A Schur transform for spatial stochastic processes

Abstract

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.