On Random Marked Sets with a Smaller Integer Dimension

The paper deals with random marked sets in ${\mathbb R}^d$ which have integer dimension smaller than d. Statistical analysis is developed which involves the random-field model test and estimation of first and second-order characteristics. Special models are presented based on tessellations and solutions of stochastic differential equations (SDE). The simulation of these sets makes use of marking by means of Gaussian random fields. A space-time nature of the model based on SDE is taken into account. Numerical results of the estimation and testing are discussed. Real data analysis from the materials research investigating a grain microstructure with disorientations of faces as marks is presented.