The dimensions of the cuboid 'voxels' upon which the
discretized field is observed.
num.vox
The number of voxels that make up the field.
type
The marginal distribution of the Random Field (only Normal
and t at present).
df
The degrees of freedom of the t field.
Author
J. L. Marchini
Details
The Euler Characteristic \(\chi_u\) (Adler, 1981) is a
topological measure that essentially counts the number of isolated
regions of the random field above the threshold \(u\) minus the
number of 'holes'. As \(u\) increases the holes disappear and
\(\chi_u\) counts the number of local maxima. So when \(u\)
becomes close to the maximum of the random field
\(Z_{\textrm{max}}\) we have that
$$P( \textrm{reject} H_0 | H_0 \textrm{true}) =
P(Z_{\textrm{max}}) = P(\chi_u > 0) \approx E(\chi_u)$$
where \(H_0\) is the null hypothesis that there is no signicant
positive actiavtion/signal present in the field. Thus the Type I error
of the test can be controlled through knowledge of the Expected Euler characteristic.
Note: This function is directly copied from "AnalyzeFMRI".
References
Adler, R. (1981) The Geometry of Random Fields.. New York: Wiley.
Worlsey, K. J. (1994) Local maxima and the expected euler
characteristic of excursion sets of \(\chi^2\), \(f\) and \(t\)
fields. Advances in Applied Probability, 26, 13-42.