EC.3D: Expected Euler Characteristic for a 3D Random Field

In AnalyzeFMRI: Functions for Analysis of fMRI Datasets Stored in the ANALYZE or 'NIFTI' Format

Expected Euler Characteristic for a 3D Random Field

Description

Calculates the Expected Euler Characteristic for a 3D Random Field thesholded a level u.

Usage

EC.3D(u, sigma, voxdim = c(1, 1, 1), num.vox, type = c("Normal", "t"), df = NULL)

Arguments

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.

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.

Value

The value of the expected Euler Characteristic.

Author(s)

J. L. Marchini

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.

See Also

Threshold.RF

Examples

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)

AnalyzeFMRI documentation built on Aug. 8, 2025, 7:16 p.m.