On the Fisher’s [formula omitted] transformation of correlation random fields

Abstract

One of the most interesting problems studied in Random Field Theory (RFT) is to approximate the distribution of the maximum of a random field. This problem usually appears in a general hypothesis testing framework, where the statistics of interest are the maximum of a random field of a known distribution. In this paper, we use the RFT approach to compare two independent correlation random fields, R 1 and R 2 . Our statistics of interest are the maximum of a random field G , resulting from the difference between the Fisher’s Z transformation of R 1 and R 2 , respectively. The Fisher’s Z transformation guarantees a Gaussian distribution at each point of G but, unfortunately, G is not transformed into a Gaussian random field. Hence, standard results of RFT for Gaussian random fields are not longer available for G . We show here that the distribution of the maximum of G can still be approximated by the distribution of the maximum of a Gaussian random field, provided there is some correction by its spatial smoothness. Indeed, we present a general setting to obtain this correction. This is done by allowing different smoothness parameters for the components of G . Finally, the performance of our method is illustrated by means of both numerical simulations and real Electroencephalography data, recorded during a face recognition experimental paradigm.