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 finitedimensional real vector space $V$. A statistical transform is proposed from such collectionsfinite spatial stochastic processesto numerical tuples using the SchurWeyl 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 DIRLab. Further applications to morphometry and statistical shape analysis are suggested.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.06221
 Bibcode:
 2018arXiv181106221M
 Keywords:

 Mathematics  Statistics Theory;
 60D05
 EPrint:
 9 pages, 1 figure