Spatial and Spatio-Temporal Log-Gaussian Cox Processes: Extending the Geostatistical Paradigm

Peter J. Diggle; Paula Moraga; Barry Rowlingson; Benjamin M. Taylor

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Spatial and Spatio-Temporal Log-Gaussian Cox Processes: Extending the Geostatistical Paradigm

Peter J. Diggle, Paula Moraga, Barry Rowlingson, Benjamin M. Taylor

arXiv:1312.6536v1Dec 23, 2013stat.ME

TL;DR

The paper addresses how to model and predict incompletely observed spatial or spatio-temporal phenomena across point-pattern and other data formats. It develops LGCPs, discusses likelihood-based inference and computation, and demonstrates applications including disease-risk mapping and real-time surveillance. The authors conclude that LGCPs provide a useful geostatistical framework for probabilistic prediction, although computational and modelling limitations remain.

Problem
The paper addresses prediction of incompletely observed spatial or spatio-temporal phenomena when observations may be point patterns or aggregated counts.
Method
The authors model intensity as Λ(x) = exp{S(x)} with a Gaussian process S, and develop likelihood-based, Bayesian, and computational methods for inference and prediction.
Results
The paper demonstrates LGCP usefulness for smoothing, spatial disease-risk mapping, and probabilistic prediction across spatial and spatio-temporal applications.
Takeaways & Limitations
LGCPs support probabilistic prediction of incompletely observed spatial or spatio-temporal processes irrespective of data format, including real-time updating of predictions.
Takeaways & Limitations
The approach can be over-elaborate compared with kernel smoothing, and disease-risk estimates may depend on an unreliable population-density offset and piecewise-constant covariate surfaces.

Abstract

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In this paper we first describe the class of log-Gaussian Cox processes (LGCPs) as models for spatial and spatio-temporal point process data. We discuss inference, with a particular focus on the computational challenges of likelihood-based inference. We then demonstrate the usefulness of the LGCP by describing four applications: estimating the intensity surface of a spatial point process; investigating spatial segregation in a multi-type process; constructing spatially continuous maps of disease risk from spatially discrete data; and real-time health surveillance. We argue that problems of this kind fit naturally into the realm of geostatistics, which traditionally is defined as the study of spatially continuous processes using spatially discrete observations at a finite number of locations. We suggest that a more useful definition of geostatistics is by the class of scientific problems that it addresses, rather than by particular models or data formats.

1. INTRODUCTION

The paper reframes geostatistics around prediction of incompletely observed spatially continuous phenomena rather than fixed data formats or model classes. It presents LGCPs as a framework for point-pattern and aggregated-count data, including spatial and spatio-temporal applications.

1. INTRODUCTION: Spatial statistics can be classified by continuous variation, discrete variation, and spatial point processes.Continuous variation is represented by a stochastic process, discrete variation by a finite-dimensional random variable, and point patterns by a counting measure.
1. INTRODUCTION: The paper argues that spatial continuity versus discreteness is the key theoretical distinction and that many natural processes should be modelled as spatially continuous.
1. INTRODUCTION: A one-to-one match between geostatistical, lattice, and point-pattern data formats and their associated model classes is inappropriate in many applications.
1. INTRODUCTION: The authors redefine geostatistics as models and methods enabling predictive inference about an incompletely observed spatially continuous phenomenon.
1. INTRODUCTION: LGCPs model a Gaussian process through the intensity of an observed Poisson point process, with the log-linear form λ(x) = exp{S(x)}.
1. INTRODUCTION: The paper develops LGCP theory, inference, computation, spatial applications, spatio-temporal extensions, and joint analysis of multivariate data observed at incommensurate spatial scales.

2. THE LOG-GAUSSIAN COX PROCESS

An LGCP is a Cox process whose stochastic intensity is the exponential of a Gaussian process. This construction provides tractable moment properties, flexible covariance modelling, and a natural representation of environmentally driven spatial patterns.

2. THE LOG-GAUSSIAN COX PROCESS: A Cox process has a nonnegative stochastic intensity and becomes an inhomogeneous Poisson process conditional on that intensity.
2. THE LOG-GAUSSIAN COX PROCESS: Cox processes suit environmentally driven phenomena with conditionally independent cases more naturally than patterns driven primarily by interactions among points.
2. THE LOG-GAUSSIAN COX PROCESS: An LGCP sets Λ(x) = exp{S(x)}, where S is a Gaussian process, transferring some tractability of the multivariate Normal distribution to the Cox process.
2. THE LOG-GAUSSIAN COX PROCESS: For a stationary LGCP, the intensity is λ = exp(µ + 0.5σ2) and the covariance density is g(u) = λ2[exp{σ2r(u)} −1].
2. THE LOG-GAUSSIAN COX PROCESS: Reparameterising with E[S(x)] = −0.5σ2 and λ = exp(β) separates first-order mean properties from second-order variation.
2. THE LOG-GAUSSIAN COX PROCESS: Multivariate LGCPs replace the scalar Gaussian process with a vector-valued one, yielding tractable cross-covariance properties.
2. THE LOG-GAUSSIAN COX PROCESS: The Matérn covariance family uses φ as a distance parameter and κ as a dimensionless shape parameter controlling Gaussian-process differentiability.

3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES

Inference for LGCPs targets both model parameters and unobserved process realisations using moment-based, likelihood, or Bayesian approaches. Likelihood-based inference is difficult because integrating over the latent intensity is high-dimensional, motivating Monte Carlo methods.

3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: Parameter estimation concerns process properties, whereas prediction concerns a particular unobserved realisation; Bayesian analysis can formally address both through conditional distributions.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: The paper considers moment-based, maximum-likelihood, and Bayesian estimation, with the latter two described as more principled likelihood-based approaches.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: Moment-based estimation minimises discrepancies between empirical and theoretical second-moment properties, but its tuning choices give it an ad hoc quality.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: The likelihood requires integration over the infinite-dimensional distribution of the stochastic intensity, approximated computationally on a fine regular lattice.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: Crude Monte Carlo averages over simulated intensity realisations are described as hopelessly inefficient, motivating Geyer’s alternative method.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: The resulting Monte Carlo likelihood approximation depends on the number of simulations and on how close the fixed reference parameter θ0 is to the maximum-likelihood estimate.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: Conditional simulation of the intensity requires carefully tuned MCMC, while Bayesian estimation and prediction can be incorporated into one MCMC algorithm.
3. INFERENCE FOR LOG-GAUSSIAN COX PROCESSES: Bayesian prediction averages plug-in predictions over parameter values weighted by their posterior distribution, thereby incorporating parameter uncertainty.

4. COMPUTATION

Computing with LGCPs approximates the continuous latent process on a grid and uses MCMC or INLA for inference. The central challenge is high-dimensional covariance computation, with a practical trade-off between approximation accuracy, speed, and flexibility.

4. COMPUTATION: Inference for LGCPs is computationally challenging in both spatial and spatio-temporal settings.
4. COMPUTATION: The continuous latent process is represented by a piecewise-constant equivalent on disjoint computational-grid cells, becoming exact in the limit of infinitely many cells.
4. COMPUTATION: The grid balances computational complexity against approximation accuracy, while covariance-matrix inversion is the computational bottleneck.
4. COMPUTATION: Grid counts are typically 0 or 1 when the grid is finely spaced, and inference targets the latent process plus intensity and covariance parameters.
4. COMPUTATION: MCMC generates samples from [S,β,θ|Y], whereas INLA uses a mathematical approximation.
4. COMPUTATION: 100,000 MCMC iterations produced more accurate predictive-probability estimates than INLA in a restricted spatial LGCP scenario with known covariance parameters.
4. COMPUTATION: INLA is faster and avoids convergence and mixing diagnostics, whereas MCMC is more flexible for extensions to standard model classes.
4. COMPUTATION: The MCMC design uses gradient-informed MALA proposals for some components, random-walk proposals for covariance parameters, and adaptive scaling.

5. APPLICATIONS

The paper applies LGCPs to smooth spatial point patterns, assess multitype spatial segregation, and construct covariate-adjusted disease-risk maps while representing predictive uncertainty. These applications show how spatially continuous models can analyze point-pattern and aggregated-count data without tying model interpretation to a particular spatial partition.

Smoothing a Spatial Point Pattern: LGCP smoothing treats intensity estimation as prediction of Λ(x) = exp{β + S(x)} from observed event locations.A smooth estimate can be obtained from a suitable summary of the predictive distribution, such as its pointwise expectation.
Smoothing a Spatial Point Pattern: 703 hickory-tree locations are used to illustrate posterior intensity mapping and identification of unusually low- and high-density areas.The maps identify areas where tree density is less than half or more than double the mean density.
Spatial Segregation: Genotypic Diversity of Bovine Tuberculosis in Cornwall, UK: For Cornwall bovine tuberculosis data, LGCP probability surfaces and dominance areas reveal strong evidence against randomly intermingled genotypes.Genotype 9 dominates strongly in eastern and smaller western areas, genotype 15 in a central area, while genotypes 12 and 20 have only small south-western dominance pockets.
Disease Atlases: The disease-atlas application models cancer cases with population density as an offset and disease risk as a spatially varying component of an LGCP.The alternative formulation uses intensity Λ(x) = d(x)R(x), with R(x) = exp{S(x)}.
Disease Atlases: Spatially continuous LGCP risk maps have interpretations independent of the partition into subregions and can incorporate high-resolution environmental covariates.The authors caution that population density may be available only as small-area counts, making a piecewise-constant surface a convenient fiction, and note that the offset estimate is not guaranteed reliable.
Disease Atlases: The Castile-La Mancha maps identify small areas of raised covariate-adjusted lung-cancer risk, including contiguous cells north of Toledo and around Illescas with exceedance probability above 0.6.The displays include predicted relative risk, estimated variance, and the posterior probability that relative risk exceeds 1.1.

6. SPATIO-TEMPORAL LOG-GAUSSIAN COX PROCESSES

The paper extends LGCP modelling to spatio-temporal point processes and illustrates real-time disease surveillance using overnight predictive updating. It also discusses computational, data-access, and confidentiality constraints affecting such systems.

A spatio-temporal LGCP models points conditional on an intensity Λ(x,t) = exp{S(x,t)}, where S is a Gaussian process.
AEGISS application: AEGISS converted NHS Direct calls for diarrhoea or vomiting into a spatio-temporal point pattern using residential postal codes.Postal codes were referenced to 100-metre precision, effectively continuous at Hampshire’s study scale.
AEGISS application: The surveillance model combined spatial and temporal baseline intensities with a spatio-temporal LGCP and separable correlation structure.The spatial baseline used kernel smoothing, while the temporal baseline included seasonality, day-of-week, and service-uptake trend terms.
AEGISS application: Overnight MALA updates retained predictive exceedance counts for risks exceeding 2, 4, or 8 times baseline, thresholds chosen with clinicians.A doubling was potentially interesting, whereas an eightfold increase was considered potentially serious.
Discussion: Modern computing enables longer MCMC runs and finer spatial resolutions, but confidentiality concerns and fragmented NHS providers complicate broader deployment.The paper identifies combining multiple data streams as a possible direction for future health surveillance systems.

7. DATA SYNTHESIS: INTEGRATED ANALYSIS OF EXPOSURE AND HEALTH OUTCOME DATA AT MULTIPLE SPATIAL SCALES

The paper proposes treating exposures and health risks as spatially continuous processes even when observations are recorded at different spatial scales. This framework supports integrated prediction from disease counts, exposure measurements, and individual-level longitudinal data.

Spatial misalignment describes exposure and health outcome data recorded at disparate spatial scales, a central challenge in epidemiological synthesis.
Multiple-exposure disease-risk models can treat case locations as an LGCP while representing exposures as spatially continuous processes.The stochastic risk component captures variation not explained by the covariate processes.
For area-level disease counts and exposure estimates at different locations, inference targets the predictive distribution of the continuous risk surface.The joint model combines Gaussian processes, measurement-error models, and Poisson observations, with MCMC used for sampling.
Leptospirosis example: A leptospirosis cohort contributed approximately 1700 subjects with repeated binary sero-conversion outcomes and time-constant and time-varying risk factors.Follow-up blood samples occurred at approximately 6, 12, 18, and 24 months.
Leptospirosis example: The proposed longitudinal formulation represents individual and residential characteristics alongside a spatio-temporally continuous Gaussian process.Residence-based covariates can, in principle, be indexed by continuous spatial location.
Inference for the joint model with repeated outcomes and incompletely observed covariates would require further development of MCMC algorithms.

8. DISCUSSION

The discussion presents LGCPs as models for predicting incompletely observed spatial and spatio-temporal processes across data formats. It emphasizes probabilistic prediction, computational feasibility, and the practical limits imposed by skewness and increasingly complex data sources.

LGCPs are presented as useful for predicting incompletely observed spatial or spatio-temporal processes irrespective of data format.The paper links this broader modelling scope to likelihood-based, classical, or Bayesian estimation and probabilistic prediction.
The applications prioritize prediction of unobserved response-surface variation over estimation of model parameters.Bayesian prediction with diffuse priors accommodates parameter uncertainty and presents model-based smoothers probabilistically.
In public health, identifying where and when disease incidence may exceed an intervention threshold can be more useful than reporting point estimates or significance tests.
The log-linear formulation offers tractable moments and a multiplicative decomposition, but can produce highly skewed intensities with near-zero regions and sharp peaks.
Future methodological work concerns principled analysis of variable-quality multiple data streams and robust computation for increasingly complex inference problems.