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Decomposition of Dependence for High Dimensional Extremes

Decomposition of Dependence for High Dimensional Extremes

Tail dependence in high dimensions is difficult to summarize and model. Via the framework of regular variation, we propose two decompositions which help to summarize and model high-dimensional tail dependence. We start by defining, via transformation, a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regulary-varying random vectors preserve regular variation. Rather than model regular-variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. Because this matrix is positive semidefinite, standard eigendecomposition allows one to interpret tail dependence via the eigenbasis. Additionally this matrix is completely positive, and a resulting decomposition allows one to easily construct regularly varying random vectors which share the same pairwise tail dependencies. Applying to financial data, we use the eigendecomposition to understand the modes of extreme losses across 30 financial sectors. Applying to US precipitation data during the hurricane season, we find that the time series associated with the second principal component is significantly influenced by the El Nino/Southern Oscillation index.

[This is joint work with Emeric Thibaud, EPFL; and Yujing Jiang CSU.]