On the Fisher's Z transformation of correlation random fields

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,R1 and R2. Our statistics of interest are the maximum of a random field G, resulting from the difference between the Fisher's Z transformation of R1 and R2, 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.