On the Fisher’s Z transformation of correlation random fields

• Published in: Statistics & probability letters Vol. 79; no. 6; pp. 780 - 788
• Main Authors: Carbonell, F., Worsley, K.J., Trujillo-Barreto, N.J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15-03-2009; Elsevier
• Series: Statistics & Probability Letters
• Subjects: Exact sciences and technology; General topics; Mathematics; Multivariate analysis; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Stochastic processes; Correlation; Probability theory; Gaussian distribution; Electroencephalography; Random distribution; Statistical method; Hypothesis test; Random field; Correlation analysis; Gaussian field; Distribution function; Numerical simulation; Field theory
• ISSN: 0167-7152; 1879-2103
• DOI: 10.1016/j.spl.2008.11.007

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.