Dimension reduction of multivariate data was developed by Y. Guan for point processes with Gaussian random fields as covariates. The generalization to fibre and surface processes is straightforward. In inverse regression methods, we suggest slicing based on geometrical marks. An investigation of the properties of this method is presented in simulation studies of random marked sets. In a refined model for dimension reduction, the second-order central subspace is analyzed in detail. A real data pattern is tested for independence of a covariate.
"Sliced inverse regression and independence in random marked sets with covariates." Adv. in Appl. Probab. 45 (3) 626 - 644, September 2013. https://doi.org/10.1239/aap/1377868532