Testing Equality of Distributions of Random Convex Compact Sets via Theory of $\mathfrak {N}$-Distances

Two-Step Method for Assessing Similarity of Random Sets

“…When the measures µ and ν correspond to distributions of random functions, then testing the equality of µ and ν runs analogously as above, only with the difference in the choice of the negative definite kernel L . Here, we use the kernel introduced in Gotovac Ðogaš and Helisová (2021), constructed especially for random functions as follows. We evaluate testing functions t (1) and t (2) in discrete arguments u 1 , .…”

“…In the simulation study, we first focus on four models which illustrate the usage of the procedure. The models and approach to simulation can be found in Debayle et al (2021), Gotovac (2019), Gotovac Ðogaš and Helisová (2021) and Gotovac et al (2016). For all considered models, we simulate 200 realisations and compare 100 vs 100 realisations of the same models as well as 100 vs 100 realisations of different models.…”

“…Note that the same repulsive model is employed for simulation studies in all above mentioned papers, similarly as the Boolean model, while the same cluster model is used only in Gotovac (2019). In Debayle et al (2021), Gotovac Ðogaš and Helisová (2021) and Gotovac et al (2016), another cluster model is considered, which is not suitable for our current study since its realisations consist of too few components. The fourth model is the Boolean model with grains to be ellipses (called the ellipse model in the following paragraphs).…”

“…New methods taking this into account have been developed in the last five years (Debayle et al, 2021;Gotovac, 2019;Gotovac Ðogaš and Helisová, 2021;Gotovac et al, 2016). In Debayle et al (2021), morphological skeletons (Serra, 1982) of compared realisations are constructed and then, similarity of two realisations is defined through a function describing mass growth around selected points of the skeletons.…”

“…A disadvantage of the method is that the test of similarity in this sense is weak, especially when the neighbourhoods are omitted, so in order to achieve better accuracy, the positions of the components must be taken into account again. An approach independent of the positions of the components is introduced in Gotovac et al (2016) and improved in Gotovac Ðogaš and Helisová (2021). The authors of the papers distinguish between two realisations via a heuristic approach based on approximation of the components by unions of convex compact sets using Voronoi tessellation (Chiu et al, 2013) with respect to a special hardcore point process on the realisations and consequent comparison of the support functions of the convex compact cells of the tessellation using the envelope test from Myllymäki et al (2017) and the test based on N -distance (sometimes also called the kernel test (Gretton et al, 2012)), respectively.…”