B0277
Title: Singular value decomposition based low-rank representations of copulas
Authors: Oliver Grothe - Karlsruhe Institute of Technology (Germany) [presenting]
Jonas Rieger - Karlsruhe Institute of Technology (Germany)
Abstract: Optimal low-rank approximations of arbitrary discretized (checkerboard) copulas are analyzed. Methodologically, we make use of truncated singular value decompositions of bistochastic matrices representing the copulas. The resulting (truncated) representations of the dependence structures are sparse, and memory usage decreases significantly. Due to the simple structure, essential statistical functionals of the copula's dependence structure are still readily available. The low-rank approximations conserve the uniform margins properties of the copulas but might lack non-negativity if the copula density has high peaks. For cases where non-negativity is crucial, we calculate the (Frobenius)-nearest valid discretized copula as a corrected low-rank representation. Copulas with stronger monotone dependence generally correspond to bistochastic matrices with larger ranks. Therefore, truncation leads to higher approximation errors than for copulas near the independence copula. We show how centering around the diagonals of the copulas compensates for this effect, leading to good low-rank representations in these cases as well. We illustrate the low-rank representation for various copula examples and families and derive some analytical results. We also discuss important general properties of the approximations and link our analysis to continuous decompositions of copula CDFs and copula-generating algorithms.
