally simple. The continuous model proposed here uses a data augmentation step to sample from the posterior distribution of the exact locations at each step of an Markov chain Monte Carlo algorithm, and models the exact locations with an log-Gaussian Cox process. A simulation study shows the log-Gaussian Cox process model consistently outperforming the Besag-York-Mollié model. The method is illustrated by making inference on the spatial distribution of syphilis risk in North Carolina. The effect of several known social risk factors are estimated, and areas with risk well in excess of that expected given these risk factors are identified.

Quick inference for log Gaussian Cox processes with non-stationary underlying random fields

Cox processes are natural models for point process phenomena that are environmentally driven, but much less natural for phenomena driven primarily by interactions amongst the points. The class of log-Gaussian Cox processes (LGCPs) has an elegant simplicity, and one of its attractive features is the tractability of the multivariate normal distribution carries over, to some extent, to the associated Cox process. In the statistical analysis of spatial point patterns, it is often assumed isotropy because of a simpler interpretation and ease of analysis. However, there are many cases in which the spatial structure depends on the direction. In this paper, we introduce new families of anisotropic spatial LGCPs that are useful to model spatial anisotropic point patterns that exhibit a degree of clustering. We propose classes of families consisting of elliptical and non-elliptical models. The former can be reduced to isotropic forms after some rotations, while the latter family goes beyond this property. We derive analytical forms for the covariance of the associated random field, and some second-order characteristics. A moment-based estimation procedure is followed to make inference on the parameters that control the degree of anisotropy. The estimation procedure is evaluated through a simulation study under a variety of scenarios and various degrees of anisotropy. Our methodology is illustrated on two real datasets of earthquakes in South America and the Mediterranean Europe.

Spatially aggregated epidemiological data is nowadays increasingly common because of ethical concern of data use as well as preservation of patient confidentiality. They are typically presented either as the count of disease cases or as an average measurement from districts partitioning a study region. In most cases, the partitioning is based on administrative convenience rather than information about the aetiology of any disease or public health problem. While inference for spatially aggregated data commonly make use of model that assumes a spatially discrete variation, we argue that a spatially continuous model should be considered when there is a scientific justification for its use, especially when the underlying generating process of the disease outcome is hypothesised to behave in a spatially continuous manner. In this thesis, we consider geostatistical methods as a framework that can be used to analyse spatially aggregated data. This thesis is a series of papers, two methodological and one public health application. In the first methodological paper, we developed a computationally efficient discrete approximation to log-Gaussian Cox process (LGCP) models for the analysis of spatially aggregated disease count data. We compare the predictive performance of our modelling approach with LGCP through a simulation study and an application to primary biliary cirrhosis incidence data in Newcastle-Upon-Tyne, UK. Our results suggest that when disease risk is assumed to be a spatially continuous process, the proposed approximation to LGCP provides reliable estimates of disease risk both on spatially continuous and aggregated scales. In the second methodological paper, We developed a model-based geostatistical approach that allows us to model the relationship between the Life expectancy at birth (LEB) and the index of multiple deprivation (IMD), when these are available over different partitions of the study region. We found that the effect of IMD on LEB is higher for males than for females. We show that our proposed model-based geostatistical approach does not only provide solution to any form of misalignment problem but also allows for spatially continuous inferences. In the third application paper, we developed a spatio-temporal model for monthly Chronic Obstructive Pulmonary Disease (COPD) emergency admissions data in South Cumbria and North Lancashire, UK, 2012-2018. We assess the relative contribution of socio-economic and environmental variables for forecasting COPD emergency admissions. In addition, we develop an early warning system that triggers an alarm whenever COPD emergency admissions exceeds a predefined incidence thresholds. The result of our analysis can potentially help NHS Morecambe Bay Clinical Commissioning Group stakeholders to define areas to target early intervention as well as inform resource allocation for healthcare system so that its limited resources can be used to maximum effect.

ABSTRACTWe consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.

Log Gaussian Cox processes on the sphere

Bayesian model based spatiotemporal survey designs and partially observed log Gaussian Cox process

Level set Cox processes

In this paper, we develop a computationally efficient discrete approximation to log-Gaussian Cox process (LGCP) models for the analysis of spatially aggregated disease count data. Our approach overcomes an inherent limitation of spatial models based on Markov structures, namely, that each such model is tied to a specific partition of the study area, and allows for spatially continuous prediction. We compare the predictive performance of our modelling approach with LGCP through a simulation study and an application to primary biliary cirrhosis incidence data in Newcastle upon Tyne, UK. Our results suggest that, when disease risk is assumed to be a spatially continuous process, the proposed approximation to LGCP provides reliable estimates of disease risk both on spatially continuous and aggregated scales. The proposed methodology is implemented in the open-source R package SDALGCP.

<p>Whilst the deforestation rate of the Brazilian Amazon has decreased drastically over the 2005-2015 period, thanks to an ambitious program to fight deforestation, since then, forest degradation resulting from logging and wildfires became the major source of aboveground biomass losses and the Brazilian Amazon turned into a net carbon source. This could be partially explained by a decoupling of fire occurrence and deforestation, historically one of the key drivers of the fire regime in the region. Moreover, since 2015, deforestation rates and associated fires are rising again, and new deforestation frontiers are opening in previously unaffected areas in the central and western Amazon.</p><p>Fires in the Brazilian Amazon are closely related to climate and agriculture: fires are used to transform forests into pastures or cropland, and subsequent burns are used to maintain grass productivity. When nearby rainforests are sufficiently dry, deforestation and agricultural fires escape and can cause large wildfires. Local communities’ fire management practices impact greatly the likelihood of these escaping fires, but also bear a cost. High mortality rates after even low-intensity fires lead to fuel accumulation and canopy damage, increasing the vulnerability of forests to subsequent burnings. Coupled with a regional reduction of precipitations due to climate change and deforestation, the Amazon forest could be threatened by a cycle of massive dieback and increased fire activity. Thus, it is crucial to understand the drivers of different types of fires in the region and how to prevent them. Of particular interest is the role played by the policies deployed after 2004 to reduce deforestation rates in the region and their recent weakening.</p><p>Building on previously published literature on the drivers of fire regimes and deforestation in the region, data were collected on potential drivers of fire regimes related to climate, agricultural expansion, ecosystem integrity, infrastructure, populations, environmental policies and land conflict. MODIS Active-Fire dataset was used as a response variable, and also classified into deforestation fires, agricultural fires and forest fires thanks to deforestation and land use data in a second step of the study. A spatiotemporal modelling approach, relying on the Log Gaussian Cox process and R-INLA package, has been adopted to assess the relative influence of different drivers of fire regimes in the Brazilian Amazon for the 2006-2020 period. Preliminary results on the drivers of fire regime in the state of Para for the last four years show a powerful influence of drivers related to agricultural expansion (especially ranching), integrity of the forest cover, presence of rural settlements and environmental policies. Different protection regimes have varying influences on the fire regime, with sustainable use areas being the less efficient. Law enforcement efforts seem to have an inhibitory effect on fire occurrence and protected area downgrading, downsizing and degazettement favour them.</p>

Forest fire management is not only an emergency task, the preventive task could be even more important, being better to avoid the risk of a forest fire ignition before it starts or minimize its hazard, rather than later trying to extinguish it. If we associate wildfires with their spatial coordinates, along with other variables, it is possible to identify them by means of a spatio-temporal stochastic process. Spatio-temporal clustering of wildfires could indicate the presence of risk factors. In fact, what is usually of interest is to assess their dependence on covariates. Two were the objectives in this paper. Firstly, to evaluate how the extent of clustering in wildfires differs across marks. Secondly, to analyze the influence of covariates on trends in the intensity of wildfire locations. We analyzed the spatio-temporal patterns produced by wildfire incidences in Catalonia, located in the north-east of the Iberian Peninsula. The total number of fires recorded in the studied area, during the period 1994-2008, was 10,783. In addition to the locations of the fire centroids, several marks and spatial covariates were considered. We specified spatio-temporal log-Gaussian Cox process models. Models were estimated using Bayesian inference for Gaussian Markov Random Field (GMRF) through the Integrated Nested Laplace Approximation (INLA) algorithm. The results allow us to quantify and assess possible spatial relationships between the distribution of risk of ignition and possible explanatory factors. We believe the methods shown in the paper may contribute to the prevention and management of wildfires, which are not random in space or time. © 2012 WIT Press.

We view the locations and times of a collection of crime events as a space-time point pattern. So, with either a nonhomogeneous Poisson process or with a more general Cox process, we need to specify a space-time intensity. For the latter, we need a \emph{random} intensity which we model as a realization of a spatio-temporal log Gaussian process. Importantly, we view time as circular not linear, necessitating valid separable and nonseparable covariance functions over a bounded spatial region crossed with circular time. In addition, crimes are classified by crime type. Furthermore, each crime event is recorded by day of the year which we convert to day of the week marks. The contribution here is to develop models to accommodate such data. Our specifications take the form of hierarchical models which we fit within a Bayesian framework. In this regard, we consider model comparison between the nonhomogeneous Poisson process and the log Gaussian Cox process. We also compare separable vs. nonseparable covariance specifications. Our motivating dataset is a collection of crime events for the city of San Francisco during the year 2012. We have location, hour, day of the year, and crime type for each event. We investigate models to enhance our understanding of the set of incidences.

Planar Cox processes directed by a log Gaussian intensity process are investigated in the univariate and multivariate cases. The appealing properties of such models are demonstrated theoretically as well as through data examples and simulations. In particular, the first, second and third‐order properties are studied and utilized in the statistical analysis of clustered point patterns. Also empirical Bayesian inference for the underlying intensity surface is considered.