  In order to understand structural relationships among sets of variables at
extreme levels, we develop an extremes analogue to partial correlation. We
begin by developing an inner product space constructed from transformed-linear
combinations of independent regularly varying random variables. We define
partial tail correlation via the projection theorem for the inner product
space. We show that the partial tail correlation can be understood as the inner
product of the prediction errors from transformed-linear prediction. We connect
partial tail correlation to the inverse of the inner product matrix and show
that a zero in this inverse implies a partial tail correlation of zero. We then
show that under a modeling assumption that the random variables belong to a
sensible subset of the inner product space, the matrix of inner products
corresponds to the previously-studied tail pairwise dependence matrix. We
develop a hypothesis test for partial tail correlation of zero. We demonstrate
the performance in two applications: high nitrogen dioxide levels in Washington
DC and extreme river discharges in the upper Danube basin.
