Two-Step Method for Assessing Similarity of Random Sets
DOI:
https://doi.org/10.5566/ias.2600Keywords:
connected component, curvature, similarity, $N$-distance, random setAbstract
The paper concerns a new statistical method for assessing dissimilarity of two random sets based on one realisation of each of them. The method focuses on shapes of the components of the random sets, namely on the curvature of their boundaries together with the ratios of their perimeters and areas. Theoretical background is introduced and then, the method is described, justified by a simulation study and applied to real data of two different types of tissue - mammary cancer and mastopathy.
References
Bullard JV, Garboczi EJ, Carter WC, Fuller ER Jr. (1995). Numerical methods for computing interfacial mean curvature. Comput Mater Sci. Vol. 4: 103–16.
Chiu SN, Stoyan D, Kendall WS, Mecke J (2013). Stochastic geometry and its applications. John Wiley & Sons, New York.
Debayle J, Gotovac Dogas V, Helisova K, Stanek J, Zikmundova M (2021). Assessing similarity of random sets via skeletons. Methodol Comput Appl Probab. Vol. 23: 471–490.
Gotovac V (2019). Similarity between random sets consisting of many components. Image Anal Stereol. Vol. 38: 185--99.
Gotovac Dogas V, Helisova K (2021). Testing equality of distributions of random convex compact sets via theory of N-distances. Methodol Comput Appl Probab. Vol. 23: 503–526.
Gotovac V, Helisova K, Ugrina I (2016). Assessing dissimilarity of random sets through convex compact approximations, support functions and envelope tests. Image Anal Stereol. Vol. 35: 181--93.
Gretton A, Borgwart KM, Rash MJ, Scholkopf B, Smola A (2012). A Kernel Two-Sample Test. J Mach Learn Res. Vol. 13: 723--73.
Hermann P, Mrkvicka T, Mattfeldt T, Minarova M, Helisova K, Nicolis O, Wartner F, Stehlik M (2015). Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process. Stat Med. Vol. 34.18: 2636--61.
Klebanov LB (2006). N-distances and their applications. Karolinum Press. Charles University, Prague.
Matheron G (1975). Random Sets and Integral Geometry. John Wiley & Sons, New-York.
Molchanov I (2005). Theory of random sets. Springer, New York.
Moeller J, Helisova K (2008). Power diagrams and Interaction processes for unions of discs. Adv in Appl Probab. Vol. 40: 321--47.
Moeller J, Helisova K (2010). Likelihood inference for unions of interacting discs. Scand J Stat. Vol. 37: 365--81.
Mrkvicka T, Mattfeldt T (2011). Testing histological images of mammary tissues on compatibility with the Boolean model of random sets. Image Anal Stereol. Vol. 30: 11--18.
Myllymaki M, Mrkvicka T, Grabarnik P, Henri Seijo H, Hahn U (2017). Global envelope tests for spatial processes. J R Stat Soc, Ser B (Stat Methodol). Vol. 79: 381--404.
Neumann M, Stanek J, Pecho OM, Holzer L, Benes V, Schmidt V (2016). Stochastic 3D modeling of complex three-phase microstructures in SOFC-electrodes with completely connected phases. Comput Mater Sci. Vol. 118: 353--64.
Serra J (1982). Image Analysis and Mathematical Morphology. Vol.2: Theoretical Advances. Academic Press.
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