Fuzzy covariogram and related concepts

Our works are dedicated to generalization of Matheron's theorem ([1]) about covariogram to the case where possible estimation error occurs.

Matheron's theorem shows crucial connection between covariogram and chord length distribution. However when we run the error in calculations due to environmental issues (for example computation in non-uniform environment), this error must be incorporated into model somehow. To do that one can take error as random variable with prespecified distribution. We rather model the concept by introducing fuzzy convex bodies.

We define the generalization of fuzzy distribution function as a distribution of fuzzy random variable introduced ([2]), then define fuzzy convex bodies by adding convex body and subtracting(in Hukuhara sense) from it fuzzy numbers in Rn, concept of fuzzy covariogram.

We then show Matheron's theorem analogue for this case ([3,4]). Next we show the relation between fuzzy distribution of orientation dependent random segment and fuzzy covariogram trying to retrieve results found in [5].

This is a joint work with Victor Ohanyan.

References

[1] Matheron G. “Random sets and integral geometry", Wiley, 1975.

[2] Viertl R. “Statistical methods for fuzzy data", John Wiley & Sons, 2011.

[3] Ohanyan V. K., Bardakhchyan V. G., Simonyan A. R., Ulitina E. I., “Fuzzification of convex bodies in Rn", Program systems and computational methods, n. 2, pp. 1-10, 2019.

[4] Ohanyan V. K., Bardakhchyan V. G., Ulitina E. I., “Random segment length distribution and covariogram for fuzzy convex body", Journal of Contemporary Mathematical Analysis, vol. 55, n. 1, 2020.

[5] Gasparyan A. G., Ohanyan V. K. “Orientation-dependent distribution of the length of a random segment and covariogram", Journal of Contemporary Mathematical Analysis, vol. 50, 2, pp. 90-97, 2015.