# spatialAtRisk: spatialAtRisk function In lgcp: Log-Gaussian Cox Process

## Description

The methods for this generic function:spatialAtRisk.default, spatialAtRisk.fromXYZ, spatialAtRisk.im, spatialAtRisk.function, spatialAtRisk.SpatialGridDataFrame, spatialAtRisk.SpatialPolygonsDataFrame and spatialAtRisk.bivden are used to represent the fixed spatial component, lambda(s) in the log-Gaussian Cox process model. Typically lambda(s) would be represented as a spatstat object of class im, that encodes population density information. However, regardless of the physical interpretation of lambda(s), in lgcp we assume that it integrates to 1 over the observation window. The above methods make sure this condition is satisfied (with the exception of the method for objects of class function), as well as providing a framework for manipulating these structures. lgcp uses bilinear interpolation to project a user supplied lambda(s) onto a discrete grid ready for inference via MCMC, this grid can be obtained via the selectObsWindow function.

## Usage

 1 spatialAtRisk(X, ...)

## Arguments

 X an object ... additional arguments

## Details

Generic function used in the construction of spatialAtRisk objects. The class of spatialAtRisk objects provide a framework for describing the spatial inhomogeneity of the at-risk population, lambda(s). This is in contrast to the class of temporalAtRisk objects, which describe the global levels of the population at risk, mu(t).

Unless the user has specified lambda(s) directly by an R function (a mapping the from the real plane onto the non-negative real numbers, see ?spatialAtRisk.function), then it is only necessary to describe the population at risk up to a constant of proportionality, as the routines automatically normalise the lambda provided to integrate to 1.

For reference purposes, the following is a mathematical description of a log-Gaussian Cox Process, it is best viewed in the pdf version of the manual.

Let \mathcal Y(s,t) be a spatiotemporal Gaussian process, W\subset R^2 be an observation window in space and T\subset R_{≥q 0} be an interval of time of interest. Cases occur at spatio-temporal positions (x,t) \in W \times T according to an inhomogeneous spatio-temporal Cox process, i.e. a Poisson process with a stochastic intensity R(x,t), The number of cases, X_{S,[t_1,t_2]}, arising in any S \subseteq W during the interval [t_1,t_2]\subseteq T is then Poisson distributed conditional on R(\cdot),

X_{S,[t_1,t_2]} \sim \mbox{Poisson}≤ft{\int_S\int_{t_1}^{t_2} R(s,t)d sd t\right}.

Following Brix and Diggle (2001) and Diggle et al (2005), the intensity is decomposed multiplicatively as

R(s,t) = λ(s)μ(t)Exp{\mathcal Y(s,t)}.

In the above, the fixed spatial component, λ:R^2\mapsto R_{≥q 0}, is a known function, proportional to the population at risk at each point in space and scaled so that

\int_Wλ(s)d s=1,

whilst the fixed temporal component, μ:R_{≥q 0}\mapsto R_{≥q 0}, is also a known function with

μ(t) δ t = E[X_{W,δ t}],

for t in a small interval of time, δ t, over which the rate of the process over W can be considered constant.

## Value

method spatialAtRisk

1. Brix A, Diggle PJ (2001). Spatiotemporal Prediction for log-Gaussian Cox processes. Journal of the Royal Statistical Society, Series B, 63(4), 823-841.

2. Diggle P, Rowlingson B, Su T (2005). Point Process Methodology for On-line Spatio-temporal Disease Surveillance. Environmetrics, 16(5), 423-434.