Spatiotemporal Prediction for Log-Gaussian Cox Processes

Abstract

Space±time point pattern data have become more widely available as a result of technological developments in areas such as geographic information systems. We describe ā exible class of space±time point processes. Our models are Cox processes whose stochastic intensity is a space±time Ornstein±Uhlenbeck process. We develop moment-based methods of parameter estimation, show how to predict the underlying intensity by using a Markov chain Monte Carlo approach and illustrate the performance of our methods on a synthetic data set.

Point process methodology for on‐line spatio‐temporal disease surveillance

“…Our point process model is a straightforward adaptation of the model proposed by Brix and Diggle (2001), which in turn is an example of a spatio-temporal Cox process (Cox, 1955). Conditional on an unobserved stochastic process R(x, t), cases form an inhomogeneous Poisson point process with intensity λ(x, t), which we factorise as…”

“…To estimate parameters of S(x, t) we use the moment-based methods of Brix and Diggle (2001), which operate by matching empirical and theoretical descriptors of the spatial and temporal covariance structure of the point process model. For the current analysis, we assumed a separable correlation structure in which ρ(u, v) = ρ x (u)ρ t (v).…”

“…Note that an error in the expression for Cov(N t , N t−v ) given by Brix and Diggle (2001) is corrected in Brix and Diggle (2003). For our model, the time-variation in µ 0 (t) makes the covariance structure of N t non-stationary.…”

Likelihood based inference for partially observed renewal processes

“…As in Brix and Diggle (2001) and Diggle et al (2005), we assume that calls are made according to a log-Gaussian Cox process (Møller et al, 1998) and approximate the rate of calls during the ith day of the two-month period by…”

“…Since c(β, σ 2 , µ 0 , n(i) i ) cannot be evaluated explicitly, we use the missing data Monte Carlo approach of Section 4 with a Metropolis-Hastings algorithm as in Brix and Diggle (2001) to sample from (6), which, in turn, can be used to sample the missing counts N(i), i ∈ {T 1 + 1, . .…”