  Motivated by the prediction of cell loads in cellular networks, we formulate
the following new, fundamental problem of statistical learning of geometric
marks of point processes: An unknown marking function, depending on the
geometry of point patterns, produces characteristics (marks) of the points. One
aims at learning this function from the examples of marked point patterns in
order to predict the marks of new point patterns. To approximate (interpolate)
the marking function, in our baseline approach, we build a statistical
regression model of the marks with respect some local point distance
representation. In a more advanced approach, we use a global data
representation via the scattering moments of random measures, which build
informative and stable to deformations data representation, already proven
useful in image analysis and related application domains. In this case, the
regression of the scattering moments of the marked point patterns with respect
to the non-marked ones is combined with the numerical solution of the inverse
problem, where the marks are recovered from the estimated scattering moments.
Considering some simple, generic marks, often appearing in the modeling of
wireless networks, such as the shot-noise values, nearest neighbour distance,
and some characteristics of the Voronoi cells, we show that the scattering
moments can capture similar geometry information as the baseline approach, and
can reach even better performance, especially for non-local marking functions.
Our results motivate further development of statistical learning tools for
stochastic geometry and analysis of wireless networks, in particular to predict
cell loads in cellular networks from the locations of base stations and traffic
demand.
