On the geometry of a generalized cross-correlation random field

Studying the geometry of random fields (RF) has become a well established tool for dealing with the statistical problem of multiple comparisons. This has led to an increasing number of applications in medical imaging and astrophysics mainly. In particular, cross-correlation RF, defined as the sample correlation between pairs of independent Gaussian RF, have been used for analyzing functional connectivity in brain images. However, in some cases one needs to deal with correlated observations for which the “independence” assumption is no longer valid. In this paper we define a generalized cross-correlation RF that allows for correlated observations. We also study the geometry of such RF by giving results for the expected Euler Characteristic of its local maxima.