Random Marked Sets and Dimension Reduction

摘要

In this chapter, random marked closed sets are investigated. Special models with integer Hausdorff dimension are presented based on tessellations and numerical solutions of stochastic differential equations. Statistical analysis is developed which invovles the random-field model test and estimation of first and second-order characteristics. Real data analyses from neuroscience (track modeling marked by spiking intensity) and materials research (grain microstructure with disorientations of faces) are presented. Dimension reduction of point processes with Gaussian random fields as covariates was recently studied in the literature. In the present chapter this research is generalized in three different ways. Marked fibre and surface processes with covariates are subject to dimension reduction, where we restrict to the sliced inverse regression method. Slicing is suggested based on geometrical marks. In a refined model for dimension reduction the second-order central subspace is analyzed. Numerical results on estimation and testing the central subspace are presented based on simulations.