CoxProzesse

Cox processes, also known as doubly stochastic Poisson processes, are a class of point processes in which the instantaneous intensity function is itself a random process. In a Cox process, once the random intensity Λ(t) is realized, the points form a Poisson process with rate Λ(t). Unconditionally, the process is a mixed Poisson process, because the Poisson randomness is compounded by the randomness of Λ.

Mathematically, if Λ(t, ω) is a nonnegative, adapted stochastic process on a probability space, then conditional on Λ, the number of points in a region A follows a Poisson distribution with mean ∫A Λ(t, ω) dt. The unconditional distribution of counts is obtained by averaging over the randomness of Λ, leading to overdispersion compared with a standard Poisson process. In particular, the variance of counts typically exceeds the mean.

A common and flexible subfamily is the log-Gaussian Cox process (LGCP), where Λ(t) = exp(G(t)) and G(t) is a Gaussian process. Other constructions include shot-noise Cox processes, where Λ is a random field generated by a superposition of localized sources. Cox processes can be stationary or nonstationary, depending on the properties of Λ.

Applications of Cox processes span ecology, epidemiology, geology, astronomy, and finance, among others, wherever spatial or temporal event data exhibit greater variability than a Poisson model can capture. They are used to model clustering and to assess intensity patterns and dependence structures.

Inference for Cox processes often relies on conditioning on Λ and integrating it out, which leads to computationally challenging likelihoods. Methods include Monte Carlo likelihood, Bayesian hierarchical modeling, data augmentation, and approximate or composite likelihood approaches. Simulation proceeds by sampling Λ, then generating a Poisson process with the realized intensity.