EC.3D {AnalyzeFMRI} | R Documentation |
Calculates the Expected Euler Characteristic for a 3D Random Field thesholded a level u.
EC.3D(u, sigma, voxdim = c(1, 1, 1), num.vox, type = c("Normal", "t"), df = NULL)
u |
The threshold for the field. |
sigma |
The spatial covariance matrix of the field. |
voxdim |
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. |
The Euler Characteristic χ_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 χ_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(χ_u > 0) \approx E(χ_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.
The value of the expected Euler Characteristic.
J. L. Marchini
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 χ^2, f and t fields. Advances in Applied Probability, 26, 13-42.
EC.3D(4.6, sigma = diag(1, 3), voxdim = c(1, 1, 1), num.vox = 10000) EC.3D(4.6, sigma = diag(1, 3), voxdim = c(1, 1, 1), num.vox = 10000, type = "t", df = 100)