  The variance, higher order moments, covariance, and joint moments or
cumulants are shown to be special cases of a certain tensor in $V^{\otimes n}$
defined in terms of a collection $X_1,...,X_n$ of $V$-valued random variables,
for an appropriate finite-dimensional real vector space $V$. A statistical
transform is proposed from such collections--finite spatial stochastic
processes--to numerical tuples using the Schur-Weyl decomposition of
$V^{\otimes n}$. It is analogous to the Fourier transform, replacing the
periodicity group $\mathbb{Z}$, $\mathbb{R}$, or $U(1)$ with the permutation
group $S_{n}$. As a test case, we apply the transform to one of the datasets
used for benchmarking the Continuous Registration Challenge, the thoracic 4D
Computed Tomography (CT) scans from the M.D. Anderson Cancer Center available
for download from DIR-Lab. Further applications to morphometry and statistical
shape analysis are suggested.
