A metric space of subcopulas — An approach via Hausdorff distance

A concept of copula robustness and its applications in quantitative risk management

“…On the other hand, the (pseudo-) metric d μ 1 ,...,μ d defined by (2.2) generates a topology that is at most as fine as O [0,1] d . It is finally worth mentioning that the metric on the set of bivariate subcopulas that was recently introduced by Rachasingho and Tasena [40] basically differs from the metric d μ 1 ,μ 2 and from its variant d ∼…”

“…For any ν, μ ∈ M α 1 (R ++ ), we haved M Höl,α ,ψ α (P ν , P μ ) = max n=0,...,N −1 sup (x,a)∈D n d M Höl,α Q ν n (x, a), • , P μ n (x, a), • /ψ α (x) = max n=0,...,N −1 sup (x,a)∈D d M Höl,α ν • η −1 n,(x,a) [ • ], μ • η −1 n,(x,a) [ • ] /ψ α (x) − a, 1 ) + a, z ν(dz) − μ(dz) + a, z ) ν(dz) − R d + v(y + a, z ) μ(dz) 1 ψ α (x) ≤ sup (x,a)∈D sup w∈M Höl,α R d + w( a, z ) ν(dz) − R d + w( a, z ) μ(dz) 1 ψ α (x) = sup (x,a)∈D sup w∈M Höl,α R d + h w,a (z) ν(dz) − R d + h w,a (z) μ(dz) 1 ψ α (x) ,where D n := D := {(x, a) ∈ R d+1 + : a, 1 ≤ x} and h w,a (z) := w( a, z ). For the map h w,a : R d ++ → R, we haveh w,a d Höl,α = sup z 1 ,z 2 ∈R d ++ :z 1 =z 2 |h w,a (z 1 ) − h w,a (z 2 )|/|z 1 − z 2 | α = sup z 1 ,z 2 ∈R d ++ :z 1 =z 2 | a, z 1 − a, z 2 | α |z 1 − z 2 | α |h w,a (z 1 ) − h w,a (z 2 )| α | a, z 1 − a, z 2 | = sup z 1 ,z 2 ∈R d ++ :z 1 =z 2 x α |z 1 − z 2 | α ∞ |z 1 − z 2 | α w Höl,α ≤ c α ∞ x α , where c ∞ ∈ R ++ is chosen such that | • | ∞ ≤ c ∞ | • | and we used that | a, z 1 − z 2 | ≤ a, 1 |z 1 − z 2 | ∞ ≤ x|z 1 − z 2 | ∞ .Since ψ α (x) = 1 + x α , the above calculation can therefore be continued with≤ sup x∈R + c α ∞ x α d M d Höl,α (ν, μ)/ψ α (x) ≤ c α ∞ d M dHöl,α (ν, μ).Rachasingho and Tasena[40] recently defined a distance d on the set of bivariate subcopulas as follows. For two bivariate subcopulas C 0 and C 0 , they putd(C 0 , C 0 ) := h d [0,1] 2 ([C 0 ], [C 0 ]) + h | • | dom(C 0 ), dom(C 0 ) ,where the summandh d [0,1] 2 ([C 0 ], [C 0 ])is the Hausdorff distance (with respect to d [0,1] 2 ) between the sets of bivariate copulas [C 0 ] and [C 0 ] induced by C 0 and C 0 , respectively, and the summand h | • | (dom(C 0 ), dom(C 0 )) is the Hausdorff distance between the domains dom(C 0 ) and dom(C 0 ) of C 0 and C 0 , respectively.…”

On subcopula estimation for discrete models

“…Hausdorff distance has also been used by Rachasingho and Tasena (2020) to define a distance between bivariate subcopulas. The idea has been extended to multivariate cases in Tasena (2021b).…”

“…In other words, we need to replace a subcopula S with its representation, say, r S ð Þ so that we may define S n → S as r S n ð Þ→ r S ð Þ. de Amo et al ( 2017) is the first one who worked in this direction where they represent a subcopula S with its graph so that we have S n → S if and only if their corresponding graphs converge under the Hausdorff distance in this case. Rachasingho and Tasena (2020), on the other hand, identify a subcopula with the class of its copula extensions. This provides a relationship between the convergence of subcopulas and their corresponding copula extensions.…”

An Optimized Model for the Local Compression Deformation of Soft Tissue

“…, where the objective function is defined as Hausdorff distance function for image matching [34][35][36][37], which is expressed as follows: , otherwise return to Step2. Soft tissue deformation can be simulated more realistically by using the optimal spring stiffness and damping coefficients which determined by the above simulated annealing algorithm.…”