  Employing the framework of regular variation, we propose two decompositions
which help to summarize and describel high-dimensional tail dependence. Via
transformation, we define 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 regularly 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. This matrix is positive semidefinite, and eigendecomposition allows
one to interpret tail dependence via the resulting 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. We illustrate our methods with Swiss rainfall data and
financial return data.
