We had an interesting talk at QUT last week from N. Leonenko (Cardiff) on the fractional Poisson process (FPP), a generalization of the familiar Poisson process (PP) with long-range dependence (cf. Laskin 2003). The name suggests an intended analogy to the fractional Wiener process (FWP), and its long-range dependence obviously means it no longer shares the characteristic “memoryless” property of the PP. When its domain is, say, R^2 (Leonenko & Mirzbach) with some underlying reference intensity field (perhaps the realization of a Gaussian process) a typical realization of the FPP should have points clustered around the peaks of the underlying intensity field, but with an ‘extra degree of clustering’.
As such it seems to me that the GP + FPP combination could be a handy modelling device for describing non-independent data sampling in a geostatistical context. But what about astronomy? Could there be a sensible motivation to apply this model to mapping the large-scale mass distribution with galaxies as tracers? Such analyses already use the PP (e.g. Kitaura et al.; Nadathur) but could they be using the FPP to take into account the role of galaxy clustering within halos (does that even make sense)?
Important to note is that the FPP does not seem easy to fit (I have the impression that its likelihood might be intractable), though it is easy to simulate from. (Sounds like an ABC challenge to me!)
Update: Two additional questions. What would be the advantages of using the FPP in continuous space over the generalized Poisson for counts in discrete cells (Sheth 1998)? And what is the practical difference between the FPP and a marked PP model (Reddick et al. 2013)?